N Unit Moving Average Of A Funktion

Magnetismus: Mengen, Einheiten und Beziehungen Wenn Sie gelegentlich eine Wundkomponente entwerfen müssen, aber nicht mit der Wissenschaft der Magnetfelder auf einer täglichen Basis umgehen, dann können Sie sich darüber verwirren, was die vielen Begriffe im Datenblatt für den Kern verwendet haben Stellen Sie dar, wie sie verwandt sind und wie Sie sie verwenden können, um eine praktische Induktivität zu produzieren. Über deinen Browser: Wenn dieses Zeichen mal nicht wie ein Multiplikationszeichen aussieht, oder du siehst viele Fragezeichen oder Symbole wie oder Sequenzen wie ampcannot dann bitte akzeptieren meine Entschuldigungen. Index zu magnetischen Begriffen Ampere Einheiten in der SI Diese Reihe von Webseiten verwendet das System der Einheiten bekannt als die SI (Systegraveme International). Für weitere Informationen über die SI, und wie es mit anderen Systemen vergleicht, siehe Unit Systems in Electromagnetism. Magnetische Mengen im SI Die effektive Fläche eines Kerns repräsentiert die Querschnittsfläche eines seiner Gliedmaßen. In der Regel entspricht dies genau den physikalischen Abmessungen des Kerns, aber da der Fluss nicht gleichmäßig verteilt werden kann, wird der Hersteller einen Wert für A e angeben, der dies widerspiegelt. Die Notwendigkeit für den Kernbereich ergibt sich, wenn man die Flußdichte im Kern (begrenzt durch den Materialtyp) auf den Gesamtfluss, den es trägt, in Beziehung setzen möchte. Im Beispiel Toroid konnte die Fläche etwa als Produkt der Kernhöhe bestimmt werden Der Unterschied zwischen den Haupt - und Nebenradien - A e 6,3-fach ((12,7 - 6,3) 2) 20,2 mm 2 Da sich der Fluss jedoch konzentriert, wo die Weglänge kürzer ist, ist es besser, den vom Hersteller angegebenen Wert zu verwenden - 19,4 mm 2. Für die einfache torusförmige Form wird A e berechnet (Snelling) as Dies setzt quadratische Kanten zum toroid echte ist oft abgerundet. Es gibt eine leichte Wendung auf die Frage der Fläche: Der Hersteller Wert für A e geben geben die richtigen Ergebnisse, wenn verwendet, um die Kernrezension zu berechnen, aber es kann nicht perfekt für die Berechnung der Sättigung Fluss (die abhängig von der engsten Teil der Kern oder A min). In einem gut entworfenen Kern A min wird nicht ganz anders als A e. Aber halten Sie es im Auge. Effektive Fläche ist in der Regel in Millimeter quadriert zitiert. Viele Formeln in Datenbüchern gehen implizit davon aus, dass ein Zahlenwert in mm 2 verwendet wird. Andere Bücher, und diese Notizen, nehmen Meter quadriert. Effektive Länge Wirksame Länge im SI Die effektive Länge eines Kerns ist ein Maß für die Distanz, die die Flusslinien in einem kompletten Kreislauf verlaufen. In der Regel entspricht dies genau den physikalischen Abmessungen des Kerns, aber da der Fluß eine Tendenz hat, sich auf die inneren Ecken des Weges zu konzentrieren, wird der Hersteller einen Wert angeben, der dies widerspiegelt. Im Toroidbeispiel konnte die Weglänge etwa gleich wie gewohnt bestimmt werden (12,7 6,3) 2 29,8 mm Da jedoch der Fluß konzentriert ist, wo die Weglänge kürzer ist, ist es besser, den vom Hersteller angegebenen Wert zu verwenden - 27,6 mm. Für eine einfache torusförmige Form wird e e berechnet. Ein anderer gemeinsamer Kerntyp, der EE, ist in Abb. EEE dargestellt. Die rote Linie stellt den kürzesten Weg dar, den eine Flusslinie nehmen könnte, um den Kern zu umgehen. Die grüne Linie ist die längste. Dargestellt in blau ist ein Pfad, dessen Länge ist, dass der kurze Weg plus vier Sektoren, deren Radius ist ausreichend, um den Weg in der Mitte hinunter die Gliedmaßen. Das ist alles ein bisschen annähernd, aber bedenkt, dass seit Fertigungstoleranzen auf Permeabilität oft 25 gibt es nicht viel Punkt im Sein genauer. Die effektive Länge wird üblicherweise in Millimeter angegeben. Viele Formeln in Datenbüchern gehen implizit davon aus, dass ein Zahlenwert in mm verwendet wird. Andere Bücher, und diese Notizen, nehmen Meter an. Uarr Top of page Magnetomotive Force Magnetomotive Force im SI F m 0,25 mal 2 0,5 Amperewindungen Vermeiden Sie magnetomotorische Kraft mit Magnetfeldstärke (Magnetisierungskraft). Als Analogie denke ich an die Platten eines Kondensators mit einer gewissen elektromotorischen Kraft (EMF) zwischen ihnen. Wie hoch die elektrische Feldstärke ist, hängt vom Abstand zwischen den Platten ab. Ähnlich hängt die Magnetfeldstärke in einem Transformatorkern nicht nur von der MMF ab, sondern auch von der Distanz, die der Fluss um ihn herum reisen muss. Ein Magnetfeld repräsentiert gespeicherte Energie und wo W die Energie in Joule ist. Sie können auch MMF auf den Gesamtfluss beziehen, der durch einen Teil eines Magnetkreises geht, dessen Widerwillen Sie kennen. Es gibt eine klare Analogie hier mit einem Stromkreis und Ohm Gesetz, V I mal R. Die Analogie mit elektrischem Potential (Spannung) führt zum alternativen Namen magnetischen Potential. Es besteht jedoch eine Verwechslungsgefahr mit magnetischem Vektorpotential - das hat ganz andere Einheiten. Eine spezifische MMF ist erforderlich, um eine gegebene Feldstärke entlang einer bekannten Pfadlänge aufrechtzuerhalten. Praktische Spulenwicklungen werden aus Kupferdraht hergestellt, der eine Strombelastbarkeit aufweist, die hauptsächlich durch seinen Querschnitt begrenzt ist. Es besteht daher eine Grenze für die MMF einer Spule im kontinuierlichen Betrieb von etwa 3,5 × 10 6 Ampere-Umdrehungen pro Quadratmeter Blende. Magnetfeldstärke Magnetfeldstärke im SI-Magnetfeld Stärke Alias ​​Magnetfeldintensität alias das Hilfsfeld alias die H-Feld-Alias-Magnetisierungskraft Immer wenn Strom fließt, wird immer ein Magnetfeld begleitet. Die Wissenschaftler sprechen von dem Feld als wegen der sich bewegenden elektrischen Ladungen - eine vernünftige Beschreibung der Elektronen, die entlang eines Drahtes fließen. Die Stärke oder Intensität dieses Feldes, das einen geraden Draht umgibt, ist gegeben durch wobei r, der Abstand vom Draht, im Vergleich zu der Länge des Drahtes klein ist. Die Situation für kurze Drähte wird durch die Biot-Savart-Gleichung beschrieben. Übrigens verwirren Sie nicht die Geschwindigkeit der Ladungen (wie Elektronen) mit der Geschwindigkeit eines Signals, das den Draht hinunterfährt, in dem sie sich befinden. Denken Sie an das Signal als die Grenze zwischen den Elektronen, die begonnen haben, sich zu bewegen, und diejenigen, die haben Noch zu gehen Die Grenze könnte sich nahe an die Lichtgeschwindigkeit bewegen (3x10 8 ms -1), während die Elektronen selbst (im Durchschnitt) etwas in der Nähe von 0,1 mm s -1 treiben. Die Elektronen würden von einer Schnecke überholt - auch wenn es nicht in Eile war. Sie können einwenden, dass Magnetfelder auch durch Permanentmagnete (wie Kompassnadeln, Türklinken und Kofferraumhalter) hergestellt werden, wo kein Stromfluss offensichtlich ist. Es stellt sich heraus, dass auch hier die Elektronen sich in der Umlaufbahn um die Kerne bewegen oder sich auf ihrer eigenen Achse drehen, die für das Magnetfeld verantwortlich sind. Dualität mit der elektrischen Welt Im Beispiel ist die Feldstärke dann - H 0,5 27,6 × 10 -3 18,1 A m -1 Die Analogie mit der elektrischen Feldstärke ist mathematisch und nicht physisch. Ein elektrisches Feld hat eine klar definierte physikalische Bedeutung: einfach die Kraft, die auf eine Testladung geteilt durch die Ladungsmenge ausgeübt wird. Die Magnetfeldstärke kann nicht in gleicher Weise gemessen werden, da kein magnetisches Monopol äquivalent zu einer Testladung ist. Vermeiden Sie die Magnetfeldstärke mit der Flussdichte. B. Dies steht in engem Zusammenhang mit der Feldstärke, hängt aber auch von dem Material innerhalb des Feldes ab. Die strenge Definition von H ist Diese Formel gilt allgemein, auch wenn die Materialien innerhalb des Feldes eine ungleichmäßige Permeabilität oder ein permanentmagnetisches Moment haben. Es wird selten im Spulenentwurf verwendet, weil es in der Regel möglich ist, die Berechnung zu vereinfachen, indem man annimmt, daß innerhalb des Feldes die Permeabilität als gleichförmig angesehen werden kann. Mit dieser Annahme sagen wir stattdessen, dass Flux auch aus einem Permanentmagneten hervorgeht, auch wenn es keine Drähte gibt, um ein Feld zu verhängen. Eine Feldstärke von etwa 2000 A m -1 liegt bei der Grenze für Kerne aus Eisenpulver. In einer idealen Induktivität würde der Fluß, der durch eine seiner Wendungen erzeugt wird, alle anderen anderen Wendungen umkreisen. Echte Spulen kommen diesem Ideal nahe, wenn die Querschnittsabmessungen der Wicklung im Vergleich zu ihrem Durchmesser klein sind oder wenn ein hoher Permeabilitätskern den Fluß direkt um den Weg führt. Bei längeren Luftspulen ist die Situation wahrscheinlich näher an der in Abb. TFK dargestellten: Hier sehen wir, daß die Flußdichte in Richtung der Enden der Spule abnimmt, da ein Fluß eine Abkürzung einnimmt, die die äußeren Windungen umgibt. Nehmen wir an, dass der Strom in die Spule 5 Ampere beträgt und jede Flusslinie 7 mWb darstellt. Die zentralen drei Umdrehungen alle verbinden vier Linien des Flusses: 28 mWb. Die beiden äußeren Umdrehungen verbinden nur zwei Linien des Flusses: 14 mWb. Wir können die Gesamtflussverknüpfung für die Spule berechnen: lambda 3times28 2times14 112 mWb-t Die Nützlichkeit dieses Ergebnisses ist, dass es uns ermöglicht, die gesamte Selbstinduktivität der Spule zu berechnen, L: L lambda I 1125 22,4 mH Im Allgemeinen, Wo eine ideale Spule angenommen wird, sehen Sie Ausdrücke mit N timesPhi oder N timesdPhidt. Für eine größere Genauigkeit ersetzen Sie Lambda oder Dlambdadt. Sie könnten vergeben werden für das Denken, dass es keine Notwendigkeit zu buchstabieren, was aktuell ist. Das ist offensichtlich sicherlich Ihr Fehler ist zu vergessen, wie hart alle Schriftsteller auf Elektromagnetismus streben, ein bereits schwieriges Thema zu verschleiern. Heres das Problem. Abbildung TMX zeigt zwei Spulen mit unterschiedlicher Anzahl von Windungen, aber die gleiche magneto-motive Kraft. Bei der Betrachtung der MMF macht es keinen Unterschied, ob Sie zwölf Windungen haben, die einen Verstärker tragen, oder drei Umdrehungen mit vier Amps oder zwei Umdrehungen mit sechs Amps. Soweit die MMF geht alles nur zwölf Ampere-Turns. Sie erhalten in jedem Fall genau das gleiche Magnetfeld. Angesichts dieser Details über die Anzahl der Umdrehungen und die Anzahl der Verstärker spielt keine Rolle, nur das Produkt der beiden. Einige Schriftsteller entscheiden zu sagen, dass der Strom zwölf Amps ist. Sie schreiben mir 12 A und überlassen es Ihnen, zu entscheiden, welches Szenario diesen Strom gebracht hat. Diese heimtückische Übung trägt auch Formeln zu. Was ist in Ordnung, solange seine konsequente und klar für den Leser was passiert ist. Wenn sich der Strom ändert, dann haben wir nach Faradays Law eine induzierte Spannung. Sie müssen sich dann merken, dass die induzierte Spannung pro Umdrehung und nicht die gesamte Spulenspannung ist. Ambiguität beginnt zu kriechen. Es hängt davon ab, ob Sie sich mehr für Physik oder Technik interessieren. Diese Seiten nehmen die letztere Ansicht und unterscheiden den Strom von MMF. Strom hier ist also, was ein gewöhnliches Amperemeter liest, und die Anzahl der Spulenwindungen. N, wird ausdrücklich geschrieben. Die Physiker kommen doch am Ende, weil man, wenn man nur von Widerstreben als Ampere-Wendungen pro Weber spricht, Induktivität als Weber-Wendungen pro Ampere ein bisschen fertig wird - auch wenn es das Konzept der Flussverknüpfung eher schön reflektiert. Aber Permeabilität als Weber-Umdrehungen pro Ampere-Meter Diese Seiten werden umgewandelt, um Großbuchstaben I sowohl für Gleichstrom als auch für einen Strom zu verwenden, der als RMS-Menge gegeben wird, während der Kleinbuchstabe i für sofortige Werte des zeitvariablen Stroms steht. Trivia-Punkt: Warum ist das Symbol, das ich für den Strom verwendet habe angeblich, steht es für elektrische Intensität, im Gegensatz zu Gesamtmenge an Elektrizität (Gebühr). Maxwell. Obwohl, verwendet das Symbol C für Strom und verwendet elektrische Intensität auf das E-Feld beziehen: was die meisten Menschen heute wissen, wie elektrische Feldstärke. Also geht es Stromdichte Der zeitvariable Fluss erzeugt induzierte Spannung (EMF) - Wenn man dieser fünfstufigen Sequenz folgen kann, dann wird der Aufbau eines mentalen Bildes einer magnetischen Komponente einfacher. Denken Sie daran, Sie setzen in einen Strom und erhalten wieder eine induzierte Spannung. In der Tat, wenn man die Permeabilität als linear behandeln kann, dann sind die Konstanten N. l e. Mu und A e können in eine Konstante für die Wicklung zusammengefasst werden, die heißt (Überraschung) Induktivität. L - Ich gebe die Basiseinheiten für alle Mengen in dieser Gleichung, die es den Nervenkitzlern ermöglichen, eine Dimensionsanalyse zu machen, die bestätigt, dass es konsistent ist. Richtig, also dann unsere fünf-Schritt-Beziehung zwischen Strom und EMF kocht auf: Sie können im Begriff zu beschweren, dass Sie wissen, die EMF auf Ihre Wicklung, aber nicht wissen, die aktuelle in ihm. Die Antwort ist, dass der Prozess dann umgekehrt arbeitet - der Strom wird aufbauen, bis die induzierte Spannung ausreicht, um der angelegten Spannung entgegenzuwirken. Sie können mehr erfahren, indem Sie Faradays Gesetz betrachten. Wie berücksichtigen Sie das Vorhandensein der Sekundärwicklungen in einem Transformator. Ein Weg ist, die ersten vier Schritte der Sequenz oben zu nehmen und sie separat auf jede Wicklung (ob primär oder sekundär) anzuwenden. Die arithmetische Summe über alle Wicklungen ergibt einen Gesamt-Kernfluss. Von der Zeitrate der Änderung des Flusses haben Sie dann die induzierte Spannung in jeder Wicklung (da Sie auch die Anzahl der Umdrehungen für jeden kennen). Es gibt weniger langwierige Methoden der Analyse Transformator Betrieb, die Sie wahrscheinlich besser machen würde. Aber sie sind eine andere Geschichte. Auf die Beziehung zwischen Thermodynamik und Konfigurationsentropie. Ursprung des Zweiten Gesetzes der Thermodynamik. Diese Seite ist Teil eines Satzes von Seiten, die zum Schnittpunkt der Interessen des Autors in Physik und Mathematik gehören. Hinweis: Blaue Links sind intern, grüne Links sind externe Links kursiv in einem neuen Fenster geöffnet. Die Frage Es gibt zwei Versionen der Entropie in der Wissenschaft bekannt. Einer ist die traditionelle, traditionelle, thermodynamische Entropie. Es wird typischerweise im Zusammenhang mit z. B. der Platzierung eines heißen Steins in einem Eimer mit kühlerem Wasser diskutiert: nach einer Weile kühlt sich der Stein ab und das Wasser erwärmt sich, wobei beide die gleiche Temperatur erwerben und zu einem thermodynamischen Gleichgewicht kommen . (Siehe Abbildung unten, auf der linken Seite klicken Sie auf die Schaltfläche, um es laufen zu lassen.) Wir gehen davon aus, dass es keinen Energieeintrag von außen in den Eimer gibt, noch keinen Energieverlust aus dem Eimer in die Umwelt. So wird der Schaufelstein als geschlossenes System betrachtet. () Die andere ist die weniger bekannte, konfiguratorische oder logische Entropie. Ein typischer Kontext, in dem es diskutiert wird, ist die Platzierung eines Gases, das in einem Behälter in der Mitte eines Raumes eingeschlossen ist, der ein anderes Gas mit demselben Druck enthält, dann wird der Deckel des Behälters geöffnet und sein Gas wird befreit. Die Moleküle des Gases, die sich im Behälter befanden, zerstreuten sich und vermischen sich mit den Molekülen des Gases im Raum, und nach einer Weile bilden die beiden Gase eine gleichmäßige Mischung. (Abbildung unten, auf der rechten Seite, simulieren nur die Moleküle des Gases im Container klicken Sie auf die Schaltfläche, um es laufen zu lassen.) Wieder betrachten wir den Raum von äußeren Einflüssen zu isolieren, also ist es auch ein geschlossenes System. Der Anspruch bei der Konfigurationsentropie ist, dass es anfangs mehr Ordnung im System gibt, da die Moleküle der beiden Gase getrennt wurden, aber im Laufe der Zeit vergeht und die Moleküle sich vermischen, nimmt die Ordnung ab und damit erhöht sich die Entropie. Die Frage ist: Gibt es eine tiefere Beziehung zwischen den beiden Arten von Entropie, oder sind ihre Ähnlichkeiten nur offensichtlich Gibt es einige zugrunde liegende Gesetz, das die beiden Versionen verursacht, oder sind sie nur durch eine oberflächliche Analogie verwandt Ein heißer Stein ist in einem Eimer platziert Mit kaltem Wasser nach einer Weile kommen die beiden zu einem thermodynamischen Gleichgewicht. Moleküle, beginnend bei