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# Statistical Mechanics Lecture 4

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#### Automatisierte Medienanalyse

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Stecher University tonight we
want to get to the real heart of statistical mechanics the Boltzmann distribution and that I was hoping to work out example the ideal gas so let's just Rick quickly review for
this we work barracks West kind kind before or a basic principle or a set of rules for calculating the probabilities peaceable I won't repeat them I was
going to repeat the with 15 minutes late for the column P Of ideal piece of quiet can't remember PR by piece of the probability of the state in any 1 of a large number of replicated identical systems just to remind quickly things called occupation numbers namely the number of these subsystems here in the eyes state and if you divide that total in the total number of them that defines the probability that any 1 of them is in state and that gives us a probability distribution over the states of the system from that we can compute an entropy which is equal mightiest summation or what are piece of our wanderer them Peace abide was our definition of entropy that we that we discussed why would go and factor we found it came from just that complicated by multinomial distribution capital and a factorial factorial product little in fact Orioles asked if we wrote that out riding the little and Sabatini begin His peace of mind we found out that Our we want to do with the find out what set of occupation numbers was the most likely set of occupation numbers turned out to be coming to maximize the entropy maximizing entropy but subject to a constraint consider Ferguson looks think of this entropy here as a function a function up that's a function on all of the piece of think of it as a function but a function of all the peace abide put Kidlington rather simple function it's a function which is the sum of terms what can be reach Peace abide so it's a simple things relatively simple which is justice some or all the variables problem on the probabilities are a little particularly simple function to survive outside that simply has a lager but that that be OK but it was important to remember but there are constraints constraints 2 constraints the 1st constraint wanna maximize s subject to constraints constraint warned that the summation of piece of is equal to 1 the next constraint is we started with a certain cultural amount of energy let's call it capital E Times and capital in white times capital in because I'm imagining a certain amount of energy on me average In each 1 of these boxes and double the number boxes I will want to double the amount of energy that I have the whole system so we assume the puddle images proportional but in and the average energy in each subsystem too summation peaceable RIT east of the probability that a system has risen state and the energy of the other by state an equal to its equal the energy per box the average revenue per box soldiers E. that's fixed fixed with determine that once and for all by saying that was a problem the amount of energy in the 80 per cent of boxers that can vary and so it's necessary that the average energy in each box is fixed equal to eat way he was total energy divided by the number boxes OK so these other 2 constraints after the right minus the entropy like that might entropy plus peace by law peace I just threw up particularly with stick reasons I wanna talk about minimizing something rather than maximizing it it's just a habit of doing away with the cab for me to talk about minimizing something so I'll talk about minimizing the
function minus subject to these constraints the constraints also involved the same set of variables so 1 on rules for finding the minimum of function given some constraint and there we talked about last time the method is a method of Lagrange multipliers a highly formal formalized mathematical construction extremely powerful our anybody does 1st or by the statistical mechanics who use it over and over and over but all sorts of contexts where you want to find the minimum of something we know some of the things this is the way to do it or what you do is you take your function which in this case was called call this of peace of mind affable appease calling it because we called because it just a function that we want to minimize I want minimize F subject to these constraint so Romario mind you were too don't you take F a new to which the constraint equation but constraint equations constraint equations in this way the left-hand side of the constraint equations the thing which is equal to 0 you multiply each constrained by a Lagrange multiplier which is the number later Ron will figure out what the numbers are our let's call the 1st Lagrange multiplier but but called the general Lagrange multiplier last time land but now I'm going to have to Lagrange multipliers 1 for 1 constraints the other for the other constraint and some recall how often Bader I'm so the Landers in this problem mouth and beta amyloid distinguish them different they play little different and so it is a 1st constraint equation Alpha times summation piece of minus 1 and then the 2nd round multipliers called beta the Serb that's summation east of all eyes Peace minus the constant each here it is a fixed constant that's been determined once and for all by the by the by the energy supply for the whole of a collection of her From identical systems so he is also a constant and he'll have now was the ruled the rule is when this is called now prime and using the same notation I did last time where you take the thing you want measure website that you want to minimize you Ed Lagrange multipliers the constraints and thank you minimize F prime let that will leave you with unknown value Of the offers baby but did you figure them out by imposing the constraint yes it is a function Raffarin beta before the model focus on the back yard is right it but why I want to focus on the fact mostly that's a function of OK a very lucky in this case each turn year old well 1st of all we could do to minimize it is differentiate the with respect to the variables and recalled a 0 that's the way you find a minimum that means of that take a set of P and differentiated with respect to the P when I differentiated with respect to he's the 1 he won't give us anything it's just a constant they want different you'd expect Pete it contains Alpha but would not differentiating with respect Alpha so for practical purposes I'm slowly this 1 he likewise when I differentiate this thing with respect to P B E won't contribute so let's just get rid of forget it it's not going to be important because they equation that we're gonna right namely that the derivative of this With respect to each of the peas is equal to 0 it's not nonsense that Delhi Arias is all we have to minimize I was ever F is also a song and it's
