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Statistical Mechanics Lecture 9

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Automatisierte Medienanalyse

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Stecher University the Vatican
remind you we will go on to do today is going to study the Ising model I was terribly unfair rising rest time that was I I looked it up and I was but somewhat basis isolated I Azinger failed to solve the model correctly or whatever not be correctly and thought it was a phase of phase transition the one-dimensional Ising model that would be totally inexcusable and that it is not exactly what happened was that bad he solved one-dimensional rising model correctly would be given to most thesis advisers a problem claims the famous physicist lines and he saw the correctly he realized it didn't have a phase transition but the basis of that he Our believed and wrote In his thesis that PIC model and indeed dimension does not have a phase transition that was that was that was incorrect although talk about that today we're not going to solve the higher dimensional rising models it too hard for us they are very hard but way will use an approximation method that physically very intuitive and was see that in higher dimensions sufficiently high damage there is a phase transition will talk about what their phase transition means our phase transition means a sudden change in the
properties of the system as you very for example will be the temperature arms but let's let's go slow what's taken steps when I go back to this simple example that we study last time we studied a problem in which they a collection of little magnets each magnate could be upward debt and it was an energy function and the energy function differed between often now that may arise again the energy of
a magnet winds up is different energy Magna went down we introduced a little variable called Cigna could be plus 1 minus 1 plus 1 minus 1 and we wrote an energy function I wrote last time as a magnetic moment new plans a magnetic field kind of thing which tells you whether the spinners up or down change notation this is too complicated New Times be they just come together and always come together so we might as just call the product 1 thing and will call the product J it it turns out that as a standard notation for what will be the ring later sold his college 80 with J. stands for a know I'm also will make it might it doesn't make a difference physics because there really is
just a redefinition of what you mean was open barrel here for the original energy favored down there changes signed it favors up but other than that the interchange of up and down there really is no different so that's the way I have now it will favor popular favor
signal positive because of signal positive energy is negative and of course lowering the energy is favorable lower energy is favorable especially at low low-temperature accidentally will we did is we studied a whole group of them we did some comminatory X the try calculate how many states Our with little in up little in inbound bowler could done something much I did that way just up to work to do it in some careful detail we do something else with already learned that you could always think of a subsystem as a system of being a small subsystem plus a heat bath and in that case the rest of the heat that is just what the theater's provides the brings 1 of these the equilibrium only could just focus on 1 of them we just focus on 1 spin the course of his long magnets and a that just assume that it is in the thermal equilibrium with its environment and environment being the rest of the in that case there would just focus on 1 is organized and we would write the partition function for 1 of them as being summation although the configurations I'm configurations are there don't seem easy tho configurations each other miners beta sirens J. them plus David J. Collins said white plaster figures as opposed to write economic is made at times the energy Barcelona calculate was sin the building near the is telling us whether it's up or down so that really just 2 terms when Turner soon up that gives us each of the beta J and turns donor fatigue reminders Baby Jane that's the partition function for just 1 the problem was my form of my this beta J. each the mind bigotry when we calculated it thinking of the whole system was 1 system memory and so we got we just got this can be and how however the Moors was we got factor like this each in Chile could really short circuit all of that stuff about come apart and just focus on 1 spend their time and their independent of each other whenever you have independent systems 5 Yucatan simply take a factor partition function factors into a factor for research I serve to each will spin River fact but that would concentrate on expanding benzene and happens to be twice the hyperbolic sign of beta J we did last time we got twice the hyperbolic costs the data Jr erased the number powers reached we took the logarithm of that which is interesting thing it's simply becomes a summer for example we energy which is just a derivative of the Library of Congress is nothing but some of the images of the individual healer concentrating on 1 the time and go warned that the time I will calculate exactly the same energy for it so let's calculate energy energy can be calculated as 1 always eat with a minus sign derivative of zany with respect day that this is the same as the derivative water the removal order of disease as well as and derivative observed OK so the Solheim a derivative of Z with respect the baby as a tool and they want to derivative of hyperbolic cosigned hyperbolic signed so we will get a hyperbolic signed a Bader times J. but they really have to differentiate argument of the call sign kosher the costs with respect the beta and that gives us some of the GATT no not yet we still have to abide by Ze until they get rid of this 2 0 here I look at it here Mr. Duchess numerical multiple Emilio and the partition function and on America multiples don't matter but now we see exactly why where defined by his with 2 Otis cancel an arrow get J. saying provided by car which is called J the hyperbolic end of this indeed an hour a day off there is a negative Jr you are we are yet it might be yes they are the dollar young so that the average energy are over spin and you can also ask what is the probability that the spinners out vs. down Wednesday average signals that thing which could take on the value plus 1 of my warm bore the average said Willie energy it minus J said this is the average energy that's pretty clear that the average stigma is just the same thing have the Jake so that's right that was 1st alright energy and this is the average energy and is now the average energy of 1 particle it's not the average energy of the whole system of a focus on and that's equal j hyperbolic tangent beta J. led the same thing and we could also talk about the average
