 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ఏంరంయతాడిమడిం అణికారంసి ఩రిలారంటా నికతికిక౸ంరినిమిలూధినంరకానిరంటితౕండ౪ రికికటతిటాకతి ఇవికాటాలలణిని. పతితిత౿తిదాాయా  and one hypothesis is the total cornerstone of statistical mechanics, everything of statistical mechanics and by extension our understanding of thermodynamics of natural phenomena everything from these two postulates and we need a hypothesis and that I will explain why need the hypothesis and what are the two postulates. The two postulates in order to understand those two postulates we would need to know few other things but let me tell you the two postulates then I will go back and forth because they are very very important then I will see to understand these two postulates we need to know some things. The first postulate is what gives introduced time average is equal to unsub-average okay time average is like the quantity x, x time average the way you do you take a long trajectory as I will tell you I need to talk of trajectory is some I am observing a system for a huge long time then time average is that I am just I am just taking one random variable I am not talking of correlation that has a little bit more complex and I am not going to do immediately that so I take snapshots I measure the values at different may be equal time intervals then I say limit t going to infinity and this is time 1 over t 0 to t ds xs this is the time average is this clear to everybody is very important thing this time average that what you are doing we are measuring the time fluctuating as we discussed is fluctuating but I have this time then I say okay I do it up to time a long time t then I take the time t going to infinity and I do that and that becomes independent of t and this is the time average ensemble average is the one that gives introduced you have a mental replica billions of billions so instead of time average now I construct and this was the you know whenever I it helps to be very relaxed so you have this glass of water identical many many same amount of volume same glass same water everything billions of billions of time now instantaneous configuration of water in one of my say this is my these are my mental replica this is my original one now the mental replica controls its volume its density its temperature say or volume total number of water molecules and its energy if I have there in isolation nve but other than that it does not control what are the positions and orientation of the water molecules so each of my mental replica position orientation of the water molecules are different is very important so I am constructing it is a brilliant construction done by we that gives he created a mental replica and he realized that if if they are identical in macroscopic sense nve I have no control over his microscope and he is a huge number of microscopic states they are something like you can say 10 to the power n or a to the power n exponentially grows exponentially number of particles so each of them are distinct microscopically very important to understand that they are exact macroscopically but they are distinct microscopically so now I can go and say okay they are macroscopically identical but they are at the same time and individual of them will have a little different value of the x but now I can average over them that average will be by this always by the angular brackets that limit n going to n is my mental replica then 1 over n sum over the values this is my ensemble average again ensemble is a mental construction with the condition that macroscopically they are the same microscopic they are distinct because a microscopic system is huge number of microscopic states and now I go in my mind I calculate the by the x if for each of my system and as I said n is billions and billions and sum over then then the first postulate of statistical mechanics is time average equal to ensemble average that the first postulate of statistical mechanics now we will do the second postulate now anybody knows the second postulate this is really a beauty exactly equal a probability now tell me since you have told me what is it exactly so but at the core of that all my systems are envy and each of the microscopic states are equally probable why did you need that okay one thing I did not so this number two number one I go back little when I did do this thing I limit n going to infinity stick sense I have to do pixi n is the my system number of system so if they are equally probability 1 over incomes here so that is the reason so gives had no other option you know he wrote this pixi but he had no other option but he was working in nve he knew both my distribution that is same energy the only sensible thing is to do is to equally probability this really beauty so now we have done the two postulates okay I will go through little bit more detail on my notes because of the trajectory and space and all these things now I have done the two postulates of statistical mechanics time average equal ensemble average and equally probability so you know realize the first postulate required the second postulate without that we do not go anywhere now gives needed that is of course this this thing is given to bulge man they are got a hypothesis because bulge man used it earlier now anybody can tell me what is our body hypothesis and also tell me or others I will tell you why we need the algorithm hypothesis this is really very very impressive really very intellectually stimulating so we have to hypothesis I said the whole statistical mechanics is based on two postulate and one hypothesis that's all everything follows from that the huge construction huge theoretical framework of statistical mechanics which everybody uses with the biology chemistry material science physics even that's the theory of many body systems there's no other but his whole thing is based on two postulates and one hypothesis and the name of the hypothesis is our body hypothesis but why do we need the algorithm hypothesis and what is our body hypothesis tell me no so what is now tell me tells me it is not enough to have equal probability but the particle system must go from one to the other system must not remain trapped in one state so our body hypothesis now tells that the system visits every state that's why glass we have the problem breakdown of our gaudicity so the second hypothesis guarantees forces the system to to go through all those states being equally probably is not enough if there's a large barrier between them you don't go from one state to other no because I cannot put ensemble average equal to time average if I do not have equal probability once I have equal probability now if my my fiancee is like that it does not go from one to here it gets stuck here that's the computer simulation all the time we'll break down our gaudicity so we need it to go and that is why you need the algorithm hypothesis now I'll go through some things so we have two