einer geordneten Struktur und Dispergieren über den verfügbaren Raum. Die Konkurrenz-Wissenschaftler sind nicht einverstanden, ob die beiden Entropie-Versionen zusammenhängen. Viele der bekannten Autoren von Physikbüchern, die auf ein breites Erziehungspublikum zielen, nehmen stillschweigend an, dass die beiden Begriffe der Entropie tatsächlich zusammenfallen, so dass das Sprechen über das eine wie das Sprechen über das andere ist. Solche Autoren machen sich nicht einmal die Unterscheidung zwischen den beiden: sie reden über Ordnung und Unordnung (Konfigurationsentropie) im Kontext des zweiten Gesetzes der Thermodynamik (wo die Rede über die thermodynamische Entropie angemessen sein sollte, wenn sie einen Unterschied zwischen den beiden wahrnahmen ). Zum Beispiel nach Stephen Hawking: Das nicht abnehmende Verhalten eines Schwarzen Löcher Bereich war sehr erinnert an das Verhalten einer physikalischen Größe namens Entropie, die den Grad der Störung eines Systems misst. Es ist eine Frage der gemeinsamen Erfahrung, dass die Störung dazu neigen wird zu erhöhen, wenn die Dinge sich selbst überlassen bleiben. (Man muss nur aufhören zu reparieren um das Haus zu sehen, dass) man kann Ordnung aus der Unordnung (zum Beispiel kann man das Haus malen), aber das erfordert Aufwand an Aufwand oder Energie und so verringert die Menge der bestellten Energie zur Verfügung . Eine genaue Aussage dieser Idee ist bekannt als das zweite Gesetz der Thermodynamik. Es heißt, dass die Entropie eines isolierten Systems immer zunimmt und dass, wenn zwei Systeme miteinander verbunden sind, die Entropie des kombinierten Systems größer ist als die Summe der Entropien der einzelnen Systeme. Zum Beispiel betrachten wir ein System von Gasmolekülen in einer Box. Die Moleküle können so wenig Billardkugeln gedacht werden, die ständig miteinander kollidieren und von den Wänden der Box abprallen. Je höher die Temperatur des Gases ist, desto schneller bewegen sich die Moleküle, und je häufiger und härter kollidieren sie mit den Wänden der Schachtel und je größer der äußere Druck, den sie an den Wänden ausüben. Angenommen, zunächst sind die Moleküle auf die linke Seite der Schachtel durch eine Trennwand beschränkt. Wenn die Partition dann entfernt wird, werden die Moleküle dazu neigen, sich auszubreiten und beide Hälften der Box zu besetzen. Irgendwann später konnten sie zufällig alle in der rechten Hälfte oder zurück in der linken Hälfte sein, aber es ist überwiegend wahrscheinlicher, daß es in den beiden Hälften etwa gleiche Zahlen geben wird. Ein solcher Zustand ist weniger geordnet oder mehr ungeordnet, als der ursprüngliche Zustand, in dem alle Moleküle in einer Hälfte waren. Man sagt also, daß die Entropie des Gases aufgestanden ist. In einer kurzen Geschichte der Zeit, 1 p. 106. Ähnlich wie Brian Greene (Hervorhebung im Original): Erstens ist die Entropie ein Maß für die Menge der Störung in einem physischen System. . Zweitens in physikalischen Systemen mit vielen Konstituenten. Es gibt eine natürliche Evolution in Richtung größerer Störung, da Störung in so viel mehr Möglichkeiten als Ordnung erreicht werden kann. In der Sprache der Entropie ist dies die Aussage, dass sich physikalische Systeme dazu neigen, sich zu Zuständen höherer Entropie zu entwickeln. Im Stoff des Kosmos, 2 p. 154. Die Tendenz physikalischer Systeme, sich zu Zuständen höherer Entropie zu entwickeln, wird als zweites Gesetz der Thermodynamik bezeichnet. Ibid, 2 p. 156. Oder betrachte Paul Davies, ein anderer bekannter und populärer Schriftsteller: Das sogenannte zweite Gesetz der Thermodynamik wird oft mit der Aussage formuliert, daß jedes geschlossene System zu einem Zustand der totalen Unordnung oder des Chaos neigt. . Ein Maß für den unbarmherzigen Aufstieg des Chaos nutzt eine Menge, die Entropie genannt wird, die im Großen und Ganzen der Grad der Störung in einem System definiert ist. Das zweite Gesetz besagt dann, dass in einem geschlossenen System die totale Entropie niemals am besten abnehmen kann, bleibt sie gleich. Fast alle natürlichen Veränderungen neigen dazu, die Entropie zu erhöhen, und wir sehen das zweite Gesetz bei der Arbeit um uns herum in der Natur. Eines der auffälligsten Beispiele ist die Art und Weise, in der die Sonne ihren Kernbrennstoff langsam verbrennt, Wärme und Licht unwiederbringlich in die Tiefen des Raumes spuckt und die Entropie des Kosmos mit jedem befreiten Photon anhebt. Irgendwann läuft die Sonne aus dem Treibstoff und hört auf zu leuchten. Die gleiche langsame Degeneration leidet alle Sterne im Universum. In der Mitte des neunzehnten Jahrhunderts wurde dieses düstere Schicksal als der kosmische Hitzetod bekannt. In Zeit, 3 p. 34. Wie wir sehen, bekunden bekannte Autoren und Wissenschaftler ihre Worte nicht, wenn es darum geht, die Zunahme der Entropie mit der Zunahme der Unordnung zu identifizieren. Beachten Sie, wie Davies, im letzten Auszug, in der gleichen Atem aus der Konfigurationsversion, die Entropie als Störung auf die thermodynamische Version, die Entropie als Hitze Tod sieht (im Wesentlichen als der Verlust der Fähigkeit, nützliche Arbeit zu produzieren) sieht. Jedoch, einige andere, nicht-wie-bekannte Autoren bevorzugen eine unüberbrückbare Lücke zwischen thermodynamischen und Konfigurationsentropie zu sehen. Sie sagen, das zweite Gesetz ist über Thermo-Dynamik, und Thermo bedeutet Hitze, vergessen Sie nicht, dass die Leute Sie denken, die Beobachtung, dass sowohl die Konfiguration und thermodynamische Entropie neigen dazu, zu erhöhen ist ein neugieriger Zufall, eine bloße Analogie, und dass wir sollten nicht zu viel in diese zu lesen Analogie. Sie weisen darauf hin, dass die thermodynamische Entropie in bestimmten Einheiten gemessen wird, nämlich Joules pro Grad Kelvin, während die Konfigurationsweise eine bloße Zahl ist, ohne dass Einheiten anhängen. (Die Konfigurationsentropie ist der Logarithmus aller möglichen Anordnungen, z. B. von Molekülen, die zu ununterscheidbaren Konfigurationen führen, daher ist es eine reine Zahl.) Dann merken sie auch, dass die Zunahme der thermodynamischen Entropie unsere Fähigkeit, nützliche Arbeit zu leisten (z ), Aber es gibt keine solche Vorstellung im Falle der Konfigurationsentropie. Die meisten lautstarken unter den Wissenschaftlern war Frank L. Lambert, ein pensionierter Professor Emeritus der Chemie am Occidental College, Los Angeles, Kalifornien, der buchstäblich jeden Eck gefüllt hat und jede Wippe, die das 2. Gesetz der Thermodynamik im Internet betrifft und gegen die Kämpfe kämpft Idee, dass die thermodynamischen und Konfigurationsentropien verwandt sind. Nach diesem Wikipedia-Artikel ist er bekannt für seine Befürwortung, die Definition der thermodynamischen Entropie als Unordnung von US-Generalchemie-Texten und deren Ersatz zu ändern, indem man Entropie als Maß für die Energieverbreitung betrachtet. Der folgende Auszug aus dem Wikipedia-Artikel zum zweiten Gesetz der Thermodynamik sieht aus, als ob er direkt von Professor Lamberts eigene Webseiten genommen hätte. Das Konzept der Entropie in der Thermodynamik ist nicht identisch mit dem gemeinsamen Begriff der Unordnung. Zum Beispiel wird ein thermodynamisch geschlossenes System bestimmter Lösungen schließlich von einer trüben Flüssigkeit zu einer klaren Lösung mit großen, geordneten Kristallen umwandeln. Die meisten Menschen würden den ehemaligen Staat als mehr Unordnung als der letztere Staat charakterisieren. Im rein thermodynamischen Sinne hat sich die Entropie in diesem System jedoch nicht verringert. Die Maßeinheiten der Entropie in der Thermodynamik sind Energieeinheiten pro Einheit der Temperatur. Ob ein Mensch einen Zustand eines Systems als ordnungsgemäßer als einen anderen wahrnimmt, hat keinen Einfluss auf die Berechnung dieser Menge. Die gemeinsame Vorstellung, dass die Entropie in der Thermodynamik einer populären Auffassung von Unordnung gleichkommt, hat viele Nicht-Physiker dazu veranlasst, das, was das zweite Gesetz der Thermodynamik wirklich betrifft, völlig falsch zu interpretieren. Nun, wenn Professor Lambert Recht hat, zusätzlich zu den Nicht-Physikern, sieht es aus wie ein paar echte Physiker (einschließlich solcher Leuchten wie Stephen Hawking, Brian Greene und Paul Davies) auch geschafft, völlig falsch zu interpretieren, was das zweite Gesetz der Thermodynamik ist Wirklich, während Prof. Lambert (ein Chemiker) tiefer Entropie und das zweite Gesetz verstanden hat. Könnte das wahr sein, aber Professor Lambert ist bei weitem nicht allein in seiner Ablehnung des Zusammenhangs zwischen thermodynamischer und konstruktiver Entropie. Es gibt zahlreiche andere Wissenschaftler, die mit ihm einverstanden sind, da seine erfolgreiche Kampagne, um amerikanische Chemie Lehrbücher aus dem Makel dieser Verwirrung zu reinigen zeigt. Doch andere ziehen ganz andere Schlussfolgerungen ab der Annahme, dass die thermodynamische und die Konfigurationsentropie nicht miteinander verknüpft sind. Zum Beispiel, Brig Klyce, in dieser Webseite. Argumentiert wie folgt: Er stimmt zu, dass die Erde nicht thermodynamisch ein geschlossenes System ist, da sie einen Zustrom von Energie von der Sonne empfängt, aber er sagt, dass es ein geschlossenes System sein muss, das konfigurativ (dh soweit Ordnung betrifft), da thermodynamisch und Konfigurationsentropie (oder: Energie und Ordnung) sind nicht verwandt: Der Zustrom von Energie kann nicht zu einer Erhöhung der Ordnung führen. So versucht Klyce, die Zunahme der biologischen Ordnung auf der Erde zu erklären, und behauptet, dass es einen anderen Ursprung als die von der Sonne erhaltene Energie haben muss. Er nutzt dieses Argument, um den Begriff der Panspermia zu unterstützen, die Hypothese, dass sich das Leben nicht auf der Erde entwickelt hat, sondern hier aus dem Weltraum gespritzt wurde (eine Theorie, die die Frage, wo und wie das Leben zuerst erschien, unbeantwortet ist, aber das ist ein anderes Thema) . Er kommt zu dem Schluss, dass das Leben solche Injektionen der Ordnung von außerhalb der Erde erhalten hat, und deshalb ist es sehr geordnet. Seine Sichtweise kann natürlich auch dazu verwendet werden, die schöpferische Auffassung zu unterstützen, dass das Leben nicht durch natürliche Selektion entwickelt wurde, sondern auf der Erde von einem intelligenten Designer geschaffen wurde. (Klyce hat mir so viel zugesprochen, in persönlicher Kommunikation). So sehen wir, dass die Behauptung über die Verwandtschaft zwischen der thermodynamischen und der Konfigurationsentropie über die Physik hinausgeht. Die Entschließung Im Folgenden werde ich zeigen, dass die beiden Entropieversionen Konsequenzen eines tieferen mathematischen, tatsächlich statistischen Ergebnisses sind, wo tiefer nicht unbedingt schwerer zu verstehen ist, dass es eigentlich ganz einfach zu begreifen ist. Also das Hawking - et-al. Die Ansicht wird gerechtfertigt sein, während der Antrieb, um amerikanische Chemie-Lehrbücher umzuschreiben, sich als sinnlos erweisen wird. Aber bevor ich einen mathematischen Theorem sage und es beweise, ziehe ich es vor, den Leser mit einer Beobachtung angemessen zu motivieren. Lets einen weiteren Blick auf die Figur, die die Moleküle zeigt, die sich im Raum verteilen (noch einmal drücken): Was haben wir hier Sind diese realen Moleküle, die sich im Raum verteilen. Natürlich nicht. Dies sind nur einige simulierte Moleküle, bloße Kreise, die lila gemalt sind, die von einem Programm bewegt werden, das läuft, wenn man diese Web-Seite lädt und die Taste drückt. In der Tat warten eine Minute: diese Kreise sind nicht einmal umhergezogen. Kommen Sie, um darüber nachzudenken, nichts bewegt sich hier. Der Bildschirm von Ihrem Computer enthält einige Pixel, und das Programm manchmal malt einige von ihnen lila, andere Male es malt die gleichen Pixel weiß, und so weiter. Pixel bewegen sich nicht auf deinem Bildschirm, sie sind einfach an festen Orten. Das Programm trifft Sie zu glauben, dass Kreise bewegen sich durch Malerei Pixel in verschiedenen Farben, zu geeigneten Zeiten. Aber ich schwöre, ich habe nichts im Programm mit dem ausdrücklichen Zweck getan, dich in den Glauben zu bringen, dass es Kreise gibt, die sich verteilen. Alles was ich tat war, dass ich das Programm gebeten habe, einen lila Kreis um jeden Punkt zu zeichnen, indem er zuerst Punkte in einem ordentlichen arrangierte Mode (100 Punkte in einer 10 x 10 Matrix) und verschieben dann jeden Punkt einzeln, unabhängig davon, wo die anderen Punkte sind, zu einem benachbarten Ort. (Nachbarschaft bedeutet eine Stelle in einem festen Abstand von der gegebenen, in einem zufälligen Winkel also die neue Stelle ist irgendwo auf dem Umfang eines Kreises zentriert an der vorherigen Stelle und mit Radius ein gegebener fester Abstand.) Ich gab keine explizite Befehl Auf Punkte zu verteilen, gab es keinen Befehl, um sie von der Mitte der Figur wegzukommen. Und doch tun sie genau wie die echten Moleküle eines Gases in einem realen Raum. Warum was ist die Verbindung Okay, die Moleküle sind physische Dinge, sie tragen ihre Masse im eigentlichen Raum, und wir könnten versuchen zu untersuchen, warum sie sich durch die Anwendung der bekannten Gesetze der Physik ausbreiten. Aber diese Punkte Warum auf der Erde zerstreuen sie auch Wir haben keine Physik, hier zu arbeiten, alles geschieht im virtuellen Raum, in einem Computer. Natürlich kann man behaupten, dass es etwas Physisches gibt, das diese Punkte umsetzt und sie als lila Kreise malt: die Bits in den Chips deines Computers Gedächtnis. Sicher, aber die Beziehung zwischen Pointcircles und Bits in Computer-Chips ist so ätherisch, dass man nicht herausfinden kann, welche Bits was machen? Infolgedessen verteilen sich die Bits nicht in Ihren Computern. Darüber hinaus sind Ihr Computer und seine Hardware völlig irrelevant, denn dieses Programm könnte auf einer Turing-Maschine ausgeführt werden, abstrakt, mathematisch, wie jeder erste Jahr Student der Informatik wird Ihnen versichern. Etwas anderes ist hier los, was nichts mit der Physik zu tun hat: Es gibt weder reale Moleküle im Raum noch Steine, die hier abkühlen. Wir haben ein drittes Beispiel der Verbreitung oben, eine dritte Art von Entropie, unabhängig von den beiden anderen physischen. Etwas wie Entropie und das zweite Gesetz der Thermodynamik können auch abstrakt, mathematisch, ohne jede materielle Umsetzung beobachtet werden, ohne physische Unterstruktur, um es real zu machen. Hmm Zu viele Entropien, denkst du nicht zu viele, um alles nur Analogien von einander zu sein, durch Zufall. Wir sollten mit so vielen Zufällen verdächtig werden und nach etwas tieferem Ausschau halten, das alle zugrunde liegt und vereint. Und weil diese dritte Art von Entropie nicht einmal körperlich ist (sollten wir sie als virtuelle Entropie bezeichnen), so können wir uns auch einen Augenblick über die Physik vergessen und versuchen, diese dritte Art zuerst zu erklären, weil sie reiner ist, ohne das Materialgepäck Der beiden anderen Fälle. Nachdem wir erklärt haben, wie diese virtuelle Ausbreitung geschieht, können wir sehen, wie es physikalische Implementierungen erwirbt, die die thermodynamischen und Konfigurationsfälle der Entropie erklären. So sehen wir: Was wir erklären wollen, ist, warum ein bestimmter Punkt im Raum (einer dieser Kreise), der einen zufälligen Spaziergang durchführt, sich von einem gegebenen ursprünglichen Ort im Durchschnitt entfernt. Diese letzte Qualifikation ist sehr wichtig: Es muss eine durchschnittliche Distanz von der ursprünglichen Lage sein, denn ein zufälliger Spaziergang impliziert, dass der Punkt wieder zu seinem ursprünglichen Standort kommen könnte, für alles, was wir wissen. Aber wenn wir das Experiment eine große Anzahl von Malen wiederholen und die Orte des Punktes bei jedem Zeitschritt ausgleichen, sollten wir feststellen, dass der Punkt seinen Abstand von seinem ursprünglichen Standort erhöht. Aber warum . Nun, hier ist eine einfache qualitative Erklärung: Die obige Abbildung zeigt einen Punkt, der mit einem lila Punkt an den Koordinaten (x, y) markiert ist. Der Punkt soll seinen zufälligen Spaziergang am Ursprung begonnen haben, der mit (0, 0) markiert ist, und erwägt nun, zufällig irgendwo auf dem Umfang des schwarzen Kreises zu springen, denn das ist, wie es sich bewegt: zu jeder Zeit ist es bei Irgendeine Stelle (x, y), und bei der nächsten Zeiteinheit findet sie sich irgendwo in einem Abstand von 1 Raumeinheit weg von (x, y), der sich in einem zufälligen Winkel bewegt hat. Hier ist die entscheidende Frage: Obwohl alle Punktdestinationen