Spain and the average spin or the average keep calling it should call the magnetization the average value of the Mr. Obama's of magnet that is just equal to Tianjin hyperbolic tangent of Jake OK this is where the leaders will it but there are only just drop picture of a hyperbolic tangent is a function which looks like now this is a hyperbolic agent X an X is rated JT so this is a hyperbolic tangent or beta J. a number we could have said they want to know why keeping J. could with it can't just something proportional to the inverse temperature these slope of the hyperbolic tangent over is 1 just has a unit slope over here and then asymptotically becomes flattened coastal constant namely 1 over here and minus 1 OK so think about what that if they beta is very very large that means low temperature and suppose Jay's is positive and beta Jerry His very large and positive way out he wants telling us is that the speedy will point up with probability very very close to 1 value 1 and the average of the Spain very close the 1 the only way to be close to 1 if only values of takeover reminds 1 1 knows where the average to be really close to 1 of 3rd the probability is almost exclusively for it to be up and that's what's going on out here I could also look at the negative axis beta we don't want to be negative use inverse temperature but taken be negative just corresponds putting others signed an order does is flip up and down so as you might expect what goes on her way out here when beta JT very negative Spain wants the point in the opposite direction Magnum wants a point in the opposite direction and that's all what's happening here otherwise the 2 sides are completely cement not as bad as the origin and nowhere is the origin here the Argentines beta J. equals 0 that means very high temperature their high-temperature baby is universe temperature high-temperature off the temperatures just constantly being kicked around and it overcomes the temperature just over condoms any bias for it to be up-or-down stormy averages just plain 0 OK so this was a very very simple system now will come to the next most complicated magnet it's the 1 that we talked about back to it the Ising model which and the one-dimensional Ising model which it is at at each point along a one-dimensional array is a stigma simmers Iraq that thing is that energy depends on the relationship between neighboring 1 energy is now summer now this is not the 1 particle energy This is the total energy enough recently we better keep in mind that this thing over here is the the energy of just 1 spared the whole thing OK now actually studying the whole shebang energy is the sum of all of them all the progress on neighbors Of course it on 8 times I plus 1 His jury each site and you multiply was spin by the spin the magnitude of the of the value of its neighbor will also put a J there Vijay is just a numerical constant which I'm so sorry introduced works with private and also that upon assigned their at the energy the minus signs means that if they lose them being Parra you want a lower the energy as much as possible the new law the energy by having them all parallel if all parallel the product of the spins is all equal plus 1 and with be minus sign that I included here with lower and used to make them more Harare also have an effect but plus J but that imagine them on the ground state the system or the grounds state of the system like the ground-state will have the lowest energy will have them all lined up but on which direction up or there is have similar saying they have the same energy or opts or back now will happen if we she signed on Jane the Rose may get plus chasing mice my plus 1 is really a different system now it's really the same system and all you have to do to see if the same system Mr. redefined every others by changing its signed on every other spins were due called up you know called them to redefine the variable and they would you find out is the ground-state once they have them or impact harrow up there Arab family but it's really mathematically an identical system and again in Europe to state this article could be up which cases wound down which cases will be up which faces will be down or the opposite so it's a it's mathematically identical a young guy dead volume Azinger bruises and plays Michael 3 of that it day but then online calculators the usual thing a calculator friend the partition function best friend the petition function and what is that that's e minus some early summer willing to be something all bull possible configuration a movies could be up there and that's a huge sums it up do it summer the minus J some nations in my eyes I 1 that looks terrible look very hard Our hope said Mr. must something here the temperature begin burst temperature is beta dash or did she is no external bowlers it is the latest in a long line of thought we could do it with an external magnet to its to discuss that at some point before the moment no external magnetic and this is a problem with 1 of them look at this 1st organizer question that we connect I would like to know the
into the following question supposing I know that they Magnolia here is not then I could not only that the overuse but what is the probability that in links in the chain it's also what that's call the correlation function it's a conditional probability that if we know that spin over here is up what's the probability of other place that also could ask a different way you could ask what we have sustained question really want is a B average of the product of Spain's at 2 different locations where the average now you might think in this case to be right you might think that will fond of them the chain that uses there spending up only he would have very little effect farther down the chain and you might expect and that the average of the product would be 0 because whatever this is here as well as equal likelihood to be up abandon they might gas but may be wrong them might be in effect on just having once been up Over here that would propagate all the way through the system and tell you that is no bias throughout the whole sample you could die announced that by looking at average property the average value of this here with a call at that point I tried in the spin on a plus it where it was In units down if it does exist memory this kind of memory where if this spinners adopted by the system always to infinity then this would not go to 0 0 with large distances if on the other hand that bias does go to 0 then this 1 being up would not biases formed an army average of product would be 0 so this is an interesting diagnostic tests of affect the lives of 1 Spain be on neighbors and how far that propagates down Cheney OK so let's let's Everett of guests what answers to this case think of this instead of spilled scenery on magnetic figure a game of telephone game of telephone where you was 2 neighbors along chain people In a message is the start of 118 some it could be the B & B over here and the messages can be a very simple message you 0 0 1 academic fancy messages like kids play this game had a heart attack in their former I'm dog food for the rest of my life I 5 acknowledges 0 1 0 1 people you pretty well and people talk pretty well in group of people but the site Delaney is not absolutely perfect as pretty good but it's not absolutely perfect and the question is How far down the chain as a signal propagate before it gets lost before and it just becomes equally a likely far down the chain that a person agrees with the starting point but disagrees these days no matter how good the fight ability as long as it's not I'm still perfect you'll lose memories sufficiently far if there's a probability In early of 1 per cent in each time the probability from nowhere is 99 per cent but then to go to India units change in its 99 per cent of 99 per cent of its . 