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Statistical Mechanics Lecture 7
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rule on this steam a university farright
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around letter where the tonight is just beginning or we're going to do is so rich we get a 2nd Wong what 2nd war means water just I will try to explain a 2nd the best Ramada Buri nor should be lots of questions which I will try to answer nor will be rebuffed a 2nd him maybe 2 or 3 people in the world know more but I've never met any shot back so will talk a little about the sector where means but before we do that our new work I just think about physical examples that show a new we've learned some information about doing statistical mechanics 1st I just want to illustrate some of it is the Maribel facts are about gasses and so forth by contemplating the speed of sound speed of sound rejects particularly fairly good gas was the speed of sound toe formulas for the speed earned but if you think about it for a moment and you start with a very dire moved well would you expect the speed of sound B U 0 let's say you make a little bit of all over advanced region how fast because that over density spread the or you get or I would guess I would guess that is sure isn't going to be a bit faster than the takes from molecule to travel up to a certain distance and sigh of relief that I would guess you might expect that look over this region just spreads out with a velocity that's not too different than the average velocity of the molecules how does it spread our spreads out by molecules moving out and how fast of a molecule of overlapping will with their velocity was their velocity are now and we have Drury after work over with a fine award of Austria's so right arrow formula and thermal equilibrium at a temperature T gas every module PARIS AP energy he Gober free Harris Katie now I'm going to put Acadian because you are some laboratory numerology Frei has cable times of temperature where the temperature is measured end Our Kelvin Joe remains wrote that the absolute 0 but the difference between the a 1 degree car sponsor 1 centigrade degree I was barring fifths of Fahrenheit degrees OK so it's pretty hands and you could see from the formula that temperature goes to 0 0 when the energy goes 0 see right here OK so low temperatures 0 where'd 0 I is an energy that energy must be the kinetic energy of mark your job you've has for an ideal Garrison a box salt rapidly onehalf restroom are 2 of forms of Austin square and now that tells us what the average velocity is are multiply right to me didn't mean free now abide by it but take the square root probably vapors velocity squared for more than just Boston squared the you might expect that that's more or less the speed of sound in In a said the because there is more exact forestry there was more exact formula more exact formula calm laws but actually studying the mechanics of lumps of gas and the forces on them the forces Water market order that would like reduce tonight we could do another nite it's not hard it's easy to to do away with our Norton was able ability well I mean job set but stopped short of a bag you take a box of India's a typical of a region in the guest and you 1st say what
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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
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stick it in 40 gets armored I get a box are stuck and I got a bar the velocity of self of Austria for hours From buyer 500 I think 500 500 meters per 2nd get out of the out of this formula Ocilla along a high side right answers about 300 meters per 2nd if you had there you get small governmentrun during their 14 needs of 14th broke force neutrons and 2 young atomic weight is 14 after number still very hat yes to a stop when well may are rigged if they it did not take how far I could report be captive but a point gap there you may be right arm he may well be right store but many cases on the orders of magnitude are correct may and grow suppose back over the first one actually do this calculation methods to parents who was a speed of white given me a comic weight of a nitrogen is given Bolton's constant of course they didn't eat in order to use it the but our where's your equation deep he might be rolled all I had to do was to measure be pressures a function of density sorry enough to Dorothy from Russia were the the first one to realize it was basically the velocity of the or right now walk on proposal Ross you're ready for the puzzle roar not ready for the proposals were gone yacht what did that show which is constant yeah I'm amazed a fairly good assumptions are Riyadh course yes we do so with temperature was constant aren't I think that's probably very good offer of small amplitudes found a way to measure was more amplitude stairways the temperature very remark Tuesday as a reserve by Brickyard again I'm not absolutely sure about that I I would guess starts for the temperature is very removed variation in the army and the near ideal gas our from very very low amplitude sound waves because of it was the the variation of pressure is also small yard the variation of pressure small variation advance abuse more variation of pressure with respect not necessarily small usually a said is a the pressure would be used best course all us the dead rock this deeply Bureau which is calculated at the pressures that you working it took PPD role is not necessarily a constant pressure depends on the town media density complicated way in particular as a molecule get really squeezed together with derivative of craft over pressure with respect the role gives very very large so you're right but be quashed because this changes gutted we are fast but something new truck which like what this said to the young showing charters is quantity depends on pressure yes no this is it this is I'm sorry our talk OK pressure this is at lower pressure or better at low density where the gas can be approximated was an ideal gas it doesn't In the ideal gas arrange it doesn't depend on pressure but as you are out of the reason is because so if you go to a formula Our pressure is equal to rolled times of temperature at fixed temperature pressure is just proportional roll bog has the pressure gets larger and larger whose formula was a breakdown for example might start varying Gross which case as you say you're on the euro arms will depend on prom pressure for just the a bad start lot has said that the law loss of that that also it probably won't be and problem got Ikea no I never thought about every March this is this is 2 was the depth of got into it but now about worldwide that's less well I've made approximation 30 musical 28 Ma I'm not if show me like
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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