Osama please by a large piece of art this is very fortunate that each Turkey is a sum each term of which contains only 1 of the peas each term Sosa policy I want to differentiate this with respect to some particular P just a focus attention away with doing what's pick P 7 P 7 we wanted differentiate this with respect to P 7 incident equal to 0 so here's a P 7 years after his warned term mergers P 7 large P 7 what's the derivative of that with respect P 7 differentiate this with respect at lower right now is writing a derivative of that Prime with respect the peace 7 y 7 no reason just 1 to special 7 is a randomly picked number now what balky number not so after finding drew that time with respect to P 7 is a derivative with respect to P 7 of B
7 log P 7 and that is the case we have P P at 2 terms in the derivatives in the 1st firm with different we differentiate P E keeping lined up a lot so that gives us just locked he said differentiating P gives us 1 was other terms will you differentiate lot what's the review log 1 over P. times p White so is pretty easy at 1st a what a derivative of this thing here with respect to P 7 really easy even easier than this 1 plus Alpha saying for every 1 of the
pieces in place and held the derivative this Plaza data he said at pretty simple wear the same thing for P & P Florence 9 P a a 152 so let's just right he might a log PART 1 blows Alpha and be fired noise isn't supposed to be equal to 0 well this is kind of amazing at this point statistical mechanics is full of things like so magic magical identities which all of a sudden you playing with them you look at that while just discovered something important I just discovered something unexpected unexpected look this is really equations the PRI it's gonna tell us what PRI is let's write it has logged PIE equals minus 1 plus outlets a number now we have to get a laugh is yet full figured out soon enough minus data Eli it was at callous P ideas with exponentially right so 1st for each of minus 1 plus offer Watson number that's a number I call it by its Standard namely its standard name is just 1 over Z minus Ali Eco 1 plus Alpha is called in statistical mechanics it has indeed for the moment I'm not going to write to but this is just what I did allow Alpha was and I don't know what is so I haven't lost the deed any information by replacing minus 1 of our friend calling it 1 was now exponentially this saying the 1st factor is Ito the minus 1 plus out for that disease but then the 2nd factor is he said the miners beta release of all eyes this piece here doesn't distinguish the different it's just a number it is important but it's not important but the relative probabilities of different states all contained in here among we found we found that the relative probabilities Of the different states are simply exponential 's exponential of each of the miners babies our bail must mean something they must have some physical significant it's the thing only variable in in here that tells us what kind of well that it's the only variable and here has any chance of telling us what the average energy nozzle this looks like he is a probability is the the energy going horizontally is just an exponential function which falls off the bigger baby is the faster it falls off the swallowed baby is the slower rate falls off where are these different curves going to be parameterized might well parameterized data but they also commit parametritis by the average energy it's quite clear that the further out this curve is like there's a larger than average energy the bigger Bader the smaller the average energy the smaller data the bigger the average energy so bidder has something to do with energy it's the thing the EU twin the change the average energy we will be more specific about that it's a thing told the change the average energy but fun or just called and as significant as a certain Lagrange multiplier where the 2 real significance a nurse temperature bathe is the inverse temperature but how the hell we proved that we already have a the definition of temperature our remind you would we get to it for the definition of temperature is but he is not probabilities in terms of 2 parameters 1 Z I want data with to Lagrange multipliers might now have to come back had only thinks the Lagrange multipliers we fix Lagrange multipliers by making sure the constraints satisfied so the 1st constraint is the sum of all the probabilities has stayed up toward that's the equation of threat about moral Lizzie Kansas summational body of each of the miners baby Esau by is equal to 1 incidentally with the Esau supplies come from 1 of well there that opinions you say some laws of physics that had to do with whatever is inside the box tells you what the possible energy levels are so gross from our point of view now was simply numbers that are unknown somebody has been smart enough to compute them and tells us what the energy levels are what this tells us is what B average with the probability of be in lawyers as a function of data bases the thing that has to do somehow with the average energy fears as we very beta we shift occurs I understand I last I know not yet not none of this is EIT yacht was a whole bunch of icy misses I equals 1 articles to write a prank right but you can also think of it as a function of the energy level but just think every level has a certain energy so the big debate is the more
rapidly this function forced 0 0 if they is huge than it is dominated by the lowest energy levels just make energy give little bit bigger and you lose a whole bunch of probability on the other hand they Disney's 0 then this curvature of all practically flat and it's going to contain important contributions from large energy so by Tony data due to new energy between the average energy Oakham at that moment OK from this from here now we see what Zia's multiplied by Z area have a formula for the Aziz for unknown Lagrange multiplier it is currently in general Lagrangian multipliers the pains are now well they are numbers but then they may depend on each other but he'll be 1st Lagrange multiplier Alpha rule strictly speaking it's easy to the 1 plus Alpha that we see is a function of Data General is a function of beta but it doesn't depend the of all the states of the system services E of data and has a name you consider working from now zeal data from purposes from here on in is by definition of some or all of the states of the system of each of the mines babies survive where does it fit into the probability distribution it fits in as a kind of normalization constant make sure the total probabilities Oedipal 1 but that's all right it is a function of beta let's keep let's remember almost Everything statistical mechanics is buried in this function Bader refined this is an extremely powerful thing is