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# Statistical Mechanics Lecture 2

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Stecher University but then a few
minutes talking about units arm arms in particular the
Boltzmann constant What is the Boltzmann constant the Boltzmann constant like most stuff like many constant fear merit physics are conversion factors are the speed of light is a conversion factor you know that the conversion factor from a distance is to work 2 times of course a few judiciously choose you units you concepts warned the Boltzmann constant is also a conversion factor and typically these things conversion factors from what I can to call human units 2 units which are some more fundamental human units units which are convenient for scales and magnitudes of quantities which people can access reasonably easily OK so we haven't talked about temperature will go on to talk about temperature but authorities say something units 1st Europe's of
temperature as invented by a I guess 19th century physicists and chemists not sure when Fahrenheit was caldron of course was late-19th-century must show who will watch Fahrenheit laws there I was 17th century appeared calm but many case the units that we tend to use these days are Calvin units or centigrade arm Calvin temperatures Calvin aura relative Absolute Zero temperatures send a grade relative to what they freezing point of water which is totally arbitrary and centigrade scale the unit of 1 degrees as you well know was simply set down by by you'd arbitrary short to divide the temperature scale between freezing in boiling water freezing of boiling water at atmospheric pressure into 100 units highly arbitrary enough to be wired it was water that was used the food of use alcohol were different unit therefore review someone but it wasn't a unit which some people could manipulate people could manipulate temperatures that measured temperatures and not to hide the measure of a
change in temperature of the degree with a human scale from a human scale and North may have a lot of Adams thing you could make in terms of your shop and so they Cal warned unit degrees no temperature change is a human Khan struck Bell was invented for convenience if you like now it was at the time it was not fully understood Ocalan problem have understood it but it was not fully understood that the natural units the real
physical units the temperature has to do with his energy that at temperature it is really and energy for example example here is that the temperature of a gas for example determines the Connecticut energy of the molecules and 1 measure of the temperature of the gas is just that a kinetic energy of the molecules will you might say wait a minute that's not so good because some of the gasses that composite of helium and oxygen of come composer helium a chose explode but there were helium an oxygen then the mean helium atoms all the oxygen atom surely they have different kinetic energy but they don't thermal equilibrium fixed temperature I will prove that eventually we will prove that all of the Adams have basically the same murmur same kinetic energy and in fact if you put a bowling
ball suspended a bowling ball at finite temperature in Tulua into a gas or even a liquid Our a kinetic energy of the bowling ball would be exactly the same as the kinetic energy an oxygen atom that seems a little bit odd but when you think about not that odd if the bowling ball has a say in energy is a single molecule molecules it was truly loot move a lot slower than a single molecule the reason is massive much larger sewage and these squared off the bowling ball is the same as and least square foot oxygen molecules the bowling ball will be moving a lot slower but the main point that temperature is really a measure of an amount of energy so the natural units the temperature a really energy but at the time that temperature 1st started to be measured nobody could measure Mark Cuomo but had the vaguest idea how molecules on mess of molecules were how much energy a typical molecule had at a given temperature the a fact probably in the beginning nobody had the idea that temperature had to do anything to do with the energy of a basic constituent so be it scale of temperatures was defined before there was a notion the temperatures Sosebee equals energy pockets of natural unit for temperature could be Jules for example for jewelers a good universe temperature at a rather large unit of temperature to see that we these are typical energy of a single molecule and thermal equilibrium of a order temperature of room temperature for 1 molecule is a caII kind kind number in jewels and so the conversion factor between human units and let's say the the basic units of energy the BMW larger a very small number of very small number so it's justifying defined by him by giving you the relationship between these energy of a molecule in lewd gas energy of a single molecule in a bad mood gas that dialog because we know why worry too much about the energies of interaction between the molecules the energy are of kinetic energy kinetic energy of a molecule and delude guests is people the 3 hands down the free is easy to understand was enough to do it dressed diminished because its number of right it has a Connecticut deeper moving along the x-axis has a kinetic energy from moving along the Y axis as a kinetic energy from William z-axis of the factor of 3 and a half well that's just a glitch definition types of 3 halves be Boltzmann constant time is temperature but this is a human scale of temperature so it could be Calvert temperature measuring caldron units in that case the Boltzmann constant here has units of energy divided by degrees Kelvin Energy divided by degrees Kelvin end at a rather small number as you might expect if temperatures 3 100 degrees at room temperature temperature Calvinist 300 room temperature the energy of a single molecule is clearly very very very
