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Statistical Mechanics Lecture 1

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there are no more questions tonight we begin the study of statistical mechanics narrow statistical mechanics it is not really modern physics free modern physics and modern physics and I assure you will be closed modern physics it's probably the 2nd law of of thermodynamics will probably last anything that that comes up time and time again the 2nd law of thermodynamics has been the sort of around a guidepost guiding light if you like uh through warm what we're talking about macula making sense statistical mechanics thermodynamics may not be as sexy as the Higgs boson but I assure you it is at least as deep Bond it's a lot deeper particle physics from distributor disowned it's a lot deeper it's a lot more general and it covers a lot more ground than explaining the the world as we know it and in fact the Web statistical mechanics we probably would not normal that he's blows on all right a solo without little starting point where statistical mechanics about well let go back a step the laws of physics the basic laws of physics Newton's laws principles of classical physics classical mechanics the things that were in classical mechanics Corish quantum mechanics of 1 of those things are all about predictability perfect predicted during point on a mechanics you can't predict perfect worry and that's true but there are some things you can predict perfectly and some of those these are the predictable 's of quantum mechanics again as in classical mechanics you can make a predictions with maximal with called maximal on precision or maximal whatever is predictability if you know 2 things if you know the starting point which is what we call initial conditions and if you know the laws on evolution of a system if you can measure be alert you know for for whatever reason to be initial starting point of the a closed system a closed system means 1 which is I everything all it is sufficiently isolated from everything else that the other things in the system bone in Florida if you have a closed system if you know the initial conditions where exactly or released with precision is necessary and you know the laws of evolution of the system you have complete predictability and that's all there is to say that because in many cases that complete predictability would be totally useless having a list of the position and velocities of every particle in this room would not be very useful the list would be too long and there are subject to rather quick change as a matter of fact solve till you could see while 1 of the basic laws of physics are very very powerful in their predictability they also in many cases can be totally useless actually analyzing what really going on statistical mechanics is what you use basically probability theory statistical mechanics Amistad 1st of all it is just a basic probability theory sophistical Mick as applied the physical system for what is inapplicable that's applicable you don't know the initial conditions with complete perfection that applicable law when you may be applicable if you applicable if you don't know the laws motion with infinite precision and its applicable when the system you're investigating is not a closed system with its interacting with other things on the outside another would just those situations where ideal predictability possible and money resort to you resort the probabilities but because the number of molecules in this room was so large and probabilities 10 the become very very precise a quick car predictors numbers when the loss of about when the loss of large numbers are applicable statistical mechanics itself can be highly predictable but not for everything on it as an illustration your Home box of gas the box of gas might even be an isolated the close box of guess it has some energy in particles rattle around if you know some things about that but I guess you could predict other things with great precision temperature you can't predict that year energy in the box of gastric a predictor pressure these things are highly predictable but there are some things you can't predict you can't predict the position of every molecule you can't predict when they might be off fluctuations of fluctuation which share your fluctuations are things which happened which don't really violate probability theory that the solar tales of the probability distribution things which are unlikely but not impossible fluctuations happen from time to to time in a sealed room every so often an extra large group of extra large density of molecules or appearance of small region bigger than the average someplace else molecules will be less stands and fluctuations like that are hard to predict you can't predict the probability for a fluctuation but you can't predict when fluctuations Gorica happened our it's exactly the same sort of thing flipping coins flipping coins is a good example but we are a favorite example for thinking about probabilities after flip a coin a billion times you can bet that that approximately half of them will come up a therefore comic tales within some margin of error but it will also be fluctuations every now and then if you do enough times a thousand heads in a row will come up can you predict the winner thousand heads will come up not but could you predict how often enough thousand hits will command yes now very often power so that's what statistical mechanics is for it from making statistical probabilistic predictions about systems which are either too small contain elements which are too small to see too numerous to keep track of usually both too small city but see it's trouble you could see some pretty small things molecules the survey remain upbeat to small to see but they want to any of them that too many of them to keep track of and that's when we used probability theory plus statistical
mechanics we're going to go through some of the basic statistical mechanics Our applications not just applications the theory laws of thermodynamics the laws of statistical mechanics and then how they apply to gasses liquids solids car whether we will get the quantum mechanical systems are not I don't know but term but just the basics of the basic idea OK and incidentally another thing which is very striking is but generally speaking over the history of of the steadily over my history and physics armed and I'm sure this goes back through work of the middle of the 19th century some all great physicists all of them were masters of statistical mechanics it may have been the sexiest thing in the world but they were all masters of white 1st of all because of useful but 2nd of all because it's truly beautiful the truly beautiful subject of physics and mathematics and it's hard not to get caught up in it hard enough to get the not the fall in love with the reason I teach it is not for you it's me I love teaching our love teaching it I've teacher over and over and over again analysts and my wife is consisted of learning and forgetting learning in forgetting in learning forgetting statistical mechanics my opportunity learn again OK let's begin with laser called mathematical interlude in this case it's not interlude that's a starting point and I'm just going to make some extremely brief remarks which you all know we set think your modem about probability just that have very just a level ground where are we talking about and arms or what I will I am not going to explain his don't think anybody could explain it is white probability works why does it work if you ask why it works the 1st Inter will be doesn't always work you may have a probability for something and you cast out and sometimes it doesn't work because of all the exception was so they had the question is why work what has always working mostly works to put us when doesn't it rarely how rarely every so often but there is a calculus of probability of mathematical theory of probabilities and talk about little better OK so uptight probability to be a primitive concepts basically primitive concept was a positive is a space of some sort of space possibilities the spacer possibilities could be the basis of outcomes of experiments or it can actually be the space of state of system the same system could be the outcome of an experiment the experiment consists of determining the state of the system that afraid the system the outcome so we have a space and let's call at spacings label the elements of that space for the little I for example of Luo flipping coins I would be the heads of tales everywhere flipping di da diced dies of died based on the RIA would run from 1 to 6 if they