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# Statistical Mechanics Lecture 5

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high-backed the thermodynamics now when I was going through my lecture notes for tonight I realize just how supple some things could be end really why it is that students eyes go like this when they're studying thermodynamics that is full of always fancy calculus tricks where sometimes every little in the Wishon about what's really going on but it's really all in the calculus tricks on new condole you a little hemostatic a particular and only 1 problem with the study the problem of pressure of an ideal gas tonight in until evenly we know where the pressure comes from a comes from the molecules bouncing against the walls of the system we calculated just by estimating majority calculated the average energy of every molecule of three-year T 3 has a temperature so we know how fast moving on the average we can estimate how many of them are moving in each direction because there projected traffic you might wonder incidentally if the molecules near the wall are isotropic after all the presence of the wall influences what's going on here opinion might think well maybe there are more going horizontal pool Waldron going arrow toward their way turned out to be true that even very close to the wall the distribution of velocities is still pretty isotropic but there you might wonder for where wherever principle we're just an accident it might also wonder whether the velocities near the wall happened to be different than the velocities and the interior far from the wall which case should be making a mistake by just taking the go 3 years K T and her and using it for every molecule matter where you are and that they are not making a mistake but had Elian state you know from statistical mechanics nor from an honest evaluation on the set of principles which don't use in Toulouse pictures of molecules hitting the wall so for but which really used these basic foundations of statistical mechanics for the foundation of statistical mechanics on mathematical very mathematical and not Compton not hard but their mathematical and has a sauna emphasized to you over and over in situations where the rules are mathematical when you practice fully give up into wish you purposefully give up and do wish him because you're not sure that during the wishes of correct such as for example the velocity distribution near the wall was the same as in the interior of the war to purposefully give up installations and you ride the mathematics when you're riding the mathematics the rules sort of his autopilot rules will you 1st figure out what you want and then you just start going with are using whatever tricks you have until you suddenly find yourself with a formula that you recognize and that's just the leaders farmers you get good at after a while and the great physicists have I know or were very good at how France and I don't mean people who just studied a statistical mechanics I need all of very very good physicist now on loan thermodynamics it's fun but it's the fun of surprises from the surprising relationships which suddenly fit together and tell you some physics that's what the attraction is for many of us Einstein was the grand master of thermodynamics Fineman was a grand master of the kind of Grand Master of it well maybe maybe dog lure could stop any way get in the end he said that the man had changed sack viewing station south seceded said yes yes but we we really doing is checking into which in the mathematics and then they could really be wrong that the distribution of velocities near the wall is saying the interior will rely on you will you don't go to all of the mathematics and then say Well I think I think I've proved that the amount that the rules of calculus work out OK 8 because they agree with my naive intuitions about way molecules behavior but not it still hate your checking your intuitions mathematics focus so you what we could we could do well little calculation of calculating the pressure on walls Gibbons the average velocities of every molecule of 3 have skate we were probably do the calculation by by saying what's not even talk about averages what's in his room is filled with molecules which are moving velocity one-half movies where equals or equals 3 educatee right but just 3 temperature use
our theory units that don't have Boltzmann's constant winner of a massive every molecule therefore we cover severely molecule Nova velocity of every molecule we can compute be forced on wall by asking how many molecules per unit time hit the wall if if indeed they distributed uniformly which we don't know for sure our and if we know that there a rumor that their direction of the velocities is isotropic we can calculate the number of collisions on the wall per unit time we can calculate the momentum delivered by each particle wall particle bounce off the wall and the perpendicular component of momentum which is really going this way using a ring this way we can calculate the momentum delivered a wall and of course Forest is who rate of change of momentum and so we can calculate the force a wall pressure is the force per unit area so we can calculate that it's fun you could try and do it yourself and you get the writer who get the right answer Bulmer In the case where a molecules are completely free so but that's not really good enough we really want a set of rules which of good enough that they could take into account collisions between the molecules my gut collisions between molecules they could they could upset the whole thing off me do do our we want a set of rules which are a is robust which doesn't depend on details oversimplified supra simplified model of we get a free get free gas means a guest particles were particles among interacting with each of the ideal gets so the ideal gasses a place the easy easy to do things with but not very general the rules of statistical mechanics are very very general and would be sufficient to calculate if you if you could do the calculations and do then you can calculate the relationship between the pressure and other variables are very general what is it that we want only called the equation of state like calculate don't know what the pressure is as a function of temperature volume but a two-year is it that determines the guests in this room basically the