cold the partition function no letter has to do with partitions I don't know partitions after walls in the room for this in the partition function it's a function of beta jazz Wilsey data was essentially the Yum inverse temperature but we don't know yet next equation summation of peace of mind east ice is the average energy right that out that's going to turn the other Lagrange more what we determined beta basically toaster are worked out we have summation of I of is 1 0 Missouri Ito the miners Bader Esau body that piece of writing and we multiplied by Esa body that's equal pay suppose we calculated ZEU we know was only calculated the rest of this equation determined they the rest of the equation having think it Woods EU is this now becomes an equation for Bader In terms of what in terms of the average energy if we know only average energy and were lucky this equation might have a simple enough form that we can find out where the average with the would be In terms of energy can also read it the other way we can say whatever they is it determines the average energy both the legitimate ways to look at it for the moment just austerity equation among rewrite it trip very famous track the famous trick goes as follows a let's differentiate now let's
begins is but sake Etowah miners baby Esau bite that's a plus the partition function This is the partition function and led differentiate with respect to bait vii and differentiate both sides with respect to the partition with respect to beta again another trick easy by debate that will yellow aside had a different shape this with respect to Bader you just pulled out a factor of minus Easter might derivative of an exponential with respect the argument with exponential just gives you a factor works in exponential here because minus sign and east of IT and Ito a miners Doretta recognize when I say that the derivative of respect 0 of each of these terms just pulls down a factor might Cesar by but now is a D B C Y the price but here's yes rather it is son of it a dish I just have the definition of Z a right I took the definition of Z top equation and differentiated and not look what I get I get something which looks like that which has a piece of the 2nd equation here so a forgo we have but divide by noisy on both sides busy downstairs here and there was this summer the sum is exactly the thing that we identified as the average energy so we have be average energy E the average energy is 1 always eat Desi by minus is a minus sign our ringers plots car so I have a remarkable formula for the energy in terms of the
derivative of the partition function with respect to base its rise about he each energy for this is suggesting tourists visit the thing Nicky work computer users partition function the partition function with going defined and everything in it but a particular has the average energy sequel toward a minus 1 Ozzy derivative of with respect to Bader so if we know the partition function which will calculate various cases differentiator with respect to beta and that houses the average energy is also of course a function of data every year is a function of beta so it is telling us Is the energy as a function of Bader you have your words that 1st trip before I answer the question is not that this is my the derivative of the logarithm of z with respect to Bader it's really the logarithm Rosie which is going top occupiers a lot while Lizzie plans a derivative is a derivative of the locker up question did is his homework have captured this equation in my life that I will part differential they will address forms of cast very well at the now it up the norm he is not constitutes a function of beta In this equation for the purposes of this equation Is he of data Was it means social find the B is closely connected to the temperature is is gonna tell us the energy as a function of the temperature was minus due Aug zebra debater this is the 1st of many so magical on formulas fans are statistical mechanics it is really really but that extremely easy and extremely hard on its Opel it's full of surprises its
foes the application of very simple formulas which they yield extra mainly surprising and powerful results makes it hard is have the kind you can guess are you can't you can look at the formulas and all of course this is obvious he is equal to derivative a logarithm of a partition function the whole story here was complicated you could get that but it sure is a simple formula turned to powerful formula death your duties as average energy part box remember yup per box but they just look at the library see it as a kind of like a lot of is this sort of EIS logarithm of some of exponential is not present it is not some of the locker revenge each individual exponential can do that facing Frost use but should say what they would bring down the house their closing its right some of the guests at a about what Is that most you have been given when they have brought work your bags and he again and use and that will be solution that year in slow like at all you were out of work here will make right now the offers made handsome interesting physical significance remember ahead of there are other things in the problem sizes to peaceable was the average energy finding is that there's a trade-off between average energy and data you cannot find their fix the average energy and then determine what day it is that would be the standard way of looking at Lagrange multipliers but you can't really be other ways you could say the average energy is determined in terms of anger and we're going defined as said the beta has the significance of don't have a life of tone the temperature of the universe temperature and that's a vexing were not quite as yet opposite lava worn away long time ago we did redefined it is a disease that disease said Ito want was offered is it just a redefinition this this is predicated on the 2nd of Asian fewer 0 that that's follows a lot he said Eric Richard here best that that a lot of stuff there's a lot definition a B average and yeah strengths the maybe but energy would do he had itself yeah there may be other things going on for example you may want to constrain momentum or some other conserved quantities Our for the moment let's suppose there are nor the conserved quantities the Muslim nothing left to constrained that you may come in then you may say got you may be interested in changing the volume of you box that you guesses for the moment we supposed to do that all the parameters which defines systems such as the volume of the box such is the shape of the box only are the things that you could imagine varying are kept fixed those called control parameters changing the volume of a box changing the shape of the box changing the magnetic field on system was a control parameters and their importance but at the moment we're imagining absolutely fixed what do they know how what do they determinant Bentley energy levels yup so when you change the control parameters you a general change energy levels and so doing you change everything including the partition that shit prior dishes out drought know Missouri you don't need mosey because because EU From arm the same thing this was just a definition of Zee it's a number was a member depends on the on data depends on the because when we worked out what it is we
