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# Statistical Mechanics Lecture 8

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Stecher University Carla
Temecula 2nd law firm in its little bits and talk about the form array recurrences conquering
side but with all of them are on the right hand side so let's just say there 2 particles of forget the momentum from minute and just draw the took X 1 next 2 or X XMY of her 1st acts in the 2nd X and say that both of them are on 1 half of the room is to say that the phrase point is somewhere in the quarter of a square sold suppose there were only 2 particles moving in 1 dimension and we start somewheres and they will system was around randomly to particles scatter off each other they do a random things what percentage of the time or both particles In EU left-hand side of the room accord with a free particles the but what if there are any particles well through to the end or through to the end 1 return to the end of the year 80 after Doha um that depends that depends on how or constraining here for several particles October left laughter of we just said Joe laughter laughter that didn't care where they were up the figure would be 1 with 2 to me is another way to think about it and that's to say let's take the place space end identified must sub region that as the sub region which are interested interested means an interesting configuration of it is very unlikely right so here a lowly region of the phase space the phase space is much bigger and the volume was the whole of volume of the phase space but forget momentum momentum is unimportant what is the volume of the phase space if there and particles it's the volume of the box raised and the order free and power depending on the volume of the whole thing is that he's because it's an end dimensional face space over 3 and order of works forget defray the volume of the whole phase the contains of volume and that's the same volume today and that we discovered we calculated the partition function integrated over position volume let's suppose this region of phase space over here has a that a much smaller size although particles have to be in and then the volume of this region of phase space with all the particles in here will be some movie and this is the volume of the box so that not the volume of faced takes a body in the face bases Vitaly and volume of box and therefore asking about the particles being in some small volume could be half the things could be half of their big volume a case the system small regions and the volume of this region of spaces will veto the what does what percentage of the time would you expect the phase point to be in here the Oviedo the end of a will be to the end of a big set little Vitaly year could be some small number of whatever is but 1 of about big vie to be a big lead to the end cult is proportional to the entropy of the whole system after the entropy of a region of that phase spaces the logarithm of the volume from that region if all you know is that the system wasn't some region of phase space the volume of that region of phase space issues the library is began to pick up the the play which was yellow not worrying about the momentum why the momentum of the moment to use the momentum roughly speaking of whatever I a temperature so it's not worry about that OK so roughly speaking this here will veto the end is entropy better gas would have if it was in this little region for particles warned that little region must Iqbal the region pretty small so that's just a number it's but but what about big Vitaly and that's entropy of the whole gas and thermal equilibrium or it exponential of the entropy Largo the veto the end is the entropy of the guests in this region and Vitaly and as the exponential and so what this is telling us is that the likelihood of finding a system trainee volume of phase space here is always proportional e problem my mind because the denominator each of them minus 3 and 4 b of um of these thermal equilibrium state the meaning of
that is heads very improbable eatery the entropy to amend the entropy is a very very small number what's the entropy of the these molecules in this room roughly speaking proportional to the number of molecules is pretty close to does being the number of molecules can and not more want and the 30th and 30th amateur exactly what something like that so Vitaly In the aura of the entropy is is enter the 30th the probability of finding yourself from a pretty little volume of a space like this is not easy to the entropy that hurry it is the mindlessly entropy the minus the entropy is is the probability of finding itself from a small region after this is saying this is 1 want to be in here is the same as this 1 or 2 to be and indeed case so we studied over here little over big was a half but and a half is not a terribly small numbers but when raised to the Pollock tend to the the idea that it is a very small number OK so the likelihood at a random draw a point from the phase space that you find yourself in this tiny volume of phase space here is in this case here 1 tool raised can not the the 30th of the end of the 30th power it's a pretty small number How long after wait on the average define yourself in that region well about what kind of water keys to attend the 30th if the fraction of time you spend in that our region is 1 over toe to attend the 30th and how long we'll have to wait for you find yourself in that region food to the end of the 30th it what units years seconds it it doesn't matter if procedures all with its worthy of jet wife interesting losers would do