were to die than than we would have road north of indices to keep track of 2 Dyson forth so I is based of possibilities of outcomes or the space of possible states of the system and if we are are ignorant statistics always has to do with ignorance you don't know everything and so you assign probabilities if our problems and so we assigned a probability P olive the outcome to and to draw questioned whether the rules should be aligned with the fee of I have to satisfy and lets out after the beginning at least in the beginning let's and imagine that I innumerate some discrete of finite collection of possibilities later on we can have an infinite number possibilities or even of continuously infinite number of possibilities but for the time being laughed I might run from 1 end In possibilities for rules our 1st of all piece of has to be greater than or equal to 0 Negative probabilities we don't like them from what they mean trip next the summation of lot 80 of peace of Iife should be wondering if that means the total probability when you Ed everything up all possibilities should be 1 you certainly should get some results next this is a kind of composers is the law of large numbers that if you even make many replicas of the same system would do the same experiment over and over on very very large number of kind which end take all of the outcomes which gave you all of the experiments which gave you I I would calm that's some number of call game of end of our i.e. that's the number of times that the experiment turned up the Iife possibility EU divided by the total number of trials phone number trials is the sum of all I will adjust the total number trials that the limit of their this is a physical hypothesis physical hypothesis that can go wrong think if it is not large enough but in the limit of large and goes to infinity and no limit on very very of course and never Goss would you never get to do an infinite number of experiments but nevertheless idealized were assuming we can do experiments that the limit then goes to infinity effectively urban reached then that is P Vice appeal controls by assumption the race of kilobyte take everybody happy with that the user saw a time frame of some countries now let's suppose that there is 8 quantity Webb scored FORT it's some quantity that's associated with state we can assign it we can make it up for example FOR system is heads and tails in nothing but heads and tails we could assign S of heads and call it plus 1 an effort tales and call a minus 1 if our system has many many more states we may want to assign a much larger number of possible and but some function of the state it's also a
thing that we imagined measuring it could be energy of a state or it could be the momentum of a state given the state of some system as an energy would be call them that case perhaps iii of IT or it could be the momentum or only be something else it could be whether or whatever you have right to work to think about that important quantity is the the average of providing and the average of vise I will it the quantum mechanical notation for it even lower not doing quantum mechanics it's a nice notation physicist tend to use all workplace mathematicians hated on just repair brackets around it means the average the average value of the quantity averaged Over the probability distribution of Hazzard definition of its definition is that it's a summer heII FOR the party waited With the probability for example if and incidentally the average of IT does not have to be any of the possible values that African take on for example in this case where of heads plus 1 in every effort tales is minus 1 and you flip a million times and the probability is a half of heads of a half tales average of F will be 0 0 was not possible outcome the experiment was no rule lightly armed for the average should be 1 of the possible experimental out but but is the average and this is its definition its each value of F is weighted with the probability for that value of F you collided another Lady Lucan right it as a summer of FOIA the number of times that you measure it on divided by the total number of measurements of sort of ideas in the limit that there are a large number of measurements the fight to be the average that's on mathematical preliminary for today or were you or I wanted to or to a level playing field by making sure everybody knows the probability is I want averages will use it forward all right OK let's go let's start with coin flips I wish that request but every single class of course slips even when I'm teaching about the Higgs boson her OK the butler but Klein a lot and the Brooklyn a lot arms and not the probability that heads is usually deemed to be one-half and the probability for tales of usually also To be one-half under that wine is still lies and a half and I have what's the what's the logic what Philip logic tells us that in this case it's symmetry it's the symmetry of the point of course not calling his perfectly symmetric even making a little mark on to distinguish heads and tails are bias is a little bit but apart from that kind kind he bias of marking the Corning maybe just hang you scratch the calling this symmetric heads and tails are symmetric with respect to each other and therefore there no no reason no rationale for you flip a coin for the year of Florida to turn up aides more often entails as Oates symmetry quite often I might even say always some deeper sense but at least on at least in many cases symmetry is a thing which dictates probabilities probabilities are usually taken to be equal for configurations which are related to each other by some symmetry symmetry means if you act of the symmetry you reflect everything it turned everything over that there were at least system behave the same way OK another example besides Corning flipping would be by slipping by slipping instead of having to states has 6 states won by and we can imagine coloring but we Coerver faces red yellow blue and then on the back green proposed orange take that's are that's our body the color and we have to keep track of numbers we keep track of colors and what is the probability that the a war we flipped the forbidden to hear hits the ground what's a probability that absorb red but 1 sex 6 possibilities also symmetric with respect to which other we use the principle of symmetry the tower key each might be all equal will and all equal to 1 6 Boer there is dosimetry what is really the die is not selected for example what efforts waited in some on the fairway all 1 been Clark with faces that I'm not that nice and paralleled cubes then what's the answer the answer is symmetry want tell your you may be able to use some deeper underlying theory and use some concept of symmetry from the deeper underlying theory but absence of some some something else B and there is no answer enters experiment do this experiment a billion times keep track of the numbers assume that things of converged and that way you visited the probabilities you measure the probabilities and thereafter you could use them you could use them if you keep a table of them and then you could use In the next round of experiments all are you may have some theories and deep underlying theory which tells you a quantum mechanics or statistical mechanics statistical mechanics tends to rely mostly on symmetry so there's no symmetry to guide you the guide Europe are my implementation of probabilities then it's experiment now there's another rent is another possible answer this answer is often frequently invoked and it's a correct answer under other circumstances it can have to do with the evolution of the system the way system changes with time so let me give you some examples on what might have to do let's take Our sticks cited Q and assumed that are 6 sided cube is not symmetric it's not symmetric love but we know our rules we know that if we put that Cuba down on the table suck you would but that by Downer table we stand back this thing has this habit of jumping to another state jumping to another state and jumping to another state called the law a motion of the system the law of motion of the system that whatever it is it 1 in stand at some next and then it will be something else according to a definite role the incidents could be seconds a be