found out where as we found out that has implicit dependence on beta Georgia beta dependent number yes it depends on the ERA which were mentioning a fixed set of numbers of the from all that when we odd things in the system volume of the box electric or magnetic field or other things that we might theory of the control parameters voltages whatever you have are you may change the energy levels when you do you changed it but for a moment was effects that get so what so far that fixed then it the we saw was actual numbers were right now that depend on May yes it all they were given 5 B's show but they made as a number depends on the average energy to fix the average energy debate yard and we know we're not very ER but not really worried the Ely a fixed set of numbers the energy levels of the system you get a wife has example run while then was the boys and now the eons adjust the 6 7 members of durables they are not the energies of the system the energy of the system depends on the PRI the Elise adjusts numbers the average energy really depends on the key source the eat up yard this I thought the 1st thing he has made his right now we're just a of a truck or baby is a function Our E or EE is a function of data right evil 1 of them can be used as a parameter that you might very well in a box of against it's easier much easier the measure the temperature than it is to measure the energy in the box have you measure the energy of a box of gas will you could know you could always news equals MC Square that's allows you wait here it's a great way but there were not helpful column will hard to measure it's much easier to measure the temperature and from Ahmadinejad measure temperature if you know the connection between he and beta you could determine the energy but as 1 hang up we haven't shown yet the debate is connected with the year with the temperature so let's do that parallel 1 step 1st Yao before we do that much talk about entropy remember entropy comes before temperature our logically comes before temperature so it behooves us to calculate the entropy I don't think we need this anymore let's see if we can get an expression for me entropy in terms of the partition function it is more magic entropy is equal might summation of IE piece of Iowa peace a bite equals minus summation I want all the Z each of them minors Beta east of IT pizza by his war was a momentous supply worries that we lost it but that's a probability fans logged piece a body logarithm of peace of piece of body equal to 1 always each of miners buyer right OK logarithm of areas might this data east of minus logs 80 Parisi what I dead lot of peace buys a product of lot of war either minus Largs and log of each of the miners baby the miners baby Shiite OK what's the 1st earlier with my our see this is as a restaurateur from reassigned starring he plus plus that's good because entropy ought to be positive OK what's the 1st turn P by i 180 Esau by is number favors of number doesn't depend on I still stake outside of the 1st turn times this is
essentially just some of peace by as a 1st term a C average energy so a 1st area here is dated times the average energy that's good we are the compute the average energy From sold given we would know how to calculate beta Pansy average energy that's a 1st everybody see anybody cannot see yell out right now you can see that the 1st as beta plans average energy the bus part of the definition piece a yet I I think that now the 2nd term of 4 the 2nd plus 1 mosey log Rosie from here logs from here and then some but what I eat miners daily the where is the sum of IEEE Evita the miners C surmise that zing so this city just cancel Lizzie this is still not fancy gymnastics year our at some point you go onto autopilot thing you just there some mathematics until you find a formula that you like but here we have some miners babies abide is equal to dizzy cancels Izzy and now we have a formula that we like but woes as saying there were calculated the entropy so we have an interesting formula for the entropy they had that is easy I could write iii in terms of a certain derivative of the logs 80 but they don't need to undiscovered leave as the average energy plus the logarithm of Z the logarithm of Z is a thing which comes up over and over again and that in a given name at the moment it does have an name at related to by a factor of temperature Helmholtz free energy but for the moment we don't need them this is that this a formulaic if we calculate the partition function which is calculated as a funk function of Bader is is a function of beta then we're capable of calculating the average energy were capable of calculating the entropy and lack of recalculating entropy and the energy we can calculate the temperature so let's go to the next step which is calculating the temperature don't they don't think of him martyr toward the autopilot again and see if we can find out something about the temperature of the find out something about the temperature we have to remember something about the definition of temperature up e tape Butcher the definition is that the change in energy Oh sorry to the change in energy when you change the entropy by 1 unit is clock temperature D E equals T DS and basically the derivative of the energy with respect to the entropy that's temperature that's the way we defined it when we used the definition of it in this way we found that he always flows from proper cold that's a pretty good indication that some do with temperature is the definition of the temperature of the system the rate of change of energy with respect to enter salt let's try they calculate after this of course of course is also equal to 1 over temperature is equal to the rate of change are s with respect energy the inverse temperature is the rate of change of S with respect to entropy let's see if we can do that derivatives reveal vests with respect to energy a pay salt let's differentiate this with respect to energy 1st help keep in mind that they would be simple if they didn't depend on energy the debated does the pair of this stood terms from the 1st there DES by D is equal to beta plus he had times debater by DVD servers I differentiated this with respect to energy and they don't want term and the other terms is the energy times debate by energy would next this 1 let's differentiate lobby Z said did this very battery that go back a step that we