small of 300 is pretty big he is terribly small cable Martin is very small and this number in human units human units now means a standard units leaders and so forth cable money is about 1 . 4 times 10 to the minus 24 there'd be no accident that I will guard rose number is also roughly 10 24 my maximum but an EU hats are Jules divided by degrees Kelvin Belieu over jewelers a unit of energy in so a unit of mass times velocity squared we could reduce it to work a standard units the jewels of we like but this is what name point is that's the really big delisting quantity year temperature In fundamental physics always comes in multiplying cable to monitor at an all fundamental formulas of physics K Boltzmann times temperature comes in and that's just be call us the fundamental quantities is really energy and so we redefine cable month temperature has units of energy will we redefine and no unit of temperature but score a capital T which is just cables swim climbs the human unit of temperature once we do that we will get rid of the greens we will get rid of cable money in all week wage will never see it again but you do have to remember that what I call temperature differs from thermometer temperature by a very small factor OK that a factor actually is there is no units of entropy memo entropy is it's a unit of information measured in measured and fundamental little bits of information at the kind that entropy was discovered as an interesting quantity which always increases the was discovered by a steam engineer by the name of Connell Bayern he was esteem engineer we did for a living he built design steam engines on is no anything about the microscopic underlying principle of entropy yet quantity quantity was so easily measured body at a quantity defined in terms of temperatures in energy users for em to call the entropy or somebody calls entered peak and got these letter AS Roma work a little s for a minute and a poet a sob carnal here so city Connell had a quantity which he called entropy it has unit but we were students from use to you over here for minute Parliament energy divided by yet temperature most temperature measured in Calverton then it's Jules 34 for Calvin jewels per Kelvin was as condos unit of entropy but in modern physics we think of entropy is measured in bits of fundamental unit entropy is logarithm of 2 for example and that's mention works so notion Of not some of the notion it's just the conversion factor again condos unit of entropy differs by what I have called entropy up till now I have to get fear OK bum remember what that appears from a statistical point of view if you have a probability distribution if you have some probability distribution over some discreet 7 states then we define be entropy as capitalist narrow the beat minus the sum of all the states of peace a Lada rhythm of peace a byword that last week a rough measure of the logarithm of the number of states which are important and under a statistical distribution here so we called repeat what's the connection between current entropy and and this is Baltimore and sent the to loans entropy are he'll what's the connection the connection is simply a factor of key boats Monday and it is such that it is equal to 1 older cable two-month books most times over times condos quantity mother words when Carmela's entropy is a normal number like 6 an error a steam engine both moms entropy is enormous Altman's entropy is roughly proportional to the number of molecules in the system or from like that Alonso others now wants to know how go back and forth between human units of temperature and human units Our entropy and more fundamental units the temperature its energy entropy it's a dimensionless quantity and from now 1 I will not introduce cables the bottom factors would effectively is gone out of all of our equations but you could put it back in the way of putting it back it is all the ways whenever you see capital and you want to get it into human units think of escape Boltzmann Prime's temperature and Calvin whatever you see an entropy S which has a huge huge value typically and you want to get into steam engines units provided by cable yes all that these guys said that figure a depends on of course now depends on what you energy using surf you're using the energy unit Jules there will be jewels some of the point is it's not known whether jaw joy is energy end MKS units MKS meters kilograms of seconds Surat so what is it it energy out a laughter 1 kilogram of object moving at a velocity of 1 meter per 2nd is a factor of 2 half-empty squared so twice the energy of a 1 killed object a 1 meter per 2nd
that he had not yet there is a a lot better get past her founder different constant review Cheney units a will change to change units instead of using for our instead of using telling units we use Fahrenheit units the cables will change your ways that that shattered all of youse was doing those an illustration right that that are so energy use energy itself is a young a quantity with units and so you have to specify what unit to talking about when I say temperature has units all Our of energy yes it does but you have to pick a unit of energy of working in the meter kilogram 2nd assembly elects a new natural unit tho you're unit of energy is called jewel but not in New York it's unit of energy in 9 MKS Europe's and that's the unit that you would use the temperature if you are a key performer was simple on the other hand you could use and I could use a friend killer tons of he and feel that's a bad thing to talk about our about some of the unit