postulates we have done now I have to do I would like to tell you about the concept of a space I'd like to tell you about the concept of trajectory because without trajectory we do not have an algorithm hypothesis or equal probability without phase space phase space is a sample space we do not have the probability distribution so these are the two things that I need to do now let us consider single atom and single and start following its position and velocity from time t equal to 0 so in order to get all its future thing in the position and momentum space we need two coordinates in a one dimension we just need two coordinates what is position and velocity they can be plotted against each other so the way one plots this phase space is the very initial is of the classical mechanics but this is the way classical mechanics is usually not taught okay so if it is x and is momentum p then this is the kind of thing that you get the particle moving through with the different velocity and different position and this is is called the phase space so the p and q or p and x this is the phase space of one single particle in three dimension this phase space of a single particle in the six dimensional okay sometimes it is called mu space now it is very important that look so now we are trying to understand the trajectory and quantify the trajectory and the phase space the location of the particle in the phase space so the this this space which is defined by the position and momentum is called the phase space so this is gives so the phase space gives you the movement of a system if I have a one particle only then it is sometimes called mu space and one particle in three dimension is six dimensional space so phase space of a single particle in three dimension is six dimensional and in two dimensional the phase space is four dimensional however in our world we are interacting system we have a n dimensional a n particle system so n particle system then we have a six n dimensional space this is something which puts off students but is need not really be difficult so the phase space is of n dimension n particle is six n dimension however you are not going to really work with it the in one of the major thing of statistical mechanics is that you have to go through the formalism understand the formalism so this is required to derive the equations and one thing of stat mac this is a from beginning to the end is a highly mathematical subject this is one of the reason probably in chemistry it is not taught that much but unfortunately for all of us without stat mac we cannot do anything we cannot do any of the studies theoretical understanding so but this mathematical grinding that we go through or mathematical kind of conveyor belt that you go through is not terribly difficult and at the end of that you have equation which is tractable ok so right now let this so the phase space is a six n dimensional system of a three n ok this phase space of n atoms I am not going to molecule yet n atoms and that is six n dimensional three n coordinate plus three n velocity or momentum now as I am considering a system n number system n number of particles here then I draw the phase space now vector n that means this is three n and position q n this is the six n dimensional phase space ok this is three n dimensional I cannot draw six n dimensional so I have drawn it two dimensional but this is three n this is three n now now I have n number of atoms and molecules here which are undergoing collisions which are moving in a solid they will be vibrating liquid they are colliding and changing positions so instantaneous state of my system is a point in this thing because a point has six n values and then that determines position and velocity of each of the particles is a reduction in description but it is still hidden nothing has been solved but we are developing a formalism so now as the system moving this particle going here this is going there this is going here this is coming here all these things as the particle is moving and my system going from one microscopic state to another microscopic state I am here moving in my phase space ok because this gives my starting position next position I this each point is important thing each point signifies instantaneous state of the system ok this thing so the six n dimensional phase space so the motion of the system in this phase space is called the trajectory so in computer simulations when you are doing averaging your trajectory you are actually following through all the your thousand particles say atoms just then you are taking care of this six thousand at any given time you are going time t to delta t plus delta t you are doing little bit here yes all the particles the same moment and they will move in one direction if all of them of the same direction then they will be translational movement right all of you I give the same momentum in the direction so you will just move in that direction ok yeah that is very easy you will have a whole system will move so this moment the position this one will move with the same velocity so it will parallel to p and it will move in q direction with a constant velocity absolutely that is what you know the equation that gives you the motion in the phase space is nothing but Hamilton's equation that we write as a level equation so Hamilton's equation gives exactly that quantity ok now it is a very good question because it brings to the essence of classical mechanics and movement of the particles in the in the in the in the in the phase space now so one particle system we can do that so now we have understood the let us guess a few examples so let us take an harmonic oscillator of a single one particle harmonic oscillator then depending on energy now it is a bound state bound state means it will be like this and is an ellipse because this is an ellipse so phase space is now highly restrictive if you talk of vibrational energy relaxation with discrete energies then you will be going from one state to another state and you know some very interesting things happen okay free particle however it is not a bound state it can go all over the phase space so this is a free particle one free particle that is moving so harmonic oscillator bound state this is a free particle then two particles are tragic now i want to consider trajectory of two particles and then they will be i will be of this we do all the time in computer simulations really interesting this trajectory say we are doing a chemical reaction and chemical reaction is from going from one bound state to another bound state or one cause a bound state to another cause a bound state then you get this guy is going around moving here in a bound state bound state character is always like that okay so then it is going like that then it escapes the barrier then goes to another bound state so character of the trajectory in the phase space tells a lot of what is going on in the system