microseconds or whatever but imagine the discrete sequence and let's opposes all law a genuine along tells us how this Cuomo's moves around for example and red now we've done this over and over times in different contexts but it is so important that I feel the need to emphasize again this is were a lot of emotion it is it's a role telling you what the next configuration will be given a cell rule of updating of updating configuration red goes to blow Lugosi yellow yellow grossed green greengrocer orange on 1st purple and purple broke back to rank giving the configuration of any kind you know what it will be next thing you know it will continue to do of course you may not know the law may be all you know is that there is a lot of this type so you normally do next only to the next I'm draw this law as a diagram forcing do this in other contexts duet we have red a subject To right to draw squares red blue old green orange or yellow yellow orange purple 5 so a low like this can be just represented by a set of Hawaii's connecting arrows doors to blow the 1st green renounced yellow Eurostar and honors purple purple goes back direct Mickey assumption now that there's a discreet time interval between such events I'm not assuming that thick you would has is a symmetry to it anymore we kill may not be symmetric at all it may have . 1 1 edge 1 face may be kind of a faith but this is the rules to go from 1 configuration to another and each step take such a microsecond I might know and I have no idea where I begin but I can still tell you if I let let's it's a microsecond from microsecond and my job is to catch it at a particularly instead an escort the corridors and amyloid study trip but I can still tell you the probability fridge 1 abuses 1 sex doesn't dosimetry well maybe it does have to do with some symmetry but in this case it wouldn't be the submitted the symmetry of the structure of the bank it would just be the fact that as it passes through these sequences states it spends 1 one-sixth of its time red one-sixth of its time blue one-sixth of the time Green if I don't work starts I just got flesh each of your flesh out of my probably will be 160 now that 1 6 did not really depend on knowing be detailed loss for example different but made up a new law red goes Green Green goes I arranged marriage goes to yellow yellow goes to prep purple goes to bloom and back array this shares of the previous law that a closed cycle of events in which you pass through each color 1 before you psycho around you may not know which will also allow nature it is for this system but you could tell me Indian the of epic probability will be 1 6 for each 1 of them solve this prediction of 1 6 doesn't depend on knowing the starting point and doesn't depend on knowing the law offers it to just important to know that there is a particular kind of law on a possible laws for the system which will not give you 1 6 yes that's right another loss red blue green yellow orange purple visceral says that is thought to read start with red gotta blow the result will you agree with you or of theater green you go back to rank or artist out purple you go yellow back purple notice in this case if you on 1 of these 2 cycles you stay here forever if you knew only upper cycle cycle if you know you started does not await you start but if you know that you started on the upper cycles then you would know that there was a one-third probability to be red one-third probability to be blowing one-third probability to be green and Zero probability the be red yellow which I Yukos of the 2nd cycle because sigh with purple might not known way started but you knew that started in the law triangle in cycle here than you would nor the probabilities of one-third for each of these 0 for each of these now what about a more general case the more general case might be that you know what with some probability that you start on the upper triangle here and with some other probability the lower triangle effect let's give these triangles name so let's call this triangle plus 1
triangle and this 1 minus 1 triangle it is giving the name catching them a number of numerical value if you hear something or others called plus 1 year something arose from minus 1 right now I have to appeal if you have to start with something you've got to get from someplace else doesn't follow from symmetry and doesn't follow from cycling through this system some probability that theory cycle plus 1 or cycle minus 1 where might that come from will flipping somebody else's Corning only here flipping a coin or he might decide which of these 2 might be a biased coin so you'll have a probability to be plus 1 and a probability to be minus 1 base to probabilities are not probabilities individual colors that probabilities individual cycles OK now what's the probability blew the probability for blue begins with the probability that you're on the 1st cycle times the probability that I the 1st cycle you'd look at from 3rd so the probability for Lula red or green is one-third the probability that you're on the 1st cycle and likewise be probability that your at yellow will be this is the probability a a red blue or green in this case ends this times one-third it will be the probability for purple yellow orange OK so in this case you need to supply from other probability that you've got to get from some ways out of this case here is what we call having a conservation law In this case the conservation law would be just the conservation of this number Ray blue and green with assigned the value plus 1 at plus what could be the energy or it could be something else 2nd call it the zilch for some reason I call everything is 2 there's no name for it young so anyway but stressed think of it as the energy firm to keep things are as familiar the energy of these 3 configurations might all be plus 1 and the energy of these 3 configurations might albeit minus 1 and the point is that because the rule keeps you always on the same cycle that quantity energy zilch whatever we call it it is considered it doesn't change that conservation law conservation Is that beat configuration space space of possibilities divides up interest cycles like cycles don't have to have equal size is another Case 1 2 3 for luck this way and then the 2 guys over here going to each other I saw oak Our red goes to blue goes to green goes purple that's the upper cycle here and the cycle is a yellow goes on and those he'll go still we have a all conservation law here it's just the number of states with won the value of the conserved quantities not the same as a Member States about you but to conservation law begins somebody would have to supply for you some idea Of the relative probabilities of these where that comes from is part of the study of statistical mechanics and the other part of the study it has to do is say if I know 1 1 1 of these tracks how much time the waste With each particular configuration that's what determines probabilities statistical mechanics from a priority probability from somewhere that tells you that tells you the probabilities for different conserved quantities and cycling through the system now question Norfolk as a course did not all the I all set the time spent in each state right so it as it's completely determine a state laws a completely determine status would be classical physics laws completely determine a stick no real ambiguity of what state except you kind of lazy you didn't determine the initial conditioned your your kind wasn't very good each stayed only last from microsecond a euro lazy guy in you you only have a resolution of Billy 2nd but nevertheless you're able to take a very quick flash picture and and pick out 1 of the state's that's that's a circumstance that we're talking about if a day too picture is feasible that if the they got what the let she sees a site yes that's a good assumption bomb yes right so we want to determine the value of conserved quantities when you know it and then you could reset the probabilities for it column on latch so lot so let's talk about let's talk about energy for a minute yet we have a closed system Our represent a closed system drop box close now closed means that it's not in the interaction with anything else and therefore thought