temperature us about it is also the number of guest molecules also sorry the number of guest molecules the room number of molecules temperature and possibly the volume work depend on a volume no sorry will depend volume and the volume we would like to calculate the pressure in terms of so we're fury we need that we need a set of principles are my room described those principles tonight and worked out I hope the pressure of an ideal gas armed with no intermission whatever no picture and our minds of what we do and I'm not saying the pictures in mind aren't good things for understanding physics but there are times when you want to suppress the pictures and simply go with the mathematics a statistical mechanics is 1 of them because you want new rules to be general update managers remind you over here on the blackboard a couple of hours a fact mathematical fact worked before and then I want to prove will see around this is 1 of the damned trickiest theorems it the drives people crazy drove me crazy they're very proved a once when I was young young man and then I come back to it every time I teach the subject and every time I teach the subject I cannot remember how you prove it and I have to sit with a piece of paper for a half an hour however are evident very very simple birth here and then will you theorems are of the calculator pressure but I wanted it to mathematics on the blackboard just so we have tried remember not to erase and so will have Needham please don't let me erased by the 1st thing had to deal with Inter b and entropy is the summation your evisceration minus a summation of p log p piece of IOR peace of mind and if the distribution of the Boltzmann distribution at minus the summation now peace of mind is 1 all over the partition function each of them minors beta Times energy of the Eiffel level of piece of right and then logarithm of peace of ice so what's the logarithm of the subject here it's miners beta Esau body of water a member of Martinez logs right so Ahlem everybody recognizes this misses a product or broader it is the sum of the logarithm of the 2 factors in the product so that minus logs 80 miners Beatty minus signs cancel my were get rid of a mouse that we don't make mistakes later the measure remember the 1st German here is just being times the average energy probabilities flight times east of our side when Saunders just to average energy so the 1st turn just
gives as data but the thing we're or B B average energy focus its
guru it got world to a 2nd here well 6 times logs age but we have a sunken Yulia zeal logs E. don't depend on IT what depends on ideas only this might receive was made as I what is a summary of each of the miners RIO biding its Z that's a definition of z salt busy they did get from the summation cancels over here and all you get is a plus clogs ASEAN could be and arms in what it was the what's warned the summation of each of them minors baby cancels assume nominator if you want now yes OK let's is rearranged a little bit into was a standard form this the formula formula but member the baby is 1 over temperature
and let's showcase sir we have s very a due 81 or temperature right soared Markowitz multiplied by temperature and write the formula and the following fashion or lately because this is sort of standard energy my he times X energy managing minus energy money peak times it is equal come minus t logs multiplied by tea so that means we have 3 times as we get rid of the T and the energy termed energy times us and that's equal to tee times logs with a minus sign
I think with a minus sign yup yes murmuring but this thing has indeed it's not important words name is works important only because you may want be using it over and over their cars over and over in physics will want to have given a brought purposes we really does want this formula here this is important formula here our February to you is good but we might as well give this
its name restore its name a it's called the Helmholtz free energy things it given names when they of Cairo over and over again are we going to find out the plays a leading role in calculating things like pressure and all kinds of weather on the Bay of Pigs this is called 30 this another thermodynamic variable Lihir energy would have where energy we have temperature we have entropy we now have a new thing called the Helmholtz free energy Helmholtz H E L & H O Elzy and at the moment it's simply they object which were overtures occurred once therefore not very interested see the current once more tonight but I tell you that occurs or replace OK that's 1 piece of information that we need in this former don't need the fact that it's called AT orders called Helmholtz free energy we need this equation for the next equation this is not a hard this is not the theorem that great the theorem which drives me crazy would go on to be doing is introducing yet another variable into a problem are like all variables of this kind control parameters and notes you will see something about control parameters control parameters are parameters that you'll as an experiment that easily can change their parameters which I'm acroscopic they're not trying to change details of 1 molecule at a time an example would be the volume of the container of it gets you have a piston but block cylinder of gas with just minute and move the piston back and forth and in the process change the volume of gaps so volume as a control parameter other control parameters magnetic field on a system of electric field on a system or a pop applied electric field applied magnetic field item of you think of any others easily you know our not pressure we must the relationship that you can but variables common payers what is usually called the control parameter and 1 is called the conjugate thermodynamical variable pressure and volume are closely connected and that conjure repairs yes you could change the pressure you could have a pressure gage in here and by moving medium by moving be distant change the pressure but of course what you doing is changing the volume and responds a change in pressure that is said more but said that it would not guests is not a system is of Adams whatever to made out of but young when not assuming that the guests Kabila Quinn the mere the biggest solid Indiana would be squeezing on a solid that's why it's important that have general methods you cannot use the ideal