this exercise just understand just understand understanding what its our systems are reversible this is if you wait long enough they will reverse themselves and we really have a sealed room here and you have evolved so let's say starting from the year from the odds state bill would prompt a thermal equilibrium or looks for what looks like thermal equilibrium would spend a long long find but every so often every 1 of told to attend to the 30th the years or whatever you would find the modules and have wrong you wait long again equilibrate again that looks conventional and then all of a sudden you find the molecules and the other half of the rumor that corner of the room and if you integrated or study over sufficiently long times you will discover that the entropy goes down b a three-year our these on this goes up and down up and down up and down in a completely time symmetrical a completely time symmetric way what's not time symmetric is if you knowingly start in a very on configuration another Lejeune knowingly starred in that a tiny volume of space most likely the next thing is to find itself out of that so if you start and odd situation with all the molecules in the corner of the room you expect the next thing the flying it the molecule spread out in fact you'll find and the next thing in next thing and next thing is pretty much the spread out uniformly and that sounds Igor violates the reversibility of the physical laws but and if you little waited long enough you'll find it reversing itself and doing everything imaginable for closest a lot necessary here said let's say that let's say a major voice but we have a seen letters rap some years of course but was forced OK so we have a number tool the the 30th OK but now a change units as part of this unit is the units of 2nd but the ureter Bowers wept as is number because of me as a food for the end 30 abided by divided by what's 3 thousand or something a perforate right OK that's say show to attend the 30th binders sorry it doesn't matter whether right center 30 of my nurseries 10 if that is if we were to be short of a slew danger Willie but they suggest time actually come back then now him because of fraction of the part how
long it actually spends in the corner that depends on how fast molecules knowing and so forth but a lot of H shares quantum preserve area of Kory recurrence their organs with that that's just and they it's more than an interesting point deep conceptual point they resolved and help resolve the puzzle are the 1 the apparent 1 win this time and be the 2 winners of the laws of motion of course were required to make sense out of it is that we going to eventually have to understand why the universe started in a little corner of phase that's a separate issue both that you know that the EU and Turkey said so have I had a change how I'll agree absolutely him stowaways opened room questions constantly being addressed why did the uterus star Romario in a small corner of space policy had that set up a a lot of Quebecers proved that a good question is 1 which will answer the universe must not be closed to close the orders we currently currently current that does not present a crow good statistical explanation of the world but they use this time if you want if you watch the closed system long enough you would find that a score some measure of the localization of the particles will earn heard decrease to start off from a corner and there pretty D localized for long time and pop up and then go back down pop up go back down but the kind scale the discovery this reversibility that work goes up must come down so speak is this toe to attend a there is is more about the comeback what does everything is everything Everything will be so our every hello in you would find the molecules in exactly the right configuration and tourists wasn't a corn are pretty much the same way these with them all years marriage that you'll see everything happened see thing happened both direction start slaughtered tho rooms but this right NASA all always yes yes but we're trying to understand why the world looks forward used the problem and thing Bultmann numerous it's a bit more recurrent problem of recent years is the the universe really were closed box and you want to ask What's the most likely configuration defined a planet with the armed with people on what's the most likely possibility he you would discover that the most likely possibility is so have you don't form gas everywhere except the smallest possible amount of gas necessary to make up a planet having condensed into the plant the chances that you would see till planets of the chances that you see 1 planet our very very very small the chances that you see 2 planets Our vastly more negligible and that so if you asking the question What should astronomers expect to see you in a world given that there are strong given that there are strong conditional probability the conditional probability that there is a planet among most plan on the planet there are astronomers around the middle astronomical observations what's the most likely thing they will say the most likely thing they will see that will look out and see nothing but may see some gets out there but they will not see convinced him to another planet by far the most likely fantasy is 1 planet if you if you know you have 1 and not so what's the probability that you'll see the universe filled with stars absolutely negligible under less you know that in the rat fairly recent past and the fairly recent past you started with some very exceptional an unusual starting point and then the flew all out from the starting point is over likely perhaps certain kinds of structure that a random fluctuations would not have anywhere it's call the problem of boats when brain call the problem Altman brains because people want a little bit the excessive on it and said the most probable astronomer who would be a single astronomers Hayden disconnected from anything else but it is a problem it's a problem of using statistics to understand the world the way it is and it's always it is always a conditional question the conditional question always given that way he had several various things what's the probability that we see X and the probability of closed universes that we see Ex God would be X would be much higher to see only X
