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# Statistical Mechanics Lecture 7

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#### Automatisierte Medienanalyse

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rule on this steam a university far-right
forces around the forces have to do with the pressure from the outside and net force if is a great deal of pressure of the pressure on the walls and the pressure on that wall was forests so this enforced due to gradient pressure you take that into account what it was you know what is In response to the variation pressure it's articles and its acceleration of a box about the move now are the acceleration depends on the mass in the box of density of mass density comes into it but in the east you find a formula and formula From the right formula before is that the speed of sound squared the velocity of a molecule the speed of sound squared is equal and this is it this is a fairly general formula this formula is more general than just ideal gesture approximately against East squared it is equal to the derivative under pressure with respect the mass density 9 will write the mass density as for product of the density of particles the thing we've already called role density of particles times the mass of individual particle massively individual particle and the density of particles is mass density Saleh the standard formula 4 square of the speed of sound is a derivative of the pressure with respect to mess tent city which is the same as derivative the pressure with respect to a particle beams to be provided by the Max let's see what the ideal gas formula has say about the ideal guest formula but we do lot it is pressure at times the volume equals 18 times but for the cable smart and kind temperature and for providing Barbara volume In all the while you're Valeo volume is a particle density so here we have particle answering this is wrong but not the mass density 5 well are accurate as we have here derivative of P with Specter wrote crimes were Max this is a formula it we have as a function of Raul was at the root of p with respect to roll the river of P were for spectral physical to cable to my team of differentiating we derived by 1 more mass we tabled smart until over and not quite the same formula but for the velocity it's were off all of this it was the formula One Luanda is wrong by a factor of square root of 3 square of 3 1 . 7 so it's a fact but order a magnitude that's essentially correct on we could let's work it out let's work it out for 80 for the area where some numbers written Baron uses for but but just and normalcy the formula so build beat the velocity will be the same 2 women a factor of our 1 . 7 whatever it price if you stick cable my it 1 . 3 times tend to a minus 20 romance well above from Paris on air molecules air molecules of mostly made up of nitrogen and nitrogen is 1 of these owned by molecular weight accord 2 Adams of diatomic molecules I'd nitrogen is nitrogen the has a comic weight 14 member and so the Mets on a young wrestler nitrogen With a massive air molecule is about 30 28 30 plans a mess of approach on a new look that up that worries are being are for plants confirm a minus 26 kilograms were so we now called a don't temperature would alone temperature from temperatures 300 300 degrees is saying units tomatoes a number he
I a but of is white well it could be what more months the month of Armagh monographs order now in Harare not free particle gas from went into the gas or into the system whatever happens to be whatever he back I'm going to put in a single harmonic oscillator is probably as part of the system limited the rest of the system as a heat bath and ask about the property of the Hamas across Oregon our oscillator could be a sprained a massage or anything else which naturally oscillate so it could be harmed an electromagnetic waves McCafferty well for example oscillator sorry Crystal all sorts of oscillations take place in crystals of tried Soros lots and lots and lots of oscillating systems in nature almost anything quo oscillate if you just disturb will bear so we say the harmonic oscillator was studying an an extremely wide clients Our systems anything which when disturbed the little bit away from equilibrium always from its ground state oscillate can I covers almost anything