gas will offer solid Valencia last season Syria young that would you could change the number of molecules there but that to control parameter track chalk if they haven't been equilibrium there's a temperature if you're at the U. of the boiling temperature exactly her steam and water in their simultaneously you just sort not even gets neither against nor liquid situation and the through the gravitational field liquid might serve on the bottom and guest might similar Top but would still be equilibrium and you could still studied by the methods of abusing OK so that's a river right and I want this theorem just because of the music stores were on the black here forest ever whether it's here on the to sit too simple to be a full-blown it supposing you have 2 functions both 2 variables and complicated 1 of functions of the so-called ease it will be the energy later and the other function would recall the entropy S core S and these 2 functions are functions of 2 other variables namely remember energy entropy over here to other variables the independent variables in the problem and those whatever core TV and I'm not there was no significant Selam lining the mark just think of these as the independent variables that we can control from outside temperature if we like the volume of those independent variables and given the nature of the system given the nature of the guests were not be given the temperature and the volume The Deep End of variables the if you know enough you could calculate it would be energy and entropy bozo there was some was in some was Indiana might yet both of these things over interested that why the following year we'll take this because it's public and who would have been even ask what a good for but a silly fear but here's the theorem and it has nothing specific to do with the particularly nature of these font of what they are too will be pending variables to independent variables it says says that the derivative of the energy or EU degree of each with respect to be independent variable VAT at fixed that's a funny thing to do and these are the independent variables you usually differentiated dependent variables keeping some of the independent variable fixed with something different women differentiate EU with respect the volume keeping the entropy fixed or see what that means a moment and that's alright arrested and that's the sum of 2 terms are actually a difference of 2 terms 1 of them it is a derivative of the energy of E which respectively with respect to their volume keeping the temperature effects while still normal normal with differentiating deep end variable with respect to an independent variable keeping me or other independent variable flex that's what we normally do calculus but as an extra term and the extra Turner's minus the derivatives EU with respect Carey and could be at fixed volume which is a little weird calculating that the river beekeeping s tricks get out of the upstart calculatingly derivative EU with respect that keeping the volume fixed but other weird thing and then derivative of s with respect to a volume of keeping a temperature fixed says my fear Of
all the stress of the servers of Acacia poses multiplied for the young this is a really weird partly to return I think if you set them to try approve it it probably wander around the New year and the year in the dark for quite a while so it takes me out I would say it takes me a half an hour a good day to remind myself what this means how you prove it you wanna see the pro but from gas power I like it so much because they figured about myself Mr. of form wobbles the things if you figure out yourself when you're young but I think I could still do it now expected did do it now I did this afternoon let's draw are independent variables after the but let the let let's hope I could do it now held temperature and log what he'll take now there are 2 independent variables 1 is called X let's draw the contour lines of S M O individual 1 can't relying on withdraw some specific lying along which S is constant and often looks like that not of the drawing a line is S S is equal to some constant because are the contour lines but I'm not about withdrawing them there's 1 contour as equals constant belong there at fixed that's important thing fixed that only 1 calculate the rate of change of eat at another variable with respect the volume of how is changing as you change the volume of but along the line of constant that with this said keeping its constant so here we are we're a point tho he wouldn't change the volume point of India ends We're mistake on a line of constant X so along that line the constant best calculating if you I don't think he might V along line of constant I use patient dealt the EU dealt the because I like to remind myself that these things are actually radios of differences now some let's right now the following formula when I it when I'm using belt that you are not distinguishing it from it's infinitely small to differential it just helps remember what you doing 3 members of their ratios of small differences so don't they is part Philip E. with respect through volume at a fixed temperature turns a change in volume in motor from somewhat of a 24 point she rate of change are e with respect to V at fixed temperature firms DVD lost by D TV fixed funds delta T these are like these 2 things are the first one is the rate of change of E with respect VD keeping the temperature fixed Georgia has to to do with moving from here he and the other 1 is the rate of change as you changed temperature keeping volume 6 keeping defects and built the tape are yet never misses this is normal nothing special yet that a 1st year is the 1st term over here it's this 2nd the 1 that we want to manipulate and fool around with and see if we can understand why quarterbacks so as to steps the 1st is to write that the rate of change of easy with respect he lets courts will depend on the blackboard give it some more class for ahead ye by DVD that constant real story beauty by deep sea that constant Dr. T are the 1st they were Mariba what was right all of this justified you go you could hurt and Eddie given point you couldn't stop playing in a number of different directions manipulating various things and server processes a branching Treaty of possible formulas that you could write a book What Italy is not that big and if you're lucky you'll hit 