 will start with the briefly revising what is this course about so the whole goal of statistical mechanics it came into being with the essentially with this phenomenal paper of Maxwell to propose the velocity distribution now why Maxwell proposed the velocity distribution because that was the time if you remember the around the same time the equation of state was being formulated you remember the name of Barthelot, Deuterichy all these names but of course the successful one was by Van Waals so there was with the kinetic theory of gases the is fair to say the atomistic view of nature put forward by Dalton but the atomistic view of nature came into being with the with the kind of theory of gases so I think that's the time people started thinking you know to understand the perspective that the time when people were doing classical mechanics people were doing hydrodynamics hydrodynamics was fully developed almost fully developed so things were mostly continuum and classical mechanics based that's what made quantum mechanics such a huge impact so so the kinetic theory was growing and where people like Maxwell realized that he must have realized almost much of what we know today that the way to describe them gases that is the simplest system as a particles who are moving colliding among themselves colliding with one and that's the way pressure comes into being that's the way heat is being conducted thermal conductivity or electrical conductivity so Maxwell went out to develop a theory of these kind of phenomena viscosity and he then realized there is not possible to calculate the whole thing but he realized one thing if I have a distribution of velocity probability that there is a that there is a this number of particles with the velocity between v and v plus dv which is called pv dv when he has that kind of thing then so the number of particles is between p between v and v plus dv was pv dv that was the basic so pv is the probability that was the first time in this game we are playing the pro distribution was introduced and he could calculate as we know the equation of state at this pv equal to nrt he could calculate pressure equal to one third mc2 and the average speed of light at a particle which is c2 is I think 3 kb by m other one is the speed is rho 8 by 8 kb by pi m all these relations that you read in your undergraduate follow from that particular ansatz that he published his description and then as I told you that their difficulty was that it was difficult to take into account the collisions and the molecule it was realized very quickly that the the system this is system what is beauty of this that the system is an equilibrium equipments which is time invariant that means state of the system is not changing macroscopic state of the system is not changing with the time but microscopic state of the system is changing with time that the basic understanding that already was evolving at that time and so what bulgeman tried to do bulgeman tried to put that into far more concrete with a expressively taking into interaction between atoms and molecules and there is a term in his theory called collision cross section where two particles come they have they can come to a distance and they colloid collision away the probability of collision when the two particles are approaching each other with a velocity v1 and v2 that probability is a collisional cross section which is determined by the intermolecular interaction he could not go very far as how he discussed where and that is a subject of time dependence statistical mechanics which we are not going to at least do this part of that this part of the course but it has a very paramount importance because it gave rise to a function which essentially became the entropy in in a microscopic definition of entropy which will be which we use to discuss the study so basic idea then was is the following that people faced with the problem of describing the such questions as why ice melts with a latent heat of 80 I think calorie per gram or kilo calorie per gram and why argon like systems or iron goes into FCC lattice while sodium goes into BCC why water crystallizes to this hexagonal ice and the heat or protein falls into the native state and why it falls into the state that it falls so or the how in your body when you inhale you live in Delhi you know you are all the pollution so you can imagine the kind of damage your DNA is made but you still survive the reason is survive because we have a efficient I accept the young children and old people immunity in our body that all these DNA that are being or getting or oxidized or the particles getting deposited on lunch are getting removed okay so and that thing that the way a DNA gets continuously damaged DNA is continuously rectified because there are certain proteins they move around DNA and correct the mutation so these are the process you need to understand so this name of all these processes that I am talking freezing melting boiling or this protein folding protein DNA interaction they are called large amplitude phenomena these are involving many many particles and as I discussed yesterday several times that we cannot even solve with a simple potential as an interaction potential a three body problem the reason bulgement failed beyond a point because of that there is no way even this time and hundred more than 100 years now substantially more than 100 years we have tried and there are certain advances but not a great deal of advances of understanding starting from really called pascal nature of matter then go to calculate the thing so faced with the difficulties that what bulgement faced will it gives decided that the other way of doing these things we must this must be abundant these ambitious project of doing the ambitious