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Statistical Mechanics Lecture 3

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entropy of a probability distribution is a storm of War of the States a probability distribution now means a probability for a system to be in the eye state where now we really are thinking for the moment about states being discreet so that we can count them off every state has a probability P of course where it came from where that probability came from will discuss later but let's suppose somebody gives us plopped down for us a probability distribution PORT then his appeal was also energy of the energy of the eyes state and the assumption now is that the sum of all the PRI is equal to 1 but and we much a definition the sample P I on a EU of our IT is equal to the average energy that makes sense to add up the call of the states Stewart and their energies but we did according to the probability of the eyes state and that's what normally called in average quantity of statistics on probability theory and never understood the difference between what statistics is what probability and polled by some of my friends or statisticians said there is a difference but I have never understood what think statistics are actually set date on you have real data take our but there about how a statistician is different than a probability probabilistic and sure that you know anyway I'll use it to terms anonymously statistics probability and but probably not the probably not but I'm not the right OK so long this one's more or less the definitional useful definition this 1 just some of probability should be the 1 hand entropy is by definition for a given probability distribution start of miners and their P library of key Of all eyes looks like it's negative but it's not the reason that all of the peas are less than 1 and for anything which is less than 1 Malaga rhythm negative so this is actually a positive quantity right so we discussed last time probability distributions fire imagined that there was 0 1 parameter family of probability distributions here's a Member X. his next warned of ending may be plotted against energy of a particular state and parameterized by an average energy obviously is a probability distribution shifts broader and broader or shifts from a ride in some ways the average energy gets larger our there's a particular probability distribution which is very special 0 wears except that equals 0 or equals the minimum value which the minimum value of the energy and that of course the ground state of assist on the grounds that the system is described by a probability which is Zero everywhere except the state of low-cost energy in which case the entropy is exactly 0 but as the probability distribution as this 1 parameter family of probability distribution shifts arrive 1st of all we energy tends to become larger and entropy tends to become larger because the entropy is qualitatively simply a measure of actually the logarithm but that's a measure on a number of states that are important underneath the probability distribution the probability distribution has broader and brought a wider and wider the average energy goes up and at the same time our the entropy goes up will always assumed that the entropy is a monotonically increasing function of the average energy now of course you could invent probability distributions for which that's not true you can invent a probability distribution which gets narrower and narrower as it moves to the right but as we will see the probability distributions that governs thermal equilibrium are not like that we'll find out exactly this picture is a picture that governs the probability distributions from estates in thermal equilibrium thermal equilibrium is characterized characterized by a temperature of course but who not characterized by an average energy or have a box of guests and you know the average energy in it and be handed to the thermal equilibrium thermal equilibrium means that would ever was gonna happen has finished happening and now it's simply quiesce and the molecules a role thoroughly mixed up and thoroughly thoroughly dispersed throughout the system and not much is happening on the microscopically of course our that state of thermal equilibrium state of thermal equilibrium he tends not the flow Woolsey will understand that soon enough he tends not to flow because all parts of the system are foreign equilibrium end and as US as a says we will say a probability distribution forgiven the average energy doing fact broad and broaden and broaden his energy goes up so that the entropy is a monotonically increasing function of energy important factor OK we'll go through now similar laws of thermodynamics 0 will talk about the 1st loss is just energy conservation the
zeal a lot in this in it axiom that there were people or accident rising thermodynamics Zero so of key before almost anything else but today I think we think of the 0 4 as a consequence of the law the 2nd warmer serves the 2nd offer more barometric says it's good entropy increases no 1 on earth is every means what it means is you have a system described a given probability distribution and it's not in equilibrium when it's not in equilibrium and limit columnist equilibrium generally speaking the probability distribution abroad would approve that eventually will approve it and a qualitative kind of way while qualitatively we will From the assertion that if you'll have a probability distribution and you will allow it to Yvonne for being a little bit Carol S. being a little bit careless being a little bit careless is called fine-grained cost Excuse me cost you will find that we probability distribution always broadens and that's not surprising for a probability distribution can narrow it means you found out more about the system the new originally known technically a few talking about a very complex system of a large number of degrees of freedom very difficult keep track of exactly the opposite ends the happened you lose track of things because you lose track of things the probability distribution tends the abroad and that increased entropy with time alright now 1st lot and energy is conserved 2nd law entropy always increases with time and it doesn't increase it stays the same but never decreases bucks assume that the entropy always increases or stasis hour and now talk about 0 4 0 loss states basically 1st of all it's a research that this is such a thing as thermal equilibrium but if you take a box much talk about a box of gas but it doesn't have to be Jessica be liquid it could be any system which is isolated and self-contained that if you wait long enough it will come to equilibrium now what does it mean for it to come the equilibrium if you imagine that the system is made up of separate parts of Kong 80 beat analyst preferred system in that assumed that they collected connected means that they could exchange energy with each other so what some pipes between them allows energy flow between them Tito we must short repertory out to Korea the 2 tour repertory that's fine right now what it says is it is a concept called temperature is such that energy always From logic temperature a small temperature energy flow from larger temperature too small temperature No. 1 and No. 2 0 it's all right so that's sorry it flows From larger temperature the smaller temperature if the system is not in equilibrium but if you wait long enough the flow will eventually come to an equilibrium with supposes only 2 systems from and energy will flow from the 1 of high temperature for the 1 of lower temperature on until what point well until low-temperature 1 becomes harder than a higher temperature war but then the energy will fall back the other way while at half what happens is the energy just flows from the call the system until they equilibrate until a equilibrate equilibrate as we will see means that temperatures become equal when the temperatures on both sides become equal he stopped flowing at cover the meaning of temperature of throughout the meaning of temperature is that tells you which way he flows No. 1 the temperature exists such that heat flow of energy flow is from the heart of the when the temperatures become equal energy stops flowing again if AT is in equilibrium with beat that after all means that the temperature of a user saying the temperature of B & B is in equilibrium with that means that the temperature B equal to a temperature of state then obviously the baby is in equilibrium with state that just says the temperature raising the temperature be in temperature B is the same as temperatures saying the temperature of a temperature of receive the same James cost be given some arise 0 loss by saying his emotional temperature which we already introduced but there are but of reviews it could illuminate this law is not temperature at No. 1 No. 2 energy flows From higher temperature poet temperature and No. 3 in thermal equilibrium the temperature of all parts of the system is saying that the temperature was not the same in all parts of the system energy would flow until it became the that's the basic 0 floor the existence of a temperature function that tells you which way energy flows and the existence of an