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have DP I took away DP that was a pretty DP is a square root of 2 m over beta Times Beach you stop pressing we do is we call out a factor of square root Road fish a court please equals square root of 2 where data if Carter I'm so 1st of all we have a factor of square root of 2 over beta that's coming from BP Amoco ID Cuomo missing just becomes Etowah minus Q Square is a definite integral work ends of the integration it was something of a my momentum from me anything ranks between this is a number if you happen to remember the number happens to be the square root of 2 pipe who cares it's a numerical constant at we don't care about happens abuse have has applied for a price Brosius where we see it before it's the same 1 that occurred in the ideal gas no integration workers rest this that I had to revert replaced by this were about this for over here we go exactly the same game let's make it this is the X J have variables x squared beta spring constant over tool we will call y square back meets X was square root of To win over Beta Kappa fans white fat we have DXC so the DX is a comedy wiII give us an additional factors square root of talk Tom Over beta With Katie spring constant Western DX part of the square root from here and then be integral just becomes each miners made a detour minus Y squared White are changing variables we turned growing just a simple and grow ecommerce was script and I is equal to the square root of party so let's all and very simple assault partition function Zee chore but they say was with sequel 1st or a to price reply is of no particular interest to us it contains also want square root Our END poet capped by a square root of em over Kappa him from me a cap before anybody orders grew campus what about BY about these frequency of the oscillator remember the frequency of a harmonic oscillator in terms of the spring constant the draft said so we could just right this is worn all over the frequency and I'll do sorrow but of course the frequency is a constant but nevertheless I wanna keep it there from now to parlay overall may get plans 1 provided by data last now let's calculate the energy of the ocelot our we do that we calculate the logarithm of Z and differentiate it with respect to beta that's a formula however we learned new trust Swansea is equal or wad of 2 part of our major but a constant source constant and then my years log beta minus Lord Vader that's equal prolong zinc and now lawyer Dill is your calculator minus derivative of logs vary with respect beta or is at a quarter of the energy average energy at Sarah minus 2 arms by debater were cruising now not thinking with just and chasing symbols constant gives us nothing minus sign takes away the sign from here and the derivative of lobby with the log beta with respect to beta is just 1 which also is better known as the temperature not so different Mary we don't a free have skates he Roberts a striped Rios key for the particle the guest now the 1st thing is wears white well because this is just a onedimensional oscillator soaring more taxes so afraid or what happened to the food down here why wasn't it 1 half and the reason is because they what tool integrals each of which gave us all half Lord made the 1st and the role here give us 1 square debate and the 2nd 1 also give us 1 or square debate if there would have been more integrals of this height for whatever reason more dimensions war degrees of freedom whatever we would have got you want he asks log data for each of them onehalf long really means 1 have tea feature so for each integration that
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we have to do which has this yet x squared called calcium integrals feature integration like that but we have to do there is basically all onehalf the temperature in the energy are what other 2 kinds of energies into that we're talking about we're talking about the energy when also Is kinetic energy potential G factor comedic image potential the average kinetic energy of potential jurors fact equal and each 1 equal to onehalf they 1 half temperature so kinetic energy is onehalf temperature for potential images onehalf temperature together New York temperature now has a couple of interesting things and all of us about their stools interesting things Norris about them both Samara 1st as a theater doesn't depend on the mass of the oscillator settles also laws also tool of the particle in now and again at a given temperature be Energy a kinetic energy of a particle did not depend on the mass of particle velocity varsity because the velocity onehalf envy squares the energy 1 halfempty squared his disdain for the From mass particles be square of course will be swore off a heavy particles off a say show here the average kinetic energy is the temperature Paris independent of the year Of The year of a massively password that's also independent of the spring constant that still surprising if you think about it because Amanda making this spring constant hue imagine making the spring constant so large that long known for on Earth care or elsewhere care stretched Spring or way from its equilibrium point want think under such circumstances but it's not spring just constrained Maine which has a fixed going perhaps only being 0 which cannot be changed from those circumstances any sensible person would say you can't excited he came give energy Norway was no wait till would start vibrating as Norway as a way to work to reduce any structure all our you can't do it any Connecticut edgy because it's absolutely the place became given any potential energy because you can't pull it away from that point on our way our force to exert on it yet a formula doesn't seem to care formula doesn't seem to care there always seems to be energy KTT for every are story no matter how on physical and how hard it is to get the house where those something wrong with question he feels did is a severe seal this case doesn't of key You're KM K appears here brokers appear here I I want to thank a locker Orem with Katie appears as a marriage of constant let's let's KB can the 500 something wrong about the limits here something there is something wrong but something that's wrong that we've ignored warned very crucial feature of nature's remind anything about classical mechanics sisters are correct classical mechanical closure on the other hand is crazy so we tell you to do example the kind of crazy would be involved week talk about a particle young boy had set watched by which now is that this time a previously that the position legally dead body like that for capsule about which side yeah yeah but did make it did make any difference over who was multiplicative constant your petition on is real we're here not only want particles only consulate or yell Allen 1 us away vessels as the world simpler system oscillating system in a heat back what we find the odd conclusion that the energy stored of our Soria is independent of the spring constant lets especially the spring constant let's take the mass to be fixed but now we just jack up the spring constant not hand over 500 enter kind of for all practical purposes it must be infinite and yet the formula 2nd city where a large number puts it here and still the derivative of logs you with respect to beta is equal to a temperature OK so some funny going on on this occupied people around the turn of the 19th century they were confused by it and the reason they were confused by it is because the ideal guests walk member energy per particle seemed to work pretty well when the particle where the particle was treated as a point particle but supposing that point particle that molecule was actually not a single point particle which molecules on what they might be of diatomic molecule Viacom molecules combined break there will oscillators Charlotte lost orders and so it 1 might have expected that caused Monday steadfast or if they are stay if they have a good solid spring constant of sort so you have to give them quite a bit of energy to where you have to give a call quite market should require knocked from our vibrating barked formula seems to say that for diatomic molecule you what have not 3 escapee but may be defined Kiki to account for an extra harmonic oscillator stopped vibrate woes known that then energy wasn't there are so we could think of to conclusions 1 conclusion is that the diatomic molecules not really by Palmer point particles but that's ridiculous the other conclusion was doing something wrong with missing some ingredient of physics year which tends to keep that postulated from us from having this much energy you of course all nor the ingredient is right good won't carry it is of course quantum mechanics right so we we