project of doing from a fully atomistic description following the trajectory and collisions of every atoms or molecules and by trajectory I mean that you have the two following in the first one tack particles positions in the three dimensional space and its momentum with time and then the same for the all other n particles so it is a six n dimensional coordinate space which you call face space so the system is executed in the motion in the six n dimensional even if we neglect rotation this huge dimension space so this is called trajectory so the concept of trajectory is very essential that I described before and that will come again and again in the discussion so so this hugely complicated thing which is the motion in face space which is the trajectory and so trajectory in in face space that is is is not possible to get now you think of that say we solve the trajectory we do that in computer simulation trajectory of this certain number of a much smaller number many of the time what you do with that information yes you know the position and atoms and molecules are every space in time for seconds and hours what do you do with that information you know that is a huge amount of information and much of that is just impossible to analyze it of course correct in getting the velocity correlation function for few particles that we know that that is very helpful these days not at the time of Gibbs but we know can do that now but this in the a huge amount of information that a trajectory has in it for equilibrium properties of the system that information is not meaningful even if we have that information it is not meaningful so we need a method a system to kind of synthesize and use these information in a much more succinct and more more clever and intelligent way and that was done by Willard Gibbs this brilliant construction of ensemble he said as I give the example that there are 10 glasses or 100 glasses you know oh water at the same half filled state then they are all the stain my glass has the same properties at at the room temperature same density same specific heat same conductivity in every property is the same however each of those 10 water system of this a few avocado number of water molecules a few hundreds of avocado number of water molecules in the glasses they are different at any time not only the microscopic state of water is changing in each a but it is one of the way to say okay I want to get idea of the microscopic state then I need to build I have to construct mentally huge number of these glasses now and each of them now at a given time each of them are in a different microscopic state so then he said if I know now if I construct just like Maxwell and Bozeman need a probability distribution that the system is in a given microscopic state then I can start talking of probability distribution so ensemble picture allowed you this transition from a trajectory based description to probability based description that was the whole idea of ensembles ensembles allowed you to go to probability and once you go to probability then you start discussing these things in a certain quantity analytical way which gives develop so that was the goal that the to make the transition to the probability and you know Bozeman was heavily criticized for the his description of probability but on the the other side of Atlantic will it be alone alone in sitting in the year university at the Stirling laboratory year university you live you can read his life in Wikipedia I used to have in some slides is a little bit about his life is amazing man who alone single handedly did all this theory of statistical mechanics you know and there is beautiful quotation by Mooliken and other people that you know how he single handedly did that now okay so now what you are going to do is so we okay now when I started doing that what did will that gives used okay he said I can now talk in terms of the total number of microscopic states in the system I have the ensemble and now I want to relate my total number of microscopic states to the thermodynamic property so we describe whenever you see whatever I am saying certain jump or inconsistency please stop P it a class becomes better only when the students ask questions otherwise it gets kind of monochromous this is what we discussed yesterday yesterday's lecture that this is the important thing that in certain form came from Bozeman and as I told you that this could be considered very much at one more postulate of statistical mechanics or equilibrium statistical mechanics that this is the equation from which entire equilibrium statistical mechanics you know was developed that this one equation so now if how how do you go from here to the next step subsequent that is now what we need to know we need to show that from entropy from this equation in terms of omega the number of microscopic states in the system I should be able to describe condition like I should be able to derive an equation of state that means I should be able to get pressure as a function of this I should be able to get for example the chemical potential or free energy so this if I know this and I know this equation then so this is the equation which gives the relation between the microscopic states which is the real microscopic of the system we discussed in the last class that the energy level diagrams it may be arrows as particles occupying different energy levels and giving rise to different arrangement of microscopic states and total number of microscopic states we worked out with energy equal to 8 that 4 states that are allowed if I have 4 energy levels 0 1 2 3 now with that information see all of them have the same energy so all of them with the same energy and I have this like in a particle in a box you have you have the remember