equilibrium when the temperature in all parts of the system are set by so let's see if we can relate to that of those ideas for the notion of temperature that I have that we introduced earlier last time namely that can't Butcher is related these derivative of energy or a change in energy With respect 2 and if you change the energy a little bit the Aboriginal juice millions of you change the average energy a little bit the entropy will change and that derivatives that rate of change is called the temperature but still see what this definition of temperature has to do with with the idea of the flow of heat which way it goes so let's begin which take a to piece of a system composed of 2 parts but court AT & be with little pipe between and let's write down all the basic equations that we know but before I start for more let's assume but the temperature of BU is bigger than the temperature of a now if the temperature of 80 is bigger than the temperature of switch them not just interchange a and B in everything I say OK but we have to pick 1 Of course you can have a temperature be equal the temperature but let's take a case with and not equal and that means and not in equilibrium nonequilibrium temperature bees temperature very focused so I tried and some equation the 1st equation tells us that when the system responds and does whatever it's going to do the total energy doesn't change that was a good deal what could do is gonna be some energy shift from 1 to the other so let's say the change in the energy of a plus a change in the energy of B when whatever happens happens tours after a small amount of time that it is equal to 0 we take our stopwatch we wait 1 2nd and we look at the energy redistributed self refined apathy and of changes we find apathy and you'd be changes but energy conservation tells us that the sum of the energies of change next the change in entropy of a C plus a change in entropy of be change Republic Bingham factor travels always greater and once in a while it's equal to 0 but only in very special cases in Genoa statement is generally greater than 0 change in the entropy of a bus a change of the entropy of beef which is a change in the total entropy theater B of a whole system no awards the probability distribution of the combined system the probability distribution of the combined system shifts in such a way as to make the distribution brought our that's OK now let's use 1 more statement that the change in energy it is equal temperature times a change in interfered with supply this the both Eddie and B U V e a is equal to temperature of a car's a change of the entropy of AT and seen fleeing from beat then we can rewrite this equation over here this is the 1st law of thermodynamics now has T. Eddie d best a plus DSB is equal to 0 or just a ride in a form which is somewhat more useful for me I'm going to work solve this equation for D S B solve it this guy over here where we get we get that DSB comma D S B is able to minus appears unwilling to transpose this turned the right hand side that would be a minus TIA and now I have to divide by TB DE s and now take this formula and stick it hear him as usual a list of people to build physics you 1st thing really hard about what you want to and a new blindly just go ahead with the equations with a lead so we have run SA and SB BSA Diaz but they connected so let's eliminate 1 of them and rewrite this inequality so we rewrite the inequality EDS L E plus but now what miners because DSP has a minus sign might he 80 TB times DS and that's great event 0 took 2nd walk tells us allow maker multiply it by TB for simplicity let's just multiplied by TB song temperatures up positive unless otherwise there are situations which temperature could be negative but not for us for so was it's as TB my last TAT tried
about she'd be minus t times D S. a is greater than 0 what if now we already assumed that the temperature of is bigger than the temperature of a you would also look at the case which would Raymonda both the temperature of indeed bigger than the temperature of a B C B minus theory is positive what is that say about BSA a positive number times a change in the entropy is Pocekay that's says the D S a is positive cell but what we conclude from this equation DE best favor for the particular set up but there is greater than 0 Millwall what happened if she would be given to be exactly the opposite NTB minus TA would be negative and VS it would be negative but we've already chosen are convention we decided to call the the 1 with the bigger temperature and that case Diaz aid is bigger than 0 but does not apply this equation harvest inequality by TAT times D S is bigger than 0 but what is terms Diaz said it's a change in energy of the subsystem this series D of the energy of Aso only found by aluminum manipulation and the use of a couple of laws or other laws the 1st lot energy conservation the 2nd lot entropy increases natural reused plus a little bit of juggling of a trend posing and multiplying by a simple algebraic operations we find that the changes energy of a is greater than 0 it follows of Qantas change in the energy of the is less than 0 another words he has flown or energy has flowed from be day so we have proved that with the definition of entropy he Assad the definition of temperature has given here that the temperature is such that it defines the direction the heat flow he will fall and when I say he'd incidentally I always our I just mean Energy Energy will flow from beat on till the temperatures become equal where temperatures become equal of course if he equals TB there is no flowers and that to equilibrium was established so equilibrium is established order will be no floor Our where TB equals T.J then there's no fetal together OK so that the that's the that's as is the might years due to 0 the law of thermodynamics namely the temperature determines the direction of heat flow when he flow stops because the system comes to equilibrium and must be that the temperatures the same in every part of the system conversely Pacific temperatures not the same in every part of the system he will flow onto it is to say yes he does Sinbad assesses data system yes and that is true that is true that is true but it is not an equilibrium it could be because when you break it up and a smaller subsystems the smaller subsystems are not in equilibrium with each other and then you just apply the analysis the subsystems right yet so bad that there was correct I did imagined that system Maine system beast separately Warren equilibrium with themselves so speak up that's a good point but now we come to the hard questions are out of his mind 5 states that old that from investors say they positions and velocities men of their single yes except that in quantum mechanics we can take those to be discrete variables basically yet that all or we simply substitute for In a integral instead of a some young but that's what just a look a state always means