and not YDY meaning some of apply plans so it's not a good theory to think we're just the result of a random fluctuation if we were a result of a random fluctuation with things just symbol themselves so
simple version in the very simple version there's is no interaction between the Adam between these magnets but there is a magnetic field is a magnetic field is pointing up pointing down Russia which way my notes searcher correspond to is a magnetic field each Adam has a magnetic moment as just a little number attached to it tells you how strongly it interacts with magnetic field has magnetic moment called Newell has to do with the the strength of the magnet basically there's a magnetic field the magnetic field as you the pointing up or down like a minimal which way chose it without worry about it but be energy of 1 of these magnets is different if it up or should bounce in particular if we magnet it is but I think we give it a plus energy and the magnet is down we give minus so let's event of variable for each magnet that's given name of scholars Sigma This is the signal for the 1st batter and as a signal for the 2nd Adam Baba and soon Izzy the plus or minus 1 it's just a label or radio variable which is plus or minus 1 of the 1st spin is up that means signal 1 is up Is plus 2nd now means Sigma tool is and so forth OK was the energy of the system if the spinners up then the energy it is positive and it's just equal commute times 8 which was spinners now what CNG then archaic supposing that In the end will in Spain's up and will end spins Varroa Roxie Energy Energy is equal to little and minus little AM at times new Times H New Times a key is this energy of 1 spin up and New Times H is suspended down is the land equals the number of ups and middle and his number of that what's full and plus little and they get so little and cross will M. 's capital but they with that Gonnella can right down the Boltzmann distribution calculate anything we want using statistical mechanics so to do that shares of each he let the dimensions 80 were Rossiter number the strength of the magnetic field it X-terminal magnetic field imposed on the magnet from outside so fraught purposes it a number and new was also a number and we might as well be the matters will put new inmates together and just call a whole thing number that's often bind sometimes called middle age but I thought I would just expose the various pieces of it Our would you prefer a Koberg each time new little H in the big each again we could do that because all right now so it was easy sorry he equals Mysidae energy and it equals but thank you frenzy equals rest thank not how configurations Oregon there many configurations are there With little in the Alps and little and barons without asking which ones a which many configurations are wary of capital in the things we want to group them into 2 groups 1 group of Leyland and 1 group of little in Herminie such configurations are family such arrangements are that's a
comedy tort problem yah yah perverse rival the number of configurations with this value of energy the number of configurations with this value of energy is capital and factorial number not number but a a number of states if forgiven in mind is in factorial Over a little and factorial real and factorial remembering that little in and little an up to begin that's the number of such configurations let's take 1 of those configurations and ask what the Boltzmann waivers for the most weight means Ito the miners beta times the energy factor be doing is is working out the partition function the partition function it is the sum of all configurations that means it's the summer already In New Guinea and such that relentless Kalamazoo could begin won't bother writing wrapped but keep them in mind each of the miners Bader times the energy riches New Age kind In minors so we just take saying we're more now for each and my and there's going to be a certain number of configurations and that number of configurations is as common torque coefficient here so we can write this your and is a not very Yarborough remember what the number of yeah that at the capital and for so long that saying that they both over the summer are good terms some in the end and making sure of was and now with the right away games and I say yes that are sorted Israeli Turkey dinner patient simple but but no each individual configuration use this and the number of configurations with be given a energy is sites you'll on the sale director formula that as some modest and land up and factorial Over will end factorial times being in fact or big and mightiest little in fact from this it is capital and factorial Wuhan factorial will and structural fines each of them minors data a let's leave it this way from minute fabrics not forcefully Let's right that's right of following each of them minors beta H In let's call that such call that X let's call