you could think about our store West those these statistical mechanics of the harmonic oscillator interesting question what Is the average energy of the Soria how was it depend on man hours of depend on spring constant for after months it would work out the answer to a question we do we begin with a Hamiltonian energy with expression for the energy where to put it into the Boltzmann distribution and then we're gonna calculate average quantities for the they was the energy the energy it is 1 payoff match branch X parte squatted where X it is the core of the oscillator away from its equilibrium position nasa's the mass plus the spring constant and X squared off over 0 this is not a Boltzmann Carson now we are this with boats Boltzmann will feed into the background would be no longer I will keep track of with constant here is a spring constant a hoax spring constant or equivalently if we go a moment of instead of velocity it becomes P squared provided by away momentum squared or right away and plus saying KX where over 2 so that's expression for energy it is our pressure for energy and now we want to write the Boltzmann distribution for this From this system here and that each of the miners Bay that they don't be embarrassed temperature now what it terms temperature beat back Trevor to will hectare they are so it was in this room there it would be 300 degrees for wasn't liquid helium and it would be 40 degrees 4 degrees whatever it was in the middle of the it would evaporate would be there our a lot Fork each other miners beta peace squared off over to I'm factorize each miners beta can a X squared off until assembly use the fact that the exponential of Osama is just a product of exponential woes so factory like this that's the expression miners based upon the energy where we do with it we calculate offering partition function partition function is our best friend in statistical mechanics so we calculated and that's equal to the sum of all configurations which in this case means DP DX not freely a situations the X integration just gave us the volume of the container of a box was in now that's not true because we're doing integration the intervals factor PPE reacts to separate the roles of stressed the product of integrals right there was caught the P in the world and the X integral approach the P downstairs circus remember room through right DP for me into repeat and will be X OK so all we have to do now was to do this to integrals remember that numerical constants in the partition function multiplicative constants don't make any difference so why not because when you take the locker room of the partition function they correspond he editor of constant and it always differentiating the partition function we don't care about numerical constants architects selects forgave numerical constants remark movies and the roles the P integration we do the same way that we did others remind you how we get it always said that the peace squared off to award and baby I think our current year we she's variables would call back you square obviously that means it he was a quarter of a square root 2 with em over beta can't once we do that Was this integral here becalmed integral now DP we
mathematics it is that a strain of vibrating strings is basically an infinite number of Hallmark oscillators the infinite number of become shorter and shorter wavelengths if each 1 of these oscillators head energy equal to the temperature and there are an infinite number of them that string would have an infinite amount of energy when I was 12 I was watching the normal equilibrium sums raw every answer is that most of these oscillations have very high frequency the shorter wavelength the higher the frequency of the oscillator and therefore most of his oscillations in any given temperature are frozen how by quantum mechanics not free partially as a temperature rose up and up more of his oscillations begin threw a pass away we do have energy but firms are so out mode Samoa paradox yup rose salt were this story it's just that I'm a specific heat of crystals very obvious fact remained it was it was obvious that experimental physicist around 1900 that there was something wrong because they're crystals not behave according to classical mechanics story Estrich extremely extremely easy to procedures yeah yacht the crystal