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wanted to and compared with this calculation is the quantum mechanical calculations corresponding quantum mechanical calculations of again quantum mechanical oscillator you know in equilibrium how much still have the not corner Kevin's very little we need the expression for the energy of a Hallmark Soria end most of the your insurance novel of harmonic oscillator energies are quantized a common discrete multiples of discrete multiples of discrete multiples of plants constant Kearns but to crime Qantas can't plans a frequency the energy of an oscillator is a unit of energy were each energy costs each bar Omega these 7 energy levels a set of state of quantum harmonic oscillator is just as discrete set of state support equally spaced and that's all we have to know about the quantum harmonic oscillator only Kwan mechanics we do know is that energy of a single harmonic oscillator calms indiscreet integer multiples of each from maker now we can calculate the partition function for the quantum mechanical are story more very seriously and was always Z is the sum of all states for the classical oscillator we replace that by integral quantum cackle oscillator it's truly Osama Hassan over of each other miners Bader finds energy of the and state and the energy of the in each bar or may go on keeping it each borrowed throwing away beyond a boss with can't I can't refuse to keep track of both of them too much so this is a formula for the partition function will set a complicated mess of exponential also song about what works hard rock Teresi Ito the mind this favor an age for major His Etowah miners beta H Omega and however so this is a serious the 1st terms is as worn parts when is equal to 0 so this is a series that has the form 1 plus a number of court X In this number in here plus X squared plus XQO cost by about 5 matzo geometric series at a geometric series which is very easy to sum up the answer is that this is equal to 1 provided by Warne miners X that's a geometric series where where this thing here just sex so now we summer up with chapter partition function seat is global 1 divided by 1 minus each other minus beta each bar homemaker amounted that's all calculation of the partition function is fairly simple if enlighten number weight is off but I think this is probably the best way to righteously justly was form our does our ways to write it but the suffice what we want Cal we calculate energy the energy is a derivative log z but also equaled its it's sold in this case it's a little easier just take it to be won over oversee plans a derivative of zany with respect data this is equal mine this it is equal combined this derivative of logs with respect just using the property of water remember that the the river water of em give your was this case areas a little easier just to calculate the map are taking a logarithm OK so wasn't easy by debate as well as some overly expression the denominator when you differentiate thing with thing of the nominating you get the anomaly the square so we get busy like debater watchers work out easy by data will contain all 1 provided by 1 players each other miners beta each all square I just coming from different cheating a denominator 1 Rex Warwick's wears on a yacht minors OK next we have to differentiate the argument we have to differentiate what's in the denominator with respect the beta that's going to give us another minor sign from here and for minus signs of the 1 might sign yacht it will be 1 might assign left a minus sign from here of a minor from here but basically it's just a derivative and enumerated he of the exponential or what a derivative of the exponential with respect the beta H far Omega each of them Bader age Bomb later by May what was it Miners here now we have to abide by that he is multiplied by 1 minus reminds beta blob multiply by that and that gets rid of 1 factor here what is that that might miss the energy we want classy energy salt from either sign here put and by sheer lock we warmed up for for the possible quantities this is the answer this is the answer was is compared with this is compared with 1 over baby so what simulation ship between them well let's 1st got by temperature but 1st guess what this is like a hightemperature heII temperature is a situation where the classical theory is going wide classical theory goes well what each 1 of these little quite Dov energy here are was very small of classical sparring has a great many armed units of energy so in that sense the classical system in quantum units has a lot of
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energy that must mean there has very high temperatures in song quantum classical systems quantum systems become classical when the temperatures very hard work temperatures high they have a lot of energy and the quantization the G becomes unimportant so let's go high temperature and see what we have restored River we 5 here because this is the energy births erase everything else and just Causey emerging hightemperature means beta they know how sorry for hightemperature means Bader very more it is worn over temperature hightemperature beta worse beta small pay a small every take the 1 at that I think it's very very near 0 this is perfectly this is just approximately warn get closer and closer to warn as baby gets more and more saw me as a dancer H Bob May got this factor here just crossed a 1 we could do the same thing with this fact easier but then we get trouble oneliners warned that a bad better from bad things to do so or what we do is we expand its its miners now eaten the mind this beta Omega is 1 plus I think data H Borel Amiga using the expansion equally X Is warned plus exports x squared off to war and so forth the only keeping track of the 1st I have to keep track of the 1st with her 1st a actually 2nd the 1st Army cancels out OK so my could to just take the 1st approximation warned the 2nd terms is so approximately exponential by just X which is is they maker which just gives us each bomb again the armory the next terms asked about are proportional the beta squared ridiculed greater falloff in store for us and them much more so they can be ignored when baby gets furry small where we get we get worn over Bader former beta to see in the hightemperature situation which is reproduces classical physics are but look has about the low temperatures