1 a tree somewhere that enormous something good and I'm really afraid that is the way it eventually you get good Avenue get in sting you give instinct for due next 5 so be eased by IDB let's just look at us we have on he had D E by D. S at constant volume but this is the baby teed constant volume of single volume Astrix area in the EU by D. T a cards volume was the same errors he by D S a constant volume B. S by a go-ahead here with those of a rumor that the E by T. right Daddy but please Tobias by also accounts volume OK so here I had my D E baby AS Carson volume all with still 1 more thing now sorry we're interested in the rate of change of EU with respect to beat that means we should the by built the Treaty on both sides to see get rid of the dope be provided by Dr. provided by Delta OK so now we're Kirgan but where so far have I put in that I've moved along the line of constant XI I have nothing in here has told me that I'm moving all along the line from 1 point to another
along the lines of constant entropy know why wouldn't arrested in country comes entropy will come out not from this formula but will come out soon enough so but a case so will move along this line how comely characterizes like this curve because point is characterized by slow that's the is just it he by Delta Street let's see what we can figure out of our let's is a little bit of a calculus to see if we can work out something about the tea by Delta we let's write down a conditional moving on a line of constant attacks that is along this line DES is equal to 0 along this line PS is equal to 0 and now we have to do is write the DES partial other s with respect volume at 6 p trends volume Delta Force the volume plus partial of S with respect tea at fixed volume has dealt take boasts learning VS is equal to 0 0 right yes is equal to 0 along with the definition of a line of would make this equal to 0 and now they be able to get a formula Fidel the T by Delta V in terms of these partial derivatives to going abide by Dr. B. because we want to stoke the by dose of solids the vibe don't think Hey we have our target of India delta of the tea by Dr. B. we can now immediately read off that Delta T size dealt the VAT is equal minus D S by V at constant TV divided by U.S. did TD at constant V this would be true for any function tests whatever on this map year that the slope of we can't lines the slope of the contour lines is related to certain partial derivatives had a off I would at that takes me and the move to see why this is I have to go through this will step you see but now we know what Delta T by Delta so let's put it that 1st of all gives us a minus sign and then delta T by Delta V partial that's what's right about partial abreast with respect to V a Council but he can't provided by a partial rests with respect to at constant as array of 4 factors in 4 factors 3 in the numerator 1 in the denominator this is in the denominator devoted VS beauty but foot cynically by a great stroke of luck that thing in the nominator happens to be the same thing as 1 of the terms enumerator get I said we could've gone off in all sorts of random directions but this taxes in there but warned because this is getting simpler now so let's get rid of the numerator sorry made and the numerous Ada I would not what done where we deliver we done believe we've proved this little limit yeah I cannot fill slimmer without drawing his picture it's something you wanna be able approved displayed mechanical steps but I cannot rule out during pictures and they're figuring out warm doing says a sure it said that there is no doubt that arm for most systems in most circumstances yet there are some special cases were much rule and then you have to be more careful but generally speaking yet Bancorp is a monotonic function of temperature and the reason is the reason is simple love the Boltzmann distribution as you see the temperature if you increase the temperature it gets narrower his you decrease the temperature it gives broader and entropy is just a measure of that with sold typically yes some very special cases but but yacht are your of things are nicely behaved OK so here we have our 2 facts the 3rd factor has to do with the definition of pressure end I want to make very clear that pressure just 1 special case of 80 responds to a control parameter were by response control prime solve hours because they want 1 act 1 thing we have to do but actually tools but they're easy firstly understand what pressure really years here's are still under his arm pistons molecules in here hitting the Pistons so for exerting pressure on it supposing we move the piston a little bit maybe pressure in the interior does work on the press only our on to the promoters and how the pressure pushes out and those who work on that work winners at work come from that workers energy the work done on the past and is equal to the change an opposite sign if they have to guess does work on the Pistons then there's a change in the energy in here on negative think they work maybe they get smarter but a brought the following way and they think about a
thief Nagin the piston rises a little bit wealth that means there's been an increase and energy in and the gravitational potential energy of the and in this case that's called the work done on the piston by but pressure and by energy conservation any work would start on the Pistons must be taken away from me you from the gaps to energy conservation tells us that we change in energy of the gas is minus the work done on the piston now let's suppose that area of the piston is but suppose the piston is moved by an unknown DX vertically DX vertically and let's and you also have to imagine something else we have to imagine 1st of all that we do this slowly arms some odd things can happen if you do it fast for example you Coppola piston out so fast that no molecule has a chance to hit the distant as is moving the knowing that had been no pressure are OK that's a rather extreme situation but it does illustrate for you that you want to define pressure by being able that average over many collision was so you want to slowly move the piston slowly slow motion and no limit of very slow motion and the something else you wanna do you wanna make sure no energy comes into the system from outside no would you want to insulate the walls of the system I remember when I was a undergraduate I learned the word adiabatic I had no idea what it my professor of engineering told me that there needs that a system is insulated From he coming into Iraq and later on when I was in graduate school another professor from me adiabatic means slowly well what abatic means is slowly and no he comes at the if you move the piston 80 directly with no heat coming into the system change of the energy of the gas is minus the work done on the Tristan and miners who worked on the testing is before my is a force on the distant pressure area pressure times area of forest pressure times areas faucets definition of pressure force per unit area force harms
