 This was the one which was worked out by one by Einstein and which is the many many things he did you know he did the Brownian motion and Brownian diffusion but this was the thing which was done somewhat later and it's a brilliant way that he started somehow he got into his idea to study the probability of a fluctuation and he introduced the concept that truth at system equilibrium in the homogeneous state it has two independent thermodynamic variables and he worked out with the and it played a very important role in the later discharge of hydrodynamics that he for example he took temperature and pressure and walked it out and this that situation isothermal isobaric is the best way to do hydrodynamics and he also do entropy and volume these fluctuations and then he found out the probability of fluctuation the way we did that he constructed a fluctuation and got the work done to this create that fluctuation and from there he did the probability of fluctuation now why do we talk of fluctuation okay and what is the reason we know that at equilibrium free energy free energy is minimum so now if the free energy is minimum then if I consider if I expand free energy as a function of say density and I say my density undergoes a fluctuation around equilibrium fluctuation so density is a not sorry row average plus delta row then the way one writes is that free energy a not at equilibrium row average now the first derivative of free energy with respect to density goes on like that now this quantity which is a derivative because the free energy is minimum these goes to zero so then I have and I make that delta change in free energy as a delta delta row then this also goes out then I can write as a first term of this Taylor expansion as so this should be just derivative partial derivative other things kept fixed now then we can say that if I want to have a fluctuation that fluctuation has this cost in free energy that means your small enough fluctuation my free energy surface is harmonic the zero average okay is that clear is a very trivial logic but at the same time this very profound so these are the then we call these quantity as the force constant of the fluctuation so you immediately see that this is the quantity which must be positive so that if there is a cost to free energy then if in as in phase transition or chemical reactions if there is a barrier and then there is another state here then to small we say to small fluctuations this is stable but your large fluctuations is goes over there when it reaches there these derivative becomes negative now so either it has to once it reaches there it has to come back there or come there so the equilibrium between the two such states characterized by minimum is determined by the difference in this but dynamics is profoundly dependent by the oscillations here and the curvature there okay now we do not want to talk of dynamics instead we want to talk about this quantity that what is in in so this is the chapter you can read where talked about the fluctuations so basic idea is that that what are these these these fluctuations are the most important quantity because this this is the determines the response the second derivative determines the response of the system to fluctuation how that is the called the rest linear response or response functions and these response functions are the most important properties of a system like when we brought the rock from the moon other than the density the first thing they calculated is the specific heat actually whenever you go to many museums they'll write down the specific heat of that below and the conductivity those properties conductivity specific heat are the response functions why because if you so now as I am telling you from the morning that these are very important if I change if I want to change temperature by delta t then I have to give a amount of heat delta q and so this is the relation cp if you bring it here then delta q by delta t is the cp so specifically is the amount of heat needed to change the temperature by 1 degree that's what you have been started in the in 8th rate at least we started in 8th rate compressibility now I bring here dvdp then is the compressibility that mean the unit pressure the pressure needed to change volume everything by unit and this is the magnetic susceptibility chi it is connected to the polarization of the magnetization created in the system by applying this amount of external field so every case we are applying an external perturbation here we are giving an amount of heat here we are giving an amount of pressure and here we are giving an external magnetic field on the left hand side we are having the response of the system and then these constant coefficients are the one which gives the response the magnitude of the response near a phase transition these response functions all diverge then a very small amount of perturbation create a huge response specifically it goes to infinity remember the lambda curve compressibility goes to zero gas liquid transition you can compress in it so then and these also diverges so stability condition of a system both mechanical this gives you the mechanical stability and these you give the thermodynamic stability both the stability are connected with these these quantities so whenever we talk of a system is stable and non-stable there is a set of stability conditions which is discussed in my book in a little later I will hope to get into that so I hope I have impressed sufficient amount on you to tell how the importance of these response functions they are the most important constitutive properties of the system most important constitutive properties of the system so and the ones whenever any matter is new things are created these are the first properties that one measures to characterize the system okay now so my next comment and very important comment to is that these are the quantities these response functions in addition also and these are most fundamental relation one of the things in my book first time I have the chapter 