0 0 9 1 behind their minds 1 squared widow 100 square twice on the force 1 . 91 Fish says but where the . 1 1 God but each time it goes down by a factor of . 9 9 and to go farther down the chain that product arbitrarily small and so the probability of a 100 units down the chain of the signal is remembered with good Fidelity's 0 about 0 100 units and changes about where it begins to fail seriously and housing units down chain no basically no memory the same is true it's the same thing here is then it is up and no signal but there is a bias bias is energy bias here the system prefer the lower energy and Zero temperature 0 temperature is like a perfectly accurate signal infinite Fidelity like a situation where there is no loss would ever and everybody just lines up this Jones up and was laid down here will set up Ms. Jones of battle this lady will say that was perfect correlation are but if there's any slight infidelity we're looking forward to this side of this House had this and you know what it will fade and that'll happen here that's all happen he and I icing got that right you OK so let's see if we we can calculate the votes How what the track here is a track the trip is a marvelous trek and I prevented was that the buyers and the trick was a focus instead on the spin on the links between so here is what I imagine rising said himself he said look there are 2 possible configured there are 2 possible values the 1st rural imagine for a moment that the chain is fine but cement imaging finite and then will let it get very big but let the Cheney finite and the 1st thing can either be or down let's start by assuming its and later on will put back the configurations words down so don't write the partition function as a sum of 2 terms in the 1st the first one was up in the 2nd term the first one was Mary let's concentrate on a 1 with this thing is up to begin with this 1 of the big the beginning of the changes are then I can I can either you lose as my information about the next thing he II can I that use the value of the spin or I can talk about a product of the spin on this 1 for example but stakes signaled wunderkind Sigma 2 0 say no 1 was not be up one-time signature would tell you everything you want to know about signature so let's call that a new 1 and what is it it's a variable has to do with the relationship between the 1st and the 2nd spit the variable which has to do with the relations but if you where they you know what 2nd what about the Mets warm what introduced cigna tool Sigma 380 and call lack that's how renewal fuel like this Of the link between Think of instead of thinking of the spins thinkable links the links have 2
possibilities and parallel array of that would tell you nothing about any individual spin unless you knew the first one was up you know the first one is up and you know new 1 billion no signature was suppose also known you want you till then you know what's in the and faxing the 3 will just be new 1 mutual why is that it knew what was up knew what kind of new to to square 1 signature was plus or minus 1 multiplying it by itself is you want and you find out of sync new 1 kinds mutual Soviet known you want and you know you to you know the 1st obviously if you not amused for all of the links between you know all the spin and has no redundancy there's no bubble camping as long as you know that at 1st and is up and his no double counting is equally good to know the newest which lives speak on the bonds between the as it is not themselves but that's very useful why because state energy is just made up out of these bonds variables on meaning beyond the relationship between neighbors so His all account like we can write that this energy is just as some although the bomb call them the bottom of the neighboring bonds some older bonds 1st bonds 2nd 1 notices 1 fueled by are particles but some of the bonds of new fight Nova words but also some alive but Cisco so I knew why the world's 1st Mon variables 2nd variable himself times by his jail we don't when that is multiplying anything anymore it's just the individual bonds and the individual bonds are all independent no relationships between them as long as you know that that 1st is up no relationship between and among them fans it is as good to know you as it is the the novel Ucon substitute you could write you could write you could substitute the song over Spain's substitute for the sum of the values of the barn where the possible values of the bonds plus 1 minus 1 now it's plus 1 if they're lying and minus 1 of their entire life so I can write this as that of the sum of spin some of the some of the bond variables the miners JT data Collins the now in fact tell you that prices why is it twice that and that get physical was found that her food yield partition function the other the other thing is to remember if the number of the magnets is very logical that much but we should remember that 1 fewer bonds and spit of those met this has now reduced to exactly this problem except that I have really accept new in this year His called have some over coupled individual Barton and has exactly the same form as the partition function for road would be simple a magnet remember now mule does not stand for the value of the magnet that stems from relationship between the 2 neighbors but still we know the partition function there it is saying they placed cost data a baseless revenue from its twice this however were extra factor to 1 outside of it quite twice cuspated Jake and now let me put in the thing which I didn't put here which was if I have a lot of spin this case raised me and Al so this is now the number of bonds which is the same as the number of magnets minus 1 minus 1 OK so are you with me or is 1st with hit by itself is not Ferreira at last 1 both are some you there is some clear this way but about yet if my some are back 5 the workers of course that the other large eaters push us there rare that some remember that some next fall just streets of Prague right so this is just a year some of each individual box that arises In a separate best some for each 1 by lawyers for each 1 them on of the editors of some of these walls that is the product of the nominal each in letter dated individually but the factor there fewer words if you have trouble seeing exactly why factor I think the door is still write it down restating To mark the right back