DX ART at times DX is the change in the volume of the gets right area times DX is the change in the volume so we can write a DX as DVD now that this is away and Mr. Farley's pressure you can make it defines pressure we in fact is where we go right we can write if that is the change in energy with respect the volume the only question is whether we keep fixed in doing this derivative where we keep fix some rate of change of energy with respect the volume under the circumstances that we do the operation AT about equity OK now was another aspect of the notion of 80 abatic anybody know another Nova meaning Tati batik for the 2nd law of thermodynamics which will come to says entropy always increases except for about where doesn't increase it stays the same never decreases never say never but never Mark OK 1 up process to use with the entropy doesn't increase stays the same Bozo the 80 batik
derivative of entropy with respect for the volume at a fixed temperature of times the derivative on the energy With respect and be have fixed while you that this looks terrible looks too complicated but fortunately this is something we seen before but remember would Debye DES is the temperature holding the volume text holding the volume fixed media holding the system fixture not changing the control parameters it's just the original system with fixed energy levels and your holding that fixed under those circumstances D E by D S it is just the temperatures and that's D E equals TBS illegals or large P E by S is the temperature so this is not quite so bad we could replace this factor over here just by the temperature it's getting simpler for I the now yep Ferdie yes but catch but thank you copy now the
only derivatives are derivatives with respect the volume of fixed temperature that some promising volume and temperature commit lovers independent variables and the only the reviews of Pierre are so familiar things how the energy changes with respect the volume at fixed temperature and how s changes with respect the volume and fixed temperature T as just a temperature 1 more step because also happens to equal might the derivative with respect to the volume at fixed temperature of energy my tee times yes I'll take the 1st considered by DVD that's devalued by via fixed temperature the 2nd terms if I differentiated with respect the volume fixed temperature I would get 2 terms from a product here 1 of them would be he times a derivative of vest with respect to a volume at this 1 what's the other 1 the other 1 is derivative will respect the volume of the temperature at fixed temperature how the temperature changes you very the volume if you keep the temperature picks it doesn't change right so the 2nd terms there a smell we have are fundamental theorem pressure is equal to the derivative with respect to the value of the Helmholtz free energy at a fixed temperature now that is much simpler than anything else you could emerge from pressure and in fact we know more we know that they Helmholtz free energy is just minus logs easily calculated them partition function was sort of finished so let's that again think its class members with plus plus the derivative with respect to a volume is fixed temperature so we can take the temperature army outside works with only the plus sign the temperature times derivative of log z with respect of volume came at a fixed temperature always need to be able a calculator is how the partition function depends on the volume and we can calculate the pressure everything I've done up till now is completely general it wouldn't matter if were liquidated against up plasma lover or a solid solid inside the cylinder it doesn't matter what the chemical composition is this is what statistical mechanics not not Connecticut Peri kinetic theory resistant to picture of molecule bouncing around this newest statistical mechanics says pressure is equal to temperature at times and delivered derivative of the partition function now you can calculate the you tell you this Our would for it basically any control parameter I just use volume as a control parameter because of familiar to any control parameter there is always a derivative of energy with respect to the control parameter at fixed entropy that is called the conjugate thermodynamical variable volley of volume and pressure are conjugate variables thermodynamically many others lack of going to the now about any kind of a control parameter then you could ask how the energy of the system depends on a control parameter fixed entropy and that the fines and another variable which is like pressure which plays the role of pressure type of let's that's now we've Lilly fast so I thought because we don't want hire and annuities were
want . you see the power of defining Tracy the power that this quantity potentially can have Our it just appears Willard in these deliver another thing the logs depend on none of of analogs derivative logs it with respect to beta was the average energy logs E has all our stuff and analogs even a lot of stuff Gary Europe as question that said this is is what matters is not use In polish just knocked that becomes a complicated nonequilibrium processed and is example more fast enough nor molecule hit the wall won't be any pressure on but of of course happened is year of you may ovaries sudden change and how will the system response the system will respond by a shock wave going through fluid not imagine that you have a flawed suddenly it's easier to think about