each of the class beta new H I could call at 1 over Expo wife from that state please numbers here call each of the miners beta stage acts the of 1 why OK so what each of the miners beta each time In that extra power and you see that consider that extra the power and and what about the other factory that's why it took power and I just using X and Y because rebukes stars a memories from high school this flageolet source expos why this is the binomial expansion this is the bond binomial expansion and this whole thing is just equal plus Hawaii to the capital in right that's that's the binomial expansion has as always we forget our what Zia's let's write down Z is just X Z is equal to X which is each of them minors Bader New Age plus why which is heated up last data new H all erased the capital in power that's it as fell was easy this function is a function have named well but multiply and divide by 2 about the structure of her name the hyperbolic costs are right so such court that as well so the answer then is to and now totally in matter be interesting it's a number of multiplicative factor in the partition function usually doesn't do do anything but will leave it there and then hyperbolic cosigned of mule in each of the power right here that's a politician fortune bowl Surrey MUH my mistake paid is awfully important 50 1st temperature without it we can't differentiate with respect to to it ride so that's up partition function now bomb supposing we're interested in the question of what is the relative percentage of ups and downs what's the relative percentage that quantity has I mean that's called the magnetization the magnetization is 0 0 if errors many ups and downs the magnetization is plus if there are more ups and downs magnetization minus the opposite situation so let's defined 1st of all it's defined magnetization a willingness and is the difference between and bounced a cell of the magnetization but its usual divided by capital so that it becomes a magnet per nite funeral so let's call the magnetization it is equal to 0 and minus M times New Age divided by capital in that definition of magnetization and what is it it's the bias for reach particle whether it's up for them if the magnetization as positive it it's so the average up Mr. Obama's of Ichiro bowl I take that back the magnetization is just doesn't have the new Age there but did then that's definition that definition of the magnetization is clearly related to energy which is right a few
1st interesting system that has a phase transition is two-dimensional Ising model but 1st do the one-dimensional Inc Hong woes not a very good student who was a student of lanes was famous for a number of things um for 1 the things you spend this fall of which was not famous floor was inventing rising model but he gave his student 1 problem could determine whether there was a phase transition rising model bond the one-dimensional rising and a student got the wrong answer he said it was a phase transition was not that all as far as I know the rising ever did so why it's called rising model was just to an end an uprising is the most famous name in all of statistical care books where goes win some you lose you win some 30 rooms some new ways you assembly rooms said but what is the eyes the interesting thing about them or is it is symmetric between up and down so therefore if there give any magnetization it's because Sun Herald the system here is spontaneously broken the symmetry in the one-dimensional Ising model that does not happen in the two-dimensional Ising model does have a well-defined all of these Ising model for you right now work the following way energy is not storage particle but particle is no we were now we have no external field no external field so the particles to interact with each other for little magnets to interact with each other there would be no energy and there's no energy all configurations are equally likely in this case the magnetic field that each Spain's sees is due to the neighbors his neighbors are up its feels a magnetic field up if 1 the neighbor at both neighbors Obama feels a magnetic field down and the 1 up a one-man refused on magnetic field self was saying that read energy is associated with Perez with Perez of neighboring spins and if the Paris are in the same direction Let's take that what's he take that to be lower energy just will you have to make a choice now armed do we want the leader of the interactions of favor alignment Orient Pioline detail byline and forces aligned this alignment and this is also on the energy is equal for this configuration as it is for that configuration and unequal for this configure she of that configuration you get it right that thought we come back to these variables said Mark always say if signal is a line in the 2 neighbouring singers are line just just focus on Tuesday if there are lying and energy is lower if their underlying the images larger so wet take the energy could be some number which is usually called J. do not I don't know what it stands for is usually just call J. it's just the number has an energy skeleton energy scale of the problem J. Curley said not of particle 1 time signals particles only 2 particles from a visit to neighboring particles on a lattice L. later on will allow the latter's and now the lattices just lying later on we can have a lattice of two-dimensional three-dimensional lattice 1 and 2 of our neighbouring sites on and this is the energy of the 1 2 parents now on this energy is due be lower if their entire life because of the entire line signal was negative I want the energy to be lowest airline