the same thing it means as much as you could possibly know about the system if you were infinitely powerful lover now there may be limits that have to do with them over the fundamental rules of physics such as quantum mechanics which tells you you can always much as you thought you might like the novels such as the position and velocity a particle but still state means as much as can be known are about a system by what's scored infinitely colorful observed last it's all a quantum mechanics they are young incontinent Canucks there are strong quantum mechanics the state camping for finite box His discreet what I have to show is that if we do the correct quantum mechanics statistical mechanics than the certainly made it becomes equal to the classical statistical mechanics of sort a certain limit you could replace some by integrals that's what it comes down to that that's a subtle businesses but that's not one-liners that's a subtle business but it is true far-right next question the open question is is equilibrium after equilibrium was established you have the box of guests areas there's some probability distributions P for the system to be image different Microsoft dates what is that probability distribution what the mathematics of it so that it would begin to study now Airy right now 1st of all it's always a useful idea not only useful they usually corresponds to the facts of unimaginative system that you're interested in is not closed but is in contact with the outside world in exchange energy with it in particular it's a useful idea if you want to think about the probability distribution of the states of this subsystem think of it as being detonated in a huge reservoir god of a much larger number of degrees of freedom but provide a heat bath that provided he left which allows energy the flow
back and forth on tilt the system in question comes the thermal equilibrium with the big he think that is still be there even a little bit of energy flows from into the subsystem the temperature doesn't change much now this is something we could also parole but just think about for minute we have a big big big huge system at a certain temperature little system bit of heat flows from the from the big system little system typically the change in the temperature will be of the big system will be that will be negligible and so we can unimagined makes system is at some temperature and simply wait until a small system is In equilibrium with a big system now particularly convenient choice of heat then you'll see but a particularly convenient choice of heat bath which often actually corresponds to a facts but particularly interesting choice is just don't imagine that the system in question is 1 of our of vary large number identical systems which are connected together by little pipes allow heat the flow back and for 1 of them is a system that was studying the rest of them simply provide heat bath actually this sometimes corresponds to the moralist the facts in particular that we have a very big system and we divided into small subsystems the small subsystems may be very much alike and the big system is just he back from small subsystems just provide a large number of replicas that altogether it up to to the big system so this is a useful trick to pretend that these but the heat that is just a repetition of the same system over over again how many such subsystems will eventually Willingham they get real argued a large number of large enough W. figure beat that is very big but let's call it capital gains from moment that's a number of these systems we have and each 1 of the systems is In a state right sorry I my demonstration these are the identical systems where I want to 3 4 5 6 7 8 9 so capital in his equal tonight these are the possible state that these are all identical I couldn't do anything to them but of course the states of the system are not identical they had different energy for example so what's to call them to remind ourselves that Miss Bates are not all like and particularly they have different energy the stakes are not real system the misstated is labels Vermont labels and some of the systems are ever be in 1 state some of them B and the other states some of them will be in the 3rd state he is particularly simple example with only 3 states each state has different energy the energy might increase from right to left your right to your left not right my right my OF your left to the energy increases in 1 direction and with lot less willing to call the number of systems subsystems I state right I say the state's going to indicate that access he Community energy access and these are all possible states of the system this is all I states will be labeled but I I equals 1 heII equals tool about that there might be an infinite number of states In fact in generally will be an infinite number states most of them have so much energy that is no chance of Paul work there are gonna be occupied and so there will be a negligible number of subsystems are negligible fraction of them will occupy enormously high energy states just because we may not have as much energy available but now let's introduce the concept of an occupation number occupation number is the number of systems in this case it would be 3 occupying the R. I state in this case the red state programs there are lowest Dolores energy state will call at will in some RIP right his eyes and this it the number of our replicas here which are found in the box labeled by bite the box labeled largest means that the system has that particular state motoring arrested for statistical purposes for probabilistic purposes is interested and how many ways are of redistributing the systems among the states for a given said I was and some bodies and so by here stands for and won a tool and 3 and 4 and forth right so there are these occupation numbers and the question is given the occupation numbers how many ways on the of redistributing the states so as to create back set of occupation numbers obviously that's get beat closely connected with the probability given values of the occupation of former Prime probabilities are up for for state I'm the move Crawford occupation numbers and the more ways there are or of distributing this system into a given set of occupation numbers the more likely that occupation numbers it that's a symmetry Oregon of all possible ways of redistributing things are symmetric with respect to each other and equally likely then the most probable configuration which means the most probable set of occupation numbers is a set of occupation numbers which gives rise to the maximum number of ways of redistributing things so let's go let's go over some simple examples who's yup yup right right we're Commack at us fears of energy constraint so what's right fielder constraint doesn't and U.S. I had intended to come to that later but to rest on top of the probability distribution that's associated with this different number ways of redistributing things they are constrained by the 1st constraint was right the 1st constraint the 1st constraint is in 1 plus and 2 plus and the this is the number of