a whole universe onto itself it has an energy energy is some functions of the state of the system whatever determines the state of the system now suppose we have another close system which is built to move identical or not the identical versions of the same thing now if they both closed systems will be too conserved quantities the energy of the system and the energy of the system and they'll both be separately conserve why because they don't talk to each other they don't interact with each other the 2 images are conserved and there you could have probabilities for a for each of those individuals but now supposing that connected it connected by a little tiny tool which allows energy to flow back and forth is only 1 conserved quantity the total energy source but between the 2 of them new command what's so condemned as was the probability give a total amount of energy you could ask What's the probability that the energy of 1 subsystem is 1 thing and the energy of the of the subsystem the other if the top boxes are people you would expect there you would expect on average they have equal energy but you could still has what what's the probability for a given energy in this box given a strong overall piece of information so that's a circumstance where may be give giving the probability for which cycle you run which cycle Iran and talking about the cycle of 1 of these systems may be determined by thinking about the system as part of a bigger system would do that that's important but In general you need some other ingredient besides a cycling around the system here To tell you the relative probabilities of conserved quantities OK so where are from flying with statistical mechanics bad laws by bad laws I don't those but said the bomber or new those kind of laws the but but sense that the rules of physics on your there are no laws violate the conservation of information most primitive and based their rule of physics the conservation of information consolation information is not a standard conservation law it's the rule that you keep that you can keep track you could keep track of going forward and backward so much just mention that again that's all work until described in the classical mechanics book is reviewing it now but let's take a bad law loss it's a possible loss by bad 1 2 3 4 1 2 3 4 5 6 so these of the faces of die again but the role wherever you are red wherever you are you go is you at Red you go away now will discuss in a moment what's wrong with this long but this loss house 1 of the features a has is not reversible it's not reversible and the stand that you can go from blow to red but you cannot go from Red back not reversible Yukon predict the future wherever you want the features very simple for this particular where the ordeal next red it going a mile complicated you could make a few would be to make it more complicated but there is always warns upward red it's is it's a bad law because it loses track where you start whereas these laws don't lose tractor few knowledge about 56 56 and a half cycles then another to start red you come back and tell exactly where you'll be an you could also tell where you came from you can tell not only will you be but exactly where you came from this law you can say where you came from the law the losers information that's exactly the kind of thing classical physics does not allow quantum physics also doesn't allow acquire canticle version of it most so the role that this type of ruled that a stack of law is on 0 our would I give the least of which is so many times there is no name for it because it's just so basically primitive everybody would forget about it so basic I caught the my 1st law of physics and I wish you would catch on people start using it that mean it is really be most basic law of physics that information is never lost that distinctions over differences between states propagate with time and you never lose track in principle if you have the capacity to part the system our goes you may be too lazy to system that's your problem but nature doesn't have that problem nature allows a principle that you could reconstruct where you came from fight there about that bad what Hattie tell Lawson better laws just by diagrammatic Sears very simple good laws every state has won incoming arrow in 1 hour going at an arrow to tell you where you came from a marrow to tell you where you got a case about about those aboard laws in classical mechanics continuum classical mechanics there is a version of this same law today another name that version of that of the that goes with that of the conservation of of information called Lee of bills 0 we studied it in classical mechanics overdue account their example of those theorem on friction as an apparent contradiction where you starter Comparex sort of like saying whatever you start to come to raid where do you start to come arrest well you may not know exactly where you are but you always come arrests that seems like a violation of the laws that carry you that are distinctions have to be preserved but was not really draw what's really going on is that when you run race through here it's heating up both these surface here and if you could keep track of every molecule you would find out that the distinctions between starting point is recorded but a manager now that it was a fundamental law of physics by fundamental laws I need of a rock bottom fundamental law our 1st
series of particles for a collection of particles and the equations of motion particles were there these 2nd acts that the position of the particles From IDT squared at score acceleration we can put a soon but the masses of doing anything as a lot of particle so label and ii reviews I the label state I should not do that but College in little in the end particle and was at equal to its equal to remind us some number gamma receive number before another context can be X end by GTE and better remember what can what this the formula represents friction of the viscous drag again it has a property that if you start To give birth to start with a moving particle it will very quickly come almost fixed no exponentially come to rest pretty quickly and so if all particles in a gas for example satisfied the law of of physics it perfectly determined stick it tells you what happens next but it has the unfortunate consequence that every particle just comes to rest on at sound on sounds like a matter what temperature you start the room it will quickly come to 0 temperature doesn't happen this a perfectly good differential equation for the something wrong with it From the point of view of conservation of energy is something wrong with it from the point of view of thermodynamics you start a closed system and you start a running start with a lot of Connecticut Energy temperature we usually call but it doesn't run 0 temperature that's not what happens in fact is not only a violation of energy conservation and looks like a violation of the 2nd law of thermodynamics Mrs. things get simpler you start with a raid bunch of particles moving in random directions and you had around and they all come arrest were Duke end up with is simpler and requires less information to describe what you started with very very much like um everything going to Ray among other things it violates the 2nd law firm of dynamics which generally says things get worse things get more complicated not less complicated OK so that surrounded by another way to say another important way to say this rule that every state has that 1 arrow in in 1 arrow out finger called you the might 1st law or conservation of information supply only have a collection states only assigned to them probabilities probabilities P of state 1 P. of state to appear state 3 and so forth for some subset of the states not all of em some subset of all the others we say probability 0 so for example Armed with detail of dying and assigned raided yellow and blue probability of 3rd and green on a pink or whatever it was probability 0 Willie got that from matter we got from somewhere somebody secretly called us in our era to the regular blue and market weight and now you follow the system you follow it as it evolves whatever kind of law of physics as long as an allowable law physics after a while and you're folly in detail you are not constrained by your laziness in this case you capable of following in detail than what is the probability while probabilities at a later time well if you don't know which the laws of physics are you can't set of course but you could say 1 thing you can say there are 3 states with probability 1 3rd and 3 states with probability 0 they immediate reshuffled which 1 is uh were probable and which ones were improbable but after a certain time there will be those same 3 of about the same restates but they will continue to be 3 states which have probability and the rest are in general you could characterize these information conserving theories by saying supposing you assign some subset of the state's but say in out of interstate there are states altogether the total number of states and now we look at some em way and western end and we say for those it and states the probability for bows and states is 1 over over any of these states and 0 for all the others you understand why I say