auf dem Umfang des schwarzen Kreises eine gleiche Wahrscheinlichkeit haben, den Punkt zu erhalten (das ist ein gegebener), wie viele von ihnen implizieren, dass der Punkt sich von (0, 0) entfernen wird und wie Viele, dass es sich näher an sie wenden wird Die folgende Figur beantwortet diese Frage: Diese Punkte, die implizieren, dass der Punkt (x, y) sich weiter weg vom Ursprung (0, 0) bewegen wird, wurde in roter Farbe markiert und diejenigen, die das implizieren (X, y) näher an (0, 0) verschoben werden, wurden in grüner Farbe markiert. Es ist nicht so, dass die roten Punkte mehr sind als die grünen (sie sind beide unendlich in der Zahl), aber die Länge des roten Bogens ist länger als die Länge des grünen Bogens, und so hat (x, y) eine größere Wahrscheinlichkeit zu beenden Oben auf dem roten (weg) eher als auf dem grünen (näheren) Teil des farbigen Kreises. Das obige ist die qualitative erklärung, aber es gibt auch eine quantitative. Wir können berechnen, wie schnell der Punkt (x, y) sich von dem Ursprung (0, 0) im Durchschnitt abnimmt, unter der Annahme der obigen Bewegungsregeln, vorausgesetzt, wir machen die Dinge formal. However, if the reader feels uncomfortable with formulas and math, please note that there is nothing essential to be missed if the proof of the following theorem is skipped just make sure you read the theorem itself and understand its statement, because it is of central importance in this whole discussion. Definition. A 2D random walk is an infinite sequence of points p 0 . p 1 . on the 2-dimensional Euclidean plane such that each point p k has a distance of 1 from its previous point p k 1 . for all k gt 0. Point p 0 is called the origin of the walk, and point p k is called its k-th step . The definition implies that we dont care about the direction of the straight line defined by points p k and p k 1 . therefore the direction of this line is random hence the term random walk. Theorem (of dispersion in 2D space ). The expected distance between the origin p 0 and the n - th step p n of a 2D random walk is equal to . (Here, expected distance refers to the mean value of distances from the origin p 0 of the n - th steps of a large number of 2D random walks that share a common origin p 0 .) Proof: Think of the 2D plane as the plane of complex numbers, and place the origin p 0 at (0, 0). Let (x, y) be the coordinates of the n - th step p n in a 2D random walk. Therefore, as a complex number z, point p n would be written as z x i y. Recall that each complex number x i y can also be written using Eulers exponential notation as follows: where z is the modulus, or distance of z from the origin p 0 0 i 0, and u is the phase, or angle between the x - axis and the line that connects z with the origin p 0 . Now, if we fix the modulus z to the value 1 (since point z p n differs by this fixed distance from its previous point p n - 1 in the random walk) and allow the phase u to vary randomly in the interval 0, 2) (since the angle between successive points is arbitrary), then the new position z of (x, y) after n steps on the complex plane must be given by the following sum: Recall that the absolute square w 2 of a complex number w (i. e. the square of its distance from the origin) is equal to w w where w is the conjugate of w, i. e. w x i y w e - i u . So the absolute square of z in the above formula, which is equal to z z . is given by: Now, lets try to compute the mean value of the quantity z 2. We will use angle brackets (lt gt) to denote mean values. We have: Since both angles u j and u k are random variables with identical means, their difference ( u j u k ) is also a random variable with mean 0 (zero). This means that the whole formula of the mean value to the right of the plus sign, above, is 0 (zero). Thus, simplifying we get: Hence, taking the square root on both sides, we find that the root-mean-square distance z rms after n unit steps is: The root-mean-square is the average distance of z (or point p n ) from the origin p 0 . which is what we wanted to show. If n represents time in the above calculations, then the theorem tells us that at time n the point that performs a random walk is expected to be found at an average distance of from the origin (point of departure). This result is actually the same as found by Einstein in his famous papers of 1905 and 1906 on Brownian motion, 4 except that Einsteins calculations and notation are much harder to follow. The notation used in the above proof was taken from Weisstein, 1997 (p. 1524). 5 The above dispersion theorem tells us why the circles that perform random walks and are drawn by the program disperse on average, even though the program does not give them any explicit command to do so. The theorem explains what I earlier called virtual entropy. Now its not too difficult to see how the configurational and thermodynamic dispersals (and their associated notions of entropy) are simple physical implementations of the above mathematical result. The configurational case is trivial to see. We assumed molecules of a gas in a container covered with a lid, placed at the center of an isolated room that contains a different gas. Suppose the pressure in the two gases is identical. The molecules of the gas in the container (as well as those outside) perform approximations of random walks as they meet and bounce off one another. When we open the lid of the container the molecules of its gas keep performing random walks, but now they are permitted to meet and bounce off any kind of molecule: either of the same gas, or of the gas outside. So they disperse in space just as the purple circles in our simulation do, for the statistical reason described in the dispersion theorem, even though they dont move at a fixed distance every time, and even though they are not restricted to move on a 2-D plane. (What was proven by the theorem in two dimensions holds also in three dimensions, except that the average speed of dispersal in not equal to but slower. The proof is considerably more complex, thats why it was given in two dimensions and with a fixed distance of movement.) People often become confused with variations in the physical details of this experiment. They consider the room empty of matter (a vacuum), so when the lid is opened the molecules of the gas in the container swoosh out and quickly spread throughout the room without performing random walks. Okay. The molecules of a gas under normal temperature move at dizzying speeds all the time anyway. (According to this article by Prof. Lambert, At the usual lab temperature, most water molecules are moving around 1000 miles an hour, with some at 0 and a few at 4000 mph at any given instant.) We are not aware of their speeds under normal pressure because they meet other molecules and bounce off before they have a chance to go too far. If, however, they find that their way is free of obstacles (as in a near-vacuum), then of course theyll rush unhindered at their dizzying speeds, and thats what will cause the swooshing in the void of the room when the lid is opened. There is nothing mysterious to explain here. Practically the same phenomenon appears when you open a soda can: the great difference in pressure between the molecules of CO 2 in the can (highly pressurized) and the molecules of air outside (in lower pressure) causes the CO 2 molecules to swoosh out of the can and disperse in the air (especially if you have shaken the can and thus increased the internal pressure, hence the speeds of the CO 2 molecules). Same phenomenon. The vacuum, or the low pressure, simply speeds up the rate at which the molecules disperse. The following simulation shows precisely that (click on ): Here, the simulated molecules make longer jumps before they bounce, as they would if they could move in a relatively empty space. The result is the explosion that you see when you run the program, and the much faster filling up of the available space. Now lets turn our attention to the thermodynamic case (hot stone thrown in cold water), because that is what professor Lambert and others dispute as having any relation with the case of configurational entropy. When a hot stone is placed into cold water the molecules of the water acquire energy (well see what that means) and vibrate more vigorously. But because they are molecules in a liquid, they can disperse in the volume of the rest of the water. I prefer to avoid this dispersal of molecules in our thought experiment, because I want the thermodynamic case to appear as different as possible from the previous experiment with the gas molecules in a container that disperse in a room. So Ill propose a small modification in our thermodynamic setup an improvement, actually. Suppose that instead of water we have a piece of concrete. This concrete chunk has a square-shaped hole at its center, and the hot stone is another piece of concrete, which goes and fits neatly and perfectly () into the square hole. Hot concrete goes and fits into the hole of cold concrete, thats all Im saying. (Our figure, below, remains identical: suppose the black area is the cold concrete, and the red is the hot piece.) Now there can be no dispersal of molecules, and yet the second law of thermodynamics guarantees that the smaller hot piece will cool down, the larger surrounding chunk will warm up, and the two will come to a point of thermodynamic equilibrium after a sufficiently long interval of time. The earlier figure is repeated below, for our convenience. Lets think now: how does that happen What is the mysterious energy that flows out of the hot piece What does it consist of How does it flow and why does it disperse In this section well see the correct explanation of how energy (or heat) disperses, and in the next section well review some bogus explanations that have been proposed by others. When the red-hot piece of concrete is placed in the square hole in the middle of the cold chunk of concrete, what kind of interaction can occur between the two bodies at the molecular and quantum level Well, what we know is that some of the vigorously vibrating molecules of the red-hot piece (those that are at its outermost region, its periphery), come into contact with molecules of the cold piece. But contact is not the right term when we talk about molecules, since touching is something that makes sense only at the macroscopic level. At the micro scopic level, the atoms of some of the hot (vigorously vibrating) molecules will approach the atoms of some of the cold (less vigorously vibrating) molecules. Fine, so atoms will approach atoms. Again, atoms cannot touch each other microscopically, so when we speak of an approaching at the atomic level we mean that the electrons of the outermost shells of those atoms will come close together (always while in vibration). So what happens when electrons approach electrons Quantum theory says that electrons that come close together exchange virtual photons. If the electrons were free in space, they would scatter at random angles after this exchange. Now that they are bound to the nuclei of atoms by means of the electromagnetic force, most probably they will continue being bound to their atoms, but their mutual bouncing off will cause their respective atoms to bounce off, too. (This mutual repelling due to the electromagnetic force is the reason that solid objects like ourselves stay on top of other solid objects, like chairs and floors, and do not pass through them.) What interests us is that because one of the two electrons (assume only two of them interacting, for simplicity) moves faster in space than the other one (because it follows the vibrations of the atom it belongs to, which is hotter than the other one), quantum theory says that there is a higher probability that virtual photons will go from the fast-moving electron to the slow-moving one, so that the former will lose some of its energy, whereas the latter will gain some. As always in the quantum world, we talk about probabilities, not deterministic events. But what concerns us is the average case, and on average the fast-moving electron will send one or more virtual photons to the slow-moving one. This is one mechanism by which the mysterious energy is transmitted from one material to the other: this kind of energy is a flow of virtual photons, which cause the electrons hence their respective atoms to recoil. But there is more. The hot piece is depicted in red color on our drawing, right And in reality, red-hot things are, well. red. They are red for the following reason. Their highly excited electrons spontaneously drop to lower-energy levels, emitting photons as they do so. (Real photons, not virtual ones.) The more excited the electron, the higher the probability that the photon will have a small wavelength (high energy). Higher energies mean photons with wavelengths possibly in the visible range, such as in the red part of the electromagnetic spectrum (and even further toward the orange and smaller wavelengths, depending on how hot the object is). As the hot body cools down (we havent seen yet how), the emitted photons are of longer wavelengths, and so are shifted toward the infrared. Thats why the red-hot piece becomes first dull-red as it cools down, then its color fades more, until essentially all the emitted photons are in the infrared, so we dont see them anymore (consequently we see the natural color of the body at lower temperatures, be it black or gray, whatever is reflected by ambient light). As I mentioned, these emitted photons are not virtual but familiar photons that would be registered by our retinas if we could see them. But we cant see them because they are in the closed system of the two concrete pieces of our experiment. So they are emitted within the molecules of the material, and cant go too far because these are chunks of concrete we are dealing with, and concrete is opaque to light (less so to infrared photons, but still relatively opaque). So the emitted photons travel short distances before being absorbed by electrons of neighboring molecules, which they might excite and cause to jump to higher energy levels. Then those excited electrons might emit a photon again and drop back to a lower energy level. The photons are emitted at random angles. So, although we cant talk about the same photon moving from molecule to molecule (or from atom to atom, or from electron to electron), the net result of all this is that there is something like a random walk of photons . Hmm. a random walk. This is clearly the case with normal photons. But the virtual photons, too, do something analogous, because they are also exchanged between electrons at random angles. So this is what the mysterious flow of energy is: its photons that perform random walks. Energy in the context of our experiment is not a substance made of some mysterious and otherworldly material, but a convention for the wavelength of photons: the shorter the wavelength, the higher the energy. At a macroscopic level its often useful to model energy by the quantity of an abstract substance (e. g. temperature), because this allows us to solve conveniently problems about objects and processes in the macro-world. But down at the microscopic level of description, in our thermodynamic experiment, energy is associated with the wavelength (or frequency) of photons, i. e. photons that, as I said, perform random walks (or behave as if the same photon performs a random walk). () Therefore, random walks are the reason why energy disperses within the cold body until there is an equilibrium, and the dispersion theorem models the situation. And this discussion tells us that the two figures that were juxtaposed at the top of this page essentially simulate the same phenomenon: the figure on the left (the two blocks of concrete) shows the situation macroscopically, with photons performing random walks and dispersing within the material and the figure on the right shows the situation microscopically, depicting individual molecules of a fluid that perform random walks and disperse in a box. In the latter case, what we have is the dispersal of matter. In the former case we are tempted to say that we have the dispersal of energy but the reason I keep putting the word energy in quotes is because I want to emphasize that even in this case we still have the dispersal of matter. For, is a photon immaterial Of course not, its a little lump of matter in the generalized sense, the