systems In the 1st level plus a number of systems 2nd level was a number of systems of 3rd level for just 5 or just a summation of Y and some i.e. physical bought capital in total number of system now has another constraint which is the 1 that you reserve developers just at about namely the total energy of the system solid tried that constrain the number of systems in the 1st state times e energy of the 1st state plus the number of systems and the 2nd state times the energy of the 2nd state altogether bust-up but that of course is equal the some additional of In the summer by east of Bali was articles the total energy right now let's assume that the total energy is proportional fit in willing to be studying the system book making more and more replicas but clear that as we double the number replicas the total energy will double at least if we want to keep things fixed within a given box we do so we can expect that the public energy is equal to capital in times the average energy per box let's just call the average energy per box boxes and boxes Meryl our news I use the word boxes forget boxes average energy per subsystem here is b and the total energy is capital In Times e and that must be simply the sum of all the energy OK so before we move onto some probabilistic things let's just rewrite this in another way if I want to tell you that the occupation numbers were in 1 and 2 and 3 and so forth the total number systems was capital in and then sq was the probability that a given 1 of these systems is and I state Beaulieu answer be welcoming the think we come to the same answer the answer is but fraction of sister rooms Wearing Iife state a good definition in this context of the probability that any 1 of them is in the RIA stayed here is simply the ratio of the number of the occupation number in some light of problem number but the probability is in this context so let's the following the probability that any 1 of these boxes or any 1 of these subsystems is in the state I could just be a fraction it is a body which is the number of them in the state by divided by the total number yacht analyzer sailors and sub life that thank you that's a living normally use probability probability is a measure of the fraction of and the expected fraction of Rome if rates and type in this case the event is measuring weather system is in the state are not focus buyers and among us opposing let's let's now use this right these 2 constraint these to constraints once said that the problem number systems as capital in and the other says that the total energy is equal to capital and Pansy energy persist on lots of I do that I can then right I can then that's 3 erectus PORT and is equal to an IRA what has become this becomes the summation of 8 times PIA mistress and are a little MORE physical the capital and now divided by capital in the more we get works we get sum of peace a abide cyclical 1 that's notice a surprise nobody that's a surprise nobody Our but it's just another way of writing the constraints of the total number of systems equal end and what about the equation here this becomes summation of Internet firms P if In terms of total energy and again divide by this is what we knew all along that he underwriting inside it is just average energy of any 1 of the system OK so there's Rice's visit to good things to remember is took a strengths and whatever the probability distribution is the probability distributions some body whatever it is it has satisfy these 2 constraints average energy and total probability or if we want to think about it in terms of the occupation numbers before you could rewrite the terms of the occupation OK but before we worry about the total number of systems being kept constant and the total energy being kept constant let's just as the more primitive question for a given set of occupation numbers in 1 through and however infinitely many of them how many ways on the air of redistributing systems to correspond to that that there are a number of you know anything about common targets you know the answer by drought and Unicom probably is a something's Mrs. this atop the hardest mathematics and course I was fined common toward very tricky but there's a small number of rules and a doctor go through them on the answer them will just check it for a couple of cases with checkered for a cup of cases see that rear stroke OK so we have a set of now clearly if capital in this fight nite and let's assume capital and a very big no interest the Limited which it really is very big but for for the moment let's take a finite and let's also poses an infinite number of energy levels Walter Scott most of the occupation numbers be 0 keep that most of the occupation numbers are 0 by the into years the number of ways and I want to just write about number auto arrange what's called a number of ways well from it in order to keep geek the occupation number 6 set of occupation numbers is In factorial the total number of them factorial provided by the product in 1 factorial Inc tool factorial in the 3 factorial
dot dot dot all of them another words capital and factorial divided by the try to miss a symbol for product but the end of factorial rejected for a few cases Richard drove but this is a comet or factor but 1st of all why Balow ones which as you around they give us zeros in the denominator not why not 0 factorial is 1 so all the ones which are unoccupied just give us 1 and we can forget all the occupation numbers which is 0 contribute the factor warned for each 1 of them was no danger of this thing blowing up because there are a lot of ins equal to 0 0 factorial 0 factorial Eagles 1 factorial pickles while now we could use mathematical induction to prove that every number factorial sequels awarded if exceeds broke yacht but he'll theory the okayed just in case you think I'm not really aren't serious I'm not door factory was to hold 1st the other regional would follow at 0 yes there are lots of ways of Ray Batchelor camping what camping how many ways not for the moment forget the constraints this is number of arrangements for a fixed set of occupation numbers with no regard to whether the idea of a constraint is satisfied will come back to the question of the constraint being satisfied will have N on top of this the effect of constraints but let's just look look at this for a moment and uh study it's interesting expression we're approximated coal let's check it for a couple of cases you look forward to patch however at that now with the old 40 also that very focused ticket for a cup of cases 1st of all supposing you take all our Bozoo were interested in the case where 1 arm V occupation numbers is capital in an all the other ones 0 0 and that was 1 of these is in fact all the others a 1 and is in and factory which is 1 watt seems reasonable how many ways out putting all of the things in the 1 cup 1 but let's try a couple Percival's cases but suppose we have wrote to wear Tools subsystems and wonder how many ways out there of putting 2 of them into 1 cup with already done that that's 1 8 the other possibility is 1 1 cup and 1 in another cup what 1st check the answer this way there at 2 systems so as to factorial In each cup narrow this is the 1st couple 2nd cup 2 3rd Cup the 1st Cup has won the 2nd cup as 1 third cup has 0 so this is 1 factorial ill-timed 1 factor real-time 0 factory over which is 1 so the answer is 2 factorial or till 2 ways not 14 to systems into given a pair of cops with the other 1 being emptied of course that Stroh Ukiah but this 1 here when he or you could put this when he and 2 ways of 1 more case on our 1st of all company ways arbiter of taking 3 subsystems and distributing them in 203 States no the the same path that that's sigh as obvious as it may get a quick answer watch Japanese to factors that its 3 factorial what is it according to Mr. 3 systems that because 3 factorial although 1 factorial 1 factorial 1 factorial which is 3 factory which is sex and the fact there are 6 ways but you know how do it for you know and in general this is a formula this really is a formula you could prove it it's not too hard to prove by waiters go look up their macabre tore expose program Dr. approved high school I think this is a high school member OK so this is the number of ways of taking capital in things and distributing them among the possible states With occupation numbers and 1 and 2 and 3 that will find is that in this situation of interest this can be thought of as a function of the injured was a function of the occupation numbers and the circumstances that would be interested who going defined that things like this hour honey lead peek at a particular set of occupation numbers namely went capital in gives very very beat with long defined that the occupation numbers cluster very very strongly about a particular set of occupation numbers or better yet a particular set of fractions cap over begin that De that distribution gets tighter and tighter than gets up so that they probabilities little in the of begin armor are well-defined OK so that that much what duel in order to do that we have to have some approximation methods the 1st thing we need to approximate is fact Orioles when I say approximated I mean approximated under the circumstances that ends a role very big Izzo imagine that we simply allow the number of subsystems to get bigger and bigger and bigger and such a way that the occupation numbers also simultaneously get bigger and bigger and bigger so all the ins rumored to be assumed to be very big and let's approximate much approximated as odd using a standard approximations effect Orioles another name the the approximation Sterling's approximation so would prove it let's prove what proved early not if everybody knows had approved the progress raise your hand if you will have approved it's approximation you will be called upon to do it I notice Michael scaling back now former said
