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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
recurrences I did not show whether Hungary was really the first one to think about it but I have a feeling that vaults again probably understood this with both from Boston was before upon really thinks of thinks so calm solely a question here's a question in my but suppose you to start out with error in this room and all on 1 side of the room this is not impossible thing to W purple Wally you evacuate the air out of 1 side of the room stick it into the other side of the room and start with initial condition will always err on the left side of the road and let it go get what happens comes as thermal equilibrium fills the room pretty much uniformly entropy increases for then you waiting you waiting waiting newly Cecilia our later sooner or later the unlikely will happen the unlikely event will happen where by accident or by just a leading long enough in time for the year will reappear in the left side of the room all of us top half of the room or whatever but Masuda left here for that's cool the currents and it's a it's really no different than saying that flip point enough times I will get a million heads Annarummo very unlikely but I do enough to flock to issue will happen OK so the question is roughly speaking how long do you have to wait for the year Purim 1 side of the road it a years 10 years as a hundred years of of age of the universe so let's see if we can get a handle on on you start out with the idea I love phase space yeah they space is the space of the coordinates of the molecules mom and the other molecules because high dimensional 6 and dimensional 6 a and B costs on Pierce each particle has 3 coordinate each particle has 3 momentum component if there and particles arrest 6 and coordinates of the high-dimensional space OK now as far as we momentum space goes that's kind of bounded bounded because if any particle has a loyalists momentum it will have a very large energy and as a citizen of the energy that you put in the box no more so I'm pretty much weakened say the momentum dimensions this box is founded and let's just With just bounded by saying the momentum is definitely within some range here and it doesn't matter how many particles we have it doesn't matter how big the system is after it doesn't matter what the temperature is high temperature the more certain the momentum it but the temperature is reasonably well then the momentum direction here is pretty much founded by on and the axis that runs from the last part of the room of right part of a room sort of and say that we started out in 1 half of the then but the day's point I was in here another words the system started in phase space somewhere in not b detailed about where it is in there somewhere so the probability distribution the spread out over here and now we wait for a while and what happens the Phillies point starts to move now it always chaotic chaotically for our purposes just means pretty on predictably end not done unpredictably because the laws of physics unpredictable in principle but because trajectories like the like the example of the billiard balls errors or slight differences can magnify themselves after a while and so even if we started about 2 very very similar trajectories they would very quickly depart and you can pretty much out of imagined that this means that phase point moves around in here very very complicated ways and pretty much fills up the phase space fills up the faced placed in the sense that there are a few Of course green it and fuzzier eyes a little bit where somebody else's glasses it will look like it's pretty much filled up the space OK so what percentage of the time would you expect that that day the point resides such that the particles are all in 1 half the wrong if this were the picture research the time as crazy we don't expect at the time the air molecules to be in on the left half the Roma right analysts think we're making it is drawing a picture and just 2 dimensions right until dimensions if we don't find the X-Base in half and um received were to the right with talking about half the volume of phase space what happens there will if we had in the coordinates and we're not talking about the part of the phase space where we want particle is on the right hand
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
accidentally always apparently accidentally go planet with people want if we would have no explanation of the coherence of history why history looks like it had a coherent past our and Hawaii yes why there's a consistency historical evidence if the universe is materialized area by random fluctuation of the universe of the planet just materialized random fluctuation with us sitting here today said he used in list cases it each other but this there's the problem finding 1 player but bigger than pointing to wear Todd is given the result was followed does not the same as before finding 1 not do you know what is it's a dish said she small the he did say will relying on his reign in the fluctuations and here we have a situation where randomly in this room and a collection of molecules randomly materialized and performed a Boltzmann's head form Boltzmann's has a very small probability there's a lot we will eat and then there may be there but a strip by the hour the probability that that we discover Altman's head is going to be a very very small