 By now, we have discussed details of Maxwell Boltzmann statistics and based upon that we have derived various equations connecting to various thermodynamic quantities. Today, we are going to discuss Bose-Einstein statistics and we will also discuss that what was the need of developing Bose-Einstein statistics. What could not be addressed in Maxwell Boltzmann statistics? We will not go through Maxwell Boltzmann statistics again here because we have done that thoroughly, but let us develop some background for discussing Bose-Einstein statistics. Earlier I was using the words molecules, particles, etcetera and here in Bose-Einstein statistics I will use phase points that particle, molecule, electron whatever you consider it represents the phase points and the various energy levels we will talk in terms of cell. For example, consider here let us say we talk about two sets of cells cell i and cell j and let there be some different compartments in the cell i and in the cell j. Let n i represent the number of phase points in the cell meaning of phase points is I will just write in terms of dot it can be atom it can be molecule. Now, why I want to represent as dot because the molecules atoms cannot be distinguished from each other let us say in the gas phase. So, I will represent just as the dot and let us say in any ith cell that there be n i number of phase points then the observable properties of the system are determined by the microstate of the system and when we talk about the microstate of the system or observable properties then we think of what is the thermodynamic probability of microstate. How do you define the thermodynamic probability of a microstate? It is the number of microstates corresponding to it that is what is the comment here the thermodynamic probability W of a microstate is also defined as number of microstates corresponding to it and when we were talking about Maxwell Boltzmann statistics then the number of microstates were defined as the number of permutations of a given microstate. Let us try to discuss in terms of some example. Let us consider the cells cell i and cell j and let us say we have 4 phase points a, b, c, d although I am saying a, b, c, d we can distinguish letters a, b and c and d but let us assume that these are atoms molecules phase points which cannot be distinguished from each other. Let us consider out of these 4 phase points 3 are kept in cell i and 1 is kept in cell j right. So, we have like you know it is like 2 energy levels and in one we have 3 molecules in other we have 1 molecule. So, what is the total number of permutations and combinations? The thermodynamic probability you remember that W was defined as n factorial over n 0 factorial n 1 factorial etcetera etcetera. So, here I will write W is equal to n factorial total 4, 4 factorial in cell i in first one we have 3. So, 3 factorial in the other we have 1 factorial and this is equal to 4 into 3 factorial over 3 factorial into 1 factorial. So, that means, I have W is equal to 4 in this kind of arrangement the thermodynamic probability of the macro state according to Boltzmann Maxwell Boltzmann statistics it gives W equal to 4 this we have discussed earlier. Now, as I earlier said that let us replace A B C D by just dot why by dot because we cannot distinguish between the particles we cannot distinguish between the phase points. So, let us represent molecules by dots as they cannot be distinguished we just see dots you see cell i put 3 dots and cell j you just put 1 dot. Now, if the subdivision of cells into compartments were not there then the only way of setting up the micro state with 4 phase points is 3 in i 1 in j. When I say the subdivisions of the cells into smaller compartments that means, within cell i I can create further compartments. If that is not there then the only way we can arrange we can have this is 3 in cell i and 1 in cell j. Now, the question is when you are dealing with molecule atoms or when you are dealing with electrons there can be some additional restrictions. For example, there are systems like electrons where poly exclusion principle applies that means, in that case you cannot have more than 2 electrons together, but in order to avoid that complication with we will worry later on when we discuss Fermi Dirac statistics. Here let us only consider the particles to which the poly exclusion principle does not apply does not apply. So, that means, that gives me a freedom to put as many particles in any cell as possible as many phase points in any cell as possible that is why here you are talking about 3 can be put in i 1 can be in j right. So, that is what the comment does there can be any number of phase points in a compartment and this is Bose Einstein statistics. Let us discuss little details of this Bose Einstein statistics. So, we started with this 3 phase points in cell i and 1 phase point in cell j. Now, consider 4 compartments in each cell that is you see here 1 compartment, 2nd compartment, 3rd compartment, 4th compartment. Consider 4 compartments per cell and now if you carefully examine all these 3 are in one cell 2 in one cell 1 in another cell and then there are various combinations. There are 20 different ways of arranging these 3 phase points in cell i and since we had only one phase point is in j and if there are 4 compartments then it can stay here it can stay here it can stay here it can stay here that means, in this case the thermodynamic probability W is equal to 4. In the upper case 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 W is equal to 20. I hope you understood that we consider compartmentalization of cell i we created 4 compartments and also in cell j and then if you consider the various permutations and combinations there are 20 different ways of arranging 3 phase points in cell i and 4 different ways of arranging 1 phase point in cell j. Remember that this was not possible in Maxwell Boltzmann statistics where the microstates were considered different only when a given particle was shifted from one to another. In this case we have W i equal to 20 W j equal to 4 therefore, the total thermodynamic probability of the microstate is going to be product of 2 that is overall W is W i times W j W i is 20 W j is 4. So, total it is a T if you compare this with the Maxwell Boltzmann statistics then in that case we got W equal to 4. Here we are getting W equal to 80 see the difference. So, in general when there are any number of cells then