what happens at low temperatures at low temperatures that means large beta where's the energy small beta Ahtisaari Laura at low temperatures large large large data this exponential a small leak memory of a large numbers malls now we're going to lowtemperature large Bader this is negligible bases a 100 million or something very very low temperature and this is negligible which is get 1 the armory of answers Hbomb gun times Etowah minors each bar Omega but again this exponential is extremely small so we see farther from getting just temperature we get something that exponentially small when the temperature rose 0 when the temperature goes 0 which means baby gets a large this becomes an enormously small because of exponentially small the miners Bader With bathers large is very very small salt Worley find we find the quantum mechanics tens suppressed the energy of a mouse Florida when the arms when the temperature is lowered off very low temperatures it doesn't behave like a classical oscillator role it has much less energy than the corresponding classical oscillator would have the question is where is the crossover where does it go from being quantum the classical look good formula by various you would know that the crossover is with the exponential goes from being a small the relaunch of the biggest exponential ever gets is 1 of the miners broke Gross where growth with this unhappy that happens when they then aged borrow make it is about equal toward the crossover between hightemperature or between the hightemperature behavior the lowtemperature behavior is when what's in the exponential hears about 1 of very low temperatures this quantity here has launched a very high temperatures as quantities more of this exponential which controls where the discrete list of the energy levels as important rock salt flats right where the crossover crossover happens when beta across all the leaders the transition from quantum behavior the classical behavior it happens when data each bar major it is about equal to 1 Nazi which way the if they had H is bigger than 1 is at quantum of classical that's quantum because of a lowtemperature forgiven worn quantum they age from a Western once it starts to behave classically which just right warned very equations date is 1 although the temperature so vast says age borrower Omega it equal to the a temperature bats or crosses over you look at it makes a last it says across although it is when when they when we energy of the oscillator classically is equal to 1 Quantum is worth of quantum return for if you try and make the temperature lower than that the oscillator has less than onequarter bombs worth of energy temperature sold there is less than what Quantum's worth of energy it doesn't wanna do that doesn't wanna have West quite bombs worth of energy and strong partition function expression for the energy used almost 0 0 exponentially more is a crossover point for what's right of Graham and temperature less
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temperature OK so now we naturally nor the answers to the puzzle about 2 have the buyer whereby go more when the temperature it is for that's where and when the temperature is to would deepen the quantum regime you're and the quantum Regina the oscillator has exponentially small energy compared to what would be class according the other hand when the temperature rose up at some point the temperature will get larger than Hbomb major at that point you begin activate the oscillator you can activate the quantum oscillator on West temperatures structure that it corresponds more than 1 more energy from the sewer Ali and very look at a global temperatures up To some temperature the diatomic molecule behaves like marked molecules you don't not store for vibration from but some temperature some temperature to raise the temperature and off so the temperature becomes bigger than the quantum of energy the diatomic molecules starts to behave White a diaphragm molecule now that which is pretty high for a good stiff molecules all this Over the years of our point this stiffness determines Omega stiffness the stiffness of the castle Rader was the square root of K over and stiffer they are so little larger the frequency of crossover point a stiffer the oscillator the height of the temperature of a crossover point wealth still molecules a pretty stiff on the scale of our of robber barons and on the temperature at which monoatomic molecule a molecule goes from looking like a mile mark your bigger diatomic molecule is behind what happens if you raise the temperature even a little more gently before often before this happened right now starter rotate the other thing happened this is just 1 example on what happens if you he did even more you activate even worse things on the Athens themselves stock provide Britain eventually they'll get ionized so it may become ionized doesn't behave like an ad on all behaves like an electron and proton so this is this is an example quantum mechanics solving these problems that physicist and very very confused about a particular property where Kmart solids are simply systems Our molecules where at arrange them alive we vibrate around the lattice there's so Adam sort of oscillators off energy of the oscillators as a function of the temperature the amount of energy as a function of temperature determines the specific heat of the euro Of The year of a crystal their specific keyed over Australia crystal whether a very very large number of oscillators qualities he capacity was estimated by classical physics to be much much larger than the actual answer here it was Einstein will basically figured out where or was going on Is that the quantum statistical mechanics of oscillators suppressed the young suppressed the oscillators onto a temperature hard so this is my where Greenstein 1 of his minor more minor contributions I fully the Nobel Prize for electric effect OK but this is neither dealers said is but what Ms. net yet North America Einstein corner character 1905 storm exactly somewhere a specific to swords those early shows early those long before people are understood the foundations of corner he just know that the beyond that the quantum energy levels I had to be proportional to frequent street dance drool and was oil what no no now division based on his it was not meant a lot death brought sought to that that the areas that off for but for a week or so long as they follows although the 1st half of that probable now what did I would say we are ultimate justification was a the quantum mechanics quantum derivation RV for example idea gasp refused an example Our where the quantum derivation reproduces in the high temperatures the classical worth classical so really the right logic the real serious logic is the start quantum mechanics and show that the quantum mechanical counting the state corresponds to the integral phase space and that are I just give you an example here below the calculation of classical physicist would ordinarily the arrears arms now we got where we get go back there Twitter partition function with salt on Yao was just look at the partition function of not even that take their Europe's a how close we get croquet this partition function add Holly temperatures means small beta it is 1 provided by what miners breeder phage from right but face borrows a Carson Carrizozo major but there are age farms in America constant sits in front of all partition functions what's what's I worry about what overpaid Omega now what was a partition function that we set for the classical case there is 1 of a bagel maker but pipe is again a constant laughter particularly interesting but the dependence on and won over beta Omega here while maybe he of quantum mechanics only comes in this multiplicative constant 1 over each bar which doesn't matter when you differentiate sold here's