priests the barn right back for expression and you see very quickly wife that the yield the ended at European missile effort we actually did both ways to do last time for singles but for the problem on top we did it both ways we did it by thinking of the whole system and camping configurations and then later we just before 1 Spain and it took a 2 billion and that's it that's a legitimate are actually read still do exactly the same problems before the only difference is the physical meaning of video of the year degrees of freedom here is a little bit different Her pick Bell what is the average let's not ask about the average of the spin letters be average of you the average of a proper niggling spilled B average of product neighboring spins anywhere along the chain is going to be exactly the same calculations that let's rather than the average weekly derided as any given you a lot we can ride average of signal I my plus 1 that's going to be pinched Baby Jake West cut but you know I think with my current set up can I know exchange rate exchange visitor Finch patented that's correct salt it's not 0 jitters the product of them the non-zero and positive With picking JB positive factors non-zero and positive tells us that a net tendencies on lined up in the same direction we II plus 1st while at the same thing as saying bias from you would be positive a bias the the 1st thing about the next Warren has a better than even chance they found and much better than even chance basically can't be reject OK so this tells us correlation between neighbors and there is a correlation between neighbors but let's go far down the line but still exist exactly the questions which has started what the correlation between the 1st I've been over here In 1 in units that change how we get that only as a very simple you just right of tulips uses backward over here is Auroora calculate the average of signal party but times signal I plus leaving overhaul a gap in there because I write something firm could put it is I plus 1 my plus 1 what's in my post 1 infants in life plus 1 1 I didn't do anything that I put high-class to save my past and so forth until I get the last 1 here but now I can write this as the product new 1 new tool new 3 . about about Bantam you'll have know harmony is at 100 have been going down there Is it minus 1 season Midas 1 right Our where where Mueller warned means the sale the next new tool is a signal that he knew 3 next 1 he and tho it badly in the chain while call normal use the average product of these but all of the mules are independent of each other the energy is the sum of the energies of problem completely factorize the average of any 1 of them you can't David J. so what's the answer the answer is Beta JT corrupt how were in minus 1 another words the number of steps between the 1st and the last 1 this formula is expressing exactly the idea of a game of telephone bad if you lose kids the GAO's forcible always lesson 1 except when beta is infinite that's is absolute 0 temperature 0 temperature perfect fight Dougherty this is 1 of the how far down the line you got me and so will be 1 of which is a statement that 0 temperature if you know that this what lined up every 1 of them was lined up a widow possibilities everybody lined up this where everybody lined up that way case the product 1 but as any loss of fidelity all which means in this case the temperature is not absolutely 0 this page Baby Jake is less than 1 and each time you go another step you lose a factor of huge beta J. like losing his 99 99 % of 99 % solidly like not go sufficiently far down the chain this correlation function will become arbitrarily small it would be going all over negligible exponentially small memories is an exponential function in number less than 1 raised to be and power so the correlation for exponentially with a with young coach of signal what types signal 1st of all types of who the average but doesn't have always want you don't come up with a good pricked put was in here so I don't want is in their white because I want to really express it in terms of things which I know which of the new system I know tracked concentrate on the abundances instead of the year and the spin themselves at a marvel buyers off this she deserves some this is a pattern and this is
cold a duality in modern physics this could be closed the 1st 2 our tea equivalent of different systems we famine equivalent between a series of issues which are connected newest neighbors with another theory of that just beans which are on coupled to each other with a change of variables clever change of variables which basically took site combined which replaced which interchange sites and bomb site other points of the lattice pardons are sites and links of the things we would say that's the 1st example of a duality between different statistical mechanical systems yet it is murder if not always urges members but this cases formally said it was my the Florida might well broadened like that foreign it's got to go Ali interviews if I thought of my life said he is has not had a long way from a private lenders like it's a matter of very intricate detail where a little elementary magnets and serial preferred a winery also the times a preferred and violent but in higher fertile lying that as a set to matter of detail laughter and those Adidas remarked winter look at OK so we found that some things about one-dimensional Ising model end we found that it is that's never magnetized magnet except the 0 temperature magnifies the magnetized and translate into the statement that long-range memory of 1 spinners up all the others will be biased stop that's cool magnetization and its equivalent in fact statement that if you put a hanging magnetic field system the bias it will cause everything lineup will see that with that earned him a little calculations a little while OK it is warning systems have not faced transition it does exactly what you might be expected it to do correlations fade as you go down the line and the reason is simple it's the same reason as in game of telephone is a statistical probability that will make an error I want to make in their you start over again with new message and you wait until you make another error and then you start over again in the end and we don't know alas there will be some clumping tendency for clumps of in deciding the telephone will be longer stretches where a letter of agreement long stretches of agreement and then statistically on average every so often a switch but then long long-chain of agreement again and the switch apec so not right temperatures and that's like Fidelity being very bad winter temperature like the situation where you just can't hear neighbor you get no information for neighbor so you wind-up whatever neighbors says making a random random yet in that case what's a that the case is made equals 0 a 0 0 0 so that's a statement that does not even correlation between the nearest neighbors Blix to OK and Wang system and he made a more interesting whats Hall wrong is the dimensions pool the one-dimensional chain once you make a mistake the next fell doesn't have any support what the kind with the writing enters the sea just gets his message from the previous version of what about higher dimensions surmount that a playing telephone with different game 5 only start a message then he sends a message correctly