go back a step was for backward and doing that what I would start over wanna start completely over again that 1st moved a better motor just went awry DES BSO wrote take a differential of S and that too terms from the 1st time they did D E course e debater misses easier From a 2nd term or I get I get the derivative of the lot disease with respect respected Bader times debate at Bob White is much easier than I thought torso easier than I wrote in works to edit I wrote DES is equal to a terms from here as this and then the Riverdale logs EU with respect to Bader plans debater this is so Louisiana they have make it so complicated and the notes energy is able to minus do logs by debater so we have energy times debater minus energy terms debated how are it was Bader D yes or was a definition of temperature temperature was remember D E equals TVs right up the equals TVs OK we only have D Eagles won over Bader DS last it follows that the temperature is equal to 1 or better for use teaching this through a complicated series of steps was must not complicated OK it's very easy that he was a debater was due largely by debated that B.S these to cancel identically and the result is the temperature is equal to 1 over beta that's wonderful because we now found out what the significance of pain is the physical significance it started out as simply a Lagrange multiplier we manipulated that use a couple of identity is a little bit of calculus we find out that data is 1 divided by a temperature is at a loss as a dozen times E. and on a depends on the or Beta said I have a D lots e normal it depends on how we were variables I or beta their collective there was only 1 independent variable and you me I could have differentiated respected iii but that would more complicated his zeal only depends on 1 variable a can I be taken to be or big physical point America but turf so that yes they all good good day it would be a Boltzmann constant work to keep track Our centigrade of Fahrenheit so for it were actually beyond the Boltzmann constant times at temperatures war over Beta Beta is war over treaty rights know and laboratory units baited is 1 overcame cable by the end of the theorists units where temperatures units of energy then now has just 1 over beta 4th Path a with ball game that an if we had a of malt malt not not well not his system is defying could be the boxer gets the system is defined to be the box again at the end of this Duva collection molecules that go back and try to defining system as a single molecule you might be able get away with a certain circumstances the Nikkei daily all circumstances because the molecules interact with each other and it was assumed that these boxes were extremely weakly interacting only in interacting enough to be of exchange energy will be right so are many cases you can take the system to be 1 molecule but will do it will do it by taking system could be the boxer molecules and will see the relationship between OK so we have temperatures well good but so but what accumulate equations here's some plants the important equations summarize very here Lisa Myers 1 was a heap of miners data Our east of mind Z a summation or why he emerges babies I is a function of
Beta he is equal to minus the derivative log with respect to Bader temperatures is 1 or more beta and more more equation interesting be entropy is equal to wear that I have paid at times post logs at now I example
example drawn from real physics approximate physics to be sure mean to say that will make an approximation the ideal gaps the ideal gas a guest of molecules a box with each molecule is a point molecule and I considered number wager could say that the molecules so weakly interacting that we ignore the interaction between them all I can say that guesses is so loot that the probability of 2 molecules being close enough together to interact with negligible our you the way the approximation is the molecules do not interact at all and no way of saying that it is energy is a storm of just in this case Judge David Connecticut energy of the molecules of nothing more inside a box of boxed his box volume beat the number of molecules in the boxes and the density is an over V density of molecules on the density of energy in over a year now call role we get to it I want calculate the partition function 1st of all what the energy the state's 1 of the state the they are the collection of values Of position and momentum for each molecule that's the way we label estates so our state and is a collection of green coordinates x 1 through X 3 why Korean annoy because each molecule has 3 court accepts Y X Y and Z just want them altogether Our are in this fashion where about the moment that the moment drop you want through P and the state of the system is just up point in this 6 individual space a set of values of X a set of values label point was a sum over states some of states is replaced by an integral only Exs and repeat no words we have to make use now of these relationship or correspondence between sums and integrals I'll not go through that in detail our I think in do that have replaced a year ago by apart by some somewhere in the world so let's just do it instead of writing a of ii Weerite an integral all over D these 3 X and d to read and all the axes and Pete our interest in the partition function for partition function as a somber begin with words some IT that is replaced by the all X isn't be next Romana right each other miners Bader time seek energy of the Ohio state 1 of the energy in the stayed abroad particle in a box is just a kinetic energy for the moment we're interaction between the particles it's just a Connecticut Energy so what is a kinetic energy of a particle its P 1 squared plus 2 squared plus 3 squared 1 over twice the mass of the particle P 1 2 and P 3 other 3 components of the momentum PXQ NBC's none the 3 directors of space and don't exactly the same way when I think all of the particles have I take all of the particles all I do is ANR a reedy contributions of this type he felt this just becomes some all over the bill and labels the particles little part of little and labels which particle I'm talking about what it labels which coordinates talked about equals 1 the free end times P and squared divided by twice the massive particle mistress each them line as Beta Beta was a baby here but energy of a given state is just songs of the squares for the the moment that all of the free and moment that divided by twice but the masses Bobby equal but figure media clean this up a little bit his beta over twice the Mets Karlheinz E sub In its a way noticed some in exponent but that means it's actually just a prop up it's a problem all in all it In a nutshell rightist 50 products signed Navy unjustly of some here but this is a fairly simple expression which a prop 1 factor fee each quarter Ito the miners P squared over to wear and data where for some from us but and Nichols 1 3 there are 3 moment altogether yes that none of end particles and particles 3 quarter right why would I have me and particles that would really be privileged does that what but Freedom Court rips OK says yes Jerome K. years so they over also moment of also the vertices measure something only in OK but said I would say with this is a devotee P 1 in each of miners data over to M P 1 squared DPP tool data over to where P 2 squared DP 3 feet of miners data over to M P 3 squared but 0 . 0 also cited is also the X integrations which I have concluded be momentum integrations factorize factorize into a product deep he won for the 1st component of momentum Ito the miners Badeau with 2 M P 1 square and in fact if you think about it from this is just the real power of 1 of these integrals each integral is exactly the same as every other integral to the role of a P 1 gives us a bank and a number and go P 1 and the repeat when he integral is done gives the same number it's just capital it's just 3 and power