sort of massless matter that we prefer to identify with energy (but which can be assigned a non-rest mass m through the relation m E c 2 , where E is the energy of the photon). The useful distinction that can be made is that the configurational case involves the dispersal of matter in the form of mass, whereas the thermodynamic case involves the dispersal of matter in the form of energy . But, by whatever name and form it goes, generalized matter disperses in spacetime when its quanta perform random walks, and the reason is not physical, but mathematical, given by the dispersion theorem. Note please: when I say the reason is not physical I dont mean its supernatural I mean that the laws of physics as we know them (including Newtons laws of motion, quantum mechanics, and everything we know about the four forces of nature), do not suffice to explain the reason for the average dispersal of random-walking particles. An extra-physical, a mathematical result is needed to explain this phenomenon. Of course, if we include this mathematical result into the notion laws of physics, then we can again say that the laws of physics in this inclusive sense explain fully the phenomenon. () I hope the above discussion explains sufficiently the claim that I made earlier: the two entropies, thermodynamic and configurational (and even the third kind that I briefly referred to as virtual earlier), are manifestations, or implementations, of a deeper mathematical result. The second law of thermodynamics () is, similarly, at work whether we talk about energy or mass dispersing in spacetime. Consequently, the drive to eliminate references to configurational entropy in American chemistry textbooks is utterly meaningless it actually disseminates knowledge superficially to students because they dont see the deeper mechanism that is responsible for material dispersal, but only one aspect of it, as it appears in the thermodynamic case. However, although the relation between thermodynamic and configurational entropy is unassailable, there is still confusion about the role of order and disorder in this context. The Confusion Let us see a couple of explanations for the second law of thermodynamics that have been given, which ignore the dispersion theorem and the deeper relation between thermodynamic and configurational entropy. First, consider the configurational case, and Brian Greenes explanation for why molecules (or other material things in general) disperse in space. Greene first presents the notion of entropy by asking the reader to imagine tossing an unbound copy of Tolstoys War and Peace high into the air, letting the 693 double-sided loose pages drop on the ground, and then collecting them one by one, without looking at their page numbers. What is the probability that the pages will be collected in their correct order Greene calculates that there are about 10 1878 different out-of-order page arrangements (and presents the entire 1878-digit number using the better part of p. 152 of his book). He observes that there is only one correct (or desired) order, so the probability to pick up the pages in the correct order and keep reading about Anna Pavlovna and Nikolai Ilych Rostov (and understanding what is being read) is about 110 1878. i. e. vanishingly small. So far so good. Now lets see how he explains a more typical experiment that is often described in the context of entropy and the second law of thermodynamics. Consider opening a bottle of Coke. (Any other plastic soda bottle or other type of plastic bottles wholesale may be used too.) Gas, like CO 2 . is initially confined in a small space in the bottle. After we open the cap of the bottle, the molecules of CO 2 spread evenly in the room. Here is how Greene explains this: When you twist off the bottles cap . you open up a whole new universe to the gas molecules, and through their bumping and jostling they quickly disperse to explore it. Why Its the same statistical reasoning as with the pages of War and Peace. No doubt, some of the jostling will move a few gas molecules purely within the initial blob of gas or nudge a few that have left the blob back toward the initial dense gas cloud. But since the volume of the room exceeds that of the initial cloud of gas, there are many more rearrangements available to the molecules if they disperse out of the cloud than there are if they remain within it. On average, then, the gas molecules will diffuse from the initial cloud and slowly approach the state of being spread uniformly throughout the room. Thus, the lower-entropy initial configuration, with the gas all bunched in a small region, naturally evolves toward the higher-entropy configuration, with the gas uniformly spread in the larger space. In The Fabric of the Cosmos, pp. 155156. But there is a glitch in the above explanation. Okay, the initial gas cloud is confined in a smaller space than the space of the entire room. But why would a molecule choose to move away from the cloud What pushes it there, to the rest of the room One might counter, the highly pressurized state of the gas in the bottle pushes it, i. e. the vigorous bumping and jostling with the other molecules of the gas cloud. Yes, but we dont have to imagine a pressurized gas. The same phenomenon will be observed if the pressures of the gases inside and outside the bottle are identical: still the gas-inside will spread in the rest of the room. So, again: what pushes molecules to explore the rest of space, as Greene puts it, even under a lack of pressure differential Merely because there are many more rearrangements available to the molecules if they disperse out of the cloud than if they remain within it is not an explanation, because molecules dont care about numbers of rearrangements and opportunities given to them to explore some terra incognita they merely move randomly in space For all we know (if we ignore the dispersion theorem), they could be roaming forever around their original location, and thus staying within the gas cloud. Greene comes close to the qualitative explanation given earlier on this page (Im referring to the figure of the colored red-and-green circle, just before the theorem), but he doesnt quite put his finger on it. Greene gets distracted (and distracts the reader) with the pressure differential, instead of concentrating on an example with a lack of such differential. If I shoot a bullet with a gun against a target, it should be of little wonder if I see the bullet hitting the target but if I simply take a bullet in my hand, then let go, and see it floating away from my fingers, then I am confronted with a phenomenon that requires a nontrivial explanation. Next, consider the thermodynamic case and Prof. Lamberts crusade to purify American chemistry textbooks from the configurational blemish by severing the relation between thermodynamic and configurational entropy, making reference only to the thermodynamic case in serious, scientifically approved textbooks. Lambert attempts to explain the second law of thermodynamics (i. e. the statistical increase in thermodynamic entropy) in this web-page by means of a dialog between a Professor (presumably himself) and an imaginary Student (perhaps his alter ego: a bombastic individual that should serve as an example of how students should not behave if theyre really interested in learning, as opposed to showing off their knowledge). Here is the Professors explanation of the microscopic reasons why energy disperses in space. First, the Professor observes that molecules of water (he uses water as a typical example) move in three different ways or, better stated, their motion has three degrees of freedom, or components: a translational component by which they change their location in space a rotational component, by which the entire H 2 O molecule rotates around some axis (which can change in time) and a vibrational component that concerns the bonds between OH atoms within the H 2 O molecule, by which the distance between such bonds periodically increases and decreases (extremely fast by our human temporal standards, of course). The following figure is from the above-referenced web page, and shows the three motional components for a water molecule. What the Professor never states explicitly in his discussion with the Student is the obvious observation that, of the three components of molecular motion, only the translational one can be suspected (held responsible) for the dispersal of energetic molecules in space. Clearly, no matter how fast a molecule rotates, or how fast the bonds between its atoms vibrate, it will not be translated in space. Thus two-thirds of the Professors description of molecular motion are irrelevant for the purposes of explaining dispersal. Then the Professor goes on to explain to the Student how each of the three motional components is quantized, i. e. there are only specific and discrete values for the angular momentum of the rotational component: its not that the molecule can change its rotational speed along a continuum of values. The same is true for the vibrational and translational components. The Professor observes that the available quantized (discrete) values of the translational component are way more (really-really way more) in number than the possible discrete values of the rotational and vibrational components. But since the latter two are irrelevant for the explanation of dispersal, I will ignore this distinction in the numbers of quantized values. Now comes the crucial part in the Professors explanation. He says, take a snapshot of the current state of motion of the water molecules. Call this a microstate . Thus, in a given snapshot, the translational component of molecule 477,275,846,375,832,218 has a certain value (I said Ill ignore the other two components) its neighboring molecule 477,275,846,375,832,219 has a different translational component and so on. Collect all those components for all the zillion molecules in the quantity of water under consideration, and you have your microstate . Here it is, in Professors own words: Imagine that you could take an instantaneous snapshot of the energy of all the individual molecules in a flask containing a mole of gas or liquid at 298 K. Remember that each molecules energy is quantized on a particular energy level. Then, each of the far-more-than Avogadros number of accessible energy levels (at that temperature and in that volume) could have zero, one, or many many molecules in it or on it. The whole snapshot showing each molecules energy of that mole is called a microstate the exact distribution on energy levels of the energies of all the molecules of the mole at one instant in time. Now consider what will happen next (says the Professor). The first of the above molecules will bounce against another one (usually a neighboring molecule, since this is water and its molecules cant move too far before being hit by others) and will change its translational component. But there are many available options for the new value of its translational component. And this is true for all molecules in the fluid. Therefore, over time, the molecules will explore the space (note: the abstract space, says I) of possible values of their translational components. Hence well end up with microstates that have their translational space widely distributed, as opposed to the single initial microstate. In Professors words: Since a collision between even two molecules will almost certainly change the speed and thus the energy of each one, they will then be on different energy levels than before colliding. Thus, even though the total energy of the whole mole doesnt change and even if no other movement occurred that single collision will change the energy distribution of its system into a new microstate Because there are trillions times trillions of collisions per second in liquids or gases (and vibrations in solids), a system is constantly changing from one microstate to another, one of the huge number of accessible microstates for any particular system. Then the Student asks a crucial question: What does more microstates for a system have to do with its energy being more spread out A system can only be in ONE microstate at one time. And the Professor answers as follows (my emphasis ): Yes in only one microstate at one instant. However, the fact that the system has more choices or chances of being in more different microstates in the NEXT instant if there are more microstates for the system is the equivalent of being more spread out or dispersed instead of staying in a few and thus being localized. . You i. e. the Student already stated the most important idea, a single microstate of a system has all the energies of all the molecules on specific energy levels at one instant. In the next instant, whether just one collision or many occur, the system is in a different microstate. Because there are a gigantic number of different accessible microstates for any system above 0 K, there are a very large number of choices for the system to be in that next instant. So it is obvious that the greater the number of possible microstates, the greater is the possibility that the system isnt in this one or that one of all of those gazillions. It is in this sense that the energy of the system is more dispersed when the number of possible microstates is greater there are more choices in any one of which the energy of the system might be at one instant less possibility that the energy is localized or found in one or just a dozen or only a million microstates. It is NOT that the energy is ever dispersed over or smeared over many more microstates Thats impossible. So, what does energy becomes more dispersed or spread out mean so far as molecular energies are concerned Simple Whats the absolute opposite of being dispersed or spread out Right completely localized. In the case of molecular energy, it would be staying always in the same microstate. Thus, having the possibility of a huge number of additional microstates in any one of which all the systems energy might be in thats really more dispersed at any instant Thats what an increase in entropy on a molecular scale is. Damit. lets see. Suppose I have a marble ball in my hand, and there is a number of holes on the ground at a distance of about one yard from me. Each hole is large enough to let the marble fall inside, and suppose at most one marble can fit in each hole. I throw the marble forward, letting it roll on the ground toward the holes. Following the Professors logic (see the highlighted portion, above, and match it with what follows), although the marble can be in only one hole at one instant, because there is a large number of different accessible holes, there are a very large number of choices for the marble to be in that next instant. So it is obvious that the greater the number of possible holes, the greater is the possibility that the marble isnt in this one or that one of all those holes. It is in this sense that the positional state of the marble is more dispersed when the number of holes is greater. Does it make any sense No, not to me. A marble can be in one hole at a time, period. How can its potential for choosing from among a large number of holes send it to more than one hole And if you think the analogy with a single marble (single molecule) is misleading, okay, think of 10 marbles. Throw them forward, as before. Arent they going to end up in exactly 10 different holes Now take those 10 marbles out of their holes, step back, and throw them forward again. Arent they going to end up again in exactly 10 different holes 10 different holes, in general, to be sure. But why would the marbles disperse among the holes Unless, of course, we modify the marblesholes experiment as follows: perhaps after we remove each marble from its hole we dont step back, but place the marble just next to its hole, and give it a little kick toward a random direction. The marble then goes and falls into a neighboring hole. Then we continue like that, taking it out, and giving it another kick to a random direction. Then, the marble will perform a random walk, and the dispersal theorem tells us that if we repeat this many times, on average the marbles will disperse. But this modification cannot apply to the Professors energetic molecules, and here is why: Each molecule has a given value for its translational component, right Thats a given value for its kinetic energy level . (Note