the back both of us laughter and then they each send the message to all of their neighbors and particularly will wind-up sending the message to the person that can Missouri sent a message to the user judgment particular as a spreads out here people beginning messages from different size figure it will make a judgment that everybody will get relief messages from all the more people around if 1 old messages happens to be wrong but 3 of them are right who don't what computer scientists call correction error correction just means it will take a majority vote this works much much better and in fact the viability is good the bias None of road initial message will spread throughout and off to infinity say thing happens in two-dimensional Ising model that the bias of anyone been will buy the rest of the year fact who will be just say putting a little magnetic field 1 Spain in 2 dimensions will biased is not completely obvious why thought about a little bit I realized I can't prove that I actually can prove by proved easy way where prove it is going to be an analysis of the
two-dimensional Ising model which the but the fact realizing that wrong it was the worst things in the world we got that wrong aphakic goes wrong in every dimension with think about it from me and you realize it's very dimension the pen wires at them into the well in 1 one-dimensional each person has to neighbors as very many them to learn much of a support system gets the message really only from 1 direction on the neighbors two-dimensional have 4 4 2 square lattice so that the war people you have a pretty good chance of getting the right answer even if the fidelity is not so good if you use some some weighty procedures such as taking majority do pretty well you do much much better then fugitive wrested from 1 person about the 3 dimensions of 6 neighbors about 10 thousand images reduced 20 thousand neighbors re-establish friend Harris Harris was the 1st black Mr. America are and estimate income hundred men can't be wrong I don't know what he had in mind but In high dimensions it's true 100 neighbors won't be wrong very often fluctuations among large numbers of variables tend to be very small in comparison with the net don't have a hundred plus minus 1 and the average the average fluctuations will be much smaller the magnitude of the talks were kind of statistics soul you have a much better a shot at being able to propagate the information on 1 Spain's throughout the entire lattice the higher the dimensionality so but leads us I asked the question what would the Ising model be like in very high dimension some any number of dimensions 10 thousand whatever it is and what we say as with we say about how to say your study said cool but not 5 religion she ties it turns out to high enough it turns out that go that to behave like that fears close enough that they have quite often where does a number he become more of a trough Ward dimensions hard 3 is very large in this in this game but that's as a matter of a computation having solved that solved it he medically or whatever we note 3 years of very large chill the the answers which is not obvious is that only the one-dimensional Ising model was not have a transition was not have this a tendency but that's not something could predicted easily what could predicted easily Nuttall hard Lugo tonight is 2nd sufficiently high dimensions awfully good argument to say that there is a transition at high temperatures rings random lower temperatures everybody wants a lineup and that lineup propagate throughout the system to infinity OK on lets us lets the trick that I'm the matrix to do this with a you something called a mean-field approximation property were each side is surrounded by how many neighbors who lured to do would be they mention Peter mention these stems from dimension each side has yet surrounded bite To be neighbors imagine these very large as a Class B is very large you cannot imagine that although neighbors the field what's called a field whose fluctuation is much smaller than its average value of a large number of variables and they tend buyers the flocked to repay all it up to something of water the number of variables the fluctuations around that will typically be much smaller then Nevada OK so readily let's focus on Aspen and only others will make an approximation about but focused on this pain and write its energy 50 of this 1 spin the energy of this 1 spin is minus J said 9 times the sum of the neighboring spends all around it this is the sum of the neighboring springs some of the neighbors and AIG HBO now let's suppose that there is a bit of bias and that the average of the screening is not 0 you gotta check capital would try check but suppose the average a formula formula would allow the to be 0 2 but suppose throughout this lavish Spain has an average called stigma maybe it could make it simpler as costs in bar average average spin we'll know what is the might be 0 maybe plus be minus we don't know what it is but I'm going to replace the summer of neighbors here by simply twice beat times the average now that's a pretty good approximation of a number of of neighbors is large and the larger the number neighbors the better the approximation we have large numbers as of typical rules were allowed to average and the fluctuations a small so we can replace this by my guests J. Cody because the D neighbors like to be Jerry and
not and neighbors but the particular spent at this point the average spin at Massey Energy of 1 particular spin sitting in the back of the field of all the others is called mean-field approximation need not being in the sense Of the SPD but being In sense of average and field now being the field experience by 1 spin In the field of all the others so this is energy of this particular spin over here and now what we can do it it is due the partition function of that 1 state and that we don't even need to do the partition function we'll the partition function exactly the same calculation we build Blix said instead of writing JTU we will write to these times Jerry exactly the same calculation was here except every place we saw Jay and the upper calculations will have to be cleansed today so I can immediately write down what the average a committee that I can immediately right there and that the average of this this 1 over here but call Sigma bubble for a minute the double bar indicates that I'm talking about what its average band is just going to be but there are not going to be beta Jerry it's gonna be beta arms tool b Javed from sir mama with said I say every place that J. we should stick to DJ not every place were C. J we should stick this whole thing to be kind signal this over here is moving in the background always others and others all constitute a constant field that we just called we so this is going to be the average of despair as hyperbolic tangent of tool D. J Mama if I thank you off Tools beta B. J signal right over again but could news but here as clear they were noble body average of this particular span is going to be hyperbolic tangent of tool Bader D. J I'm simmer water now this mean-field approximation is also