water of 1 of these integrals but who make it very easy Barbie doll before we do it yellowed strive for most of my as a writer but did it's a dead it's a definite integral all momentum all possible moment the particle could have definite integrals it's an end to go over all possible momenta all possible states some of what I is a sum of all possible states of the system the into goes deep P and the X Our integrals or what all possible PNX now the moment they in the box can be anything where about the X integration they have to be inside the box constrained to be inside the box so let's focus on them 1st of all I think the 1st particles as of the 1st part just interval 3 x 1st particle and the city in the world the box what his value the volume of the box right so the X integration series each article each particle separately VAX integral dues a volume of the box altogether Ex integration of the integration factors not connected together was nothing in this in the grand depends on X we and the only depends on Pete so the into over X is very simple into rollbacks gives a factor of volume of 3 years now Ball returned to Retief for pig I'm the put an extra factor in here if you like Ricky areolate
while let's not deliberate rivet what's what's what's not will come back it no reps the factor 1 away and factorial is a little bit contentious whether it ought to be there or it wouldn't be fortunately it doesn't matter whether you put it there or not I will keep track of it I will put it there but I would tell you it comes it comes from identifying is a state with a molecule 1 over here and molecule to he is another state with molecules tool over here and 1 over here are they the same state or not wealth it's a little bit ambiguous the particles carry labels which are which have names attached to them so that their putting Larry over here and fret over here is different than putting fret over here and Larry over or are they nameless things where you could just say the state's correspondent the particle a particle only here just 2 of them 1 here and when he the into is in classical physics it doesn't matter In quantum mechanics it doesn't matter and in quantum mechanics particles do not carry electron over here electron over here is the same as electron reviewing electron over here so that means if it was too articles you would say you counter by a factor of To walk by considering all possible configurations giving the particles names and then allowing them that all possible configurations gives you to a factor of 2 too many states about 3 if they were 3 particles what's your camping factor 3 factory 6 was particles and factorial self the usual argument goes Aziz said it doesn't matter and and injured in the end it would make no difference to any of the formulas our Treasury where goes here but is a common practice to divide despite and factorial and to say that we older account did everything by Perry a factor of war over and factorial you that that was a nice thing OK so that's the volume integral what about the P integral here that gives us something to the free power forces 3 interviews integrals gives us something to the free and power namely integral DP each of them as they go over to when he squared off of that of the free and power looks terribly complicated but it's not just into roses than in the world definite integral what could depend on it could depend on data pickup depend on and it doesn't depend on please let's see integration variable so all we have an integral to I don't do the Indigo here on the side of the blackboard molecular relief but don't that let's take the integral in duo DP each of them minors Paola to M P square OK for a similar change variables I take a factor was a complicated expression the exponent here sung the doubt over to M P squared the Beagle 2 Q square ass 1st a change of variables then the integral becomes each of them minors Q. square I was easier get rid of all the stuff in here but not quite because what happens DP what Abbas to P is equal to the square root of To win over Bader firms killed but at so the becomes the square root of to out of a beta DQ but now this year and the growth Justin number this really is just a number of D Q each of them minors Q. squared is really just a member from minus infinitely painfully Alioto square replied where come from you look at up a book I so you some ways to do the integral but the mean thing is it until was just number it converges very quickly because he minus Q. square goes 0 over a Class B in the grandest finite everywhere goes from my innocence and veto infinity it's just a number and that number happens to be square root of pies we calculated to disperse Querida Pyay inside beast square root Kia and that's your integral Lazio only calculated the partition function the partition function for 3 guests from big ideal gas what's right down now volume to Lee and all end factorial times the square root for is now To motivated to em over beta head so when trial data for what our through so we can call that we can take the square root away right that this is a power a free hand them over to her 1 half a square root its find in this form but let me just show you what this and factorial does for you if you if you include you don't include you just just go through the calculation the same way you find the difference but still what's worked word growers were armed West Egg Vitaly over and factorial and make were talking about Joe 23rd molecules and is a huge number and factorial is a huge number only can approximate and factorial by the bestowing formula so Vitaly N. Becher stays veto and factorial that becomes a the end each of the miners in that for vectors S to the end upstairs NYTimes vie for power NYTimes VOA and power in at well easy has nothing to do with that I wish you were there but Tuesday swords with the brewer March but what of the race you of veto and better yet with the ratio and the density of particles so we worry about being able to replace all of this volume dependents by E. divided by a rogue 0 this is not electric charges this is stupid No. 2 . 7 provided by the