so here with this Energy Energy is alone if the spins are aligned with each other and it's higher if their entire aligning supposing we have a line of them and each was interacting with its neighbors then we can write that energy is equal to or some minus J of stigma of In the current plus 1 0 but I am a stay order means the product of the spin spin-echo expand product of the magnetic moment at each site terms neighboring site each 1 each parent neighboring pair account of what services are expression for the energy there was think about it from moment moment where they expect happened and infinite temperature the general role is it and the temperature adjustable random chaos everything is equally likely 0 magnet magnetization worry rugby and you prepare a firm yacht right everything's is random and sold particularly 0 magnetization a flood of letters every product will be on the average 0 at sites where above you will get energy Estero arrows negatives to starting with a negative bias are you starting with the ground state having negative energy
soros 0 of energy service because a big negative number so having 0 energy is effectively having a lot of energy relative to the year grams steel push us is an average therefore the neighbors are randomly distributed yard the average the average the chaos that means there is likely to be found Powell's and so the average energy will be 0 which is a lot of energy but what about 0 temperature would you guess was different temperature a beeline right the 1 label which whaling this all of everybody aligned this way for everybody aligned this way he can telephium there ought to lay down states ground states means states of minimum energy and they will both come in with equal probability of both in with equal probability but now let me and 1 more thing that me suppose that a magnetic field external magnetic field but it's only acting on 1 particle walked out of take the 23rd has a little stray magnetic field analysts say that magnetic field along 1 axis then what is the grounds the ground state has a definite orientation even if that magnetic field is very small the ground state still has a definite orientation and at 0 temperature strictly 0 temperature Baltimore and distribution always favors infinitely strongly be lowest energy state so that means that even the Zionist little magnetic fields stray magnetic field will the Boltzmann distribution will favor all of the spins pointing along 1 axis if you want to apply that kind of magnetic field that the system come to equilibrium at 0 temperature and remove the magnetic field the system will remembered everybody's holding everybody else in place and the possibility of them all simultaneously jumping Tula the opposite state remote its is enough of them so that's called spontaneous symmetry breaking that is what spontaneous symmetry breaking news in this case simple and this example has asymmetry we cannot Tilley said From quantitatively are mathematically what that symmetry is a symmetry is usually represented by a mathematical operation on the degrees of freedom what mathematical operation would you duel on the stigma 0 change from up to multiply by miners wanted right OK let's go back the earlier case over here where the energy in this case we could say the energy was proportional to just Cigna itself not segment and the neighboring said but signal by itself possess a symmetry not be energy itself changes sign when you change stigma and that some symmetry actions that you could do that don't change the energy and what about the system's opposing you change the sign of 1 Spain 1 of them symbol 1 anoxic is ever symmetry not reaIly energy changes boy seem both a change in both the energy doesn't change so take go from 2 opted to their own that's a cemetery now we have this whole vast array of them and while we change all of the sickness simultaneously right formerly the equation sigma of I for all I closed my eyes my innocent of ice we replaced every by minors argue that the energy doesn't change India's energy doesn't change and that's what it means to have a symmetry an operation that you could do on the coordinates of a system that don't change the energy the matter what state prayer for every state whatever the state is our there's another corresponding state which has the same energy and which suspends overall reorient opposite to what you started with and then a sure that in some sense there's no bias up or down if the system is not swap itself all simultaneously and pop at 0 temperature that means it could also flopped itself into now there's no way to predict in advance on less you know that kind little stray magnetic field is no way to predict the advance which way it's going goal but could go 1 way or the other because the Boltzmann distributions says you gotta be in the ground state of 0 temperature and Aziz said the little tiny stray magnetic field which will determine which 1 is but it will be 1 of them quite so clearly the next thing we want to do is to solve the one-dimensional Ising model want defined do we want to calculate exactly the same about the center or a calculator petition function for persist so will do that next time difficult quit for tonight will do that next time we'll work it out we will see that there is no doubt that does not have faced a magician at a finite temperature nothing funny happens of final our contrary to rising 4 it took a few more years for a couple of physicist named climbers and 1 to prove but if took two-dimensional letters was beautiful story for more
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