Robert Lee does Merle the that's a visit here and if you know what's Cherilyn's approximation is raising about book is a good so don't do Sterling's approximations but at 1st I'll write out what it is code and factorial as it gets larger and larger and larger becomes better and better approximated by the number in high running out of ink here now that day a truly area each reminders and now that all sorts of corrections to us which become important win in the not too big but to begin and gets the cost of this gets to be incorrect at sea of a control freak Crowley but stake in Thailand and factory and approval wouldn't do is work with the logarithm of the function were interested and factorial but instead will calculate lager and factorial now a factorial of course this is approximate equality but what is in fact and factorial His One NYTimes to it time dropped end and the logarithm of factorial it is a lot of 1 plus log to blast out back up log Our and another words it's this summer log let's call it acts for the moment some of log X where X is an integer From we has take it we could figure from 1 to capital and play them all up and that uses the logarithm urban factorial OK so login factorial is a sum written pair aside from which take it from Michael's 1 legal status from Michael 0 it doesn't matter if our were name gives very logic doesn't matter exactly we we begin the summer but From articles 1 day and logarithm authority from X equals 1 Largo acts were for the moment decks it is a integer OK so we can draw a picture of the locker 11 years warned library is a function which reflects on Mrs. X blogger X grows forever never butter ever progressively smaller and smaller rate and now we want to eat up all of these integer the logarithm at all of the integer starting at 1 2 3 4 5 6 and so forth the distance between integers here of course is 1 the difference integers all integers after all that cut of obvious that you can make as a 1st approximation and this is the approximation will make the try 1st approximation used to say this is just given by area killing them all out with a small area and the Arab can be estimated it modifies a formula a little bit but we can all this up and replace it to a good approximation by the integral it is still good approximation particularly when a gets large integral BX of logarithm of X from where away from X equals 1 well-ordered 0 2 capital and market Soloway have to do is learn how to compute the integral wars rather than bad our heads against the wall and try computer the integral of lager X let's take the answer and differentiated and see really does give La decks below the opposite direction saw the integral just be indefinite integral of large X is X Log AX minors X but check that how the hell I know that I know because I want I want by trial and error will be a trial and error and you can make it does guests in this is it OK let's always do is differentiate different Watson here and show the slot next OK so 1st of all the derivative X Log AX you 1st differentiate X misses derivative Exs is 1 that gives you a lot X a assorted White in the main times a derivative of log X what's a derivative Logic's right so derivative so good so X times a derivative of lager Exs 1 what's the derivative minus X minus 1 where we are we're proved that the integral Beebe IDX Beebe ID X this thing here is log X or are we In the rule of law X is X lager X minus so there we don't integral part of this is X 4 objects minus X is evaluated between X equals In an X equals 1 right now an ex Eagles warned this does vanishes so we doctoral X vanishes log expenditures of all vanishes so the whole answer is In the Alar in the minors we are now computed a certain approximation with no computed logarithm of In a factorial equal did or didn't minus 8 How we find in factorial from Nova lava remember something was a certain operation we do to it defined the thing itself exponentially expert OK so the exponential both sides the exponential of the left side is of course and factorial the exponential on the right side it is 1st of all the exponential of analog end and then in each of the miners and the exponential of Osama is the product of exponential xx in this case the exponential of difference is erased your exponential 's Vitaly the analog and times Ito the miners but I write them Sterling's formula somewhere where here but we've located Ito the miners In but they spend a minute or 2 you'll see that Ito the end login is the same as in I'll leave that to you to prove the easiest way to probate Mr. proved that the lava rhythms saying the logarithm of each and the and logins what an organ what's the logarithm of Anthony and an organ that so here we have it here is Sterling's formula crossed over an approximation of said what what the right visit yuck there is a there's a little discrepancy here but it doesn't matter big is indeed bigots negligible were any yard
idea comes from Ozal pieces here yet but thus quoted young right is infinite number is an infinite number of a the square root of incomes from some approximation we replaced by straight lines but is still missing a little bit is an infinite number of factors but are fight this is what I should do this is close enough here now this is close enough for our purposes and don't go don't work out a hundred factorial and try it it's not too bad after the Cambodians is actually slower than I thought that once work about a couple 1st out of our 1st when you're so it was not stresses a thought but does could produce reasonably rapidly and tend to the 23rd it's an awfully planning which is the number of Adams animal OK so here's our approximation intake in each of the miners but not so sure while the other some leading factors canceled important we can do that right here OK so now let's let's take monstrous Kombat Oracle coefficient here and that's what courts Comer variable remote areas members keep in mind what it is it's a number of ways of rearranging capital and things into little and no uh state Dillon and incidentally is maximized it will look for Lula said little in which makes this the number of ways of rearranging as big as possible go do we look for the sake of little ends which makes this is because possible the fact is that the that we will find a very sharp function but not yet we're not there yet so 1st we need to deal with the subject yet OK Wyoming going to be interested in that can ordered with Michael want maximize various but subject to constraints the took constraints are constraints we rode out before namely the summation and some by capital and summation of and by east of I is a fixed number he would like to find out what is what is this set occupation numbers which makes this as biggest possible given but the sum of the and supplies as capital in the sum of the energies is that's mathematical problem on a workout once we found that we effectively not all of the probabilities this equation he wants with these lanes the occupation numbers than we know the probably OK the 1st thing to do is to say we have a problem with the big drop a file had a little honey maximizes thing incidentally you differentiated differentiator with respect to what will in will end not continuous variables here but a little while will become