density of the power that's a factor coming from here if I have had this in factorial where I would enable the makers nice trick the answer would have simply banned and you know yet OK will stay that way right so altogether and fears of Formula Z is equal this factory carries ER overall for power in career work doesn't rule that z right so we'll pregame we will either calculate Lu aunt or a pilot started doing in the roles and the end of the day he was Z and know what it is 1st of all depends on the density that city for a fixed number of particles In a fix box the density is just a memory so it's interesting that depends on the density but the density is a number Tool and and part numbers capital in his number only Day Day here is and it's also a number of course but it's an interesting number it's 1 that we may want to change the course of the problem after example we may want to differentiate with respect to it true blue momentarily this is let's calculate lager a murmur of Zia told the logarithm Rosie is a thing which comes up over and although it comes up over here so let's calculator logarithm of z incidentally whenever you get a product in you want differentiated Tara produces an old saying 2 from 0 to stress must they exert toaster it's easier to work to differentiate us on the proper time so that that's why you take lot of things this is a product partition functions are products typically it's good to take their water them because they are easier to differentiate OK so what's lot of z is equal to end kind 3 the larger love to end apartheid over Bader plus ER over a role I believe no plus over 0 or better yet minus lager role Ricky that's a lot of Missouri what I did now closed off where while he what beat it is indeed Liberia's warranted I think you're right I think is a plus 1 Loughborough plus 1 male Waianae doing things like differentiating de Malaga of disease things like that this 1 isn't good do anything when you differentiator warning you don't get anything so it's not it's not anyway important in the formula logarithm of 2 pie that that's logarithm of Piper slaughter them and aim for slaughter them off to mind slugger base so it contains a bunch of terms only 1 of which was in it which is important when you differentiate with respect to bathe it is a free minds 3 years Lord Vader everything else functions as a constant so we can write visitors 3 hair of my history logarithm of data plus a bunch of constants constants that include the density which were not differ not changing pie and tool it worked a lot by it is also a constant sorrow indeed this is a constant and end but for our purposes just a rubber constant I think you're right things like 1 of our know that the greatest words here remind has made a lot of 1 over beta everybody at my assigned because of slog warm over beta 3 hands with the 3 come from 3 dimensions a space oppose it was 711 dimensions space 11 11 have waited the 1 have come from both here but it didn't come from it came from the square root which was down gaps in integral it came from the change of variables from this thing he Q we had to take a square root so now I came from now ballot income from their take a back that did did it did it did it did that's Zachary came from the yacht Romero Goodell ceiling Kabul's Coverdell integral Delcine intervals the square root of this thing here OK so that's way all the pieces of all parts come from his logs each What can we calculated we can calculate calculate the entropy but even more interesting for the moment and simpler used to calculate the public energy of the system so let's calculate the total energy of the system we have the partition function slide this 1 alarmed but still early yet the energy is the derivative of logs EU with respect to Bader With a miter I'd be important turn hearing only important terms his minus 3 heads and are beta so we just have to differentiate logs with respect the data and that gives us the derivative logs With respect to data vehicles from minus 3 over to what's a derivative of log data with respect to beta was 1 of a beta temperature all about them Igor signed now minus sign is the energy is actually miners this thing so the energy news twice 3 hands 3 to came from did well certainly it makes sense to save a total energy of the box of gasses proportional to the number of particles and this is telling us bad for an ideal guess which is not competent nominally general although it's often a good approximation for an ideal gas a 3rd almost ideal gas exact a good approximation to save energy per particle energy per particle 3 have energy per particle it 3 Have the temperature where would bolt worms
constantly whenever you see a formula like this in your your converted a laboratory units temperature becomes cable temperature cable my was of course a very small number and so temperatures 300 degrees you multiply 300 degrees by this terribly small boats factory and you get the number of jewels our both mountains of order tend to remind 23rd they have no yup yup yup would you shouldn't fret but without this cable also you might say Hey wait a minute the energy proprietor goes a huge number of 300 300 300 video news a conversion factor k Boltzmann which makes it a small number of the energy per particle but this is where the idea that the particles move around with the kinetic energy which is proportional to the temperature USA the another since since a Connecticut Energy is a sum of 3 terms PX square BY Square PGA Square you could safely each direction for each direction the space the energy stored in the momentum in that direction is 1 half Kate so for each direction space the energy use one-half to if there are 3 directions a space so we don't think escape to that's all our young players that can that right because that would that that would factor off the the partition function you took the logarithm it would be In independent constant and so we can differentiate it would have made the difference right now it's just a neat little trekked to include the in fact Orioles and then things wind-up depending on the density in an elegant way but that's right it would be the numerical factory start in front of the partition function never makes any difference because when you take the low makes an editor of constant the logarithm in them when you differentiated goes away story classical physics there's no particular content 1 over and over I think Quanah mechanics it is important to users the stained by U.S. if part of that that would be saw while the army and the West but the neutron stars or they will have reader a hurled yes top off this they got time to use