1 over an affair and states equally probable anti-choice has probability 1 of any and all the remaining half probability 0 did the number of states which have nonzero probability will remain constant and the probabilities will remain equal to 1 over and a clear obvious that should be obvious states may reshuffle but that that there a number With nonzero probability will remain fixed that characterization a different characterization of some of these information conserving laws for the information not conserving laws everybody goes to rent a red you may start with a probability distributions that 1 over 5 red green and purple orange and yellow and then I will bid later there's only 1 state that has a probability and acts that's a bit but OK so that's that's love is another way to describe probably information conservation quantify that we can quantify that by saying Let it Be the number of states which all have equal under the assumption that they all have equal probability led and be the number of dividend occupied states states which have nonzero probability with equal probability and then it wasn't characterizing and this characterizing you ignore the bigger images end is equal to that means equal probability for everything maximal ignorant of them is equal to 1 half and that means you know that the system is in warned however the state is still pretty ignorant but not battered US what's the maximal what's the minimum America ignorance she could have but you know precisely what stated soon in which case and was wife was 1 you know better to 1 particular state but still end is a measure of your ignorance really and in relation to and is a measure of your ignorance and associated with it is a concept is the concept of entropy would come to the concept of entropy noticed entropy is coming before anything else Anteby is coming before temperature exceeding coming before energy entropy is more fundamental certain sense than any of them will a certain sense a little who will discuss a trip minute but yes it is a logarithm of logarithm of the number of states that have an appreciable probability more or less also equal for a specific terrorist circumstance talked about that entropy is conserved all that happens as states which are occupied reshuffled but there will always be end of them with probability 1 0 record OK so that that's where we are and that's the consolation of entropy if we can follow the system in detail now of course In reality we may be again lazy loose track of the system and we might have a after a point lost track of the equations end the lost track of our timing devices and so forth and so on knowing me wind-up we may of started with a lot of knowledge and warmed up with very little knowledge that's because again not because we equations From course information to be lost but because we just weren't careful perhaps we can't be careful perhaps or two-minute degrees of freedom to keep track of sold when that happens we entropy increases but simply increases because our ignorant has gone up not because anything has really happened in the system which has come if we closed it we would find a B into HP's cursor I'll take that the concept of entropy in a nutshell going would expand on it to expand our what would redefine it with a more careful definition father but what it measure it measures approximately the number of states however have a mom 0 probability the big it is the less you know what's the maximum value of S offers largesse lagend now of course In could begin sinecure might have an infinite number of states and if you do is Knorp abound the amount the are very ignorant you could have a on a world Orleans states your ignorance abounded sorrow the
notion of maximum entropy is a measure of how many states there altogether nonsense that entropy is deepened fundamental and so it is but there's also an aspect to it which makes the certain since last fundamental it's not just a property of system it's a property of the system and viewers state of knowledge of the systems it depends on 2 things it depends on characteristics of the system and it also depends on your state of knowledge of the system or a state of knowledge of the system our soul keep that in mind OK now which talk about continuous mechanics mechanics of particles moving around with continuous positions continuous velocities early described them only described the space of states of mechanical systems are real mechanical system particles report we describe it as a points In phase space we learned about phase space space consists of positions and moment that momenta and simple context momentum was mass times velocity so roughly speaking it's the space of positions of velocity what key his momentum Gravatt weight and this axis is a stand-in for all of the momentum degrees of freedom of Eric Cantor the 23rd particles there at 10 to the 23rd peas but I can't draw more than 1 of them by the drug 2 of them but I would never be room for these accused the exit and horizontally the position of particles which we can call X X peace I'm a point here is a possible state of the system if you're not pointy in opposition and velocity and you could predict from that to if if you know the forces OK let's start with analog of a probability distribution which is 0 0 0 for some set of states and constant or the same 4 some other set for some smaller set wealth from fracture states all have the same probability and the other states have 0 probability weakened represent that by drawing a patch and he a sub region in the phase space and say if not sub region there's equal probability that the system is at any point in here and 0 probability outside Aceh so the situation where you may know something will you may not hold something about the particles that they're in some some region here for example are you may you know that all the particles in this room are in the room right so that but some boundaries on what Exxon you may know little of the particles have momentum which are within some range spectrum finds service for self a typical bit of knowledge about the Rome might be represented at least approximately by saying that 0 probability be outside this region and the a probability equal homes they wanted but equal probability to be in there OK now what happens is a system evolved as a system involves X and Pete change the equations of motion say that X change if you start over here you my daughter he Estonia by voters some nearby point himself and the motion of the system with time is almost like a fluid flowing in the phase space if you think of the points of the phase space as fluid points and like time go the phase space was like a fluid and in particular this Hatchell review what's called the occupied patch the Arctic badge becomes a mother Pat that other patch after a certain amount of time the system now is known to be in here after a certain amount of time we now know that the system is in here here anymore and that it has equal in some sense equal probability could be anywhere in the furor maker's Columbiaville sphere and know what it says is that the volume of phase the volume in phase space the amount the volume of this region in the X P keep in mind BXP space may be high dimensional not just to what two-dimensional think the area it'll Fuller humble face bases never three-dimensional even dimensional it has a P for every acts so the next more complicated system would be four-dimensional when I speak of the volume in phase space I mean the volume in whatever dimensionality phase spaces if you follow the phase space in this manner he and leader will steer you go back Toledo spirits in the year of the classical mechanics of lecture notes that occupies a thing called love lectures you follow tells you whether this evolves into evolved into something of the singing of volume another Woods roughly speaking the same number of states immediate analog of discreet situation where if you start within states and you follow the system according to the equations of motion you will occupy the same number of state after Woods as a starter with be different states but it will preserve the number of them and the probabilities will remain equal some of of the wrong the rover fewer shares the volume of this occupied region will stay the same and a little bit better it says that if you start with a uniform probability distribution and here it will be uniform in here so this is a very very close analog between the discrete case and the continuous case In this issue would prevents this kind of fundamental equations from be through this kind of equation equation where everything comes arrest that can happen OK why not let's see why can't happen assist programmers backwards you why a magic imagine that no matter where you started you ended up with P equals 0 Bellamy every point done year got trapped the X-axis it would mean that this entire region he would get mapped to a one-dimensional region one-dimensional region has Zero area so evil Stirum prevents that what it says in fact is blob squeezes in 1 direction it must expand in the the other direction situation for removing a racer is that it basis Of racer gets shrunk in means somebody else's some other components in the phase space the probability distributions spread out what are the other components in this case if I could it's the P is an exodus of molecules the in the table sold for a case of the eraser is really a very high dimensional phase space end can be erased and they come rest almost rests with the phase space squeezes this way it spreads out the other directions 0 the directions having to do with the other microscopic degrees of freedom OK so there we are with information conservation my 1st law of physics and go at a Paris was that go to the zeroth what John zeroth law will come back 0 loss anybody note 0 law says what they says we haven't defined with thermal equilibrium is but it's is