specific starts to behave as if there was a certain number of Katie earned a certain number of not keeper keeper ocelot that serves really easy look for a while what that what they saw as the after right the partition function would be the product of war partition functions the logarithm would be disarmed absolutely nor are grooming song would mean the energy use a somber short grows nor was voters are transition according those whose frequency is tool Heidi from above the temperature Vos earlier quantum Regina and not appreciably excited talk door Sarge he would already what have did the article looking good good good good good good on this that doesn't matter rewrite ignore it makes a multiplicative adding a constant energy constant every energy level is a quantum multiplying the partition function by constant and the result is no change in anything of interest the broad Air bomber show restart program the 2nd his 2nd puzzle the 2nd waters a puzzle recourse it it it says the world is irreversible and something called entropy always increase for Arafat you have to be infinitely careful with a big system to keep it from increasing on the other hand no news the equations of motion are completely reversible so anything can happen in 1 direction from half from the opposite direction so the tension there is a tension between reversibility on 1 hand of the fundamental equations and irreversibility of these observational properties 0 of complicated systems parents that took some time to sort out and people are still confused about a soul let's talk about Bush who who who who who who knew who let's imagine some phase phrase base space of states classical system and its color into the Golob particles lot of whatever it is that it has borrowed
after a little bit time or something really weird over all the energy will go will go toward him for whatever modes in your very chaotic things happen the Yusuf race-based spread
of a hill lawyers Her if you go over here this way your here your but that's just 1 point of instability of the February Copperhill distort anyways else nearby lets it was your of Austria the phase trajectories don't separated that much that quickly top of a hill heroes appointed instability In these chaotic systems basically every point was a point of instability a reporter Murfreesboro Strother on stable new a formal our please visit ass at
Stamford . EDU
Vergaser
Gasdichte
Modulation
Kaltumformen
Magnetisches Dipolmoment
Längenmessung
Kraftfahrzeugexport
Mechanikerin
Besprechung/Interview
Proof <Graphische Technik>
Niederspannungskabel
Weltall
Satz <Drucktechnik>
Rootsgebläse
Gas
Druckkraft
Thermalisierung
Avro Arrow
Stapellauf
Nassdampfturbine
Koffer
Lineal
Fahrgeschwindigkeit
Frequenzsprungverfahren
Unterschallgeschwindigkeit
Temperatur
Gewicht
Höhentief
Masse <Physik>
Direkte Messung
Luftkissenschiff
IKEA
Dienstag
Neutron
Jacht
Schwächung
Nanometerbereich
Teilchen
Schiffsrumpf
Regelstrecke
Fiat 500
Gasturbine
Walzmaschine
Fahrgeschwindigkeit
Amplitude
Molekülmasse
Energielücke
Stunde
Schall
Gasdichte
Kaltumformen
Waffentechnik
Mark <Maßeinheit>
Behälter
Windrose
Proof <Graphische Technik>
Niederspannungskabel
Urkilogramm
Gas
Rootsgebläse
Übungsmunition
Elektronische Medien
Druckkraft
Ziegelherstellung
Atmosphäre
Lastkraftwagen
Ideales Gas
Druckfeld
Scheinbare Helligkeit
Drosselklappe
Temperatur
Unterschallgeschwindigkeit
Greiffinger
Magnetisches Dipolmoment
Intervall
Grundzustand
Boot
Sicherheitsbehälter
Monat
Fahrgeschwindigkeit
Passfeder
Knoten <Astronomie>
Multiplizität
Eisenkern
Pulsationsveränderlicher
Hydraulikleitung
Elektronenschale
Rootsgebläse
Übungsmunition
Werkzeug
Weiß
Vollholz
Druckfeld
Elektromagnetische Welle
Jahr
Betazerfall
Leistungsanpassung
Temperatur
Relaisstation
Masse <Physik>
Bergmann
Tonfrequenz
Tiefgang
Dreidimensionale Integration
Schraubverschluss
Teilchen
Kristallwachstum
Kopfstütze
Spiel <Technik>
Gasturbine
Stunde
Windrose
Behälter
Waffentechnik
Proof <Graphische Technik>
Niederspannungsnetz
Platzierung <Mikroelektronik>
Gas
Thermalisierung
Sturmgewehr
Negativer Widerstand
Flüssigkeit
Amplitudenumtastung
Ideales Gas
Hintergrundstrahlung
Gummifeder
Ersatzteil
Greiffinger
Gesteinsabbau
Fahrzeugsitz
Drehen
Summer
Bombe
Parallelschaltung
Mechanikerin
Spannungsabhängigkeit
Energieniveau
Satz <Drucktechnik>
Woche
Schlauchkupplung
Regelstrecke