the G word gravity did get a job focus so if only do neutron star let how this might have changed a farmer and if we having guests in his room and we worried about the fact that the energy of the guests has a potential energy was a potential energy wiII armor I always use the vertical axis y X Zelia ones Our so what's a kinetic energy demands firms G times the height of the another term up here which would be 1 for each particle each particle we haven't eaten the minus M G E times y some of in again sorry some the Knicks bond Sowerby be part of this product structure summoned the export got so let's see what that would do 1st of all that X and Z integrations would be unchanged this only depends on why it would not change Nixon's integrations buddy Exs integrations instead of giving the volume of and would give the area of the basis so it's safe area the basis of the box and then we'll have another integral here would be eaten them I guess M G Inter will be worried From where to wear off what let's call bottom a box Waiko go 0 it doesn't make any difference but like likely 0 to infinity and what is isn't it will city and Oh yes sorry I write it 13 year you're right for the height of the box but was 0 1 with us he said box white of Europe where is what is the use of OK so you're right we know you're absolutely right we should integrate from 0 for the height of the box which will call him either boxes shelled and the area of the box giraffe to do the integral and Surrey and this comes into and parlor what's the integral of each of the miners NGY wine Ng no matter what you do no it's easy being interval the miners NGY is minus are 1 of energy to minus Angie white pretty Arthur evaluated between the 2 limits and this 2 terms of Rubin nuisance arm I but you do their research work is gives you another factor but noticed that the other factor does not depend on data pool sorry opposed a pair of beta ropes Beta somebody didn't touch me it's each of mind as beta times the energy Ito the miners beta Times energy so there is a factor of there is some additional dependency yes there is additional depends their farm and you who work Yukon work it out they will do it next time getting tired of moved too late but we could work out what the partition function is for the box of guests in the gravitational field column that has a same factor that had before plus some other factor from here OK so armored car go over to work that out since this term is dated dependent it will give some contribution to the energy per particle but of course it will it would just be giving you on the potential energy particle that roof look f what the hell what's called restaurateur it's not that hard but
they are a simplified a little bit let's let the room be infinitely height so this is literally the problem of the atmosphere between the earth's surface Annan docket Our OK so what's the integral who we think the grow a movie the go already is just 1 provided by data M GE Power right end and then the other 1 is exactly what it was before anybody remember what it was before the hour the area to the end over and factorial that's 6 such as the number and what about the other stuff square root To what end pie a Overbay of a power of 3 yet OK 0 to notice it Everything is silly and power that's nice because when you take the library tells you a lot of z z what was it is contains 3 and not over to tool minus log data I'm keeping only the things which depend on an now minus 3 an over to log beta the remote plus 2 Bonjour constants my end water beta plus locker-room area blah blah blah but don't care about that so only differentiate with respect the beta will get the same terms as before 3 temperate 3 have strands temperature offer reach for each particle and another term which looks like it differentiate this respect to baby again In all over and all verbatim which is kind to and times tiered looks like in there must be the potential energy carry you must be the average potential energy 0 dancer Igor such true looks like any case sets said that's a procedure that's an example a couple of examples very very simple calculations of partition functions and they show you among other things all incidentally as a homework problem calculate the entropy of the memorable the entropy busy the entropy is s physical debate AP plus logs calculate the entropy and the entropy per particle our just as an exercise and corset first-rate forward 3 feet dual use it for something new and better finishes up or supper tonight so it these places as the both this fall the division as equilibrium and maximum entropy subject to the constraints of given total energy which has to be computed from here and the constraint that it'll probably would example 1 which is hardly a constraint but you have to include a formal mathematics will the pretty nifty emptor you know I get you never get to the point we're not surprised by the equation but after a point you get familiar with the simple tricks and you know which equations to use 1 following the other but it's always the case but always a case that you follow a set of procedures in Aurora 7 new look at me and say Well that's a that's an interesting equation ones before with Pope William uninteresting but a very curious subject is due to fuel the reviewers we plan to start his disease head and be said that amenable black hole yes that's easy to Munich collapsing star or star Star the hard but not because complicated details not because it's conceptually well I don't think that I had talent that has and more like book you 1 them these headache fidgety does go up to a point to a point to it gets walking back European can go part but not Warren for durum amount of energy in a given volume that Tibet but there's a yes that is just 4th tuitions in energy for example young appear we're Barry I had intended to do that and I'm not going to work we would your honor calculate the fluctuations in the quantity would you want a 1st calculate is the average of the square of that quantity we're prepared to the calculate the square of the energy you want to differentiate twice with respect to beta and you'll find a formula involving some 2nd derivatives and I worked out maybe next time the population and is a very nice have a beautiful for knew of fluctuations of energy fluctuations in energy means the width of the energy of the distribution and 2 important formula relates fluctuations In energy specific heat and former worker fluctuations were not a part of the original thermodynamic fury tourist revealing but it was really Einstein and Gibbs who really really understood that a way to find out if what you talking about is the statistical mechanics of a finite collection of Adams versus complete continually with no basis statistics which the study fluctuation world lucrative next Mellor please
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