sometimes called self consistent field approximation oneself consist of self-consistent because hopefully self-consistent because this Spain over here is no different than all the others it's just 1 of the many spends in the system and of all of these have an average equal Mama the physical and torsion would require that you would say that the 1 here also has the same thing that's got to do with translation invariant do with that everyone was really the same as every other 1 for big sample and follow the self-consistent approximate self-consistent field theory is to say signal bar single double Mars if the when equation we equation its Irma Morris to equation a little their world could cultural mobile workers not moral over we've had since I've I put the bright Kanaranzi it to indicate that this was a constant OK it's a constant with proportional to be immersed temperature but yes purposes yes this is inside the Brackett said mama's inside the bracket so this is an implicit equation for similar Martin who are solid this this fall this equation would apply Richard but when there when variable almost the that all there not not finishing left 0 . 1 finnish after occasional anywhere from my 1 more prepared so the quiet have solved wishes to play got so 4 rare angle threat saying he had not bathing you're wearing a bright that access by either side of the border of patient back talk of war talk about that no you're there is always a dark years later the velocity dimensions and one-dimensional and two-dimensional for 3 three-dimensional sex race whatever it is still the EAS got a day hired man got a next nearest neighbors a work of saying directions gone they are here chart so we picked the particularly predicted particular situation and studied as a function of dimension we could study is a function of the nature of 2nd nearest neighbor couplings 3rd nearest neighbor couplings always things is just 1 way of making this spin how a lot of nearest neighbors the physics of high dimensions not something we realize in the laboratory we all we want to prove something about a system global hard we can get our hands on it but we can go to some limit where becomes much easier are we probe that in the limit we haven't proved it for the physical case of interest but at least we can prove that somewhere between in this case one-dimensional and a 100 thousand dimensions of certain new kind of behavior happens that was present in 1 dimension once we see how that works we can then ask Where does the change happen in this case it happens between 1 and 2 OK but this is a nice simple example where you could see what happens physically without having a solid told a medical problem with these very large OK so let's try the solve this equation the 1st thing to do
is to change variables I know you hate to change variables Her but there are those of the reasons can't show all complicated things I want complicated things here at that time so I'm not saying told beta DJ is a number 2 beta DJ their own numbers for our purposes right now is just the number I'm going to change variables so that 2 beta DJ Prime said Nevada is itself a variable McCalla White recalled and on the left side was not the right side the right side is just why divided by 2 data from D J. pick now the solve this equation and is going to graph both sides and see where they intersect crews way to solve the equations does draws a graph on both sides and see where they intersect Ms. is white this vertical axis it just parents' potential white and Is it can't function not the left-hand side is a linear function of everything else in here instead of calling his beta let's take 1 over it the temperature in the numerator its temperature OK so let's plot the right the left hand side of different values of the temperature to another color and plotted for different values of the temperature 1st of all let's go the very high temperature and very high temperature this is a very steep function Of course he is a large number so how big large temperatures may depend on B but eventually you get a large enough temperature that the slope of this side is large so we go to some very large temperature and a slope crying and there's not only section except the intersection of white equals 0 the the only possible answer to the self-consistent field theory is that white people say robber was why why is proportional to the average of so we learned 1st of all there's very high temperatures that the average of Sigma must be 0 as expected that I surprised when now following a temperature as you'll all the temperature me slope of this curve decreases there is a point where the slope is actually equaled the 1 and the red line here is clear engine with him ends and the Pinto a lovely idea tangent cheer with that happens when slope that happens when the temperature about when temperatures although twice d J. is equal the 1 that's when the slope or when the temperatures twice B. J something happened temperature is equal to twice B. J. when you go beyond that is now some new solutions so as you get past warned you've could be over here just as this line a rogue states still is a gift past Loeb 1 who is now solutions alone here there's also a solution over here and then he ever let's let's assume the GAO's positive just To be acrylic Jay is positive so we only interested in the right hand side here now that the solutions and the no solutions y it is not equal to 0 and therefore are the average signatures not equal to 0 another words the system has all move on global average of Spain's point they all the same direction and that's the phenomenon on magnetization what happens here is called a phase transition beyond that point net magnetization average magnetization average you that just permeates the whole system forever my wait a minute when you say hole but what about the solution how I know that the right solution was the 1 the origin for any value of the constants red curve does intersect the black fervidly origin so how I know which 1 is right at which although calculation comparable good tonight maybe next time we're going to end this problem a key little magnetic field the biases system discovered that has little magnetic field will tell us that we should be on this French on this branch of the solution and not a buoy conceive another way let's go 0 temperature 0 temperature is way out of here we know what to expect at 0 temperature 0 temperature we expect online we expect everybody toll line and so we expect these call locked into a parallel configuration was way out here everybody alive and has to lower the temperature is sliding along those lines until you get to the transition point here and beyond the transition lower temperature than the transition point the only solution was at the origin so that that's the nature of a phase transition which will talk about more but I think I won't try to do it tonight next time we will put it's pretty easy pretty easy from the point of magnetic field so they field have a whole lot of just with the only guaranteed is that sufficiently high dimension at work we know I see doesn't work for 1 an analyst said but it is mathematical beautiful sight cats want something most of it but both arms the court not to but took absolute 0 temperature 1st absolute 0 temperature everybody wants the line no it's around Madam writing the partition function as a sum of war all the configuration was how come we enumerate the configurations Starting with this configuration or normal by the number of spins of flipped relative to this configuration so we thought liberal in writing the partition function in writing partition function we would start this configuration Ito the miners Beta times of configuration at low-temperature starting at low temperatures this would be the dominant configuration and we raise the temperature will kind bid and some no configuration start to become poor could sample 1 Spain flipped let slip 1 spin In fact from withstood the same thing