pretty continuous but we have a big product and differentiated big product is more of a nuisance than differentiating Osama might well we want and maximize something but maximizing it is same as maximizing its logarithm you connect a function by maximizing its lovers if you found the place where the function as biggest that also words lager than his biggest logarithm is a monotonic function and so were ever that function as biggest its logarithm was also big cell let's take a lotta remember vest log unmanned factory or water of the target function here try private card Procter and that's equal log event factorial unsubtle I'm doing bit is going backward but get out but I've got None of the prying eyes I forgot I want to use darlings approximation before holding his wits use difference approximation OK let's use steroids approximation before we go ahead of what is this this is able to capital and factorial each other minus capital a now madam factorial Pinto that's what I want mentally and eat the Marines and then downstairs in open 1 in the 1 9 9 of his Ito minus the amount of prepare or here the miners In than this a tool to end 2 of them minors end To this end furry the end furry each of miners in 3 months and 3 and so forth right now look to EU lose the fact that we know that the sums of the half-day at a capital in they Ito the miners and 1 to and 3 himself or adjust to the capital and so those factors canceled this and this cancel and in fact all the other some leading things that are not quite calculated carefully enough they also canceled their expression that the number of arrangements to Ojai approximation is into in capital into the end divided by the product of supply now with stickers library was a lot this deception are very hard to this story is longer than it really needs to be both a bird with figures locked OK so what was close C for around common talk was the waters that's equal to end log airliner this was minus the summation of what I love and Zebari log in now you might start to smell something interesting here maybe not but I do go back and remember that deserve that the little can be interpreted as probabilities this is looking awfully interesting here the doesn't quite look like peace a buyer P supplies but somehow related to it because the ends are related to the PUC and what is the sum of piece of my love peace of the entropy entropy if the probability distribution really is given by this so let's carry this out what's carry this out however stepped out and see the fact that what we're doing is actually maximizing in trying to maximize the number of ways of rearranging the coffee packets here while we're doing is maximizing some entropy were finding the probabilities which maximize the entropy under certain constrained circumstances OK so let's just take this because of the poor with video blogs city is equal and my summation of of by login survived and now plug-in that millions of I can be identified in terms of the probability piece times the total number of thing I've heard that he In login minus summational OIE In is he a times
capital right but what about the in survive hurts a lot of peace and RE plans and right but want peace a times His walk piece of art plus OK let's take this 2nd term refers piece a bite In the log noticeably is that appear here began to use those adults and functions of I just end eliminating what is a sum of a piece of ice 1 right so they look carefully at this you'll see that the 2nd turn he had a 2nd term here is In a log in terms of summation of i've peace of mind some bike peace a body found in login bowlers some peace a bind is is 1 so the 1st thing we firing is that this a log and cancels this in-laws there's an organ here White because some of peace a buys up toward so we can forget we're free yet yes take the NOT outside the war we have we have capital in the total number of systems Kindersley entropy of any 1 of the systems remember piece of ice now on the probabilities for a given the system the B in the state so apart from this factor in which is just subsystems altogether William E. asking about his neck summarizing the entropy if we want to find out what these occupation numbers which maximize the number of ways it you could rearrange the system keeping their occupation numbers fixed it simply corresponds to maximizing the entropy that may or may not be surprising to you if it's surprising to good it's a pleasant surprise to not you not are hopeful for a blood cells just curious a a a total of said Wednesday city that but laughter a over there but I doubt that will maximize again now that it has been eases is the most probable situations just too were all old him is all occupation number distributions are equally probable but we have that something important we have he the constraint trying to do is maximized this month subject to constraints refers is that the total number of some of the ends His although equal to capital and and the 2nd is the total energies 6 then it's no longer ruled that the other thing which maximizes it his uniform Mara arm is uniform most yet both imported boss important both work now he used her creation the rate lawyers just as capital and large who will intrude it large in the same proportion did you double the Tonga systems every average occupation number were also bubble From as you have to do to really do this are mostly is of mourners paying known that I don't have more energy the girl and nor everything above some point in start making capital and bigger and bigger and bigger all of these important that our contributions will come from will land which also get bigger and bigger and 6 proportion of that not not not not coarse grains entirely different course granules what happens when you were when you look at the world through a move on to somebody else's glasses but it left see a lot and related but set of data set a member that is a positive dab you don't want to minimize this not long got you want know about bring it back to his former here this fall over here won't P is always negative so this is a positive quantity user has this go from being negative deposit well when we subtracted off capital in organ works out but not so this is the entropy This is the entropy harrowing after due is maximizing entropy but subject to constraints writers real complicated by drug do it without delay North Sea Scientific American version of The Pajama steel works we have to do it now well but as retailers all said so if you dont constrained by energy let worse is physiology of different states of the art you are not using what job and systematic this subject to constraints maximize it with respect to what maximize it would respect the choice of the probabilities every go back were maximizing with respect to a possible occupation numbers but that translates Inter maximizing entropy remember entropy Kazakh value that depends on the probability distribution and now we come to the conclusion that that the most probable distribution of Our from occupation numbers corresponds probabilities which maximize the entropy but maximize the Interbase subject the constraints in the till constraints are summation piece of ice equals 1 and summation piece a Easter by is equal to you fix you imagined fixing the energy fixing the average energy and of course fixing go total probability said equal to 1 and look for the peace of buys which maximize the entropy that's the role the rigor very happy to maximize you find the probability for the ice state by maximizing the entropy given a certain total average amount of energy for the subsystem and given the probabilities at up what is this 1 here would break the symmetry between the different states that some have big