6 I had fluctuations and I said realization of the promises later my students said they noticed that is bit too much but the whole statistical mechanics is quite a formidable structure as you have seen you have to go quite a bit before you start getting the results which is somewhat different from quantum mechanics because you start with starting equation which is also a postulate and then you start getting results which are connected to spectroscopy rather quickly not in statistical mechanics here you will get a huge amount of results but you have to plow through certain amount of things you have to understand systematically how just going to partition function is itself has this this process and but then at the end of the day much of these are for example if you are going to do soft matter you are going to do phase transition you are going to do polymer liquid crystals everything is nothing but statistical mechanics so you are going to put this extra the nucleation everything you are going to put this but at the end of the day you are starting phenomena see I have again again telling one thing in chemistry the way we do quantum we don't study phenomena rarely we study phenomena we study numbers which are then connected to spectroscopy so there is a huge difference the way these two disciplines work you go to a quantum chemistry conference here at least India they hardly talk of they talk of formalisms they talk of second decimal place sometimes but they don't talk of any phenomena so their main delight is the publication in jackson and come on today okay now so the so the important thing of these quantities here my point I'm going to make most important point are essentially the second derivatives of free energy specifically to the respect to temperature this one is with respect to density on number or volume and susceptibility is with respect to these external field so these quantity density temperature mediation sorry these are external external control parameter we'll introduce a term called order parameter little later okay so to summarize this part that these important things specific it compressibility these things are the second derivatives of free energy first derivative is zero and you can easily see why because if I give a small amount of heat or small change I put a little bit of pressure then since it is a minimum how much it is going to displace is determined by this quantity and then this is given by this which is a spring constant is a harmonic surface it is spring constant these are extremely important because these trivial apparently trivial things are basis of theory of landau's theory of estimation is a wonderful theory that we'll do after this we'll do monatomic gas then we'll do diatomic then we'll jump a little bit we'll do both Einstein later we'll go to mayors theory interacting system then we go to landau theory so I'll do a little bit back and forth kind of thing I'll not follow the textbook completely now when you write a textbook you are kind of constrained by the kind of established by Macquarie or all other people so now okay now so I'm going to now do the next part so these quantity is a very interesting quantity now how do I get that okay this is the another very important response that if I study this thing even at equilibrium even in the absence of an external perturbation is the most important thing the particles are moving this is a very important language which our ego Kubo introduced called the natural motion of the system the system is undergoing continuous the thermal motion that thermal motion gives you diffusion dynamics that gives you resistivity that also give you specific it now I'm making this far reaching observation that these fluctuations contain the information of specific heat and thermal compressibility so natural motion of the system determines and I am telling you I cannot cannot overemphasize it is so important here my student has done these fluctuations in a total energy by simulation and volume and you can see this content as I was telling in the morning they continuously fluctuate they continuously fluctuate they are just natural system remember castellan gave a wonderful example of castellan said okay I now and I have put some colored liquid here which will now go into here and say this is the up to this so same level here they are your meniscus but I make this pipe is very narrow so this meniscus is visible in a microscope so now I look at it through a microscope I'm a wonderful artist as you can figure out so now this height I call this height h and now I plot that fun age as a function of time I'll find that is continuously oscillating without anything it's an equilibrium with the atmosphere so the reason is that there is a natural motion of the system you know the way that is the nature's way or the system's way to interact with the external pressure okay everything is that equilibrium so I am in completely equilibrium but my system is undergoing these fluctuations it is very very important to realize that this is a natural fluctuation okay so equilibrium factor will define average values originate from thermal motion of atoms and molecules stretched in this from my book stress them apart is the profound result that the response function that I am discussed here in three things are given exactly by mean square fluctuations of respective conjugate thermo and properties so I'm going to now tell you what is the specific heat how do we calculate specific heat and what is the microscopic definition and microscopic minimum meaning of the specific heat okay and so this is the way we remember specific heat okay so I do the DDT I go to this DDT I do I bring one E so it become a square another term comes from here but Q is again sum over e to the bar minus e by kb kb t so that brings it so Q right so I take a derivative it become 1 over Q square it become 1 over Q square because it's in the denominator and then I take the derivative it brings a EI out and there is one EI before here so same thing EI by kb t very so