1:02:33
his his an example on the derivation of the classical statistical mechanics from acquired mechanics not really the right way to go but I have to tell you this formula it was not long before quandary carrots our so with doing something right they into it that on these base was right the aerospace X space momentum positions base was right there of the right way to think about possible systems OK now so Place 7 Natsios rarely right do you want us to do so but what is while fish do make closeup queer warmer air raid already active what we hope the energy staging equal to a temperature wanted above their corn from threshold the image is just be cooled to a temperature it does not matter what to spring constant fears what he say you bosses she got it screens up a mix water oscillations renew quite temperature point is that as you had you heated from lowtemperature hightemperature charm show up the paradox saying do you know that the oscillator always has that energy independently of house if here's which is the place where Pegeen's have that energy analytic temperature at which the classical behaviors city and gets to be are higher and higher temperature as a function of the spring constant the value of the universe tend the and the Center and the spring constant yes sufficiently high temperatures it would have an energy equal to its temperature but at any kind of ordinary temperature surgery lower threshold for it to start behaving classically would get harder and harder acquirers spring constant of different stuff that's kind of interesting because when he means that as you he system you start to discover that it has warned more degrees of freedom but so what makes crazy Marchall now crazy molecule consists of a spring Adams connected to it and then each 1 of these Adams a very stiff bring it back and then connected this he had cans out the Tom particles will really stiff spring are much much stiffer springs are has a look at low temperatures as low temperatures there's not enough temperature comments were quantum threshold for of this ocellated star afterward so his point behaves with 3 asking your heated up point where ocellated he of the frequency of the oscillator is starch but the the temperature such that the army that you would activate the oscillations are still not enough temperature Park activity this super dope a strong spring he yeah his or her by atomic mark you more starts to behave as if it had 1st 3 modules connected by spring instead of our 5 escapee Aurelio said Mr. would numbers Song is a lesson is as you keep system more and more degrees of freedom become activated and you start to discover the complexity of of our affairs you don't discover complexity of things of lowtemperature are you discover the quantum behavior of things as you he did you start to discover complexity and you start to discover degree of freedom by degree of freedom beyond themselves from a quantum constrained stop behaving classically are some of you may realize dog with a black body radiation but will come where we have a right to related there were really right now a black body radiation what's related something else Violin Strings of Ireland string of violence consists mathematically of a large number of afterward the oscillators army violin streamed parse oscillating units lowest harmonic plus on top of that Yukos was 1st on my 2nd our just but her and harmonics and fundamentally the
1:08:05
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
1:13:38
just brought underrepresented simply as twodimensional onedimensional will be there of the coordinates of the of direction will be the moment was a so there is a lot of this is a high dimensional specs we start out with some probability distribution we saw let's say for example would start out with a probability distribution which looks like back me just say this 0 probability and outside constant probability on the inside of this Bob needs to say is we know nothing except that the system starts out inside that we don't know anything more of our ripsaw cards probability density is constant will remember what were the entropy of actress from normal man somewhat maximum the maximum would be if the probability distribution was uniformly spread over the phase space tourists who knew nothing of the entropy is the logarithm of volume the Bob yacht but Osub this trainee just happen to know that what we did our best possible measurements now we discover that the system is in that block whatever whatever measurement but also on the of Nora Sputnik not and so are not talking about the entropy of equilibrium with talking about the entropy of a certain probability distribution and the probability distributions handed to us somebody tells you somebody upstairs tells you the Vermont moment after the 23rd particles and the positions of the 23rd particles Our old within this block preserve for nearly an hour to snuff out unrealistic you out might start with slowly air molecules in some kind of America drop in a corner of the menu although the air tanker out well originally the molecules were in that corner in that tank confines of a small space and water over you had a pretty good idea of what their energy woes because you know a temperature Watson there's a pretty good idea of what a moment of all particles were that might not quite corresponds to a uniform probability distribution blob but it was certainly correspond somewhat configuration the more you know that corresponds to the smaller me media volume of this region although I say these are the entropy was 3 volleys of a library volume might slow as volume gets smaller and more were not your here the volume gets more and more the entropy ghost which ways my support arborvitae case in any case an entropy which has to do with sign Of this blob in the phase space I know I have this a lobbyist time goes on your member of Purim called Rios fears rules theorem says what volume stays therefore the entropy status this is just another way of saying there if you follow the state each state goes into unique state has it and all are removed discrete will games we played in classical mechanics Corning wherever where we talked about how states involved warned the next next next any given state coastal unique state next know the unique state of Mexico In pursuit you have some probability distributions the probability distributions just on translators if there are sex state provide Rivera states and you'll know that 1 of toward them is occupied the words of their bodies in 1 of 2 states and you also know the rule for going from 1 state the next day after the next step your also novel 1 of 2 states not the same 2 states but you also nor 1 of 2 states so I the entropy and stays the saying probability distribution may change because the states which are occupied becomes different for the fact that they were