a few suddenly squeeze the flawed you different volume in either direction is this squeeze applauded and you make a shock wave going through the system will also make a shock wave of hippos people away and very complicated things happen and it's not controlled by a simple statistical mechanics of control by a holder dynamics the whole nonequilibrium dynamics the only thing you can't be sure of is that you are physically entropy increases that you could actually work out both for the special case that I said a moment ago you move the Pistons so fast that no single molecule hits the wall then you know that that there's no work done on the Pistons and the change in energy is absolutely 0 2 changes energy 0 but the volume is changed Sukumar some things about that situation Newcomb work out some things about to the figure out the final temperature is it it solvable by darned if you a squeeze the guests of Marie then all kinds of very complicated things have With he thought but we urge US said that there a lot best shot but question is what temperature was a return of equilibrium and that can be a very very complicated problems not sometimes once in a while it's easy but got last year with the team leader EADS a yacht what does other young so that there is no doubt that there is no liberty and dealt with given that you are moving along the line of constant entropy French we used to we calculated delta T by built be-all longer line of constant entropy by saying DES equals 0 Young what do you think that it is straight yes capped bread basket but let me say slowly physical that then there's some the violence of course that itself may depend and the details but In roughly speaking in means Mini Mini molecules or collided with the wall so that you can average over the details of a molecular collisions with the war you know I you could we could we could examine specific cases some questions have very very general some questions depend on the details the question of exactly what time scales constitute flow Comey 1 that depends on the details the question of what happens if you move sufficiently slowly does not depend on details yacht so a questions and Vera what's we can have an act that way off key the river logs you with respect to volume at constant temperature now when we when we work with the partition function of course we fix the temperature we say what they did is so typically when you calculate a partition function it is the partition function is a function of other variables at a fixed temperature you worked at a fixed temperature on which go back to it partition function for the ideal gas this is a very easy from this point on he is the ideal guest and a petition from function was the girl are just remind you were it was an integral all the X is an all the pieces each of them minors beta times the energy which was P squared off 2 M right now not a in great detail this was an edible over how many Exs on Korean peach free and right now if remember this factory B. P. integration doesn't depend on exit all Annex integration just gives you what was only 1 particle we give you a volume of sand particles gives you the volume silly and power was volume to the and power Hollywood an integral which depends on the temperature but not on volume P integration does not depend on a volume so workers work able was he ate some function on the reason doesn't end of the week that may or may not stick in the factory and factorial denominator make any difference wife because we're going to warning that and this factors make any difference the reason is because we have to calculate logarithm of Z analog over the rosy don't be a sum of 2 terms as selects work Apple logarithm of Ms. Izzy but but but but equals analog re from Vitaly then visit that that's constant it depends on number particles doesn't depend on the volume put minus 0 minus lagend factorial and then this something here bridges plus a lot of function of beta beta being here the inverse temperature only want to earlier depends on the volume of its this war the 1 that just seen from the fact that we had to integrate the the position of the particle although the whole volume nothing else depends on the bar so when we differentiate with respect to the volume we can forget Everything else class on
interesting stuff OK what is a derivative Ottawa a derivative of the loggers you with respect to the volume is it it through a log we wish victory while reveal right were that was opposed to multiply that by temperature however yes re-address puree dash key equally and this it's him there's not much lower says Dr. said to be we don't know with fish that with that was just a trick to calculate the distribution yup may be ideal get my sister number particles right notices NOV is the density an overview starts Omro crazy things sources say says In light of various ways of writer pressure time volume equals In the number of particles in the sample times take 90 may be used to seeing this as PD Eagles Inc 18 quick Cayzer Bolton constant and of course there is a Boltzmann constant here if you work in the laboratory units all you can't abide by In right P times the density of particles a number particles per unit volume is equal to the temperature was a variety of area answer it 1 of the volume of Bayern P E equals in divided by which is rolled fans temperature then City Times temperature recalls volume factor bald constant but we're grownup so we're used to getting torn away from constant this is the equation of state of argued yes so it is exactly what you would have computed on if you were just used into a picture of molecules bouncing off the wall but here it's justified and more than that this power in this time you could do the integrals necessary to work to calculate the petition function you have a precise way of calculating sorrow for this was just the simplest of Mini Mini examples with people calculate here we did it in order to see how the year how these have basic concepts fit together have a basic concept together and that's it that's the ideal gas where finished early today so we can we go back over questions I think I was fairly clear tonight but have not said it is a whole lot of Vols folk Welch who sent young star hasn't come of the population they really Notre tour the winner that was a flocculation farm if you look in the boxer guest like theirs and it's an equilibrium of another system room so that's exchanging energy will be edging won't be a definite value the same weaving calling images the average energy you will fluctuate on the added to the fact that the energy fluctuates in certain ways is evidence that the system has