the KE in Prof. Lamberts figure, above: it stands for kinetic energy.) So it is reasonable to imagine that any given molecule changes its energy level with each kick that it gets from other, neighboring molecules. Its energy value performs a random walk in the abstract space of quantized kinetic energy levels. But why would this kind of random walk in the abstract energy space imply that the molecule will perform a random walk in physical space This is like saying that a person who is some times happier than other times (i. e. performs a random walk in an abstract happiness space) is expected to visit more places in the world than another person who stays at the same happiness level all the time. Why It doesnt compute. Its a non sequitur. The translational components of the moving molecules can indeed acquire many different values, i. e. kinetic energy levels. But it is not obvious at all that this will cause the energy of the molecules to disperse in physical space not unless we take into account the dispersal theorem, and the observation that, in the context of our experiment, energy is transmitted through photons, which are the agents that perform the random walks, from electron to electron. (It is interesting that the word photon does not appear even once in Professor Lamberts microworld explanation of energy dispersal.) What about order and disorder Prof. Lambert makes it very clear that all talk about order and disorder in the context of entropy and the second law of thermodynamics is wholly unjustified, an example of sloppy thinking, a Cracked Crutch For Supporting Entropy Discussions . This, in spite of the fact that well-known authors such as Hawking, Greene, and Davies (among others) seem to feel no compunction to talk in terms of order and disorder, as shown in the excerpts at the beginning of this page. Prof. Lambert begins as follows in the above-referenced article: To aid students in visualizing an increase in entropy many elementary chemistry texts use artists before-and-after drawings of groups of orderly molecules that become disorderly. This has been an anachronism ever since the ideas of quantized energy levels were introduced in elementary chemistry. Orderlydisorderly seems to be an easy visual support but it can be so grievously misleading as to be characterized as a failure-prone crutch rather than a truly reliable, sturdy aid. Prof. Lambert later proceeds with an example that, according to him, shows why the talk about order and disorder is a cracked crutch. He asks the reader to imagine a bowl with water and chunks of ice floating on its surface, (figure below, on the left). After some time, the ice has melted and the bowl contains just water in liquid form (figure, on the right). Ice floating on water in a bowl (left), and same bowl with plain water after ice has melted away (right) People who are being introduced to the notion of entropy perceive the water-plus-ice-chunks as a disorderly collection of objects, says Prof. Lambert, whereas they perceive the later plain water as a uniform substance, an ordered form. So they might think we have a counter-example here: a disorderly collection of objects turned into an orderly soup. So they get confused (he claims). Yes, I agree that people would get confused if what is order and what is disorder in a situation such as the above is described to them in the manner suggested by Prof. Lambert. But thats a wrong description. Learners can always be confused with wrong descriptions. A good educator must explain things in the right way. The order in the bowl with the floating chunks of ice is to be found in the configurations of H 2 O molecules that form the ice crystals within each chunk of ice. There are many fewer configurations of molecules that form ice crystals in chunks (thats order) than configurations of molecules that float free in the soup of pure water (thats disorder, see more below). So, having heard the proper description, the learner will see an order-to-disorder progression and no contradiction with the 2nd law of thermodynamics. This is not a unique case of an initially wrongly formed perception due to intuition. Without proper tutoring in physics, people tend to think that heavy objects fall faster than light ones even Aristotle was fooled on this one that the Sun turns around the Earth once every day, and that mass is identical to weight. But a wrong first impression cannot be a reason for abandoning the more informed physical description. Why do we say there is more order in the bowl with the floating ice, and less order (or more disorder) in the bowl with the plain water Because the former situation is analogous to the molecules of a gas being restricted in a small volume, as in the examples discussed earlier in this text, whereas the latter situation is analogous to the gas molecules being spread out everywhere in the available space (see figure, below). Ordered molecules, akin to ice crystals floating on water. Molecules in disorder, akin to a soup of molecules in liquid water. There is a subtle issue, however, when we say that the image on the left, above, is more ordered. The way I arranged the molecules (dots) in a square 10 x 10 grid, of course appeals to our sense of order. But why How can we make the notions order and disorder precise, so that even Prof. Lambert (and like-thinking scientists) will have no reason to claim that orderdisorder is not an un-physical notion, a mere psychological thing, a cracked crutch for the understanding of entropy What can be made precise are not exactly the notions of order and disorder, but the very closely related notions of compressible and incompressible configuration. The reason is this: to say that the molecules of the image on the left, above, are ordered we need the judgment of a person, who would notice that the molecules are arranged in a perfect matrix of 10 rows by 10 columns. I might have arranged the molecules in a diamond-like shape, or along the circumference of a circle, or make the 100 dots form 20 crosses of 5 dots each and place the crosses themselves on 4x5 matrix, or arrange them in an essentially unlimited number of other ways. In each such case, a person with enough intelligence and patience might see the pattern, and come up with a short description of the 100 dots. The person might, or might not see the pattern of dots that I selected, depending on how difficult the pattern would be. In some difficult cases the person might fail to discover the pattern. Consider for example the picture on the right, above. It might be that I placed the dots on such x, y coordinates that the expression xy 1 is a prime number. I didnt, but I could have done so. And nobody can guarantee that a person would succeed in discovering that relation. If a relation for the position of dots (a pattern) is discovered, we say the configuration can be compressed . The notion of compression refers to the fact that the configuration of dots can be described in a short way (e. g. 20 crosses of 5 dots each) otherwise we say the configuration is incompressible . But the subtle issue is: if it is incompressible, is it so because nobody succeeded in compressing it (although there might be some yet unknown way), or because there really isnt any way to compress it, even if God so to speak, i. e. a super-intelligent being attempted to do it The notion of incompressible information is directly related to the notion of randomness . We can think that the dots on the right-side figure, above, are randomly placed . Disordered, randomly placed, incompressible all these notions seem to refer to the same concept. But some mathematicians, physicists, and philosophers, would say that randomness (a. k.a. incompressibility, a. k.a. disorder), cannot be defined, because we can never know if a configuration of things (dots on the plane, numbers in a sequence, etc.) defies any description through a rule (making it patterned, or compressed, or non-random, etc.), or it just happens that no human being (or computer algorithm, etc.) has succeeded in compressing it yet. Since we cannot always know, they say, we dont have a definition. For instance, the physicist Heinz Pagels says the following: Andrei Kolmogorov, the great Soviet mathematician, thought he could define randomness by the criterion that if it took as long to state the rule, suitably transcribed into numbers, for the construction of the numerical sequence as the actual length of the sequence, then the sequence was random. However, finding the construction rule for the sequence depends on human cleverness, and we can never be assured that the rule we have found is the simplest one that gives the sequence. . A precise definition of randomness for finite sequences simply does not exist. In The Cosmic Code, 6 pp 8687. And yet, I want to make a proposal for avoiding this trap (the trap of not knowing whether a human might succeed compressing the configuration), and propose an objective mechanism that can serve as a definition of randomness (and of incompressibility, and of disorder, etc.). Take a very specific compression algorithm, as implemented in programs that zip our computer files. I have and use WinZip v. 8.1 on my PC, but the particular implementation is not important whats important is the algorithm . Suppose the algorithm is fixed once and for all, for this definition of randomness, and we disallow any tinkering with it otherwise we must understand that we tinker with our definition. It suffices that the algorithm does a pretty good job in attempting to compress any file (which is merely a sequence of bytes, i. e. numbers). Having fixed the algorithm, my definition of randomness is a function that takes as input any sequence of numbers (which can also be the coded form of a configuration of dots, or molecules, in space), and outputs a number in the interval 0, 1), which is proportional to the percent of compression achieved. (Divide the percent by 100 to convert it to the interval 0, 1)). The parenthesis after 1 means that the number can never be exactly 1, because that would mean the sequence would vanish completely, and such total compression is inconceivable. But the closer the number is to 0, the more random the input sequence is. The bracket before 0 means that a compression of exactly 0 is possible, and it means No compression at all was achieved. (OK, I admit the above definition of randomness is not useful at all for theoretical math purposes, since it relies on a commercial algorithm or on some compression algorithm in general, a specific one and so it is algorithm-dependent. But my purpose wasnt to provide a theoretically useful definition, but to show that a definition can exist, its simply not true that randomness is an indefinable notion.) Thus we see that, in this definition, random is not a black-or-white notion (its not that a sequence either is random or is not), but has a gradation: a sequence (or configuration) can be more random than another one, which can in turn be more random than a third sequence, and so on. Nor does the definition require human cleverness, since it is all done by a fixed algorithm. Of course, an intelligent human might come and say, Now look, this sequence that you declared quite random is really quite compressible, thus not as random as you think, because I can use the such-and-such rule by which it is compressed quite a lot. Fine. That means merely that the said person did not abide by our definition, but used a different approach, a different algorithm for achieving compression. One can always imagine a different algorithm. For instance, consider again the disordered dots on the right-side of the most recent figure (or the one below). One can state that the coordinates on which those dots stand are precisely the coordinates that, in that persons numbering of coordinates, correspond to an N x N matrix, and thus the dots are perfectly ordered. The person altered the algorithm by which we number coordinates, and thus arrived at a non-random configuration. But thats not impressive at all. The point is, given a fixed compression algorithm, how much can a sequence be compressed The less it is, the more random it is declared to be, with respect to the given compression algorithm. To be meaningful, the above definition depends on the assumption that the algorithm indeed performs reasonably well in compressing its input. But most commercial algorithms do have this feature. For example, consider the following three snapshots from our molecule-dispersal simulation: I started with an initial number of 50 x 50 2,500 dots. Snapshot (a), on the left, is taken shortly after the beginning, snapshot (b) a little later, and snapshot (c) well after the dots occupied the entire space. The original uncompressed size of each of the three images (i. e. stored as bit-mapped files) was 441,654 bytes. After compression with WinZip, the files acquired the following sizes: snapshot (a): 4,823 bytes snapshot (b): 6,442 bytes snapshot (c): 8,660 bytes Thereafter, any further essential compression was not achievable. So we see that the last of the three snapshots is the least compressible one, and the dots can be said to be the most randomly distributed in it. Note that even the last, least compressible image, is quite compressed in absolute numbers (from 441,654 bytes its squeezed down to 8,660). Thats because the 2,500 dots that I placed in the available space are still too few, and the space ends up having large expanses of emptiness (white color) even when the dots are nearly evenly distributed in it. Had I used 73,600 dots, which is half of the available pixels in the above space, I would end up with an essentially incompressible late snapshot. Also note that I didnt start snapshot (a) at a perfectly ordered configuration (a 50x50 matrix) because I wanted to show that compression is independent of our human conventions about when a configuration is intuitively called ordered. An excellent article describing the relation between randomness, compression, and the 2nd law of thermodynamics (among other fundamental notions) is Gregory J. Chaitins, Computers, Paradoxes, and the Foundations of Mathematics. 