taught them and they have equal probability that stays the same Seoul on 1 hand at seems to say that entropy doesn't change Her use is this stroke is a correct view of what called microscopic entropy or armed Iowa called called microcode Micro entropy but there's another concept of entropy which does increased and the other kind of cuts of entropy has to do with something called course screenings coarse grain thickly you're experiments such that you cannot resolve points in the face you don't have even if your experiments were ideal perfectly precise experiments you still could not tell 1 state from another state of the was to cope with some resolution so resolution number of states are a quantum mechanics it turns out that resolution was naturally built in the smallest but you can never smallest on certain is given by plants can't but other situations it may just be the resolving powers are not good enough so instead of talking about points of phase space you talk about resolved or around which call costs green points of phase space now let's make an example of often we started with this Balaban phase space and has a certain volume let's follow it I tell you what it does not because sovereignty mathematics problems from described make up 80 dynamics which preserves the volume preserves sea area on the plane here but takes this phase volume here and turned warlike snake which looks like the same total volume the snake is extremely thin and for that but unfortunately because of our limited resolution powers weekend that tell a phase pouring from a neighboring cities point so we do when this happens is we take these phase points and we blob of fire we think of them instead of as being points in the phase space bobs and now we start going through it Graf and guess what blobs cover usually Labrador there are a lot career over yet overlapping had not it doesn't violate Leonardo which is losing information because because we can't keep track of this fine structure actual truly dynamics is such that the snake here has exactly the same volume was the yarn bobbled over here but because we have fuzzy resolution will looking at these things with somebody else's Hi glasses where we see we see if we were to be asked what is the volume of this blot on other worlds where we're appointed facetoface
1:22:28
with us if they exist will come to a conclusion various uniformly distributed over here because we couldn't follow the system carefully and after freak of followed with infinite precision follow with infinite precision and defined the probability density by defining it with them the procedure of volume would not change when every real situation every real situation we can't distinguish 1 state from the advice so we do this process of course grains coarsegrained is replacing points of by blobs once we do that I guess what the volume increases highways a magic big your piece car crime was a volume or a piece of cotton very well if you need the volume of the cotton fibers or up give the squeeze down the something small duet that codified goal the volume of the carbon fibers doesn't change merchants event with it pretty rigid don't but the car of growers out and now you character who has as big a bargain if you look at it through somebody else's glasses and you can see the father cotton fibers your say the volume of that piece of fiber staff has increased our arms and battled that's what happens as what had happened so young saying then that class B a video game if plant had it does just of which of such slips to get the quarter just limits of resolution yes limits of resolution on nor I'll remember that and is not really up property of system property of the system and would you know all about the store what if you could follow in infinite detail every point and the phrase space all you know in the beginning was ever was inside some here but not exactly what where then you would say after afterward inside the snakes but you don't know exactly where then entropy would be considered but to take to take into account but you can't tell 1 point from because of limits a resolution want the saying Look I know much less about where it is but I did to begin with so that we might be able to see if we were to say this not a foot yacht yes that's right but get me out of the problem is given any resolution given any degree of resolution reveal resolution is let's suppose it was good enough to resolve the snake after 10 seconds for good after resolve snake but then the way systems Yvonne phase space as they start grow even Lewis snakes Lewis makes makes would only in degree of resolution this snaking there's will give a frightening chaos is were made our way systems Eve although the phase space is they start to bread and starter spread out tentacles we're about changing the volume Agrawal fight the fibers grows more more fighters grows more fibers and eventually it no matter how good you resolution you'll eventually stopped seeing that the entropy increase that's 2nd that's what's more now let's a wedge a manager and some phase space we start out knowing a great deal let's talk about how big the available region of phase space whose firm but think of it as a gas on a large number of particles In a box of In a box that means the extras rebounded so in a onedimensional order would save the phrase faces boundary between here and he and his a face commandment me anything you know usually be energy constrains how much maximum meant be sold even if all the energy of the system when Inter 1 particle 1 particle which still itself have a finite energy and so the effect of regional face space which is available to you as some big box may be a Saab crescentshaped it's fine it's fine it doesn't extend to to infinity in a new direction OK now the phrase point start someplace exact precise please point what kids perhaps is the subsequent motion is extremely sensitive to where you started with they start over here this system will evolve over time if you started very close to it the system when evolve a very similar way for a while but there was sought to depart that would go off and some different directions for an example would be on billiard balls idealized billiard table with no friction and member of people know your pocket who Chicago straight for you plug with a cake a threecourse should restore on North Terrace 5 young so you know that if you balls of certain fresh and barbecue ball and you are extremely accurate start with Cuellar balls exactly the same on 2 different occasions if there are exactly the same thing you should do exactly the same exactly the same thing will happen now 1 of your high he fighting fighting I need bit off year even everything else being kept perfectly reflects well if you really just off by truly truly premium narrowed the difference in the trajectory will be the whole system will be small for a while but sooner or later with scars stock exponentially bearers compound errors comparisons and sooner or later that trajectory nearby trajectory is going to deviate that's the character was called chaotic systems of most systems in nature of chaos so that means that no matter how close to start together you'll eventually the part and firm and threw up a lot of phase space if you got few infinitely precise watching these 2 points seemed to spread our you know a volume of phase space strictly speaking stays the same I'd This is the origin of the fact that you start With a small belong in the phase space he'll stay pretty coherent for a while but then will start probe develop Sandra some pieces that won the wrath of you'll wait and ends and our ones will stop her grow out these are the different regions of phase space beginning to depart from each other because errors compound and points always here which are close together nevertheless eventually will depart from each other and the phase space varied the phase space volume gets extremely