electron volts plant masses or whatever you cheesy units of temperate SRU changing units of energy the temperature will also change would you units of temperature in energy should be kept the same entropy on the other hand is dimensionless this appears dimensionless our parents doesn't have got his classes sell-off that yes yes can at the bottom of self like Newt never knew what the value of his own comes the usefulness I made the actual numerical value and that's because nobody had measured the properties of Adams masses of Adams ready a on thank came along you have it arms Calvin got died just before a if the value of his constant was really I'm quite sure that they have a rough idea the rough idea and is certainly no those connected by got his number Mark he probably never knew with any detail what the value of his number was the same was true Newton had a rough idea of what is constant Wuerzburg Noah Precision OK but cannot but but the bottom on deed understands he did understand that if only he could measure the properties of atoms that energy would be related to temperature through his that he did understand 5 but bus far IMF Hollywood temperatures you have until you sense of temperature that's because when you touch a hot thing it does something to in their shells and so forth stormed an and you have an actual by intuition about cold but temperatures around arrived quantity it's a variety of mathematical quantity that begins with the notion of entropy begins with the notion of energy and the notions twin notion of energy and entropy so works but because your temperatures when system is in thermal equilibrium and we have defined that yet if we get to it tonight will do it but it just means that a closed system has been allowed to evolve for a long enough time just settles down to some equilibrium configuration is and it may be that tax system it doesn't have to be a closed system back close stop and a system in contact with a lot of other environment if it there have been environment is not changing with time the system under consideration will be equilibrated a certain sense of good become clear off with its environment energy may flow from the system for the environment that case we say this the systems Coveney environment and you may flow the other way on the average against in equilibrium on the average energy is flowing nite away in
thermal equilibrium what characterizes the thermal equilibrium of system so here we have a system that's our system consists of a lot of degrees of freedom and the they may be interacting with the environment down here exchanging energy with onto it comes to equilibrium it has a collection of state let's label them but and each state Ali hasn't Energy East's Lisa barring aren't so OK now when a system comes to equilibrium it doesn't come to a particular state fight Betsy interacting with its environment energy is going back and forth complicated things are happening but all average in equilibrium there are some probabilistic ideas some probability just probability that the system is not quite state with energies body that becomes probabilistic now because when interacting with the environment of William you could point some piece of I that piece of satisfy some rules the first one is the sum of all probabilities 1 another royal not definition is out sight take PER of alright multiplied by the energy of the RAI state and some of that's average energy District Court E a C average energy of the system will use fearlessly average we really never mind that the energy is constantly changing fluctuates some extra energy goes out into the environment and the comes back and so forth and constantly fluctuating Bober short time it averages and this is the equation for the average energy back and finally we have the notion of entropy and the entropy is just as raw emotion OK now what is the probability distribution P was it depend on well 1st of all among other things the mostly work depends on busy average energy in this system no the average energy may depend on the nature of the environment the environment is very high up the average energy in here may be hiding via leader a environment as cold the average energy here may be called but some average energy curators average energy and can be different depending on the precise nature of what's going on the average energy could be thought of as a parameter you could change it or you can change the average energy of a pot of water but putting on a slow wicket call down but putting out not so you could change the average energy and that's in the absence of probability on a depends on the average energy that later think about this is for every value of the ad and that would put the complicated bars around the series now stands for average energy meet for edit every given average energy in thermal equilibrium is a probability piece of art Of course if you increase the average energy the new film the probability distribution toward larger energy values Of these survive but so that means that there's not a single probability distribution to call equilibrium is a family of them the family of them feature value of 4 is blacks could be the plot we are state of the system of Oracle plot energy of the ice state buffet each value of the average energy this some probability distribution em leave value of the energy output in he'll put in here each common share for every energy the total probabilities half-day up to 1 end their warm-up this equation here on the right inside this each year
is just the average energy is