whatever the hell thermal equilibrium its if you have several systems and system a isn't thermal equilibrium with B & B isn't thermal equilibrium with city than a season thermal equilibrium with state we will come back that just put out of your mind for the time being because we haven't discovered describe what thermal equilibrium meters but we can now jumped the 1st law 0 minus 1 0 and 1st law and the 1st lot is simply energy conservation is simply energy conservation nothing more it's really simple the right down its simplicity belies its power it is a statement that 1st of all there is a conserved quantity the fact that we call at conserved quantity energy you know who plays for the moment not such a big role right now but there are let's just say there's energy conservation plot was a lot is at stake but simply says D E that whatever the energy years did D is equal to 0 0 now this is the law energy conservation for a closed system is a system consists of more than 1 part in interaction with each other and of course any 1 of the parts can have a changing energy but the sum total of all of the parts will conserve energy so therefore system is composed as a jewel before too parts with no link between them and this is called 1 and this is true that this reads he wanted by deed case of physical come might this be tool by 2 I really written he won by GTE plus Tito by duties is equal to 0 but that I trends follows 1 of them the right hand side just indicates just demographic that you lose energy on 1 side and on the other that's the 1st law of thermodynamics all of 1st book chromodynamics says it says energy conservation now in this context here is a slightly hidden assumptions we assume river system is composed of 2 parts these energy some of the two-part that's really not generally true if you have 2 systems and interact with each other they may be for example forces between the 2 parts so there might be a potential energy that a function of both of the quarter for example are we energy of the solar system and being very naive and solar system orbiting a Newtonian particles energy consists of the kinetic energy of 1 particle plaster kinetic energy of the other particle plaster term which doesn't belong part or belongs both of them sent the potential energy of interaction between between In that context you really can't say that the energy is sum of the energy of 1 thing plus the energy of the other thing energy conservation still troll but you can't divide the system into 2 parts swear many many contacts with many context where interaction energies between systems is negligible compared to the energy systems themselves have armed if we owe to divide this table top up in the box but was provide tabletop up the box how much energy is in each block will the amount of energy that's in each block is more or less proportional to the volume of each block how much energy of interaction between the blocks the energy of interactions a surface effect they interact with each other because surfaces touch and typically surface area it is Schmidt said by comparison with volume so many many will come back that will come back many many context are these energy of interaction between 2 systems is negligible compared to energy you either of them when that happens you could say to a good approximation the energy could just be represented as the sum of 2 energies of the 2 parts of the system plus keel thing which has to do with their interactions among the most circumstances the 1st law of of thermodynamics misses is the top always true this 2nd um has real caveat that we're talking about systems where energy is strictly and we're and energy you don't always they energies it sometimes energies editor that the deal possible from the top of the list is key ally usually lose is it Yao Yao OK so all context which we talk about the died if it's yellow it can't be but see orange origin of yellow and red while we don't count that were each yet so that's correct that the no that's that's absolutely correct remained the assumption that what I called state what I called states are mutually exclusive try to look OK let's come back entropy when larcenous with trippy with Dunn had to be with done energy we haven't done the temperature yet noticed at temperatures comes in behind entropy and even energy comes in behind entropy but temperature was a highly derived quantities by death by highly derive their mean hits despite the fact that it's a thing you feel with your body so it makes you don't really feel like it's something intuitive it is a mathematically derived concept less primitive and West fundamentally energy or entropy but will come it as comeback of entropy armed we defined entropy but only for certain special probability distribution less layout on the horizontal axis just just to be schematic on the horizontal axis we will put down all the different states his is articles 1 news articles to his articles 3 this axis just labels the various states and of course 8 4 and vertically met a Plot Probability OK well the probability of course it only defined on me Money integers here some some probability but dialects but I don't wanna have to draw such a complicated thing every time I wanted draw probability distribution program some probability distribution was some probability for each position that what we did was we defined entropy for a very special case the special case being where some subset have equal probabilities and the rest have 0 probability for example if subset consists of his group only and whole grouping and then all of these have the same probability and the probabilities after it up 1 solve the probability is 1 of a religious drug that are dry but then we defined the entropy the logarithm of the number of states in here don't speaking we don't have probability distributions like there's generally speaking we have probability distributions which are more complicated in fact they can be anything as long as their positive and all it up to 1 so the question is how we define incurred be in a more general context with a probability distribution looks like there's a right down the formula and they under the check that it really gives the answer when no when
it should you the Santa ah today I'm just gonna write it down and tell you this is the definition you get familiar with thing you'll start to see why it's a good definition is representing something about the probability distribution and lawyers representing is In so average said the average a number of states which are important lead contained inside the probability distribution of the narrower the probability distribution the small the entropy will be the broad probability distribution the bigger the entropy will be I'll write it down you now will write down and explore just a little bit s for a general probability distribution is 1st of all minus that's fine because positive but nevertheless the formula begins with my guess but some is some all out of the state of the possibilities some of what is a contribution for each place here the probability of IT Friday's or of the probability of I'd be remembered forever so tries something Austin remember that the average of F is equal to a summation although F of icons POI this is actually the average of log peace a bite it see average of log peace of mind are left workers out let's see what this gives in the special case where the probability distribution is 1 all the and for it In the state's altogether as with and because it has with and it must have heightened 1 over him because all probabilities have dipped 1 so let's work this out let's