number bottom and had his former Boltzmann looks around and says I wonder what the probability is that my wife is here also very very very cranky but much much time of his death is just bought OK but that's not what you're asking what was the best time that he a place on Tuesday what has 1 that was out there says what's what vote it doesn't matter what was left of the probability that the discovery of a as white in the world is very very tight Eds equally tiny if bolts 1 happens to have been discovered in the room right equally probable equally improbable whether that's a shut very improbable so other words Boltzmann ought to be very very surprised that his wife he 1st of all the surprise if are maybe there's another theory very well he does not much about little about the theory and discovered is still What a wonderful accident how happy I am that my wife I would be too much of a good thing he looks around he finds he says what is his conclusion his conclusion as you know I think my world is probably not a world of random statistics it's probably not a world they close the volume of molecules which is randomly assembled he could say that before he could say that before where would say before is luck I'm here I know that if we wait long enough I will be here when will I be it is a fantastic piece of luck that I happen to be here right now know at some point in time I be here this happens to be the primary here if that's the right theory that I'm here just because of 4 of the guests in the room eventually assembled into me and when I'll sky be here except when I'm here it's from here but I think the best prediction I can make it is an affair look around they will find the rest of the room pretty much in thermal equilibrium with no fancy structures that particular my wife or be here if we discover his wife he will say that's extra ordinarily unlikely even much more unlikely so my fury is probably longer and the result of a random fluctuation that this right is as good as they it he says this is a list of textbooks says George Washington trapped the territory and renewal got another textbook it says the same thing but the world was just a fluctuation urges Member fluctuation and 1 textbook happened to say George Washington trapped the cherry trees would be no reason to expect although Intel the tree the world just materialize accidentally in a configuration in which the textbooks said each up tree thought that this is not the right theory of nature that this is a little bit of a blow close closes the loss of years but it would still be true if not you could say that but then you would never nevertheless said if you wait long enough there will be many many many replicas of you in the future and almost all of them will not see a coherent history so if you make best guess you find yourself here today and U.S. How did I get here be overwhelming majority of people wake up in the morning in his question would be ones who came out of his random fluctuations now is is a serious concern before should we don't worry about that the 2nd that cosmologists don't worry about it serious theoretical cosmologist but I'm not going to try so you anything it is something that is of concern if we want to use statistics and asked What's the most probable thing we should we should we should see given that we're here we have to take into account all the ways we could've gotten here most of the ways we could have gotten here would be by random fluctuations and history would not be coherent further OK that's a problem of Altman membrane is a it is a lessee's To decrease century not cost does not of course it doesn't government life does not desperate always at the cost of something else increasing its entropy bombers the 2nd orders not say that some subsystem of the world can decrease its entropy there will always be at the cost of some other subsystem increasing its entropy even more so then there's the work for a long time but I will see and the flow of energy from the sun the earth is not in equilibrium stationary configure stationary means it stays the same because of floor if you have a system which has a flow moving through it it can create interesting structures for example a full of water through a pipe can create vortices the spin-off In all of those waters have a structure and little like eddy current so forth 430 currents have a structured the water from flowing through the pipe creates there and you could imagine the little any currents could have enough structured perhaps some interesting properties but if you stop the flow by your sealing off the end of apartheid in what happens to be eddy current disappeared and it just returns to work with us boring equilibrium sold was the fall in the case of life that would allow was because it's kind of a parent Armed violation of the principle always in search of course even in that flow situation the total entropy of everything is increasing even knowing making it up pomp sink cover what is coming in 1 going up the other end comms out warmer at the other end king in this and so altogether the stopping violated the same thing is true on wall with flow of energy from the sun Our if we sealed up the earth with sunlight let sunlight note Everything will eventually come to thermal equilibrium that would be would be no life and those who just have featureless futurists thermal equilibrium