just average energy that defines the average and change the average energy retained the energy of the year of what's in the box the probability distribution shifts of ice to drop out of the area under it is always 1 the area under it or the total probability shows want so maybe like this which 1 corresponds to more average energy the water which is keen further out and lamenting the energy increases go in this direction he of our as removed to the right energy increases and so the the probability distribution which is Pete further to the right of not doing as well be area under each 1 of these should be the same but with a 1 parameter family of equilibrium distributions like this which we have calculated yet we don't know what they are how but as a matter of experience for each value the average energy is a probability distribution and that's called PR of eat every single 1 of them satisfied with some role here and the average energy is equal to quantity here we are saying is that he had dedicated can be properly leadership has it will it but said ways which would put down where he said there would be I get us there is a fun person so short but said he would have a crime so it's like Brad gag rack fire on let's for the moment just so from experience there are a sequence are 1 parameter family of probability distributions so here's a 1 parameter family parameters called E for the moment but does not recall the equal you want a family of different probability distributions each 1 of them has an average energy so once I know the average energy of it I can't think of it as the probability distribution for that given the average energy yup so there is reason most reason these days after days of yes because some imagining that this system a system onward we're talking now about the system as opposed to the environment and there's a constant interchange of energy back and forth between the system and the year and the year of the system will call the system the rest of the is call the heat back so it could be the atmosphere could be a box of gas with permeable walls that he can leak through sitting in room temperature in this room that is constantly exchanging energy back and forth energy fluctuates so the energy is definitely has a probability distribution of you go into the system at 1 instant fame measure the energy you get 1 value to go another instance you get another value and if you do Mini Mini Mini measurements you get a probability distribution exactly that yet I thought he was a piece that it is true that an area that day these men and they say it could be like that why said but still looked but for the moment that is a 1 parameter family that that's correct yes of course such there are many probability distributions that you could write down that have the same average energy way only getting this 1 parameter family from will come later the sale as a matter of experience if you know the average energy of the system and some probability distribution associated with that of 50 dividend if the Boltzmann distribution had turned from actual bought books to distribution fight both analogist
or mathematical exercise a 1 parameter family of probability distributions parameterized by the average energy no 2 of them have the same average energy by assumptions and the higher the energy is of course the broader this problem with the broader the probability distribution is likely to be take that as a given but said function that each state that each of these types of that figure out but just a case of what we Carlos if is all we are asking what happens if there's more than 1 state with exactly the same energy what is correlated Khalid is a very technical term and if you going to try transmit information back and forth between us to act we have to do it with precision so what does it mean to say that a correlation is a very definite meet me and doesn't apply to this follows will be needed for them and the other half declare asked you could ask What is more than 1 state with exactly the same energy not average energy but uh but there isn't that he's buys Aesop's 7 niece about 6 exactly the same the that's a first-order never happens again it's not particularly important column D generous that means equal energies for different states it is a very rare and unusual things we afterward every state has its own energy every energy has from
state but that summer working assumption has gotten up there was a serious lack is he's muscles this is my style description of the state of precise exact microscope every molecule position and velocity or whatever it takes to describe the exact microscopic state of the system is Marjorie the Dodgers choice what I am meet there are thought that let the state while they say all how we label them how label them OK so let's the good guy so let's agree label them in the ass sending order or all of the energy I right end let's take it is given that no 2 states have exactly the the same energy which is typically true except in very unusual situation OK so then we have Our clear notion of what we mean by I we have notion of what we mean by P White and P of I food different values of the average energy the nature of this thing will become clear enough tonight we don't have enough time to tonight a workout Boltzmann distribution but we do have enough time to define the concept of temperature are so consider the following call 1 more thing each 1 of these probability distributions each value of the EU has it's own interpret but that the entropy S because now a function of peanuts not universal functionally depends on these probability Gershon's but given this 1 parameter family of probability distributions SOB is equal to minus the sum all I of P Peel of comic these osha the bracket myelography and sell for this reason ends becomes a function of the energy average energy in this case they enter a plea which is a feature of a given probability distribution now becomes a function of the energy because