if you have worked in the particle in a box and it is done in the in the in the context of the solid state physics that you go to density of states in omega e in a particle in a box and that is as a energy e to the power half that means it grows exponentially omega e it grows exponentially the energy and that's a very important thing so this itself is a very important relation but here we are saying a very fundamental thing that this number of microscopic states uniquely determine a thermodynamic function entropy so now given that I have to now discuss that from then the total number of microscopic states how other macroscopic properties like pressure chemical potential and follows one important thing about micro chronicle ensemble very important thing about micro chronicle ensemble that they only are talking in terms of nve so this s is a function which is nve and omega is a function that is of nve so there is no concept of temperature in this framework micro chronicle ensemble no concept of pressure no concept of chemical potential nothing is there in micro chronicle ensemble so now it our task is to evolve this relations to show how this thing hangs together how the micro chronicle ensemble leads us to thermodynamics so that is the next goal and this is the one there are lots of equations there and so I will not write to them okay now we can start these from the book statistical mechanics so this would start so the way we now think of this equation here that we give a small fluctuation in the variables nve now let us see what would happen if I increase the energy then omega will increase right because that you know already in particular box that energy if we increase the volume v keeping other things constant then again number of states will increase can you explain why if I keep everything same and then increase volume v then number of states will increase yeah so based way to think of a lattice and the lattice you develop into okay a system you develop into different grids and each molecular size you give it is a grid and then you make it larger keep n fixed then there are more grids and more places to place the particle okay and of course this is a dicey thing because it is not always a monotonic function it could be very large density it could actually be start decreasing but v and e it will be a monotonic function in probably with but interacting system at some level what will happen you would not be able to pack it anymore so it will go to suddenly it will go to zero that is the glass transition kind of scenario that we talk okay so now we consider a small variation of fluctuation and then my variation in entropy from Boltzmann formula I can write like that okay there are two ways of doing it and I will do both the two ways okay ds is this this is a that is a variation with so I take a derivative partial derivative because it is a function of three variables and I have three terms dsd dsdv dsdn dvd now we know this is the the I call the differential form of this function that means d equal to tds minus pdv plus mu dn as the fundamental equation or fundamental relation of thermodynamics I think following Castellan and these people so if this form then from there under these conditions I can now show so d equal to tds minus pdv minus mu dn you bring the tds on the side then you have ds equal to divide everything by t then what you realize now that d e ds is become t d e dv is d dv will become minus p d e dn is chemical potential mu right okay by using the cyclic rule which essentially dsdv into d e ds dv d e this three product equal to minus one okay that we use in thermodynamics constantly now I know that d e ds equal to t I know d dv equal to p then I know now combining this thing I know dsdv equal to p by t right no yeah it is a good question actually this causes always this this one causes confusion you are looking at the three independent variables the three independent variables are nv e others are dependent variable no p of course dependent is you are talking of this equation no the way we are doing it the d the s at constant v and n no but p is dependent variable this is my independent variable my system depends on three variables that that three variables is three variables are e v and n so they are the independent variables so I I think I do not fully understand your question yeah p dependent p dependent and and and all the three p is dependent on nv e all three p is dependent all three variables the nv e all three see these are the things so only there are three three independent variables system is fully determined by all three okay everything else comes out from this three okay this is a question always has because the the question this one comes if we do it will come again what the question you asked the first time I had this confusion let me spend one minute on that when we do the stability conditions in thermodynamics there we expand the free energy in a similar form in volume and temperature then the way we prove that specific heat is positive there is one through two fluctuations but another proof is through thermodynamics that for a system to be mechanically stable thermally stable we have to say specific heat is positive and mechanically stable we have to show pressure is dvdp that is negative so these are the stability conditions so there you expand your free energy exactly in these variables and the exactly same conditions say that's the first time the confusion that you have that is I face that confusion when I was doing but you think a little you have to keep track of only the independent variable dependent variables doesn't matter okay all right so then I have this is the equation this is what I am saying is essentially the what is called fundamental relation of thermodynamics