really made up of molecules of but we have to come to a dead where we needed definitions Of the fluctuation of a quantity when you have a do when you have a probability distribution OK so standard definition Joe let's begin with the quantity whose averages 0 begin with the quantity was averages 0 that means a probability distribution for it but doesn't literally mean it doesn't it doesn't necessarily mean that it seemed appeared in any sense of the origin but they ever but but the averages that doesn't look like a 0 average for it it it but they definition of the population and is and it it's not a complete description of the probability distribution it's just a description of war aspect of it which is kind of a approximation Toto with odd the average Cisco's quantity acts the average of x 0 about telling us and we know the averages 0 because we set up that way but the average of X squared is not 0 why not because it squares 0 positive on the side of positive on this side and the average of X is a measure of the width of the distribution of flocculation usually called dealt acts the uncertainty in excellent we like to call it is the the square root of B average did we used a symbol for average X squared or but here the square wrote the square meters squared fluctuations busy average of X squared OK now let's suppose that it's not centered at 0 so the averages knob 0 I always have to do is shift the variable so the average does become 0 and do the same thing in other words we define a new variable called X minus the average
wrecks West C average of X murder CF richer text 0 0 prove that is of was a subject of proved that the average of this quantity here is 0 prepared something should broke might somehow I have something unknown quantity which is just shifted so that averages 0 and watch the fluctuation fluctuation in it is where is the average of the square of this thing so we want to take this thing Norris square on average the whole thing that's and 32 good quantity let's calculator see what we can do with it puts Suva can make some sense out of it 1st of all we square before we average so were averaging x squared twice Banks the average of X plus average Rex square are want average Flynn now the average of next quarter that the number of some number and the probability distribution but the number was the average of the average Rex were it is a sequel which the average of the square the average of the average of over Vioxx the average a averages just average OK so this is just the average erect square What about this 1 he yet not quite minus 2 times so the average if I average this quantity here we minus 2 times the average of x squared a robot disorder it is what it is but it is not the same why not it's not the square of the average is square the average the average of the square this is average of x squared minus twice this plus 1 times that is my years average of X squared that's called the fluctuation every other main idea is another name for this thing variants salary right is Escobar variants of boat means is thought fluctuation away from the north away from the average keep that now bag of tricks that be uncertainty in X or the fluctuation enacts square wherever it is by definition the average of the square minors Mosquero the average now let's see if we can
calculate fluctuations in the energy will calculate the fluctuation energy in terms of the partition function where we have armed with 1 quantity always 1 goes back to the partition function that really does contain Jaffe said that the were went the definition left inside the definition of the square on fluctuation now when my right the belt the X is a square root tor root mean squares and stuff like that but this is just a definition but what it does is doesn't measure the averaged on measures the width of the distribution of how broader distribution any way the public or the root mean square there is a bit of a beer if your best vision about yeah what was right but now actually want calculator let tells us is how much I was real experimental fluctuation we expect In the measured quantities from which would vary when measured over and over again why does it very very energy coming into and out of the system is really fluctuating that is a mathematical definition energy is not box does fluctuate it's in equilibrium with a year with bad worried that OK so let's see if we calculate we've already calculated that this is now what do the energy or a new energy so it's apply this energy the uncertainty in the energy the square of it average of the energy square minus the average of the energy square receive the prison officer what was the average of the energy remember in terms of partition function minus D logs eBay debater from rack the average energy so we have the allergenic GE already we don't have to do any more work on merit but what about a B average of the squalor of energy let's go back 1st of all what was the trick that we used to calculate the average energy remember we said that the average energy summation 1 oversee each of them minors beta Esau by Lisa bite and that we recognize that multiplying by Esa by here is the same as taking minus derivative with respect to beta saw what I believe this is equal to 1 always Debye debate over the minus sign of z deferred jersey it was more about b by debater is a somewhat with Western Rev. control but suppose would get the average of the energy square where we do average of theology squared museum due squared here we differentiate twice we take a 2nd derivative trading is 2nd derivative now we don't need a minus sign because when you differentiate twice that things might assign will go away you will get a square Pansy to America's Beatty all the average this average of the squared energy that's equal to 1 Z looks like it's the 2nd derivative of z with respect square do I have their right help out former right what's and then my no miners was squared Beyond Z