each energy we have a different probability distribution so that tells us that those emotions the connection between entropy and energy if we know this 1 parameter family resisted were discussed later where that comes from no let's until the question interesting question how much energy bills are each 1 of these probability distribution parameterized by given average energy the higher the average energy more spread out the right To do a pariah we could How much energy bills of how much and how much you have to change the average energy how much you have to change the average energy in order to change entropy by 1 bit 1 bit means a lot To how much you have to shift the energy here in order that we entropy changes by 1 bit toward and the enters Goodby disappointing because some discover write a formula the black which is totally obvious the answer that the change in the enter be such receipt or revert yup sort the shaky energy is going to be equal to the derivative of the entropy with respect To the energy sorry derivatives the change in meet energy the change in the energy when I go from 1 probability distribution through the next will be equal to the derivative of the energy with respect for for the entropy kind the change in the entropy This is an identity and this is only making this is assuming nothing more that things are smooth functions that could differentiated the change in the energy going from 1 probability distribution for the next is driven the energy with respect to the entropy times a change in the end the demise Maine a little more sense to February the sale of a video of the alternative way I'm going from 1 distribution for the next the change in me entropy is equal the change in energy divided by these by DOS and sadly is no different from now between partial derivatives of probable derivatives just a derivative of you with respect debts In some problems amid maybe more variables only met to keep track of what variables here it's just delivered a movie with respect as times don't tests or can arrived in this fashion here Mr. trivial episode nothing insightful here it's just the definition of what I mean by differential changes question see you thought about the variation With respect to air their very best option would be what he has in a way B the essence of the inverse of BST Dow down our This is also equal to be by deep d of all the same things are OK if for for example I want to know how much the way I have to shift the probability disturbed how much laughter shifty energy in order to change the entropy by 1 bit that would just stick in the best 1 bit namely log too so would tell me how much energy it takes the Cheney entropy by 1 bit Benhaim point here is this subject over here this object is cold temperature that took temperatures 2 derived quantity and it tells you what how much you
have to change the interpreter for me the way you the way you say how much you have to change the energy in order to change the entropy by a certain amount as a fixated on 1 bit but doesn't matter it's 1 as as small as long as small our or using differential formulas I'm a wire right dealt with just D the definition of temperature is such that D D E equals TV temperature yes and that's the only definition of temperature that's universal general and the Eagles TBS sometimes written BSE equals 1 firms DDT OK 1 point that's interesting a little bit interesting if I change my units of I change my units from these physical units to human units in the formula doesn't change and the reason is there does change of course but it changes To become equals P measured in Calverton kind D S of the condo and why because the Boltzmann constant cancels capital T is cables London Times little tree and capital analysts is my seemed lost it not be reduced capital S it is 1 overcame Balboa times as so they can't and this is why cardinal who wrote this formula here he wrote the energy Peoples Loop PBS the cable 2 months the boat 1 factors cancel out of that 1 gets a book came radio when gets a Cayman denominator just telling you this because you may wonder what happened to the Cabletron If you go read a book on standard thermodynamics Tuesday where what happened the cable from electron they they cancel out in the Definitions start it we are definite of temperature bits Odin abstract definition of temperature that only speak clear from what it has to do with temperature as you as you experience it from tell you what do with temperature yacht up so that it is a function of dress did it in with the full not this thing soul it is that our answers the Asian obvious usually from heroes their account examples are character examples and I will also say it is typical of normal physical systems that entropy and the energy on account that when you increase energy usually increase the width of the probability distribution and so doing increase the will take as a given amount that energy and entropy are monotonically increasing functions of each other single-valued we can come later ask is that always true was no but when it's not true unusual again unusual situation OK so we're going to assume that the energy is a monotonically increasing function of entropy or entropy is a monotonically increasing function of energy and that means that temperatures past B E by S S by the EU are positive assumption for more as I said what it means is that when you would increase the parameter EU you spread the distribution over a larger number of states we feel as if they were going to approve from most of them but what taken as is given for OK so then the question is what In what way this definition temperature in any way related to the familiar concept of temperature no was so what is a familiar Constable out burned myself but only if we do