this become a square now this square but it has Q denominator so I put the Q inside and complete this square so that thing is just average energy square this quantity on the other hand is this E square kb t square and by it is it's kb t square both the two places and then 1 over Q and so this quantity is nothing but E square okay so I have E square minus average E square exactly so specific heat is then this quantity so that's why the relation that E square minus so this is specific heat CV 1 over kb t square in both the two cases I bring it upstairs so I get kb t square CV and this square is this quantity on the left hand side so specific heat is nothing but mean square fluctuation of energy this very simple thing that's what I in my book I had the original title that realization of promises this is really such a wonderful result which nobody anticipated that that specific heat is nothing but mean square energy fluctuation is a I as I said I cannot over emphasis the beauty and the importance of this relation okay now I that I did a specific at constant volume now I want to do specific heat at constant pressure specific at constant pressure will come from a our NPT ensemble and h then it become E plus PV we discussed in the morning and then you can exactly play the same game and you find out that specific heat at constant a is again mean square mean square fluctuation in the enthalpy okay that is CP CP and CV can be quite different experimentally we work with CP but CV is the one theoreticians work with that reason is that the other one much in canonical ensemble is easier and that's what we do all the time canonical ensemble okay now I go into compressibility again the same thing that I go to the NPT ensemble and I again do the V square delta V and you know dvdp here dvdp I do here by P and P here brings out of V square so I get V square mind and delta V square which is the mean square fluctuation then dvdp is the mean square fluctuation in volume and this is my compressibility so I get the relation is that the isothermal compressibility means nothing but sigma V square or here is the well done relation important thing is that isothermal compressibility is giving by mean square volume fluctuation so that's why this was given here these mean square energy fluctuation these mean square volume fluctuation these are real simulations of I believe of water there is reason to talk of water and I will talk about water in the context in a little bit in a greater detail so so compressibility we have done specifically we have done we have not done I so there is an exercise that you can easily do do this you have to do in the Grand Canonical Grand Canonical you have to consider you can do exactly same game you play same game you play you know but in the Grand Canonical remember that we are screwing it up little bit having the same notation but Grand Canonical you do you can do the number fluctuation right and now you can write the compressibility isomer compressive as a derivative with see it is dv then you write dvdp and replace v by density n by v and say I want the number fluctuation not the volume to fluctuate then you will get d d a d and that means I have a here in density I have n instead of volume and if this will be if you work it out you will be able to get this result the isomer compressibility is the the final value is the same you can calculate in the Grand Canonical you can calculate as the work ensemble two different things two different way we calculate that if we want to calculate in Canonical ensemble then I just do the volume fluctuation sorry if I do NPT ensemble I do the volume fluctuation because you know Canonical ensemble volume is fixed doesn't fluctuate but in isothermal isobaric NPT ensemble volume fluctuates from that fluctuation I can get the mean square fluctuation and that gives the compressibility but if I consider that as I told you we discussed this is difficult to do in a computer simulation the end that's the unbend Gibbs ensemble and that's I bought that I I'm not going to go into because I'm not very clear myself about it so the one of the questions that was raised why different ensembles have the same result okay the reason is the following let me see if I have the graph here the reason is the following that now let us see the relative fluctuation so sigma e square this is the width okay and this also a question that was asked we'll come to that answer both the two questions so remember when we do we always talk we need to talk of the relative value why you have to consider relative value that because you cannot compare hundred thousand and you cannot compare thousand with ten thousand say we want to describe the distribution of salary in a concern then what you have to do you have to find out the maximum salary now in concern to concern the maximum salary is changing if you want to start the dispersion you should better scale by that make that equal to one and then see how the dispersion goes here also so this is the relative fluctuation that we need to look into and relative fluctuation then sigma e square sigma e square is CV kpt square so sigma e is kpt square by square root I divide by then the energy these scales as n because this average energy there's an extensive property these scales as n to the power half the specific it is also an extensive property so we have root over n by n so it scales as one over root n that's true everywhere so the scaling of fluctuations so now if I want to compare the results of canonical and canonical or canonical an isothermal isobaric ensemble then the results will be the same when the fluctuations are small so the fluctuations are these things that is moving around here when n goes to infinity these fluctuations relative fluctuations become very very small that goes to zero the fluctuations go to zero as one over root n this is the reason why whatever ensemble you study you get the same result this is also intimately related intimately related with the stability conditions that we will we will discuss later