say the contribution from all on occupied states from all Lee of unoccupied states piece a 0 so you get nothing got us right OK so now what's right trip so 0 Webster consider the limits of key sort of log P it will repeat well let's just say to limit or not that's not right the limit of key log P E S P goes to 0 a calculate that OK I owe it to you Will calculus exercise to calculate the upper limit as P goes 0 of p log p 0 the point is that he goes 0 a lot faster than log egos infinity log P as 0 growth is very slow P goes 0 4 those goes 0 you're absolutely right that has Serbia that has to be strong but log in the limited be goes 0 is 0 he sold out with that it is knowledge the contribution from states with very very small probability will be very very small and as a probability for those states goes 0 this quantity contribution will go to 0 or not the ones here which have significant probability they all have the same peace of mind and they all have the same lot peace of mind want peace body but want body is logarithm of 1 in both of them to the supply is also war not a lot warmer and warmer and so each contribution is 1 of a and times log Oregon howling contributions are there like that so we multiply by him and get rid of on the world and is a minus sign he carried along all times and because there are in such Tour's so the and cancels and we just like left would log 1 over I am What's Love and log right so that's why the minus sign was put there in the 1st place is no miracle remind was put their biggest probabilities of less than 1 and several Largo rhythms of always negative soak up that negatively overall may assigned an entropy is positive but this is exactly the same answer that we or back the original definition just as Eagles log logarithm of the member states but this is a definition now that makes sense even when you have EU more complicated probability distribution and it is a good and effective Ste average of log P E for the special case with a probability distribution is constant like this then them and then all of the probabilities and hero 1 0 him and calculating the average of 1 of Robert mistress log 1 over and just give you or and this is a general definition Of the entropy that's associated with a probability distribution inodorus entropy is associated with a probability distribution it's nothing like energy which is a property of a system that something like momentum it's a thing which has to do with the specific probability distribution probability distribution on the space of possible states so that's why it's a little bit of a more obscure quantity from the point of view of you into it invoke definition as I said its definition has to do with both the system can be your state of knowledge of the system let's examples as calculates a mantra for a couple of simple systems software system is just gonna be now single coin but a lot of course he thank you for cell where of capital and Corning in a van and each 1 can be heads of tales etc. so as a matter of fact we have no idea what the state of the system because we know nothing of the probability distribution another words Is this same for all state complete ignorance absolute ignorance whether the entropy associated with such a configuration there and the all the probabilities are
equal under the circumstance or all of the probabilities are equal we just get to use larger them Of the number of possible states the answer here is that a logarithm of the number of total state Connolly state other altogether Italy and right to In all that's Amsterdam will change definitions from charisma to that the cause little a number of points is little and Over here begin stood for the total number of states of ice matched terminology is big In the total number states each of the 2 would and not totally and tired to Italy and totally in states altogether to states for the 1st scoring to states of the 2nd quarter fell 4 totally and altogether and the total number of states capital and what's the mat what's the entropy given that we know nothing the large the high end of our 2 s is equal in log that's a slaughter a of 2 to ISO Healy see an example of the fact that entropy is kind of a additive over the system that's proportional to the number of degrees of freedom in this case in times log and we also discover a unit of entropy you event is called a bit that's what a bit it is In information theory that the basic unit of entropy for a system which has only 2 states upper down heads of tales or whatever the entropy is proportional to the number of bits of in this case the number of points times the logarithm of troops a lot to plays a fundamental role in information theory a unit of entropy does not mean that in general that entropy is an integer multiple block to receive a 2nd it does mean that harms OK so that however that this case over here let's try another case let's try another state of knowledge here are state of knowledge with Zulch we know nothing this is the maximum entropy maximum entropy logarithm of capital and logarithm of 2 the little land which is analog to 1 bit of entropy fridge point if you like OK let's try something else let's try in state and we known is all will who something else supposing we know the state completely another words that's the case with In is equal to 1 that would be the case where we know sure the probability is only known 0 4 1 state and it is 1 in S is 0 logarithm of 0 still absolute knowledge perfect knowledge complete knowledge corresponds to 0 and repeat the more you know the bigger the more you know entropy excused OK let's say picket case an interesting case let's take our heads in coattails again and he is only now we know that all of them are heads which begin with that of the heads the entropy then 0 except for 1 of them which is tales as opposing we know which warned his tales what's the 0 but upon we don't know which 1 tales equal probability fault all were all states which contain 1 tail and in minus 1 heads what's the entropy cancel wiser organ how many states aren't there with nonzero probability the answer Little and it could be sonic abuse Tonica because 1 another words for this particular situation capital and the number of states they have nonzero probability is as little all the states at the same probability Soren exactly this situation except that and is a sequel to little and and possible states this 1 could be tales is with the details as we tales have equal probability of capital and end and the entropy now is for this situation the entropy is equal from a library of real and noticed that that's not an integer multiple of log to in general and general entropy is not an integer multiple to nevertheless lot too was a good unit errors but a key basic unit of entropy called a bit s in this case is equal to lodge yes the the 2 luck to them from the effects of the viability of not 50 yet they ate love they were going to absolutely now computer scientists of course alike to think in terms of tool for a variety of reasons for mathematics of is nice is a goal is the smallest number which is not warranted it's also matches got too small to do 1 arm but there it's also true that the in the physics that goes on inside your computer's connected was switches which are either on or off or so capping units walked to lose their useful yet a they but doesn't yet standard de Quebecers at the a definition of courses definition definitions the debate is OK God I'm good it depends on who you are if you're an information the theorists a computer scientist offered on a disciple of Shannon then you like walked to the base too In which case visitors 1 and then the entropy here just measured in units a bit if of your physicist then usually work in basic he but the relationship is just a multiplicative a block to Ian Walker base to which is related by a year numerical factor it's always the same OK so where I ride log LOG being 1 of the easy but very little would change use mother Basler logs OK so there we are mainly just tell
you what the definition is out of the entropy In phase space if we're not talking about if we're not talking about