7 The article is written for the general educated reader it doesnt contain a single mathematical formula. By the way, in that article, Chaitin writes (my emphasis): Entropy measures the degree of disorder, chaos, randomness, in a physical system. A crystal has low entropy, and a gas (say, at room temperature) has high entropy. p. 169. Professor Lambert gives one more example in his cracked crutch article that requires special attention, and a bit more thought. In his attempt to show that in reality order does not always result in disorder, but the opposite can also happen, he brings up the example of some solutions that start out as a uniform soup, but in the course of time crystals are formed within the soup, and so we get the opposite situation of what I showed earlier, when I discussed the ice-crystals-to-uniform-water example. My most familiar example of this (which in fact I demonstrated to my daughters several years ago, as part of their world-of-nature education) is to dissolve as much salt as you can in water (resulting in a saturated solution) and then hang one or more little threads from the top, so that the tips of the threads are immersed in the solution. After about one day, cubic crystals of salt will form on the threads in the solution, and the crystals will grow as the days pass. So, there you have it: a disorder to order demonstration The subtle but essential difference of this experiment with what we have been discussing so far is that the molecules of water and salt in the solution do not perform random walks . Instead, there are inter-molecular forces between the molecules of the threads and the salt molecules that cause the latter to go and attach themselves on the former. Then more molecules of salt come and attach themselves on top of the previous ones. Salt has the property of forming cubic crystals in its solid state, so we see the crystallization of salt along the threads. If we do not allow material objects to perform random walks, then of course we can get around the order-to-disorder rule. This is precisely what natural selection has been doing on our planet for billions of years. Thats why we ended up with so many biological crystals, such as elephants, oak trees, human brains, and even the lowly bacteria. Even if the Earth-Sun system were a thermodynamically closed system (which it isnt), still evolution could occur and result in exquisite biological crystals, because molecules on Earth are not allowed to perform random walks, but obey an untold number of lawful (non-random) interactions due to chemical processes (and later due to biological, and later due to cognitive processes). Prof. Lambert is distracted by these non-random events, because in the thermodynamic case (where energy disperses) there isnt anything that can serve as an attractor of the carriers of energy, i. e. of photons, and force them to form photon crystals. When photons are let loose, loose theyll go, and there is nothing to keep them from spreading at least not anything in our immediate experience. But our imagination does not have to be limited by our immediate experience. Its possible to imagine a world in which even photons get trapped. For instance, imagine a small universe with a single regular star. The star has been shining for quite some time, so its photons have dispersed practically everywhere in that small universe. Trillions of years pass, and the star dies, turning to a tiny mass of non-luminous matter that withers away, particle after particle. So there is nothing but a uniform soup of photons in that universe, which of course is assumed to be a closed system. Now suppose we insert a black hole in that universe. (Dont ask me how we do it, by what physical means without violating the laws of physics, because this is not a thought experiment, but an exercise in imagination-gone-wild) What will happen then to the photons (to the energy ) in that universe Will they keep being distributed evenly in space No, because those photons that fall into the event horizon of the black hole will get trapped and disappear forever. Some other photons, in the vicinity of the event horizon, will be disturbed and swerve around the black hole before escaping from it for good, perhaps after first rotating a number of times around it depending on how close to the hole they happened to be. In any case, the even distribution of photons in spacetime will cease to exist. But an uneven distribution of energy, compared to a previously even one, means a decrease in thermodynamic entropy, at least for a while. Violation of the second law of thermodynamics In principle, yes. Unfortunately, the universe that I described is so contrived that we cannot call the above a thought experiment. But it serves to remind us that the only reason that thermodynamic entropy seems to be always increasing in our world is because there are no photon magnets around, so energy does not get trapped, it disperses unhindered, and we observe no violation of the second law. However, mass is different: mass does get trapped, so we observe both its dispersal (when other forces are weak enough to let it roam free) and its accumulation (when other forces take the upper hand). Therefore, configurational entropy, which concerns mass, can be reversed in many situations (such as in biological and material evolution). Another objection that is often raised by those who see an unbridgeable gap between thermodynamic and configurational entropy is that the former is measured in specific physical units: in joules per degrees Kelvin whereas configurational entropy has no such physical units to measure it: its a pure number. The objection goes further in saying that, even that pure number is not definite: it depends on how finely we partition the space in which the molecules (or other carriers of mass, or information) move. Consider first the objection that the thermodynamic entropy is measured in specific physical units, whereas the configurational one is just a number. Well, so what Nobody claims that the two versions of entropy are identical, or even isomorphic. If they were, we wouldnt need to distinguish the one from the other, and there would hardly be any contention among physicists. There are many other examples in physics in which two phenomena are consequences of a deeper law, and there are features that exist in one phenomenon that are measured in some particular physical units, but are absent from the other. For instance, consider the orbit of the Moon around the Earth, and the free fall of an apple toward the surface of the Earth. For a very long time the two phenomena were considered as different as two phenomena could be. But after Newtons era we learned that they are implementations of the same fundamental law, the law of gravitational attraction. Now, in the case of the Moons orbit, there is a specific property called angular momentum L , defined as L r x p , where r is the position vector of the rotating body (for the Moon, this vector has its origin at the point of the center of mass of the system EarthMoon and points toward the Moon), p is the linear momentum of the rotating body (for the Moon, its a vector with origin on the Moon and direction tangential to the Moons orbit), and x is the cross product of the two vectors. L is measured in kilograms times meters squared per second (kgm 2 s -1 ). Now, in the context of the falling apple, there is nothing of all that zoo of properties and units, because the angular momentum is always zero ( r and p turn out to be parallel, so their cross product is zero). So what Does the nonexistence of angular momentum (corresponding to the nonexistence of temperature in configurational entropy) and the irrelevance of the units kgm 2 s -1 in the case of the apple destroy our conviction that both the falling apple and the orbiting Moon represent implementations of a deeper law of physics Then the objection that the value of the configurational entropy depends on how finely we partition the space is also misguided. The value of the thermodynamic entropy varies, too, depending on our units of measurement: it comes to one value if we measure temperature in degrees Kelvin, and to another value if we measure it in degrees Fahrenheit. So what Keeping our units fixed, the important observation is that a later measurement of thermodynamic entropy yields a larger value than an earlier measurement of it, when we measure it in the same closed system. Likewise, keeping our partitions of the space fixed, a later measurement of configurational entropy yields a larger value than an earlier measurement of it, assuming the particles (or carriers of information) perform merely random walks, and no attracting forces are applied to them. What is important in both cases is that we have a fixed scale that yields values in total order, () so that we can compare measurements, and the results of the comparisons be consistent with the total-order relation. Prof. Lambert, and others whose papers he references, mock at examples like the following: a college student organizes their closet, so objects appear to be orderly in it for a while. After a few weeks, or months, things in the closet look just as disorderly as before the organizing attempt. This, according to Prof. Lambert, is a completely flawed application of the second law of thermodynamics, and an example of how wrongly the fundamental notions of physics, such as entropy, can be perceived by beginning students. And yet, there is something in this example that, once again, escapes from Prof. Lamberts horizon. The reason is that this is an extra-physical example, one for which the low physical level with its laws is insufficient to explain fully. Its not that there is anything non-physical (immaterial) in this example, but that the laws of physics as we know them simply do not suffice for its full explanation, just as they do not suffice for the explanation of biological phenomena (and thats why we have biology and dont resort to quantum mechanics to explain evolution), or for the explanation of cognitive phenomena (thats why we developed psychology and cognitive science). There are plenty of examples showing that a higher level of material organization has copycated phenomena and principles that hold at a lower level, and the objects-in-the-closet case is just one of them. Consider another, more familiar one: the notion of force. Forces are understood literally by physicists as those interactions that result in the attraction or repelling of material objects: the strong force, the electromagnetic force, and so on. () But there are also biological forces, such as those that keep the fish of a school close together, so that the fish dont disperse in the water and dont go each its own way. Such forces can definitely be reduced to the low-level physical ones, but the reduction is not trivial. Further, there are psychological forces, such as when two people feel attraction that causes them to stay close together and form a family. Again, the reduction first to biological, and then to physical forces exists, but is nontrivial. A similar case could be given for the notion of wave: there are physical waves (e. g. of electromagnetic nature), natural waves in the macro-world (sound waves, water waves), but even social waves, such as waves of fashion and of culture. The more detached from low-level physical reality a notion is, the more prone it is to become the target of ridicule by some physicists who think that the real thing is the object of their study, and everything else is there by mere analogy and sloppy thinking by laypeople. But theyre wrong. The material world has become more complex than what physics can conveniently describe. It can be shown that even though phenomena are in principle reducible to the lowest, quantum level (in the sense that no mysterious immaterial forces or magic is needed for them to occur), there are some that cannot be reduced in practice . For example, the problem of figuring out how two people are attracted to each other (and stay close by the force of love) is not necessarily solvable in terms of quantum physics. (It might be, I am only surmising.) A similar case might be the objects-in-a-closet example. The student goes and puts objects in order, but what happens thereafter is random walks (figuratively speaking) of the objects, because forces that move them are applied to them, and such forces do not have the order as an objective, but are of varying strength and random directions (as far as the order in the closet is concerned). So we get disorder. Hard to translate to low-level physical language, but not completely irrelevant to it. Such analogies from higher levels of material organization are not completely useless, contrary to what Prof. Lambert believes, because they help the beginner relate to something familiar before plunging into the more unfamiliar, low-level physical situation. The Conclusion We saw that each of the two implementations of the dispersion theorem, the thermodynamic and the configurational case, depend on the quality of the material dispersed: when the material quality is energy, we deal with thermodynamic entropy and the notion of heat whereas when the material quality is mass, we deal with configurational entropy and the notion of orderdisorder (or in-compressibility, or randomness). We also saw the mathematical explanation, or justification of the second law of thermodynamics, through the dispersion theorem . One important difference between the two implementations is that, in our familiar corner of the universe, energy is nearly impossible to contain, to keep from spreading. As a result, we observe the familiar thermodynamic notion that, no matter what, energy will disperse in space-time. Whatever feeble attempts we make to contain it fail: if we try to concentrate energy, well spend more energy to achieve the concentration than the energy well collect, and so we conclude that entropy increases inexorably. But with mass its different. Objects with mass can be disallowed from performing random walks. When that happens, we observe a reversal in the increase of configurational entropy, which we interpret as an increase in order, or a decrease in randomness and chaos. One familiar and very important case of decrease in configurational entropy is the biological evolution on our planet, and even more generally of material evolution: considering the original primordial soup of quantum particles, which was the state of the universe shortly after the Big Bang, and the later lumpy texture of it (clusters of galaxies, galaxies, stars, etc.) created by gravity, we see that, although thermodynamic entropy increases in the universe (assuming it is a thermodynamically closed system), its configurational entropy probably decreases on a large scale, primarily due to gravity, but also due to the other attracting forces of nature. For corrections, suggestions, comments, etc. consider contacting the author. Hawking, Stephen W. (1988). A Brief History of Time. New Work: Bantam Science. () Greene, Brian R. (2004). The fabric of the cosmos: space, time, and the texture of reality . New York: Knopf. ()Davies, Paul (1995). About Time . New York: Simon amp Schuster, a Touchstone Book. ()Einstein, Albert (1956). Investigations on the Theory of the Brownian Movement. This posthumous publication of Einsteins five papers on Brownian motion is translated by A. D. Cowper, and edited and annotated by R. Frth. Dover Publications. ()Weisstein, Eric W. (1999). CRC Concise Encyclopedia of Mathematics. CRC Press. ()Pagels, Heinz R. (1983). The Cosmic Code: quantum physics as the language of nature. New York: Simon amp Schuster, a Bantam Newage Book. ()Chaitin, Gregory J. (2002). Computers, Paradoxes, and the Foundations of Mathematics. American Scientist, v. 90, MarchApril 2002, pp 164171. () Footnotes: (Clicking on the footnote number brings back to the text) () The thermodynamic entropy S is defined by the relation S Q T, where Q is the amount of heat absorbed in a reversible process, and T is the absolute temperature at which the process is occurring. Conceptually, thermodynamic entropy is the amount of energy that cannot be used to do useful work, the useless energy. () Take this perfectly as an idealization. Weve made other idealizations, too, such as that we always deal with perfectly closed systems, therefore this is not the only one. () The same convenience of thinking leads us to treat electricity as a substance in the macroworld, or as a fluid that flows within objects and yields an electrical current. But we know that, in microworld terms, this is only the emergent statistical result of a vast number of particles (electrons) that move within the atoms of the material. () Alternatively, we can think of physics as nothing but mathematics, down at the lowest level of description. Therefore, every mathematical result can be considered as potentially part of physics, in this view. Indeed, theoretical physicists often express the view that when the world is examined at its most fundamental constituents, human-made notions and reality disappear, and what remains is only describable by mathematical equations. () Thus the term Second Law of Thermodynamics is somewhat misleading, because it refers only to one manifestation of a deeper law: the thermal manifestation, where what disperses is energy through photons. People seem to use this term to refer even to the deeper law, for which there should be another term, such as The Law of Dispersal, in which the entity that disperses can be energy as in the thermodynamic case, or matter as in the configurational case, or even abstract information, as in the virtual case. () Total order is a mathematical notion that refers to a set in which any two of its elements x and y can be compared, and the result of the comparison is either that x lt y, or that x gt y, and if both x lt y and x gt y then it must be that x y . The relation must also be transitive, see here . () There are four low-level (physical) forces in nature, agreed upon by all physicists now (early 21st C.): the strong, the electromagnetic, the weak, and the gravitational interaction. A fifth force has been stipulated, believed to be responsible for the expansion of the universe, but its status as a force is currently debated. Naples, Florida Mean prices in 2015: All housing units: over 1,000,000 Detached houses: over 1,000,000 Townhouses or other attached units: over 1,000,000 In 2-unit structures: over 1,000,000 In 3-to-4-unit structures: 682,936 In 5-or-more-unit structures: 839,004 Mobile homes: 44,474 Median gross rent in 2015: 1,225. Profiles of local businesses Put