something that's a little bit more physics oriented basic faggot well OK but that the pens on the properties of light and so forth content and that such true but and really a statistical mechanical concept and we can have a box of against them without having like our servers statistical mechanics notion but they were in the middle of thousands things that temperature has to do it but really youthful definition definition that close 2 were to general is that temperature is measure or door differences of temperature if you have 2 systems let's take systems but for simplicity now supposed to systems are isolated Mr. system AT and system B and that connected by a different quite fin type simply means that they can exchange energy with each other but we quarries that a little bits of energy can go back and forth between them and course energies of fluctuate on average each 1 has an emerging outlets start them out without having been in contact with SARS not being a contact each in equilibrium but at a given added different temperature a different temperature but close 20 a TB we know what temperature means temperature is the rate of change of energy with respect entropy an abstract definition to be sure but each 1 of these boxes has prepared with given temperature by whatever it takes to make them have where temperatures have and now connect them and connecting instantly means all allowing them to interact with each other and exchange energy back
and forth which way does the energy flow that's the most basic concept of temperature or are for the most basic a concept of temperature is that he or Energy Energy will flow from the the harder in other words range system of higher temperature to the system of lower temperature elegant throwback back arbitrarily let's say that TB is bigger than T a there is way that we just had to change a and B because it's possible exactly equals TB and see in that case that he doesn't slow either way on the average of he'd hate me energy OK solely to doing it the right down a couple of simple year old rule left out 1 step I left them 1 step I left them out again something which will not become completely clear to leaders it's the 2nd law of thermodynamics the secular thermodynamics which I have given you no reason to believe that the you could take it as a past due at for the moment but we will come we will go lot better the 2nd law of thermodynamics said is that when you allow the system the come the thermal equilibrium when you allow a and B to equilibrate he will fall 1 way or the other way we don't know which energy will flow but eventually will come to some sort of compromise equilibrium the 2nd law of of thermodynamics says the entropy increases so let's try the follows that of a line of reasoning Merrill the Cobble anchor peak of the system S is equal to entropy of aII plus the entropy of beat white wine is entropy does anybody know from the mathematics because is a lot of young because because the logarithm of something probabilities multiply if you have 2 systems each 1 has some sort of probabilistic description the probability for the whole system some property multiply if you take the logarithm of probabilities today so entropy is if the entropy of the whole system is SA process be another factor useful factor I found some useful facts 2nd useful fact is that when the system equilibrate exchanging energy the total energy is conserved the 1st law of thermodynamics 1st loss yes they energy Ave a increases the energy of B decreases by the same amount that says that a musical Tom minors d he beat chain the change in energy change in energy of AT and be Casale up ulterior 0 mature make it through this with whom they rural next the 2nd law of thermodynamics the change in SA plus the change in SB is greater than 0 entropy increases again something we Lulita justify finally each 1 of these the change in energy baby it is equal to the temperature of a kind a change in the entropy of a right that each that's D E equals TBS definition of temperature likewise D B is Eagle TB terms DSB I wholeheartedly support what the because of the will of the rules call for more please visit us
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Grundfrequenz
Konverter <Kerntechnik>
Direkte Messung
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Maßstab <Messtechnik>
Lichtgeschwindigkeit
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Minute
Computeranimation
Temperatur
Druckgradient
Relativistische Mechanik
Messung
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Direkte Messung
Gefrierpunkt
Maßstab <Messtechnik>
Besprechung/Interview
Tag
Proof <Graphische Technik>
Armbanduhr
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Nassdampfturbine
Halo <Atmosphärische Optik>
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Messung
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Kugelblitz
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Wirtsgitter
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Weltall
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Woche
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Montag
Monat
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Jahr
Lineal
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Temperatur

### Metadaten

#### Formale Metadaten

 Titel Statistical Mechanics Lecture 2 Serientitel Statistical Mechanics Teil 2 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 10.5446/14937 Herausgeber Stanford University Erscheinungsjahr 2013 Sprache Englisch

#### Inhaltliche Metadaten

 Fachgebiet Physik Abstract Leonard Susskind presents the physics of temperature. Temperature is not a fundamental quantity, but is derived as the amount of energy required to add an incremental amount of entropy to a system.