peace by nite discrete systems like Chris we're talking about continuously infinite systems phase space and let's begins supposing the probability distribution is just some blob where the probabilities are equal inside a blonde 0 outside the blob another with the simple situation that the definition of the entropy is simply what you could say the logarithm of the number of states but how many states are buried here clearly a continuous infantry and theory of them so instead you just say it's a lot of the volume of the probability distribution phase space tests for a continuous system is just fine b the library of the volume in phase space nullify wrote v e you might get the sense that I'm talking about volume in space now I'm talking about the logarithm in phase space volume in the face the phase space high dimensional wherever the functionality of the phase space it is the volume is measured in those kind of units 1st momentum times velocity to the power of a number of coordinate system act but as is equal to the logarithm of the phase space volume of scored P faces these space that's under the volume of the region which is occupied and has a nonzero probability distribution this is the closest analog that we can think of the log in and represents the number of equally probable states of the discrete system more generally if we wanted we have some arbitrary probability distribution the arbitrary probability distribution he would pee probability what would be a function of it will be a function of all of the coordinates and all of the moment all of the moment and all of the quarter in queues of peas an X is whichever you like all of probability for the system to be located at point of phase space Pemex incidentally will you have continuous variables like that do you write that the sum is all equal 1 that would make sense you can't some Marlboro continuously infinite server variables becomes beatable book on bacteria spur Brooks sir a let's just a start at right now that if you have a probability distribution arm of phase space the role as the integral other equal 1 P is really a probability density phase space to probability per unit the sale and face what would you expect the anchor Peter B. I'll give hit the stock sale of minors 2 some of the result was a formula we had before 3 right formula we had before a S minus summational of Peace a viable larger piece about to got to that continue you simply replaced some my integral strong In the row over the phase space that's like the sum or what I and then probability of P and Q a of probability but in 1st approximation only in 1st approximation America leonine 1st conceptual approximation its measuring the logarithm of the volume of the blob the probability blob face all I could not Lu Lu stake fee classical mechanics Exs in his accuser quarter hp's a moment that OK so we know of the find entropy wait as we seen depends on a probability distribution and broadly Morris provide defined temperature or we could stop here I think that's probably enough for next time mobile do temperature and then now and then discussed the Boltzmann distribution riches a probability distribution of thermal equilibrium we haven't quite defying thermal equilibrium yet but we will please Besson question that on my questions now are just I just have the feeling that I probably done enough for right after war he or she must stop the ball at that's why Maine any such assumptions I made any such assumption only assumption in calculating the various entered was either that you know nothing when I told you what you know How you gotta know that and what we reasoning was whether it had to do with knowing something about the dynamics of the independents and so forth Morris calming fears
calculation of what he hears barked and saying you know nothing of the implication was that all states are equally probable were about asking how you know that on to say that all states are equally probable is closely related to saying that there are no correlation does say that if you are ever much script let's talk about correlations for a moment I did you know nothing means you know nothing some particular their freedom they use you begin knowing nothing and you measure 1 of the Corning when he know about the other 1 was nothing nothing you started knowing nothing about them you measured 1 of them used nothing about the other 1 course you know about the 1 2 measures now that's together the other case we studied supposing arms we know that all Kaunda heads except for 1 which is tales and we now measure Hey 1 of them and we find that it's what we say about some other ones it's surely ahead what we measure that that want heads what we know about the other 1 the changes the probability Watson no probability that 1 or the other ones as tales right swerve and 1 over and minus 1 that of 1 over sir won over got 1 of my 1 solo that's correlation that's correlation where when you measure something you learn something new war or the probability distribution for the or things is modified by measuring something that's core correlation arms for the complete ignorance there was no correlation any other kind of configuration in general is very likely to be summer not necessarily because are likely to be some correlation correlations said means you learn something about the probability distribution of other things by measuring the first one or you or you modify the probability distribution that's what U.S. Beaumont young that rid of their our system they make How do we had a surplus there must be a slight barrel and said I braked enter Pete is entity of whatever carrots some of the entropy use of all the individuals of proportional number of things it's editor of whenever there is no correlation no correlation to now uncorrelated systems Have a host of entropy I will come back with that's team that will combat troops it is an interesting question suppose Inc he is it you know you that if you measure coin pop them with three-quarter probability that To near the laid out role if measure 1 of them up than the probability for its neighbors is 3 quarters to be down allegedly get that lets all you know that's all you know is that if 1 of them is up at any 1 of them is up its neighbors three-quarters likely we found adjusting think calculate the entropy of such a distribution that is correlated of course I was correlated because when you measure warned you immediately know something about neighbor or make makeup that make up your own example like that make up your own example like that a computer you learn something from that OK any other questions Italy home where a after which the war was a more work before not particularly the with of the the formula here is I think it is the 1 that's on Bolton's told sure Nowhere what some Hauptmann still faces the year is slowed my what he meant buddy did he did write this is because both formula final formula for Inter entropy and the only difference between the Shannon and Altman entropy is at the the China News log till now of course Shannon discovered this partly by himself Fiji didn't know Our Boltzmann's work and a off from an entirely different direction from information theory thermodynamics but none of it would have surprised both Monday nor do I think gunboats wins a definition would surprise Sharon so they really the same thing no real point in knowing comparing them because they're comparing and there was no real difference Shannon they have I don't know whether you do or not the miners Saarinen here if you don't put the minus sign in there is called information if you put them assigned it's called Lack of information or entropy crew and so arrived a witch Sharon wrote them but anyway a it for love later later but none
of it a separate issue doesn't have to do with quantum mechanical uncertainty arm has to do with the uncertainty implicit in Knicks states not and certainty implicit fewer states end but if it is 8 0 0 0 boy yup there's a converge in fact there the Euro city age bar equals duty equals Boltzmann's constant want but Boltzmann's constant was a conversion factor from temperature to to energy the natural unit fatality Richards really energy but a mail molecule for example is approximately equal to its temperature in certain unit rose units contain a conversion factor cable our remind you to talk about table for Moor please
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Niederspannungskabel
Niederspannungsnetz
Elektronenschale
Band <Textilien>
Übungsmunition
Dezimeterwelle
Angeregtes Atom
Gleichstrom
Römischer Kalender
Zylinderkopf
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Metadaten

Formale Metadaten

Titel Statistical Mechanics Lecture 1
Serientitel Statistical Mechanics
Teil 1
Anzahl der Teile 10
Autor Susskind, Leonard
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DOI 10.5446/14934
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract Leonard Susskind introduces statistical mechanics as one of the most universal disciplines in modern physics. He begins with a brief review of probability theory, and then presents the concepts of entropy and conservation of information.

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