not so wife's it is a corner of 84 80 current rules would sees the pier in moving fluid the fluid being energy from the sun OK have Rao spent an hour talking about interesting things I can get back to some bowl things magnets magnets our our whom we talk about magnets incidentally in statistical mechanics we usually not talking about pieces of Botha we usually talking about mathematical models of a certain kind of system that a certain features that resemble magnetism 1st of all what is a magnet where a magnet is an ordinary magnet is made up of lots of little magnets magnets could even be as small as a single Adam or could be crystal grains for Aaron magnet is made up of the life of the loan magnets and typically at room temperature at some ordinary temperature particularly room-temperature being rebuffed high in this context could be high but certainly a very high temperature thousand degrees or whatever Of those more magnets are of randomly oriented and such a way that net sample doesn't have a net dough orientation and the orientation is random and not just as the orientation of the whole thing random but wrote the orientation of the parts of the random so there's no net magnetization amount to magnetic come acroscopic magnetic field if you call up down end if these energies Store Inc pairs of these little magnets is such that the magnets like to line up in the same direction as a starter cool the parent you find out that lumps groups of magnets groups of the magnets tend to be in a line but other little groups of magnets will also tend to be in alignment but no directions and you'll find so don't mean means which are magnetized which means they can't the point of the same direction but these domains are still fairly small if you call it down it experimental facts and not not completely hot dog stand as you call it that more and more energy the energy consideration that things like to be in the same direction like me that these energies lower if magnets apparel at the edges lower magnets a parallel then and you suck energy of this system more and more of them will want to come and alignment and these domain will start to grow old and eventually you may be only not fit of pouring at finite temperature non-zero temperature you may or may not be the point at which all of a sudden these domains become infinitely big sorrow that the magnets tend to be somewhat lined up everywhere is in the same direction that call the pharaoh magnetic transition a please transition that the simplest kind of phase transitions are certainly and Zero temperature you expect them to be all lined up that because Zero temperature the only state of importance and Boltzmann distribution is the lowest energy state temperatures 0 only the lowest energy state and the lowest energy state all year microscopic Adams like Harvey microscopic magnet liner yet he has to be a thinking when they heard that depends on it depends on the details in the he's a minor intellectual life into a knowledge is thinking sinking North Pole once the grabbed the southpaw a little more complicated than that and that's that's partly why there aren't that many magnetic materials right there is a tendency for them toward Durant line but also competing things going on they have to have the next general manager liner notes but which direction they lined up in random and that that so called spontaneous symmetry breaking with October talking about simple magnetic systems tendency toward orders you call them order remains parallel Miss His case ended the year OK service make remark about what you just starting to just asked if they could all lined up this way political this way political lineup that way I endeavor which way delay lineup which way do they wind-up lining and that itself might be defined or determined by the Cornelius Little stray magnetic field just 1 molecule just 1 little elementary and to be in a magnetic field which tends to line up a little bit and they govern the whole thing about the way the whole system lined up is a symmetry the symmetries which way things point if they warned appointing a direction that symmetry is broken score broke breaking the symmetry there's no more symmetry looks like there's no more cemetery but it's spontaneous there's no magnetic field pushing everything in that direction it just had to pick a direction picked the direction it may be because of that kind kind kind will stray magnetic field but there were really talk right abuse of things we talk about ended the point at which the symmetry is broken the point at which the magnets align themselves up in some direction that's a phase transition and that's phase transition this called the magnetic phase transition so our 1st of all I don't think about literal magnets because the model systems that there are studied often they're not quite quite unrealistic as theories of a ferromagnetic chants of iron what makes them interesting is of course that they resemble a lot of other things in nature and that their mathematically farm simple enough to study an interesting enough to exhibit features like phase transition makes an interesting OK so let's start would be very very simplest right at the very end as a state this figure room at this kind of magnetic he the points up points found it doesn't get the point in random directions you would think of it as hedge entails a few right but very simple mathematical magnet ii points stem cell and will love it doesn't matter how they laid out on a