over here on the same thing over here into a dimension here we have plus plus plus plus plus plus upsetter as our starting point everybody's pointing in the direction of not desk with the next state is next in energy is the lowest energy state the energy is stored in the market signal about signal which is the energy of a pair of neighbors what happens if you flip 1 spin instead of being classed let's make this 1 negative how much energy do we get timely bonds we broke a broken bond means that instead of having parallel we have and that profit right on right end How many places or there that you could do that basically on each side a 2nd number places you can do that is not total number of sites will be so that means that the number of configurations To and with 1 with 4 x 4 units of energy that's a funny want to break OR flip to spend the next configuration Watson next configuration we flip to spinners what's the energy of that configuration amid remedy broker moms Arbour Tokyo seaward mom's a broken is my mom was not broken have this was broken this was broken that was broken wrist was broken this was broken this 1 1 2 3 4 5 6 5 increase the energy by 6 units on each broken bond cost a certain amount of energy and now increase the energy by 6 broken bottles what would happen if I would have put the minus sign way out he of far away but I 8 so how each time I increase the number of spins it costly something in the multiple fact Ito the miners beta Pansy Energy Energy goes but each time I flip another spin the energy goes OK notes compare that with the one-dimensional case how much energy does it cost the flip 1 spend bonds a broken toe right about a foot to spend but thank you and the adjacent adjacent it still does too about 3 swimmers I knew look and any number of and still only call 2 units of energy that yes that victim adjacent that means is a lot of configurations or with the same energy much much more new here in this case the number of configurations with with 4 x units of energy would be just proportional to the number of sites about here the number of configurations with 2 extremes of energy proposal to the square of the number of site is proportional to the square of the number of sites because you compete and 2 bonds and flipped all the spins in between them the the foul line of configurations where a lot of that's why it's unstable with respect the flipping want tomatoes and the temperature turned because very little in a jeep a foothold big wolves of their wares here the flipped 1 cost is an energy OK that's with the foot to low-cost more June 1 that's the basic mathematical things going on in Barrow Jesus in his equation because he is black set the factor beer over the difference a coefficient of not but said that is no exact you don't see the difference between 1 and 2 in the fact that white computer this only makes sense he was born does not show us formula remakes and abuse for the whole physics the home the whole hundreds makes it was for a long time so now let's turn on a tiny magnetic field could do back where's our energy function here the RAND function over here that a I'm still concentrating on 1 spin by now in addition we imagine the whole system has an external magnetic field that means that each Spain has an extra energy not related to its neighbors but just whether it's up or down let's call the magnetic field beat time said Meyer would be kind Signora favoring down these positive is favoring Obama would command and eventually would be is very small is the extra energy not related to its neighbors but any each 1 of them each 1 of them each of the spins has sex return here and that's a whole different whole difference is in a instead of change be inch I 2 D J signal bar its could clinch 2 P. J similar Bartels Norris simmering if simmer factors out soon everything but brackets around here it's exactly the same problem except we replace tool DJ said about being a number Rick requested by giving the beat of what that does is it against the variable why he had a heads plots he plans beta I believe want eat kind um Urry which is a sign which signed or I think I wanna fail I guess I wanna a favor up spin looks promoting remind us that favors up spin How could yacht al-Qaeda says but I guess I guess it looks like it's now so this is a no equation no equation let's compared with the old 1 draw a picture plot again the right there is you again
said gap we constantly argument fear at 3 symbol OK was that it would cover can't function while a 8 dead they know what's left a suicide note left inside was the calculation of the average spin in the background field is a background field but we calculate the average In that back has report and they J. all policy the exact address what we did was we took this newly energy function calculated the average of city of stigma what do exactly the way we did before being inside this will bracket is just the number again this the number so we calculate the average spin unit in this expression for the energy and then we set that equal the average we we see we the calculational we get this number he can't wipers database that's got by saying do the spin over here feels neighbors you say you need to find out what it as equal I always prospered baby right there has why is still the same as it was before I might change the definition of white still to VGA services pinch of why plus beta but I don't know doesn't affect left inside inside the the to say yes why is exactly the same thing as it was before it is the average speed we calculated the average considered equal to the average right inside we did as statistical mechanics calculation of the average was storming the average a certain value on left them so I wasn't well OK but must be the average as was Kohlberg self-consistent approximation if he got Schiffer Caressa Westchester curve shift that that's exactly right where it should occur so let's say from Bader BE is positive that shifts curve backward Bucharest no curve and now withdraw the records the records original outside exactly the same as it was before the right side of the curve has just been shifted to left by how much I babies this gap year Our His beta be be here that we started the agenda saying curves but now there's no ambiguity about which solution we should take