your BampM business profile right here for free. 50,000 businesses already created their profiles Business Search - 14 Million verified businesses Races in Naples, FL (2015) 18,990 88.0 White alone 1,139 5.3 Hispanic 1,030 4.8 Black alone 174 0.8 Two or more races 86 0.4 Asian alone 28 0.1 Other race alone 15 0.07 American Indian alone Mar. 2016 cost of living index in Naples: 105.3 (more than average, U. S. average is 100) Recent articles from our blog. Our writers, many of them Ph. D. Absolventen oder Kandidaten, erstellen Sie leicht zu lesende Artikel zu einer Vielzahl von Themen. Recent posts about Naples, Florida on our local forum with over 2,000,000 registered users. Naples is mentioned 6,602 times on our forum: According to our research of Florida and other state lists there were 229 registered sex offenders living in Naples, Florida as of February 25, 2017 . The ratio of number of residents in Naples to the number of sex offenders is 88 to 1. Median real estate property taxes paid for housing units with mortgages in 2015: 5,163 (0.5) Median real estate property taxes paid for housing units with no mortgage in 2015: 4,713 (0.6) Nearest city with pop. 50,000: Cape Coral, FL (35.5 miles , pop. 102,286). Nearest city with pop. 1,000,000: Houston, TX (866.0 miles , pop. 1,953,631). Single-family new house construction building permits: 1997: 70 buildings , average cost: 618,000 1998: 101 buildings , average cost: 565,800 1999: 117 buildings , average cost: 863,400 2000: 131 buildings , average cost: 837,200 2001: 122 buildings , average cost: 868,200 2002: 118 buildings , average cost: 912,700 2003: 128 buildings , average cost: 819,800 2004: 173 buildings , average cost: 912,200 2005: 192 buildings , average cost: 1,457,100 2006: 96 buildings , average cost: 1,144,000 2007: 82 buildings , average cost: 1,225,000 2008: 83 buildings , average cost: 1,536,600 2009: 39 buildings , average cost: 1,864,700 2010: 63 buildings , average cost: 1,257,500 2011: 81 buildings , average cost: 1,344,400 2012: 111 buildings , average cost: 1,715,700 2013: 146 buildings , average cost: 1,314,300 2014: 185 buildings , average cost: 1,474,100 Number of permits per 10,000 residents Management occupations (25) Sales and related occupations (20) Construction and extraction occupations (8) Health diagnosing and treating practitioners and other technical occupations (7) Business and financial operations occupations (6) Production occupations (4) Food preparation and serving related occupations (4) Sales and related occupations (26) Office and administrative support occupations (17) Management occupations (14) Personal care and service occupations (8) Health diagnosing and treating practitioners and other technical occupations (7) Education, training, and library occupations (6) Food preparation and serving related occupations (5) Average climate in Naples, Florida Based on data reported by over 4,000 weather stations Tornado activity: Naples-area historical tornado activity is significantly below Florida state average. It is 54 smaller than the overall U. S. average. On 1191968 , a category F2 ( max. wind speeds 113-157 mph) tornado 1.2 miles away from the Naples city center killed 2 people and injured 17 people and caused between 5000 and 50,000 in damages. On 6101962 , a category F2 tornado 17.2 miles away from the city center caused between 5000 and 50,000 in damages. Earthquake activity: Naples-area historical earthquake activity is significantly below Florida state average. It is 99 smaller than the overall U. S. average. On 9102006 at 14:56:08 , a magnitude 5.9 (5.9 MB , 5.5 MS , 5.8 MW , Depth: 8.7 mi , Class: Moderate , Intensity: VI - VII) earthquake occurred 299.9 miles away from the city center On 3311992 at 14:59:39 , a magnitude 3.8 (3.8 MB , Depth: 3.1 mi , Class: Light , Intensity: II - III) earthquake occurred 244.5 miles away from Naples center On 4181997 at 14:57:35 , a magnitude 3.9 (3.9 MB , Depth: 20.5 mi) earthquake occurred 296.7 miles away from the city center On 2221992 at 04:21:34 , a magnitude 3.2 (3.2 MB , Depth: 6.2 mi) earthquake occurred 181.1 miles away from the city center On 2102006 at 04:14:22 , a magnitude 5.3 (4.2 MB , 5.3 MS , Depth: 3.1 mi) earthquake occurred 531.1 miles away from Naples center On 4132003 at 04:52:53 , a magnitude 3.2 (3.2 MB , Depth: 6.2 mi) earthquake occurred 266.2 miles away from the city center Magnitude types: body-wave magnitude (MB), surface-wave magnitude (MS), moment magnitude (MW) Natural disasters: The number of natural disasters in Collier County (26) is a lot greater than the US average (13). Major Disasters (Presidential) Declared: 16 Emergencies Declared: 5 Causes of natural disasters: Hurricanes: 11 , Fires: 7 , Tropical Storms: 5 , Floods: 2 , Freezes: 2 , Tornadoes: 2 , Heavy Rain: 1 , Wind: 1 (Note: Some incidents may be assigned to more than one category). Birthplace of: Chris Resop - 2005 Major League Baseball player (Florida Marlins, born . Nov 4, 1982) , Cleannord Saintil - Football player , Crafton Wallace - Mixed martial artist , Earnest Graham - 2005 NFL player (Tampa Bay Buccaneers, born . Jan 15, 1980) , Fred McCrary - 2005 NFL player (Atlanta Falcons, born . Sep 19, 1972) , George McNeill - Professional golfer , Jesse Witten - Tennis player , Spencer Adkins - Football player , Tina Wainscott - Writer , Chris Johnson (baseball) - Baseball player. Main business address for: BANCSHARES OF FLORIDA INC ( NATIONAL COMMERCIAL BANKS ), SUMMIT AMERICA TELEVISION INC TN ( RETAIL-CATALOG MAIL-ORDER HOUSES ), BEASLEY BROADCAST GROUP INC ( RADIO BROADCASTING STATIONS ), FIRST NATIONAL BANKSHARES OF FLORIDA INC ( STATE COMMERCIAL BANKS ), TIB FINANCIAL CORP. ( STATE COMMERCIAL BANKS ), HEALTH MANAGEMENT ASSOCIATES INC ( SERVICES-GENERAL MEDICAL SURGICAL HOSPITALS, NEC ). Hospitals in Naples: AVOW HOSPICE INC (1095 WHIPPOORWILL LANE) DOCTORS OUTPATIENT SURGERY CENTER, LLC (1005 CROSSPOINTE DRIVE, SUITE 2) NAPLES COMMUNITY HOSPITAL (Voluntary non-profit - Private, 350 7TH ST N) PHYSICIANS REGIONAL MEDICAL CENTER - PINE RIDGE (Proprietary, provides emergency services, 6101 PINE RIDGE ROAD) WILLOUGH AT NAPLES, THE (9001 TAMIAMI TRAIL EAST) Nursing Homes in Naples: ARISTOCRAT, THE (10949 PARNU STREET) BENTLEY CARE CENTER (875 RETREAT DRIVE) CHATEAU AT MOORINGS PARK, THE (130 MOORINGS PARK DRIVE) HARBORCHASE OF NAPLES (7801 AIRPORT PULLING ROAD N) HERITAGE HEALTH CARE CENTER (777 9TH ST) HERITAGE HEALTHCARE AND REHABILITATION CENTER (777 9TH ST N) IMPERIAL HEALTH CARE CENTER (900 IMPERIAL GOLF COURSE BLVD) LAKESIDE PAVILLION CARE AND REHABILITATION CENTER (2900 12TH STREET N) MANORCARE AT LELY PALMS (6135 RATTLESNAKE HAMMOCK ROAD) MANORCARE NURSING AND REHABILITATION CENTER (3601 LAKEWOOD BLVD) PREMIER PLACE AT THE GLENVIEW (100 GLENVIEW PLACE) Dialysis Facilities in Naples: ARA - NAPLES DIALYSIS CENTER LLC (4529 EXECUTIVE DRIVE) ARA - NAPLES SOUTH DIALYSIS CENTER LLC (4270 TAMIAMI TRAIL EAST SUITE 1) BMA - SOUTH COLLIER (12703 TAMIAMI TRAIL EAST 121) KIDNEY INSTITUTE OF NAPLES (878 109TH AVE N SUITE 1) NAPLES ARTIFICIAL KIDNEY CENTER (3699 AIRPORT PULLING RD N) NORTH NAPLES DIALYSIS LLC (1750 SW HEALTH PKWY) NRI - NAPLES (6625 HILLWAY CIRCLE) Home Health Centers in Naples: AMERICARE HOME HEALTH SERVICES, INC (5020 TAMIAMI TRL N SUITE 200) GENTIVA HEALTH SERVICES (5050 TAMIAMI TRL N UNIT B) MOORINGS PARK HOME HEALTH AGENCY (111 MOORINGS PARK DR) UNS UNITED NURSING SERVICES (5644 TAVILLA CIR SUITE 204) WEST COAST HOME HEALTH CARE AGENCY INC (2590 NORTHBROOKE PLAZA DR UNIT 203) XL - CARE AGENCY INC OF COLLIER (2640 GOLDEN GATE PARKWAY SUITE 206) Airports and heliports located in Naples: Amtrak station: NAPLES (I-75 AT RTE. 951) - Bus Station . Services: enclosed waiting area. CollegesUniversities in Naples: Hodges University ( Full-time enrollment: 2,132 Location: 2655 Northbrooke Drive Private, not-for-profit Website: hodges. edu Offers Masters degree ) Lorenzo Walker Institute of Technology ( FT enrollment: 539 Location: 3702 Estey Ave Public Website: lwit. edu) Wolford College ( FT enrollment: 250 Location: 1336 Creekside Boulevard, Suite 2 Private, for-profit Website: wolford. edu Offers Doctors degree ) Ave Maria School of Law ( Location: 1025 Commons Circle Private, not-for-profit Website: avemarialaw. edu Offers Doctors degree ) Other collegesuniversities with over 2000 students near Naples: Florida Gulf Coast University ( about 22 miles Fort Myers, FL Full-time enrollment: 11,165) Edison State College ( about 29 miles Fort Myers, FL FT enrollment: 10,649) DeVry University-Florida ( about 92 miles Miramar, FL FT enrollment: 3,674) Florida International University ( about 93 miles Miami, FL FT enrollment: 41,234) Florida Career College-Miami ( about 94 miles Miami, FL FT enrollment: 10,133) Florida National University-Main Campus ( about 94 miles Hialeah, FL FT enrollment: 4,106) Nova Southeastern University ( about 97 miles Fort Lauderdale, FL FT enrollment: 25,621) Biggest public high schools in Naples: Private high schools in Naples: THE COMMUNITY SCHOOL OF NAPLES ( Students: 726, Location: 13275 LIVINGSTON RD, Grades: PK-12) SEACREST COUNTRY DAY SCHOOL ( Students: 508, Location: 7100 DAVIS BLVD, Grades: PK-12) FIRST BAPTIST ACADEMY ( Students: 506, Location: 3000 ORANGE BLOSSOM DR, Grades: PK-12) ST JOHN NEUMANN CATHOLIC HIGH SCHOOL ( Students: 216, Location: 3000 53RD ST SW, Grades: 9-12) NICAEA ACADEMY ( Students: 193, Location: 14785 COLLIER BLVD, Grades: PK-12) ADONAI ACADEMY ( Students: 59, Location: 2590 NORTHBROOKE PLAZA DR STE 205, Grades: KG-12) INTERNATIONAL LEARNING ACADEMY ( Students: 56, Location: 2248 AIRPORT RD S, Grades: UG-12) CORKSCREW CHRISTIAN SCHOOL ( Students: 14, Location: 22022 IMMOKALEE RD, Grades: PK-12) Biggest public elementarymiddle schools in Naples: Biggest private elementarymiddle schools in Naples: THE VILLAGE SCHOOL ( Students: 488, Location: 6000 GOODLETTE RD N, Grades: PK-8) ST ANN SCHOOL ( Students: 310, Location: 542 8TH AVE S, Grades: PK-8) ST ELIZABETH SETON SCHOOL ( Students: 256, Location: 2730 53RD TER SW, Grades: PK-8) ROYAL PALM ACADEMY ( Students: 245, Location: 16100 LIVINGSTON RD, Grades: PK-8) NAPLES CHRISTIAN ACADEMY ( Students: 142, Location: 3161 SANTA BARBARA BLVD, Grades: PK-8) MONTESSORI ACADEMY OF NAPLES ( Students: 91, Location: 2659 PROFESSIONAL CIR STE 1118, Grades: UG-5) NAPLES ADVENTIST CHRISTIAN SCHOOL ( Students: 63, Location: 2629 HORSESHOE DR S, Grades: PK-8) ABLE ACADEMY ( Students: 40, Location: 5860 GOLDEN GATE PKWY, Grades: PK-6) GRACE COMMUNITY DAYCARE SCHOOL ( Students: 30, Location: 5524 19TH CT SW, Grades: PK-4) WAVES OF WONDER MONTESSORI SCHOOL ( Students: 29, Location: 7740 PRESERVE LN STE 1, Grades: PK-3) Library in Naples: COLLIER COUNTY PUBLIC LIBRARY ( Operating income: 9,409,013 Location: 2385 ORANGE BLOSSOM DRIVE 533,096 books 27,441 e-books 34,433 audio materials 51,567 video materials 18 local licensed databases 62 state licensed databases 867 print serial subscriptions ) User submitted facts and corrections: for the hospitals listed in naples, you should change the cleveland clinic to its new name of Physicians Regional Medical Center Public high schools in Naples: Golden Gate High School needs to be added to the list Add Palmetto Ridge High School to Naples, Florida Notable locations in Naples: Kokomis Ferry (A). Miccosukee Golf and Country Club (B). City of Naples Wastewater Treatment Facility (C). Naples Depot Cultural Center (D). Collier County Public Library Naples Branch (E). North Naples Fire Control and Rescue District Station 47 (F). Collier County Emergency Medical Services Station 24 (G). Collier County Emergency Medical Services Station 2 (H). Collier County Emergency Medical Services Helicopter Operations Center (I). Collier County Emergency Medical Services Station 1 (J). City of Naples Fire Department Station 2 (K). City of Naples Fire Department Station 3 (L). City of Naples Fire Department Station 1 (M). East Naples Fire Control and Rescue District Station 24 (N). Naples Police Department (O). Federal Bureau of Investigation (P). Displayhide their locations on the map Main business address in Naples include: BANCSHARES OF FLORIDA INC (A). SUMMIT AMERICA TELEVISION INC TN (B). BEASLEY BROADCAST GROUP INC (C). FIRST NATIONAL BANKSHARES OF FLORIDA INC (D). TIB FINANCIAL CORP. (E). Displayhide their locations on the map Churches in Naples include: Jehovahs Witness Spanish Congregation Church (A). The Salvation Army Church (B). Unitarian Universalist Congregation Church (C). Episcopal Trinity By-the-Cove Church (D). Baptist Church of Estero (E). Catholic Church Catholic Sisters Guadalupanas (F). Charisma Chapel (G). Church of Christ (H). Church of God in Naples (I). Displayhide their locations on the map Tourist attractions: Conservancy of Southwest Florida (Museums 1450 Merrihue Drive) (1). Collier County Museum (3301 Tamiami Trail East Building J) (2). Kelseys Collection - LLC (Art Museums 567 Park Street) (3). Aviary Zoo of Naples (Cultural Attractions - Events - Facilities 9824 Immokalee Road) (4). Holocaust Museum of Southwest Florida (Cultural Attractions - Events - Facilities 4760 Tamiami Trail North Suite 7) (5). Bird Gardens (Cultural Attractions - Events - Facilities 1060 Purple Martin Drive) (6). Educational Services (Cultural Attractions - Events - Facilities 9824 Immokalee Road) (7). E Group Inc (Cultural Attractions - Events - Facilities 9824 Immokalee Road) (8). Collier County Historical Society (Cultural Attractions - Events - Facilities 137 12th Avenue South) (9). Displayhide their approximate locations on the map Hotels: Chalet Apartment Motel Condos (844 Wiggins Passage Road West) (1). Broadwells Restaurant (851 Gulf Shore Boulevard North) (2). Best Western Naples Inn (2329 9th Street North) (3). Chickee Hut (11000 Gulf Shore Drive) (4). Baymont Inn Suites (185 Bedzel Circle) (5). Best Western Naples Plaza Hotel (6400 Dudley Drive) (6). Bellasera (221 Ninth Street South) (7). Charter Club Resort of Naples Bay (1000 10th Avenue South) (8). Baymont Inn Naples (185 Bedzel Circle) (9). Displayhide their approximate locations on the map Courts: Paladin Financial Court (1767 Knights) (1). Florida State - Judicial - Circuit Court Twentieth Judicial - Clerk Of Court Off (4715 Golden Gate Parkway) (2). Collier County of CONT - Clerk of the Circuit Court Satellite Office - Greentree Shopping Ce (2376 Immokalee Road) (3). Displayhide their approximate locations on the map Collier County has a predicted average indoor radon screening level less than 2 pCiL (pico curies per liter) - Low Potential Air pollution and air quality trends (lower is better) Air Quality Index (AQI) level in 2013 was 45.0 . This is significantly better than average. Likely homosexual households (counted as self-reported same-sex unmarried-partner households) Lesbian couples: 0.3 of all households Gay men: 0.3 of all households People in group quarters in Naples in 2010: 261 people in nursing facilitiesskilled-nursing facilities 11 people in workers group living quarters and job corps centers 7 people in hospitals with patients who have no usual home elsewhere 267 people in nursing homes in 2000 4 people in religious group quarters in 2000 Banks with most branches in Naples (2011 data): Fifth Third Bank: 18 branches . Info updated 20091005: Bank assets: 114,540.4 mil , Deposits: 89,689.1 mil , headquarters in Cincinnati, OH , positive income . Commercial Lending Specialization , 1378 total offices , Holding Company: Fifth Third Bancorp Bank of America, National Association: 15 branches . Info updated 20091118: Bank assets: 1,451,969.3 mil , Deposits: 1,077,176.8 mil , headquarters in Charlotte, NC , positive income , 5782 total offices , Holding Company: Bank Of America Corporation Wells Fargo Bank, National Association: 15 branches . Info updated 20110405: Bank assets: 1,161,490.0 mil , Deposits: 905,653.0 mil , headquarters in Sioux Falls, SD , positive income , 6395 total offices , Holding Company: Wells Fargo Company Regions Bank: 13 branches . Info updated 20110224: Bank assets: 123,368.2 mil , Deposits: 98,301.3 mil , headquarters in Birmingham, AL , positive income . Commercial Lending Specialization , 1778 total offices , Holding Company: Regions Financial Corporation SunTrust Bank: 8 branches . Info updated 20100527: Bank assets: 171,291.7 mil , Deposits: 129,833.2 mil , headquarters in Atlanta, GA , positive income . Commercial Lending Specialization , 1716 