DVD-Player
Passfeder
Fahrgeschwindigkeit
Kristallgitter
Wasserfahrzeug
Multiplizität
Abwrackwerft
Kaltumformen
Schemel
Übungsmunition
Gleiskette
Werkzeug
Angeregtes Atom
Schiff
Vollholz
Jahr
Feldquant
Betazerfall
Temperatur
Wärmequelle
Gewicht
Masse <Physik>
Direkte Messung
Mechanik
Festkörper
Bergmann
Tonfrequenz
Begrenzerschaltung
Dreidimensionale Integration
Chirp
Farbton
Jacht
Steckkarte
Teilchen
Gasturbine
Buick Century
Speckle-Interferometrie
Waffentechnik
Druckkraft
Thermalisierung
Negativer Widerstand
Nassdampfturbine
Zeitdiskretes Signal
Gummifeder
Ersatzteil
Kaltumformen
Schusswaffe
Behälter
Bombe
Waffentechnik
Windrose
Mechanik
Bergmann
Quantisierung <Nachrichtentechnik>
Besprechung/Interview
Energieniveau
Cross over <Teilchenoptik>
Chirp
Schallaufzeichnung
Gleiskette
Negativer Widerstand
Rungenwagen
Feldquant
Absolute Datierung
Betazerfall
Theodolit
Temperatur
Schaft <Waffe>
Drehen
Wärmekapazität
Kraftfahrzeughandwerk
Mechanikerin
Sturm
Filzherstellung
Energieniveau
Cross over <Teilchenoptik>
Woche
Schwellenspannung
Kristallgitter
Neutronenaktivierung
Jukos
Modulation
Hydraulikleitung
Makroklima
Rootsgebläse
Übungsmunition
Angeregtes Atom
Weiß
Atmosphäre
Jahr
Feldquant
Betazerfall
Windpark
Brutreaktor
Dotant
Temperatur
Blende <Kleidung>
Direkte Messung
Stringtheorie
Bergmann
Maßstab <Messtechnik>
Festkörper
Tonfrequenz
Stoffvereinigen
Postkutsche
Bildqualität
Chirp
Teilchen
Kristallwachstum
Digitalschaltung
Oberschwingung
Glasherstellung
Gasturbine
Speckle-Interferometrie
Flugbahn
Klangeffekt
Kraft-Wärme-Kopplung
Behälter
Windrose
Waffentechnik
Front <Meteorologie>
Elektrizität
Blechdose
Weltall
Schallaufzeichnung
Elektronik
Schiffsklassifikation
Negativer Widerstand
Nassdampfturbine
Gummifeder
Theoretische Mechanik
Zentralstern
Schubumkehr
Wärmekapazität
Mechanik
Stringtheorie
Tonfrequenz
Energieniveau
Chirp
Jacht
Bombenflugzeug
Computeranimation
Teilchen
Kristallwachstum
Breitbandübertragung
Spezifisches Gewicht
Schwingungsphase
Randspannung
Theodolit
Wellenlänge
Werkzeug
Angeregtes Atom
Thermalisierung
Negativer Widerstand
Grundfrequenz
Nassdampfturbine
Gleichstrom
Atmosphäre
Feldquant
Öffentliches Verkehrsmittel
Buntheit
Temperatur
Schaft <Waffe>
Messung
Drehen
Magnetisches Dipolmoment
Kugelblitz
Siebdruck
Durchführung <Elektrotechnik>
Fehlprägung
Leistungssteuerung
Regelstrecke
Prozessleittechnik
Schwingungsphase
Keil
Kristallgitter
Raumsonde
Grün
Standardzelle
Abend
Elektronenschale
Kardieren
Elektronische Medien
Übungsmunition
Gleiskette
Angeregtes Atom
Regentropfen
Atmosphäre
Jahr
Lineal
Luftstrom
Kampfflugzeug
Starter <Kraftfahrzeug>
Temperatur
Garn
Verbunddampfmaschine
Kraftfahrzeugexport
Sensor
Mechanik
Erdefunkstelle
Hobel
Begrenzerschaltung
Jacht
Feldeffekttransistor
Teilchen
Textilfaser
Zylinderblock
Spiel <Technik>
Edelsteinschliff
Fuß <Maßeinheit>
Klangeffekt
Stunde
Windrose
Waffentechnik
Behälter
Tag
Proof <Graphische Technik>
Glasherstellung
Gas
Nivellierlatte
Trajektorie <Meteorologie>
Schiffsklassifikation
Thermalisierung
Nassdampfturbine
Gleichstrom
Innere Reibung
Ersatzteil
Totholz <Schiffbau>
Frequenzsprungverfahren
Toleranzanalyse
Potenzialausgleich
Schaft <Waffe>
Messung
Bombe
Summer
Kugelblitz
Erwärmung <Meteorologie>
DHC-6 Twin Otter
Siebdruck
Leisten
Minute
Schwingungsphase
Trenntechnik
Fahrgeschwindigkeit
Knoten <Astronomie>
Blasenkammer
Kaltumformen
Modulation
Explorer <Satellit>
Elektronenschale
Gleiskette
Übungsmunition
Angeregtes Atom
Werkzeug
Atmosphäre
Jahr
Betazerfall
Lineal
Direkte Messung
Telefax
Analogsignal
Temperatur
Gewicht
Schubumkehr
Kraftfahrzeugexport
Wetter
Besprechung/Interview
Tonfrequenz
Schnittmuster
Feldeffekttransistor
Teilchen
Umlaufzeit
Textilfaser
Phototechnik
Automatikuhr
Fuß <Maßeinheit>
Pendel
Speckle-Interferometrie
Umlaufbahn
Niederspannungskabel
Weltall
Reif
Nivellierlatte
Trajektorie <Meteorologie>
Thermalisierung
Flüssigkeit
Gleichstrom
Doppelpendel
Initiator <Steuerungstechnik>
Großkampfschiff
Kaltumformen
Kraft-Wärme-Kopplung
Nyquist-Kriterium
Seitenleitwerk
Leistungssteuerung
Trajektorie <Meteorologie>
Computeranimation
Angeregtes Atom
Umlaufzeit
Mikroklima
Bark
Trenntechnik
Öffentliches Verkehrsmittel
Buntheit
Winter
Frequenzsprungverfahren
Speise <Technik>
Energielücke
Kommandobrücke
Rundstahlkette
Steckverbinder
Stunde
Nyquist-Kriterium
Schwingungsphase
Trajektorie <Meteorologie>
Computeranimation