now if I go back my earlier equation so there are two things going parallely please note one thing that is going is that this this equation from there we are so one is this and then all the other thermodynamics we are using however we know that that this a is is kb ln omega so when I do these things I do d kb comes out dln omega de dln omega dn plus dln omega dv so that immediately tells me dln omega dv will be temperature dln omega dv will be pressure and dln omega dn will be chemical potential now these are equations of very kind of profound importance because you are getting a definition of temperature a microscopic definition of temperature of a system in terms of the density of states now kinetic theory gives you a much simpler and exact relation between temperature what is that equipartition theory that rmd is equal to infinity that's the one we use computer simulation of the time okay so but you see those things you have are developed kinetic theory of gas that way is developed in the what we will let us let us see is a canonical ensemble where temperature is given from outside I have a system at fixed temperature gas of a certain number or number density and a given volume d that why called nvd and there I know temperature is given from outside the particles are undergoing collisions they are exchanging energy but they are exchanging energy also with the bath and there is a temperature is maintained in under that condition half mv square equal to kbd so it is supplied from outside but here I do not know the temperature at all I have an isolated system because my ensemble my canonical answer is purely isolated it is constant n constant v constant this is something I did not emphasize yesterday so this is the epitome of isolated system it is not allowed to exchange particles n is fixed it is not allowed to any exchange volume or become bigger or smaller constant v by putting a certain barostat or whatever pressure and it is completely isolated it is not allowed to exchange energy so since it is completely isolated we do not know the temperature of the system we have no control of the temperature of the system that is why in one second just I will I will I might that is why in computer simulations real though they are they are perfectionist they will do all calculations nv but will start with the npt get a constant pressure they then remove barostat then we will start with canonical ensemble npt and then they will remove the thermal bar nozehubar or whatever nozehubar thermostat and all the things and then you get you simulate in the nve so you simulate is this beast and you use the conditions that we use are essentially these conditions so we use the fluctuation formula but that is different yeah tell me not at all yeah this is again the same question he asked this is a huge country so this is what in if you do classical mechanics goldstein you will see so I have this system nve then I take a small variation I make the volume slightly bigger slightly very small amount but then I start the response then I start the small energy I change I study the response I study what I am saying by the small variations I now study the change in omega what I am telling my small variation executed mathematically gives me a variation omega and that variation is the utmost fundamental problem there is a variation that is why we do fluctuation problem in variation that is impossible to your noise will be more than what I am doing here if you look if you do classical mechanics a book like goldstein or Arnold they the goldstein divorce one full section on virtual displacement on virtual work this is exactly same thing this is a mathematical construction we are doing a purely analytical work or but the highest never because you are thinking you have to be like that okay so now that so with their isolate system I mentally take little change of volume and I say how is omega is changing I say okay that gives me the pressure and then why it goes off so quickly so now you can see this is what I have been telling that from the omega and it is a wonderful exam you will look at these equations they are just wonderful total number of microscopic of this system you make a virtual displacement in energy and that variation is temporary so how much the log omega will change with energy determines the temperature now this is we call the conjugate variable energy and temperature is the conjugate variable they are connected in a mechanical in a n ve ensemble now similarly pressure and volume are conjugate variable right and that exactly works out I take the derivative of l n omega with s2v I get the pressure and similarly energy and number of particles the they are conjugate variables and they give you chemical potential so this is the one so then we if you want in your micro canonical ensemble to calculate these things we do not do it this way but this is the way we use these three relations I give you we use it extremely and really using now in economical ensemble these equations to get the get the subsequent and developments so these are you know these are remarkable because they give our familiar thermodynamic function in terms of changes in total number of microscopic state of the n ve system so then you know this is the things I know you can write that this is the micro canonical ensemble is not realistic because you know none of the real system is n ve real systems typically are in chemistry is mu p t or n p t if the number of particles change then we do the mu p t means chemical potential p is pressure t temperature and many other things like phase transition many of the times we do n p t that means pressure temperature quite far from n ve so Gibbs went on to beautifully construct and this was I think the marvelous thing that he did