1:31:20
complicated and eventually if you wait Wong enough to assist gives thermal equilibrium This can't really staff basically fills up the whole thing fills up as as Cottonwood for opera of volume the actual volume of these fibers stays equaled whatever the initial based Of The year of the phase Bob but spreads out in a way that if you do it indeed course screening of it all it basically fills a sitting with a uniform probability back so now we watch a particular phase point a particular phase point start here that means we start with a particular configuration of the cue ball an thorough 1 through 3 15 which backed Mark and we run it starts and also Morris cervical somewheres else miracle somewheres else are go somewhere else eventually if there's no fracture on billiard table eventually you will sample something close every possible configurations to wait long enough you'll see it every possible configurations face based visit did not exactly Avery course your passed through just about every point and the phase space very very close to it and so on what happened is on the 1 he here you or something in the beginning like for example you might know that the initial phase point it's something very special its found here corner although molecules in the corner of the room where certain definite velocity going off from some direction I the bond dental each other the phrase pointing a separator phrase point the separator eventually fills up more thing he said himself so how anything anymore reached its maximum entropy became beginning entropy cerebral anything but nevertheless In a completely unpredictable and ran the way that you can't foresee you because you don't have enough detail this phase point will eventually come back to this point here the Moors cue ball valuable analogy stock UQU stock Europe right there are racked up certain very very special triangle cue ball Byron Dafoe were issued a very precise but after a a certain amount of time they begin to or from from directions and begin to fill the begins the filled not backup tho pulled cable but they fill the space of possibilities but you wait long enough if you wait long enough we can talk about how long it takes both falls welcome back and a former triangle Our Wall what's not answer it that's not in its it does depend on how precise you require reproduce the year of the original configuration if you allow even the very small car new saying this is the original configuration picks to within a certain tolerance the depending on that tolerance toward fight kind for the phase . com back about original configuration if you require if precision it will never come back to require precision within this summer Epsilon and the time that takes for pest fax through that phase point within columns epsilon is fine young billiard right yeah yeah yeah energy I'll see what knots a starting point with the image of the 3rd pillar that is over a practice that has right OK Saleh on 1 hand the entropy increases and decreases in the squash screens stand because you lose track on the other hand that does not saying that you can't return to the original configuration you could try estimate how long it will take N may move next time will make an estimate of how long it takes for a system are many degrees of freedom are say there will try out this problem the air in the room grew all the corner with its energy has attempted hazards kinetic energy RO modules are going in the same direction very very special state issued after Waller bounce off the walls or start hitting each other they go cruelly go crazy you know the system very chaotic and after a while fool room under normal pattern over from liquid rearm and we said that we wait wait wait how long does it take we can be so we should never shared a room last warm enough Marovich a really truly close system how long will not be course just orally molecules our butts kept constant temperature How long will it take on the average before WalMart tools will be a prom Corrigan or automatic corner on that corner and we can I question can be answered the answer is you warned the last major it's not reversals for news because of that goes photo completely reversible anything we can happen theocracy care at bar the novel I will talk about
1:38:04
will talk about the 2nd law more for reversibility and how paradox all cars paradox of reversibility reversibility get forget resolve no it was 1 of the great there clashes paradoxes of physics the puzzle lots of people that you undoubtedly nor the history of the mixture of draw were Baltimore to distraction and all of the joy of suicide but certainly depressed eventually came up with the right answer the right answer of means that not that entropy always increases but given any configuration its most probable with next configuration will have a warrant for interpret probably increased probability or Ferrara configurations soft back in far usual now configurations are expected for which more well I'll give you some more examples of troublemakers more intuitive next time but we do all the 2nd borrows heavily some kind of exploration scientists for a minute if so well limit into a number of chaotic systems mint number among Kabir systems now if a talking about measures on her measures on what is alarmed Hamiltonian so things are to be question I'm sure that mathematical people have could probably answer what is the right measure of the chaotic systems richest among chaotic systems on barked early but to request a Renault system Zacatecas assured In many cases you could prepare them a little bit you could change move will still remain on chaotic it's not the sort situation where you have a chaotic system and you change the rules Hamiltonian trainee kind a bit it will be calm chaotic Norwood usually some on chaotic chaotic means that the phase trajectories tend to seperate like this after a long enough time bad jargon that that I didn't want to defined it with any precision right now but it is this phenomena are close by points in phase space following each other for why wouldn't exponentially party so a system becomes effectively unpredictable Our to situations in which predictability effectively breaks down because In order to predict a system for it a length of time t your precision with initial conditions and your precision and knowledge of the evolution of the system has to get better and better and better as that kind of which you want predictability to be predictability so you take away justified as a courtroom modules doing whatever they're doing instantaneously NUS how long can I predict the weather bomb how long could you predict aware depends on how precisely in awe initial conditions if you want to predicted from all the period of time you you have to have more precise initial beta