blackboard but let's lay them out a line some of them are up some of them are down and we want to make a statistical mechanics and ask such questions are what the relative percentage of ups and downs what's the energy of the magnet and so forth right so before we begin the food we talking about statistical mechanics the Boltzmann distribution we have to have an energy function Ito the miners beta times nor the energy is still police say they are show you could think of them as Adams McChrystal for example if you could think of was Adams in a crystal the Adams have electrical current so maybe they have spin electrical currents make little electromagnets so each Adam is a mad most South Pole but but for the simplest model the study which room begin with the Adams point up 0 . and key point any other way again the purpose of this is to be simple OK so the lot of them how many of them capital in 1 of the energy will start with very very
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
equations here and then will be able to use the partition function the energy His equal energy it is equal to end times the magnetization new which all I've done here say that in mind the and the magnetization times and at this as we your it till users will come back From a definition of magnetization important warrant a that is roughly speaking the probability of being up by probability being bound for a given forget OK how we calculate the magnetization well 1 easy way he is to calculate the energy of system if we know the energy of the system and we know the number particles a window age we can calculate the magnetization so the 1st thing we will calculate using the partition function is a B average energy from that we can read off the magnetization leg he is a situation that they not we're talking about average we're talking about an absolutely right absolutely right this is a particular configuration now we should say that the the magnetization is average that you're absolutely right it's the average of the average over the statistical distribution of the Boltzmann distribution right of the and of course this is also the average energy now right OK help would their rights or go to calculate the average energy we calculate that passed through the city has energy I'm not visitors year but has said now that's theology given and given amnesty that's that's the that is this in the energy sector this is the energy for a given configuration average energy is the average value of in miners and so pilots with Is it with a probability now now Is it is obvious that that that it will involve heritage mad about From groups call color and say cities it is an equation that configuration by configurations borrowed it is up you think about it is obvious for every configuration be energy is proportional to end minus him if you average both sides the average energy will be proportional to the average of minus the average energy Wycarbah averages around all of these something is equal to something else configuration by configuration there will also be equal in the average so that plays a shawl off of him absolutely yes yes yes all of these quite everything statistical mechanics the average kid not our capital in but yes yes of course and arrived this end began around 80 begin begin begins is a number began as that Arts Arts rotatory wanted the average energy is equal to 0 and although at a fixed number of times the average magnetization scores not strictly speaking you would be usual definitions You don't have to put an average you because the definition of magnetization is average it's also true 0 them statistical mechanics now average over the probability distribution average Over the same exact thing we do would be ideal gas we have a probability distribution we calculate averages chapter it averages from the probability distribution deadline there are buyers or sellers again not least death owner of this is a model in which by definition these things can only point upper deck is a mathematical model life they may be biased obverse is down but not by but there's no such thing as East and West will look that poet fairy the energy prefers the molecules to be back remember the energy if there are up minus if systems like Toby Lorena G meeting to say that the Boltzmann distribution favors lower energy this is most definitely biased by the presence of a magnetic field no symmetry here this is a problem that has no symmetry is biased for further Adams point down and cost synergies could tip a mop which way with other likely to be a real ceiling and make some guesses which way it will they be at 0 temperature right now not definitely not right that rights of 0 temperature of what we expect the magnetization be we expect everybody to be down and that means the magnetization will be minus war where we expected infinite temperature thought down 58 50 50 expect that about average book infinite temperature everything is maximally random all states are equally probable and so at infinite temperature we expect the magnetization be 0 so it goes 1 at 0 temperature 0 0 infinite temperature this is what we expect is a right this is correct cycles minus 1 0 and no point will be be positive because average magnetization not positive because of the bias down it's the infinite temperature will defeat the bias and temperatures so random a but a magnetic energy use is unimportant and so it will be a brand but at no point where the average magnetization br or deposit appeared so I hope I'm right shed a switch the magnetic field the way he could that at that that make the magnetic field negative right Bert right switch the OK so well instead calculating the magnetization calculate the average energy we not a calculate the average energy from of politicians from should remember the average energy on destroyer right he now averages is equal time minus the derivative of the logarithm of disease with respect the bigger itself is a little bit of algebra to do here we might as well do