total offices , Holding Company: Suntrust Banks, Inc. JPMorgan Chase Bank, National Association: Naples Branch, Aston Gardens Naples Banking Center, Naples Office, Radio Road Santa Barbara Blvd Bank, Us 41 And Rattlesnake Hammock Road B, Airport Pulling Pine Ridge Banking . Info updated 20111110: Bank assets: 1,811,678.0 mil , Deposits: 1,190,738.0 mil , headquarters in Columbus, OH , positive income . International Specialization , 5577 total offices , Holding Company: Jpmorgan Chase Co. Branch Banking and Trust Company: North Naples Branch, Pine Ridge Branch, Downtown Naples Branch, Pebblebrooke Branch, Davis Boulevard Branch, Naples Branch . Info updated 20100329: Bank assets: 168,867.6 mil , Deposits: 127,549.5 mil , headquarters in Winston Salem, NC , positive income . Commercial Lending Specialization , 1793 total offices , Holding Company: BbT Corporation PNC Bank, National Association: Naples Newgate Center Branch, National City Pcg Branch, Galleria Court Branch, 5th Avenue Naples Branch, Radio Road Branch, North Naples Branch . Info updated 20120320: Bank assets: 263,309.6 mil , Deposits: 197,343.0 mil , headquarters in Wilmington, DE , positive income . Commercial Lending Specialization , 3085 total offices , Holding Company: Pnc Financial Services Group, Inc. The Iberiabank: North Naples Branch, Pine Ridge Road Branch, Downtown Naples Branch, Park Shore Branch, Airport Road Branch, Orion Bank . Info updated 20110608: Bank assets: 11,676.7 mil , Deposits: 9,387.9 mil , headquarters in Lafayette, LA , positive income . Commercial Lending Specialization , 187 total offices , Holding Company: Iberiabank Corporation 30 other banks with 50 local branches Educational Attainment () in 2015 Naples government finances - Expenditure in 2002 (per resident): Construction - Air Transportation: 8,361,000 (398.75) Sewerage: 3,890,000 (185.52) Water Utilities: 3,890,000 (185.52) Protective Inspection Regulation, NEC: 1,588,000 (75.73) Parks Recreation: 1,445,000 (68.91) Regular Highways: 932,000 (44.45) Police Protection: 510,000 (24.32) Housing Community Development: 185,000 (8.82) Financial Administration: 130,000 (6.20) Fire Protection: 99,000 (4.72) Current Operations - Police Protection: 6,536,000 (311.71) Air Transportation: 5,637,000 (268.84) Sewerage: 4,887,000 (233.07) Solid Waste Management: 4,573,000 (218.09) Water Utilities: 4,155,000 (198.16) Fire Protection: 3,548,000 (169.21) Parks Recreation: 3,174,000 (151.37) Regular Highways: 2,149,000 (102.49) Protective Inspection and Regulation, NEC: 1,477,000 (70.44) Central Staff Services: 1,379,000 (65.77) Parking Facilities: 1,178,000 (56.18) Sea and Inland Port Facilities: 1,012,000 (48.26) Financial Administration: 912,000 (43.49) Judicial and Legal Services: 434,000 (20.70) Housing Community Development: 36,000 (1.72) General - Interest on Debt: 1,725,000 (82.27) Total Salaries Wages: 18,613,000 (887.69) Water Utilities - Interest on Debt: 1,361,000 (64.91) Naples government finances - Revenue in 2002 (per resident): Charges - Air Transportation: 8,498,000 (405.28) Charges - Sewerage: 8,890,000 (423.98) Solid Waste Management: 4,725,000 (225.34) Parks Recreation: 1,430,000 (68.20) All Other: 1,387,000 (66.15) Sea and Inland Port Facilities: 1,369,000 (65.29) Parking Facilities: 413,000 (19.70) Federal Intergovernmental - All Other: 473,000 (22.56) Local Intergovernmental - All Other: 1,338,000 (63.81) Miscellaneous - Interest Earnings: 3,349,000 (159.72) General Revenue, NEC: 701,000 (33.43) Special Assessments: 540,000 (25.75) Revenue - Water Utilities: 9,413,000 (448.92) State Intergovernmental - General Support: 3,491,000 (166.49) All Other: 132,000 (6.30) Tax - Property: 8,718,000 (415.78) Public Utilities: 5,614,000 (267.74) Other Selective Sales: 2,823,000 (134.63) NEC: 2,130,000 (101.58) Motor Fuels Sales: 1,846,000 (88.04) Naples government finances - Debt in 2002 (per resident): Long Term Debt Beginning Outstanding - Water Utilities: 22,685,000 (1081.89) Long Term Debt Beginning Outstanding, NEC: 9,260,000 (441.63) Long Term Debt Issue, Unspecified - Water Utilities: 7,275,000 (346.96) Other NEC: 3,205,000 (152.85) Long Term Debt Outstanding - Full Faith Credit - Other, NEC: 8,270,000 (394.41) Long Term Debt Outstanding Nonguaranteed - Water Utilities: 20,545,000 (979.83) Other, NEC: 3,207,000 (152.95) Long Term Debt Retired Unspecified - Water Utilities: 9,415,000 (449.02) Other, NEC: 988,000 (47.12) Short Term Debt Outstanding - End of Fiscal Year: 71,878,000 (3427.99) Beginning: 67,299,000 (3209.61) Naples government finances - Cash and Securities in 2002 (per resident): Bond Fund - Cash Deposits: 8,136,000 (388.02) Other Funds - Cash Deposits: 35,806,000 (1707.65) Sinking Fund - Cash Deposits: 1,565,000 (74.64) 7.35 of this countys 2011 resident taxpayers lived in other counties in 2010 (104,488 average adjusted gross income ) Strongest AM radio stations in Naples: WNOG (1270 AM 5 kW NAPLES, FL Owner: MERIDIAN BROADCASTING, INC.) WVOI (1480 AM 10 kW MARCO ISLAND, FL Owner: ALL FINANCIAL NETWORK, INC.) WCNZ (1660 AM 10 kW MARCO ISLAND, FL Owner: ALL FINANCIAL NETWORK, INC.) WJNA (640 AM 38 kW ROYAL PALM BEACH, FL Owner: SOUTH FLORIDA RADIO, INC.) WWFE (670 AM 50 kW MIAMI, FL Owner: FENIX BROADCASTING CORP.) WAQI (710 AM 50 kW MIAMI, FL Owner: LICENSE CORPORATION 1) WVCG (1080 AM 50 kW CORAL GABLES, FL Owner: RADIO ONE LICENSES, LLC) WWCN (770 AM 10 kW NORTH FORT MYERS, FL Owner: WJPT LICENSE LIMITED PARTNERSHIP) WQBA (1140 AM 50 kW MIAMI, FL Owner: WQBA-AM LICENSE CORP.) WRFX (940 AM 50 kW MIAMI, FL Owner: CLEAR CHANNEL BROADCASTING LICENSES, INC.) WSUA (1260 AM 50 kW MIAMI, FL Owner: WSUA BROADCASTING CORPORATION) WNMA (1210 AM 49 kW MIAMI SPRINGS, FL Owner: RADIO UNICA OF MIAMI LICENSE CORP.) WPTK (1200 AM 10 kW PINE ISLAND CENTER, FL Owner: FORT MYERS BROADCASTING COMPANY) Strongest FM radio stations in Naples: WTLT (93.7 FM NAPLES, FL Owner: MERIDIAN BROADCASTING, INC.) WRXK-FM (96.1 FM BONITA SPRINGS, FL Owner: WRXK LICENSE LIMITED PARTNERSHIP) WARO (94.5 FM NAPLES, FL Owner: MERIDIAN BROADCASTING, INC.) WAVV (101.1 FM MARCO, FL Owner: ALPINE BROADCASTING CORP. INC.) WBTT (105.5 FM NAPLES PARK, FL Owner: CLEAR CHANNEL BROADCASTING LICENSES, INC.) WDRR (107.1 FM LEHIGH ACRES, FL Owner: CLEAR CHANNEL BROADCASTING LICENSES, INC.) WSRX (89.5 FM NAPLES, FL Owner: SHADOWLAWN ASSOCIATION, INC.) WSOR (90.9 FM NAPLES, FL Owner: THE MOODY BIBLE INSTITUTE OF CHICAGO) WWGR (101.9 FM FORT MYERS, FL Owner: RENDA BROADCASTING CORP. OF NEVADA) WGUF (98.9 FM MARCO, FL Owner: RENDA BROADCASTING CORP. OF NEVADA) WSGL (104.7 FM NAPLES, FL Owner: RENDA BROADCASTING CORPORATION OF NEVADA) WAYJ (88.7 FM FORT MYERS, FL Owner: WAY-FM MEDIA GROUP. INC.) WINK-FM (96.9 FM FORT MYERS, FL Owner: FORT MYERS BROADCASTING COMPANY) WXKB (103.9 FM CAPE CORAL, FL Owner: WXKB LICENSE LIMITED PARTNERSHIP) WMKO (91.7 FM MARCO, FL Owner: BOARD OF TRUSTEES, FLORIDA GULF COAST UNIVERSITY) WNRW (98.5 FM SAN CARLOS PARK, FL Owner: BEL MEADE BROADCASTING COMPANY, INC.) WJBX (99.3 FM FORT MYERS BEACH, FL Owner: WJBX LICENSE LIMITED PARTNERSHIP) WJPT (106.3 FM FORT MYERS, FL Owner: WJPT LICENSE LIMITED PARTNERSHIP) WCKT (100.1 FM PORT CHARLOTTE, FL Owner: CLEAR CHANNEL BROADCASTING LICENSES, INC.) WTLQ-FM (97.7 FM PUNTA RASSA, FL Owner: FORT MYERS BROADCASTING COMPANY) TV broadcast stations around Naples: W56DW ( Channel 56 NAPLES, FL Owner: TRINITY BROADCASTING NETWORK) WWDT-CA ( Channel 43 NAPLES, FL Owner: RUSSELL R. WEDDELL) WXDT-LP ( Channel 23 NAPLES, FL Owner: GUENTER MARKSTEINER) WYDT-CA ( Channel 32 NAPLES, FL Owner: GUENTER MARKSTEINER) WZDT-LP ( Channel 39 NAPLES, FL Owner: GUENTER MARKSTEINER) WTIG-LP ( Channel 2 NAPLES, FL Owner: TIGER EYE BROADCASTING CORPORATION) WZVN-TV ( Channel 26 NAPLES, FL Owner: MONTCLAIR COMMUNICATIONS, INC.) WINK-TV ( Channel 11 FORT MYERS, FL Owner: FORT MYERS BROADCASTING COMPANY) WTVK ( Channel 46 NAPLES, FL Owner: ACME TELEVISION LICENSES OF FLORIDA, LLC) WFTX ( Channel 36 CAPE CORAL, FL Owner: EMMIS TELEVISION LICENSE CORPORATION) WRXY-TV ( Channel 49 TICE, FL Owner: WEST COAST CHRISTIAN TELEVISION, INC) WBBH-TV ( Channel 20 FORT MYERS, FL Owner: WATERMAN BROADCASTING CORP. OF FLORIDA) WBSP-CA ( Channel 9 NAPLES, FL Owner: CALOOSA TELEVISION CORPORATION) W22CL ( Channel 22 FORT MYERS, FL Owner: ARKANSAS MEDIA, LLC) WDPX-LP ( Channel 18 FORT MYERS, FL Owner: TIGER EYE BROADCASTING CORP.) Fatal accident count (per 100,000 population) Drinking water stations with addresses in Naples and their reported violations in the past: GOODLAND WATER COMPANY ( Population served: 770 , Purch surface water): Past monitoring violations: Monthly Turbidity Exceed (Enhanced SWTR) - In SEP-2006 , Contaminant: IESWTR . Follow-up actions: St Public Notif requested (SEP-25-2006), St Public Notif received (SEP-25-2006) Monitoring, Turbidity (Enhanced SWTR) - In SEP-2006 , Contaminant: IESWTR . Follow-up actions: St Public Notif requested (SEP-25-2006), St Public Notif received (SEP-25-2006) IGLESIA GETSEMANI CHURCH ( Population served: 78 , Groundwater): Past health violations: MCL, Monthly (TCR) - In AUG-2013 , Contaminant: Coliform . Follow-up actions: St Public Notif requested (AUG-31-2013) , St Public Notif received (AUG-31-2013) TEMPLE BETHEL ( Population served: 40 , Groundwater): Past health violations: MCL, Monthly (TCR) - In MAY-2012 , Contaminant: Coliform . Follow-up actions: St Compliance achieved (JUN-05-2012) Past monitoring violations: Failure to Conduct Assessment Monitoring - Between JUL-2012 and SEP-2012 , Contaminant: E. COLI Failure to Conduct Assessment Monitoring - Between JAN-2012 and MAR-2012 , Contaminant: E. COLI . Follow-up actions: St Compliance achieved (MAY-01-2012), St Public Notif requested (MAY-04-2012) Failure to Conduct Assessment Monitoring - Between JUL-2011 and SEP-2011 , Contaminant: E. COLI . Follow-up actions: St Public Notif requested (OCT-27-2011), St Compliance achieved (DEC-20-2011) 4 routine major monitoring violations One regular monitoring violation NAPLES EQUESTRIAN CHALLENGE, INC. ( Population served: 30 , Groundwater): Past monitoring violations: One routine major monitoring violation One regular monitoring violation Drinking water stations with addresses in Naples that have no violations reported: COLLIER COUNTY REGIONAL WTP ( Population served: 134,780 , Primary Water Source Type: Groundwater) LIVING WORD FAMILY CHURCH WTP ( Population served: 500 , Primary Water Source Type: Groundwater) AURORA ACRES ( Population served: 170 , Primary Water Source Type: Groundwater) HIDEOUT GOLF CLUB SYSTEM ( Population served: 100 , Primary Water Source Type: Groundwater) STAR QUICK MART ( Population served: 60 , Primary Water Source Type: Groundwater) RANDALL CENTER ( Population served: 50 , Primary Water Source Type: Groundwater) SYNGENTA SEEDS, INC. ( Population served: 28 , Primary Water Source Type: Groundwater) NAPLES BINGO PALACE GG PKWY ( Population served: 25 , Primary Water Source Type: Groundwater) UNITY FAITH MISSIONARY BAPTIST ( Population served: 25 , Primary Water Source Type: Groundwater) 2006 National Fire Incident Reporting System Incidents: Incident types - Naples Fire-safe hotels and motels in Naples, Florida: Whites Motel, 11238 E Tamiami TRL, Naples, Florida 33962 Ramada Inn Of Naples, 1100 Tamiami Trail N, Naples, Florida 34102 . Phone: (239) 263-3434, Fax: (239) 262-7439 The Naples Beach Hotel Golf Club, 851 Gulfshore Blvd N, Naples, Florida 34102 . Phone: (239) 261-2222, Fax: (239) 435-4360 Sea Shell Motel, 82 S 9TH St, Naples, Florida 33940 Glades Motel, 3115 E Tamiami TRL, Naples, Florida 33962 Tiki Motel Apartments, 2486 Tamiami TRL E, Naples, Florida 33962 Gulfstream Motel, 4520 Gulf STRM Dr, Naples, Florida 33962 Best Western Naples Plaza, 6400 Dudley Dr, Naples, Florida 34105 . Phone: (239) 643-6655, Fax: (239) 643-4063 21 other hotels and motels All 29 fire-safe hotels and motels in Naples, FloridaMost common first names in Naples, FL among deceased individuals Naples compared to Florida state average: Median household income above state average. Median house value significantly above state average. Unemployed percentage significantly below state average. Black race population percentage significantly below state average. Hispanic race population percentage significantly below state average. Median age significantly above state average. Foreign-born population percentage below state average. Renting percentage below state average. Number of college students below state average. Percentage of population with a bachelors degree or higher significantly above state average. Naples on our top lists : 6 on the list of Top 101 cities with largest percentage of females in industries: Real estate and rental and leasing (population 5,000) 7 on the list of Top 101 cities with the highest cost per building permit(population 5,000) 24 on the list of Top 101 cities with the most local government spending on current operations of parking facilities per resident (population 10,000) 25 on the list of Top 101 cities with the highest percentage of workers working at home, population 5,000 27 on the list of Top 101 cities with largest percentage of females in occupations: Sales and related occupations (population 5,000) 32 on the list of Top 100 cities with oldest residents (pop. 5,000) 34 on the list of Top 101 cities with largest percentage of males in occupations: Management occupations (population 5,000) 42 on the list of Top 101 cities that people commute into (largest positive percentage daily daytime population change due to commuting) (population 5,000) 61 on the list of Top 101 cities with the most people born in other U. S. states (population 5,000) 62 on the list of Top 101 cities with largest percentage of males in industries: Real estate and rental and leasing (population 5,000) 67 on the list of Top 100 cities with highest ratio of median house value to median household income (pop. 5,000) 69 on the list of Top 101 cities with the most full-time park and recreation workers per 1000 residents (population 5,000) 70 on the list of Top 101 cities with the most full-time firefighters per 1000 residents (population 5,000) 73 on the list of Top 101 cities with the most full-time financial administration workers per 1000 residents (population 5,000) 12 (34102) on the list of Top 101 zip codes with the highest 2012 average taxable interest for individuals (pop 5,000) 17 (34102) on the list of Top 101 zip codes with the highest 2012 average net capital gainloss (pop 5,000) 18 (34102) on the list of Top 101 zip codes with the highest 2012 average Adjusted Gross Income (AGI) for individuals (pop 5,000) 34 (34102) on the list of Top 101 zip codes with the largest charity contributions deductions as a percentage of AGI in 2012 (pop 5,000) 42 (34103) on the list of Top 101 zip codes with the smallest percentage of returns reporting salary or wage in 2012 (pop 5,000) 65 (34103) on the list of Top 101 zip codes with the most beauty salons in 2005 83 (34102) on the list of Top 101 zip codes with the highest average reported salarywage in 2012 (pop 5,000) 84 (34102) on the list of Top 101 zip codes with the most offices of physicians in 2005 1 on the list of Top 101 counties with the largest number of people without health insurance coverage in 2000 (pop. 50,000) 18 on the list of Top 101 counties with the largest decrease in the number of infant deaths per 1000 residents 2000-2006 to 2007-2013 (pop. 50,000) 23 on the list of Top 101 counties with the lowest number of infant deaths per 1000 residents 2007-2013 (pop. 50,000) 25 on the list of Top 101 counties with the largest decrease in the number of births per 1000 residents 2000-2006 to 2007-2013 (pop 50,000) 50 on the list of Top 101 counties with the highest ground withdrawal of fresh water for public supply There are 865 pilots and 408 other airmen in this city. Top Patent Applicants Peter J. Dreyfuss (47) Reinhold Schmieding (35) Ricardo Albertorio (22) Jeffrey Wyman (22) Thomas Dooney, Jr. (21) Jacob A. Jolly (18) Philip B. Harris (16) Scott K. Mitchell (16) Arthrex, Inc. (16) Brandon L. Roller (14) Total of 1099 patent applications in 2008-2017. Recent home sales, price trends, and home value evaluator powered by Onboard Informatics copy 2017 Onboard Informatics. Information is deemed reliable but not guaranteed. City-data does not guarantee the accuracy or timeliness of any information on this site. Use at your own risk. Website copy 2017 Advameg, Inc.