there is no solution at the origin or social origin if anymore alcohol only Rees points over here for positive jury proposal jury it Internet temperature OK so let's see what happens and temperature at infinite temperature this gets very close to vertical right no no no no 0 temperatures co-star example got into the temperature is his eye and are intersection is right over here and at that point wife is equal to 0 that data that Be that time the what that Eagles maybe those goes no no I would like to 0 temperature side with lots of adjusting yeah that yuck yuck yuck doesn't matter but not all brutal way we would do a lot temperature low temperature but a large temperature pockets beta goes 0 yes the curve of the curve shrinks back tops gap 1st out it was a L O free right but that are you sure she would have you write whatever happened is as point as shrink down here removed they here right so that she can not more the novel now we're of throwing out infinite temperature but and the temperature of random solve the solution the curve looks like the solution is found near here as you have made again smaller and smaller you find no magnetization as you should worry large temperature now as you move away from 0 can't write infinite temperature is seen only solutions lie on here no matter how small the boats got a small temperature small temperature you move his way back and research here and even as you go higher and higher temperatures you stay on the curve we stay on this point of the curve all the whole thing shrinks down here is no question of being stuck at this point all temperatures randomly temperature which is not infinite is only the solution of P matter how small you make temperatures Wong of the finite how large make the temperature of large or small the only solution was up here it is this year it now young but 1st calls by professional differed 4 of their or dueling capital of your it just where they they were hiding near while the 1st 3rd more than but had a bit of both however want you that worse still what is and of course but character of the heart of the heart disease that a it's yet not is thought to be an entity what do not what do just it is massive will white line now they were going intercept while so it there were also look at the place where across but it's getting more swarms of which no doubt through that it is not this was a that that worker by every year call and I say what is
below my car mechanic will come every single time but why because you know help cars were some of the how and where you can but what Exxon there were also son at From while and aunt so will be moving at best tho if you have a federal later it's moving this way all of it both things are going in the same direction both and that right now trip at architect but so we see just a the tiny Escobar the magnetic field well but we only solution up he magnetized solution but I which he yes but nobody is feel Comey made as small as you like we will you yes yes before arbitrarily look think about arbitrarily small magnetic field which means bringing this intercept here To be arbitrarily small kindly magnetic field much smaller than the magnetic field of the Earth you still have only the solution are for fixed temperature fixed temperature means for fixed Loepere you let the magnetic field 0 to beyond could the others right right this isn't in the village that kind of like kind of a how small will still why is it called but do villagers far more measure as a look at the top of which is the without not that center for the 0 your making it he said that some of the less you see that Her intersect tests will up on it while yet what never had that shows that there is a bias if it is is normal that even if does she but look at it but what is the solution but is it possible that possible return stable praised the but were just before while at dumps us all be but dinner on a set that White House life on so 1 sector right what they we got on here at the same place already done it yes but is the lady Italy and the White House you want to keep bad provocative right gay shit so which way to better although less right right right all a lot gossip that the right prevent about which but what is clear there there's no way way I that line of severe chest pain so that nothing they are the don't is answer solution where all the spaces are equally likely to be up and down is unstable and what that solution is is sort of 50 50 probability of everybody being knocked and everybody being down the whole world being filled with plastic bins slightly over plus the whole world being with my spin is that has an average of 0 but turn on the tiniest magnetic field the difference of energies will be enormous because you have a billion light years worth of spins or pointing up and you put in that trainee little stray magnetic field the energy of up will be much lower than that just because you have is great great big number of spins sold the solution over here is just exactly what said all basically all spins pointing upward about all of them but on average everything pointing up equal probability with ever average everything pointing out that unstable situation just a tiny perturbation the magnetic field will biased and of course it really happens with Amendment right and you could imagine that you have a state of referral Magna real magnet which as equal probability of . everywhere errors and then you bring it into the field of the Earth trainee magnets it's not at all but there you bring it into the field of the Earth and pretty quickly that magnet novel was a from our dearest magnetic field as there and oriented South America error-correction so with this this configuration equally likely to be in every direction that unstable small stray field will and the stronger the magnetic the more unstable of the bigger the magnitude away stronger or the bigger the magnet more stable show run so burst but I don't know how that even more pushed for 3rd place but some of the men and women for him or her the a half Corsica From while 1 5 4 1 while some instability but want don't know why other than the value of symmetry could be going away but why why would say Europe the are that the billion from Lon Iowa sober up a about 11 but by that's the idea dish start that's for sure but is rather than the flip side is below it now some antebellum clearly not some instability 1 of the greatest look about world Garrett careful about what I know put sort OK the Vantage young follow among about prayers and then maybe being met the levels of know they might drive for more please visit us
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Metadaten

Formale Metadaten

Titel Statistical Mechanics Lecture 9
Serientitel Statistical Mechanics
Teil 9
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 10.5446/14942
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract Leonard Susskind develops the Ising model of ferromagnetism to explain the mathematics of phase transitions. The one-dimensional Ising model does not exhibit phase transitions, but higher dimension models do.

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