energy some of small so we we've reduce this now to mathematics problem find the probability distribution the find the probability distribution because you the maximum value for minors some piece of dialog given that summations piece of buyers 1 consideration piece sobriety he survive is some giving energy the donors we need a little more mathematics we want more mathematical concepts to do a mathematical concept is called were grunge multipliers Lagrange multipliers come up all over and over and over both the probability of a come up in any problem we are asked to maximize something subject constrained so I think will finish up the evening by talking about Lagrange multipliers and the next time we will apply the method of Lagrange multipliers to exactly this problem many of the thermodynamic quantities such as temperature will find it temperatures nothing but were granted multiplied so let's talk about around multipliers because a lot for 1 evening but er ever want to get to real statistical mechanics we have to get through this so to do it OK with set aside now back this is our mathematics problem right down new notebook with so we have the figure out maximize something with constraints where she was killed by a seaside can using energy and is all more than ever the fact is that what follows is really going on as an average energy we have capital and replicas of presumably the whole thing has given energy but there it is constantly fluctuating back-and-forth in such a way that the average energy in each 1 of these is the probable urging divided by him so the thing I called iii Z total energy divided by air Red Oak Tate George pick up a mathematics problem given a functionary et set of coordinates x basic coordinates are really gonna be the probability is but for the moment was just think of it as an abstract function which is a function of some instead of exit will work out 1 of 2 cases with a with 2 axes 3 Exs but the problem maximize death fighting the stationary point find of the Exs which make this function maximum right that's easy we don't do that we solve the equations partial of death with respect to x 1 equal 0 respect text tool equal 0 about that back this uses some equation reach X and we solve them simultaneously that published solution but we wanted something now we want to maximize it no what works but draw picture 1st this is set a back Exs this is a case where there are only 2 axes of 6 on Mike enjoyed the contours of an lets your other countries that paranoia by the contours of of the contour map the level surfaces of the function that typically as stationary point is usually a point well on that at a minimum or maximum minimum and maximum always appears as a point which choose their own surrounded by circles of street maps circles closed contours of summer shows you the contour map and you want to find the extreme 0 points positive and negative just look for places that are surrounded by a while ago you would assess the amount and I sliced by level surfaces the peak mountain look like Darryl were circles and also the values also report OK so that that's where the maximum number of function is where the circle workers here but now these level is level surfaces of death but now don't and some constraint but and 1 constraint was 1 of 2 constraints the problem interested in the woods warned constraint the constructors we look for the place where epa's maximum but giving that something else is true and that's something else Is that Jedi of X is equal to 0 with a genius and other functions but it said takes they had given that detects a Jubilee Exs some other function is equal to 0 noise he other functions you go free or just subtract offer 3 there are there are you could always whatever the constraint there's you can always say that some functions equals 0 so his a constraint and for example the constraint equations GM Mexico 0 might correspond to some curbs here some curve and now looking for is a lot this curve where is the function f This is courage equal 0 no longer curve we'll look at half half is 1 thing here another thing here another thing he another thing here when and where it is maximum along the constrained surface well the 1st thing is that it will always be at a point when the level surface is happens to be engine G Neukom Mukund you could see why that is armed at that that's not award received but it's not import point here OK so naturally want a firmer waterfinder tool a trick for solving beyond the constrained minimization of a constrained maximization this case is all we do he is a solution if we only ratifying it that's a place where reaches its maximum value on lawns equal 0 let's your custom let's draw a current look which passes through here and which is perpendicular through levels surfaces of F perpendicular to Solis steepest descent the army the direction you move in on the side of the mountain of full of most rapidly as to take that service like that and you look at the value or who want hear what's varying as you move from here to here she is moving as your genes vary but after a different Jesus is Jaco 0 someplace else G is something Duclos 1 someplace else Jews the minus sworn murders 1 and so forth and let's look as we move along here was look at f is thousands more years Is is as Goodby maximum at that
point not not the jittery that is not MB maximum long pouring road almost by accident this is not the place where f is maximum unless we also remember that we want G to be equal to 0 but with giddy equals 0 we move along here than F has shape but plot F and what plot against G G is changing as we move along here and as we move along path is also changing so airforce on functional here through curvature but this is Duke was 0 arrive every year Duke was 0 lying but if this not stationary at this point as we very G this is not stationary why shouldn't be our we're not looking for the place where the stationary bike looking the stationary subject to the fact that living on this line OK what can we do with change this curve server we actually make this point a stage true stationary point always have to do is enter to ask something proportional G books why is that Is this obvious it gets death yacht is a derivative over here Of f with respect to G's is you move along rare interview aidid something proportional the with the appropriate choice you can make this law 0 if you and your function like there's something proportional something here G. you could cancel this derivative and make a the spawned sober appropriate choice of gaining something proportional to you can actually make this pouring on minimum of no function no function is efforts court as prime F Brian musical death plus something proportional GE now this land is a number hidden number which is chosen To make good the river flat he I don't know what it is or think would out later but put that question off for wino leave until later but it meant something proportional GE soon we chosen so it is current which does not have a stationary point over here is shifted so it most of the parishioners mother was a statement is to take an annual To add some multiple of g than this some value of land that somebody that you become later which will make this whole function here stationary at exactly the point you're looking for this is it this is to abolish the method of Lagrange multipliers land that is called Lagrange multiplier it's a devilishly clever trek X we give you a militia you some examples but 1st before we talk about examples but says we another unknown into the problem originally we had a collection of extras that we were trying to determine what we're trying to determine them by some set of equations like but we've now he added another on unknown namely for value of land that will make at will make a little make F prime stationary at just the right point so or not but we also have another equation was other equation GE who 0 I was all point right the whole point was to maximize the function subject to Duke was 0 so always do this Standard method weak something proportional G we minimize and in a later we chose a land so that the constraint is actually satisfied on this show you an example of it if you haven't followed the details of this go back learn about the ground multipliers petite but these but these the machinery of doing it is very easy and give you an example of rule how to minimized minimize or maximize Of course it depends on the size of but that that suffered