it no it tends to put people sleep watch me do algebra the logarithm mosey as a constant from here that's going to go away when we differentiate so excited about writing it is equal get in log on the hyperbolic signs a terrible function of Beta new which in turn a library Norris 1st of all that is proportional data that's a good thing because typically energy easily differentiate will be a proportional and natural on let's differentiate us with respect busy ah sorry would respect the Bader the derivative of logs e with respect data 1st of all will have ended now derivatives logarithm of an argument of a thing is 1 over that things so that will give us in a denominator hyperbolic call sign of Baden New Age and then in the numerator we have to differentiate conch beta UH with respect to data so what happens when you differentiate Koch wears derivative of Koch cinch so that saying each they each side your age and then you have to differentiate the posh sorry that differentiate this thing with respect to Bader that gives you another Age Age outside we are set energy not quite minus this is the energy might OK so we have the energy from want the magnetization so we won't do it is divided by OH and divided by In so the magnetization is equal to my and as I said in the 1st place comes on my the rioting by ending with dividing by New Age and so it's just exactly cinch data new-age over Koch that function also has a name to cope with temperature chief of Baden New Age we will fight for the right focus in all we have to deal 0 to understand the system is understand the pinch function looks like incidentally Bader is 1 over the temperature so it is what Marcus this function as a function of the temperature new kinds H that system number it's not so interesting we could absorb it in the beta baby With a plot a thing as a function of Baden New which they common together OK so the question is what does it pinch function look like you you can work out with a pinch function looks like go by yourself I will show you what looks like well 1st of all I think Shea and Koch for very of a larger values of the argument become equal to each other that basically both exponential so they New Age but try barrel Koch of exercise is equal to eatery ex post Etowah minus X only change each of X because eagerly X minus Ito miners excellent ruled when x gets a large Escobar large X 1 X gets large what happens to them either sex it is goes away so far larger X they both are equal to Ito the X and the ratio was 1 of very far away a cinch I'm not including minus sign justice until overcast can't function Gore 1 incidentally often negative effects it goes to minus 1 of a story about that X is going positive this problem Bell wasn't the only the origin beauty origin and conch is equal to 1 x equals 0 near the origin Ito Exs 1 is 1 plus 1 2 were about this 1 then Zero but what about the linear court about the correction to it if we expand Ito the axes warned plus cuts Peter the acts will be 1 plus X minus Ito minus taxes minus 1 plus text of either by tool and the answers just X the 1st derivative here is more Roberto starts out as looking like X and very quickly does bends over very boring function starts out linear and that it gets tired quickly and it just go the flattens out that's a pinch function right and horizontal axis it is beta new which I'm keeping in mind the baby is 1 that over temperature What is the magnetization when the temperature is small steps data is large beta large a way out here and the pinch function as 1 of the magnetization is minus 1 0 temperature all spins aligned themselves what about infinite temperature infinite temperature is beta equals 0 they was 0 1st of all the magnetization is 0 around the magnetization 0 as expected and is just fills in the details for us this just fills in the exact details of the problem of how the magnetization goes from 1 at low temperatures and goes 0 temperatures U.S. me how could you get Media magnetization Degonia opposite direction while the answers to allow each to go negative if H. goes negative are so the magnetic field clips sign everything different made this the Gators they said they did would not system by but this this system does not have a phase transition is too simple the
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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Metadaten

Formale Metadaten

Titel Statistical Mechanics Lecture 8
Serientitel Statistical Mechanics
Teil 8
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/14943
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract Leonard Susskind continues the discussion of reversibility by calculating the small but finite probability that all molecules of a gas collect in one half of a room. He then introduces the statistical mechanics of magnetism.

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