even longer more precise now that's quite different than the harmonic oscillator a harmonic oscillator in phase space just move around the circle with constant frequency you start essays point over yes it travels around here start of phase point over here they just go together and they go together forever and ever off course by phase points never get far from each of the sobering non chaotic system Our 80 our particles or did there are around Osama drool inverse square law where are we do exact Tony in physics is not to be too close by trajectories To caused by orbit will remain closed by Richard forever the 3 body problem users chaotic general that of America he may bring more bitch eventually tend to depart from each other so that means predictability is a function of how accurately you know the initial condition to predict for a weight of time t universe no initial conditions to within epsilon of feet to get former unique normally initial conditions better for me to be chaos although losses but also a also yes you also meet normal wars or laws parent B and and the initial conditions for what chaotic mathematical property about do seem chaotic pendulum the bubble pendulum there's a chaotic double pendulum is the order a pendulum is not chaotic it's very very predictable and where a 2nd pendulum white back that's rings otters forms and you started going and overseen the bubble pendulum action but quite fascinating you could follow it a little bit but then
1:44:47
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 racebased spread
1:45:06
U.S. on you're asking very hard questions why some systems are chaotic systems are chaotic I would say the state of the art is almost but it's very very hard the look at a Hamiltonian nor whether it's chaotic power except for the fact it's complicated it's usually chaotic bark about out there are complicated ones which lot chaotic so if this is not an easy question something mathematicians and mathematical physicist study and we don't have answers the governor general do a little friction over With no furniture correct for the precision of dish the precision and all for which she half so the procedure so that we were right that but if you did this vicious gave her Saginaw of conditions winter bridges and whatever that means you could predict Pa now I think that probably amino stayed there but the Reichstag and is some connection between given the climate Yornup predict and how precise your half an hour the initial conditions yeah show gap so that has to do with how fast these trajectories apart from each other if trajectories tend to stay close to each other for a long period of time the system was rather predictable for a long period of time but eventually will and so what systems are unpredictable merchants William Ryder me another lawyers say is armed given any link giving any length of time you want to predict there exists colorants precision and epsilon are Brett were allow you if you knew the phrase point within that precision epsilon would allow you to predict about ways sorry during see there exist of epsilon such back and so forth Yao Yocum of culture former chart that's exactly what what's that dapper off ADP ON HAITI Norway and only of a lack of Riyadh on all or warrior wore a hero the capital since then but that those are February of coefficient is the exponential are our growth of a separation between trajectories safe with those of the which half up more they pull your off yet for him that should From chaotic system certainly don't have closed yeah called fullmotion for upon their Exxon was a starts growth have straight up badly are here back sigh leaders Colorado there from a John stable normally on stable Barra work the symbols caiman stability is just the top
1:49:31
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
1:50:24
Stamford . EDU
00:00
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
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05:15
Gewicht
Höhentief
Masse <Physik>
Direkte Messung
Luftkissenschiff
IKEA
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Neutron
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Teilchen
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Fiat 500
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Amplitude
Molekülmasse
Energielücke
Stunde
Schall
Gasdichte
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Mark <Maßeinheit>
Behälter
Windrose
Proof <Graphische Technik>
Niederspannungskabel
Urkilogramm
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17:05
Greiffinger
Magnetisches Dipolmoment
Intervall
Grundzustand
Boot
Sicherheitsbehälter
Monat
Fahrgeschwindigkeit
Passfeder
Knoten <Astronomie>
Multiplizität
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Werkzeug
Weiß
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Jahr
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Masse <Physik>
Bergmann
Tonfrequenz
Tiefgang
Dreidimensionale Integration
Schraubverschluss
Teilchen
Radialgebläse
Kristallwachstum
Kopfstütze
Spiel <Technik>
Gasturbine
Stunde
Windrose
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Proof <Graphische Technik>
Niederspannungsnetz
Platzierung <Mikroelektronik>
Gas
Thermalisierung
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30:34
Greiffinger
Gesteinsabbau
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Satz <Drucktechnik>
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FaradayEffekt
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Masse <Physik>
Direkte Messung
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Festkörper
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Tonfrequenz
Begrenzerschaltung
Dreidimensionale Integration
Chirp
Farbton
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SpeckleInterferometrie
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Ersatzteil
46:28
Kaltumformen
Schusswaffe
Behälter
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Windrose
Mechanik
Bergmann
Quantisierung <Nachrichtentechnik>
Besprechung/Interview
Energieniveau
Cross over <Teilchenoptik>
Chirp
Schallaufzeichnung
Gleiskette
Negativer Widerstand
Rungenwagen
Feldquant
Absolute Datierung
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Theodolit
Temperatur
54:25
Schaft <Waffe>
Drehen
Wärmekapazität
Kraftfahrzeughandwerk
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Cross over <Teilchenoptik>
Woche
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Weiß
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Blende <Kleidung>
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1:31:19
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1:44:45
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1:49:31
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Metadaten
Formale Metadaten
Titel  Statistical Mechanics Lecture 7 
Serientitel  Statistical Mechanics 
Teil  7 
Anzahl der Teile  10 
Autor 
Susskind, Leonard

Lizenz 
CCNamensnennung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/14940 
Herausgeber  Stanford University 
Erscheinungsjahr  2013 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Physik 
Abstract  Leonard Susskind addresses the apparent contradiction between the reversibility of classical mechanics and the second law of thermodynamics, which states that entropy generally increases. This topic leads to a discussion of the foundation of chaos theory. 