for print supposing but let's stake a simple function X squared 2nd rooms Westlake X Webb was why squares as and I always like to put a factor of 2 downstage in a white differentiated every time a differentiated next squares to exercise always prefer to put hamsters unimportant but I want a new mines this function and where we could you minimize functional exactly where the numerous to Mexico 0 1 1 0 0 right but I want minimize it DVD that X plus Y equals 1 sources draw a picture this is death Short f musical text showed that is equals correct square postwar were true was GE is equal Till 2 x plus Y minus 1 right F is a X where force war square over to G is plus Y minus 1 was seeking to minimize and subject to the constraints of the gene is equal to 0 5 Mathilde Lagrange multipliers tells us what was 1st drop picture the contours of FR circles in this case right he constraints surface of a constraint line is Expos Wykle's warning wears Expos Roddick was 1 1 1 Expos Wykle's warns line like that we're looking for a place all along this line where x where postwar squares you could want us there were X workforce were squared is minimized the by inspection you so I guess where it is no we could make a little harder by Eureka well the make it harder making it to experts y physical toward but assist check so what's the role the wrong layers In the F land turns G do that f trying is equal to X squared plus Y squared over 2 plus Linda times X plus wiII minus 1 misses and this is where the Times next step thinking of Loveland the president loans saying it's not known but thinking known thing minimized F price that's rule price so that means instead of lighting the F by D X is equal to 0 rewrite Diaz time DX is equal to 0 and likewise for why but still we get through but the removal of Prime with respect to X is equal to X. From here and then toss slander right the
derivative of asked trying With respect Hawaii physical toll y plus early supposed to do with these things was supposed to set them equal to 0 that's a process Our minimizing if Prime setting is derivatives 0 that tells us wrote is equal to minus land and why is equal to my slams x equals slammed the Hawaii equals my always fund that 1 piece of information incidentally we found out that the solution we know 1 thing that Exs eagled wife but so far we haven't we Aleksandr warriors where we do next week at the 1 more equation that GE is equal to 0 and that's just the equation that X plus Is equaled awarded salt X plus Y is equal toward residuals for Amber minus to land there is equal to 1 land that is you from minus payoff given the land minus a half we now know that the solution is X equals plus a half In 1 equals plus a half another words right over here that's the pouring which Max which minimizes excuse to make minimizes X squared plus Y squared giving and that your along was lying someplace so that's a trick of mortgage multipliers why have is if you have more variables and more constraints but suppose you have 3 variables like I work at marker workout example but suppose you have a functional several variables instead of just 1 constraint you had several constraints will lend the rule very similar to each constraint each constraint can be written as GE supply he was 0 0 with etc. to constraints G-1 equals 0 G 2 equals 0 I'll give you a while give your problem ahead more problems using Lagrange multipliers that is a function of 3 variables happens to be X squared plus Y squared miners squared G-1 now the 1st constraint is that X plus Y is equal the 1 we've done that with done something like that and the other constraints G 2 0 is that don't know let's take it why squared pluses East squared is Ziegler said some ridiculous Turkish want defining the minimum of Air Force so object to constraints that distillate would make it simpler more tragic crazy but just make it Z plus Mexico 0 because equals wanted took constraints go constraints find the place where Athens minimum given that both of these equations are true there is a place you can find it in the way they do find it is you can't get half a Lagrangian multiplier for each constraint you call overhead at and then you minimize just priority pretending that you know what the lenders were so you're right derivative that primed with respect to XI for all the eyes is equal to 0 but you still have unknowns 1 that you have specified yet land the woman when the 2 where the equations that govern when the 1 the 2 due more equal 0 and G equals 0 now you have enough equations to completely solve the problem this seems like a rather complicated thing to do but it is by by far the easiest way to minimize something or maximize when you have constraints YOU question it because in many problems of statistical mechanics we want to maximize the intrepidity subject the some constrained constraints might be better call of electric charge has some value of total energy has some value something else has some value our those of kinds of constraints reimposed in particular in particular energy is fixed and told probabilities 2 1 hours 2 constraints solve let's let's now works now right the mathematics problem we wrote before I love With and you can spend a little bit of time trying to solve it won't solve next time and what is it a function of Now it's a function of all of the probabilities P of alive Peace b a life as a function of P 1 P To without about that whole bunch of probabilities to allow variables where constraints constraints all were G 1 is equal to the summation of piece of minus 1 our words the constraint is G. Warren is equal to 0 or the summation of peace a equal 1 and the other constraint summation piece of bike ease of is equal to a fixed number the average energy given the average energy and given that the total probabilities a up to 1 problem maximize F as a function of all of the probabilities What does it give you that gives you the probability distribution as a function of all that corresponds to the most likely possible values for the occupation numbers the occupation numbers tend to cluster around these values is not easy but wants to have the rules I want to know the Royals the rest is pretty just the British forward or incidentally all of us but With as what what branch of mathematics or we do branch of mathematics probability theory the these singh's set of rules would govern Mini many probabilistic problems from How the ship not going the fear of gasses during the theory of superconductors we're doing basic probabilistic theory of maximizing but disarmed entropy number ways of rearranging things subject to some constraints that call amount of this or that is equal to summary fixed died ahead it's not special any particular kind of physical system that special on the special to the kinds of physical systems which have so many degrees of freedom that you are forced to deal with them statistically improbable probabilistically want to know that the rules are always the same next time we will solve this problem was at 5 in the right F ones yes there are Ross barriers that was the entropy which accord S S brochure P E R which is logistical
summational vii peace by a large piece of life with my maximizes would even vary what boiler variables that shift around maximize the PRI of probability distribution Olivia vary the probability distribution until you find that probability distribution PRI which maximizes the entropy given but the sum of all probabilities is 1 and the average energies E budget but that all statistical mechanics sorry the problem of the caucus for more please visit us
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Formale Metadaten

Titel Statistical Mechanics Lecture 3
Serientitel Statistical Mechanics
Teil 3
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 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 10.5446/14936
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract Leonard Susskind begins the derivation of the distribution of energy states that represents maximum entropy in a system at equilibrium.

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