structures rioters were worn overseas squared Desi by debate a squared so this is a full Georgia emerging but With sea again and the thing with that 1st of all it tells us we can calculate B of the flocculation if we know again this shows the power of the partition function of statistical mechanics is about the power of the staff of the partition function and the power of differentiation OK let a this sort of books the catalog foreign quantity the 2nd derivative with respect to Beta square of logs E when I was young at that puts you ears so his arm so what want show culture of us were were from but these Baidoa see deeds of Warri although Z Desi by debate I think that's a synthesis is yah yah is 2nd alarms E is the 1st arugula logs different she began us focus on this singer over here on it has 2 terms the 1st turn it 1 of his 1st term has gotten by differentiating easy by debater does as well as 8 times a 2nd derivative of C with respect to bear squared and then what about the 2nd term the 2nd turn is obtained by differentiating 1 each times easy by debater what's the derivative of 1 always easy with respect to beta minus 1 overseas squared terms Desi by debate exactly the gets sort we haven't even simple formula further fluctuation misses the fluctuation squared coach but he likes this is even simpler was what I was by debater there is a lot of roads the average energy is no minus the average energy is minus the average energy Serbia we have another formula that is Debye debater our minus the average energy fuck anybody noticing is what does not ENG changes with temperature daily basis center lets us free ride a bus ride this is Debye ID temperature times deep temperature by debater what steep temperature which is what did the temperature is what baby is 1 of a temperature square the other way around Beta temperatures warmer beta sewer sneaky by debater they have temperature is 1 over beta the derivative is equal to minus 1 beta squared sir I just might well bigger squared the stinky baby Bader which is temperature squared bladder right yet so I think this is just temperature square easier Friday I don't have these things
memorized former teammate hyped I think that's right temperature squared was a minus signs of son goes away its temperature squared times a derivative of temperature of energy with respect to temperature anybody know what the word is for the derivative of energy with respect to temperature specific specific he heat is the question of how and how much do you have to shit change energy to make a certain change in temperature how much you have they heed the talk of water the change of temperature by 1 degree this quantity E. derivative EU with respect to specific heat another symbol for specific heat Causey subs B. Lara said that's fine keeping fixed diet yes you do this is keeping the volume up with keeping the system fixed would not bearing the control parameter here and this his derivation arrived service Is the during this is the beat This is the specific heat at constant volume in general constant control parameters c sub show what specific he might be defined as a year and a dividing by the total number of particles of system or something what b rate of change of the energy would respect the temperature This is a famous formula I think a NDB do not stand assure our I signed Gibson know ruler was probably Einstein ON Gibbs so the fluctuation in energy to know the specific heat you multiplied by the temperature swear that will tell you how much fluctuation now I left out something like bitterly about something where left out was a Boltzmann constant Bui was a kid balls the whole thing is 1 factor of cable from there with my definition to what was CVC VCV was a derivative of energy with respect to temperature but Yao but I think that I think is a from a change of variables here I think is a key downstairs to powers now so I think if you working in laboratory units I think this would have a cable money from them into small fluctuations a small we do we expect fluctuations are a lot of particles you expect the fluctuations before it was small and that's where small was comes from so those are the fluctuation energy of radio radio about any system this is not specific to to any particularly in this specific heat but it's not specific to any particular a system of him as a specific heat as a function of what review measures Pacific he period and multiply by temperature squared and then you sit around measuring energy in the sample you should expect to see an energy fluctuation equal to that of a life that is thought that this is it cousin with respect to what he called James while b constant it's not constant is at this temperature the IOC is a function of temperature and the fluctuations are a function of temperature now nor sovereign right so this is again part of the beautiful structure of statistical mechanics this goes beyond thermodynamics with you actually see the end of a formula like this a key beyond statistical mechanics who really does depending on demand of the fact that systems have made up of molecules so and some are program number of for cable T is closely related our guns number 0 0 foot is not if you get there the vault and therefore while this this was the derivation of the fluctuations no doubt there dying off and I don't remember the connection between all specific keeps during for young lady the wrong answer a CPU is this this is closer to a 3rd said they calling specific heat I think is maybe not were narrowed because the normally the specific heat would be a constant and column Pirro unit the mass of the material or something right so actually this should be the specific he kindness well what what definitions Pacific he the rate of change of energy per unit temperature per unit mass for example OK so at me this this specific heat over here is really the specific heat times the mass of the sample With Bob got call the heat capacity that that's a temporary measure you it's the heat capacity but the heat capacity dependence on the masses of the mass of material that you'll have died
of other words roughly speaking this is proportional from number molecules on a sample is proportional from remember molecules but it is the measured specific heat times the mass of the sample or what is called the heat capacity right drive yes are so were a show you lose Javier the questions go home and look up all the heat capacities and so forth and see how big the fluctuations of the idea of a leader of a water the important thing here is I said is that these formulas you don't the pinned on the details of what things on made us want summarizes measure specifically Huneault populations without having to worry work hard to really formal or please visit
us Eckstein for down
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