overall W that is the thermodynamic probability is the product of thermodynamic probability of each microstate ok. So, I hope this is clear. Let us get into little more details. Suppose we label the compartments now consider these compartments we label these compartments as 1, 2, 3, 4 and we can go up to G and the phase points phase points are being represented by A, B, C, D, E, F according to this compartmentalization in which the compartment in the ith cell are numbered as 1, 2, 3, 4 etc up to G. Let us say the first compartment has A, B molecule, second has 1 molecule, third is empty, fourth has 3 molecules or 3 phase points etc etc. So, what we have here is we have number and we have letters right. We have number we have letters and if the numbers and letters are arranged in all possible sequences then each sequence will represent a microstate provided the sequence begins with the number. I repeat if these numbers and letters are arranged in all possible sequences then each sequence will represent a microstate provided the sequence begins with the number and we had these compartments up to G that means there are G ways of arranging the sequences. What I mean is there are G ways in which sequences can begin because there are G compartments one for each of the G compartments. So, there are G compartments there are n i number of phase points the total is G plus n i and if one starts with one number that means remaining G plus n i minus 1 numbers and letters can be arranged in any order try to understand. There are G ways in which the sequences can begin because their total number is G here. Then if you count G plus total number of phase points if you are starting with the one compartment that is fixed the remaining G plus n i minus 1 numbers and letters can be arranged in any order. Therefore, how to calculate the total number of sequences that can begin with the number is G into G plus n i minus 1 factorial. Although this each sequence in a this represents a microstate many of these such kind of sequences will be same microstate. For example, if you compare this with this here you see third is empty third is empty, but it is starting with number 3 1 a b 1 a b 4 d e f 4 d e f 2 c 2 c thus the number of different sequences of the blocks is obviously G factorial. There are G substances which can be arranged in G factorial ways alright. Now, we have taken care of sequences, but remember that all the phase points that we are talking about they are also not distinguishable they are indistinguishable. Therefore, we need so that in order to not over count the number of states we need to divide by n i factorial. So, therefore, W i is equal to G into G plus n i minus 1 factorial we divide by G factorial we divide by n i factorial and then your G factorial is equal to G into G minus 1 factorial. So, this gets transformed into this you are getting rid of this G. Now, let us apply we will apply it to cell i let us say W i this is equal to G plus n i minus 1. So, G here is 4. So, I will write 4. So, what I will do is I will write G is equal to 4 plus n i n i is equal to 3 over here minus 1 factorial over over G minus 1 G is 4 4 minus 1 is 3 3 factorial into n i n i is equal to what n i is equal to here is 3. So, what I have here is 4 plus 3 7 minus 1 6 6 factorial 6 factorial over 3 factorial into 3 factorial. So, which is equal to 6 into 5 into 4 into 3 factorial over 3 factorial into 3 factorial. So, 3 factorial 3 factorial with cancels with 6 this cancels with 3 factorial that means W i is equal to 20. This is what we counted here. Now, let us say let us say W j W j here G is 4 plus n i is 1 minus 1 factorial divided by G minus 1 3 factorial into 1 factorial and this is nothing, but this is 4 into 3 factorial over 3 factorial which is equal to 4 1 2 3 4. So, therefore, this derived formula is correct. So, now we have an expression which can be used to determine the number of microstate for any i th or j th cell. Now, you will see the next treatment is similar to that we discussed in Maxwell Boltzmann statistics. So, what we have now is for each one of the microstate cell i we may have any one of the microstate of cell j there is no restriction right. So, therefore, how do we calculate the total number of microstates including all cells cells i cell j then the thermodynamic probability is you take multiplication of all these. If you take multiplication of all these then you have multiplication of the ratio of these factorials. Now, let us go back to our discussion in Maxwell Boltzmann statistics what we talked there was that sometimes it is easier to work with logarithm of w rather than just will w. So, therefore, when you take logarithm of w I take log of w then this is you know log when you take log of a product it can be written as summation I am not going to get into details because this we have already discussed in Maxwell Boltzmann statistics. So, it will be summation log g plus n i minus 1 factorial minus log g minus 1 factorial minus log n i factorial because g minus 1 and n i factorial appear in the denominator. We can ignore 1 because generally the number of compartments in the cell is larger than 1 much larger than 1 the number of phase points is also much larger than 1. Therefore, I can drop 1 here I can drop 1 here you drop there and then you apply sterling approximation. What was sterling approximation we remember that log x factorial is equal to x log x minus x fine. So, you apply here to all these once you apply to all these then rearrange then by using sterling approximation you have log w is equal to summation n i plus g log n i plus g minus g log g minus n i log n i going by the same method because we have to now find the state of maximum thermodynamic probability and what is that maximum thermodynamic probability since we have decided to work on log w that means derivative of this log w we should set equal to 0 and then find out the corresponding expression. So, d log w is equal to summation log g plus n i naught by n i naught into d n i when you take the derivative of this you will get this kind of expression. What is this n i naught now here this is the number of phase points in the ith cell in thermodynamic equilibrium let the equilibrium be there. Now the question is can we again treat this and this independent same way as we discussed in Maxwell Boltzmann distribution this cannot be treated independent therefore, you can treat this independent by using method of undetermined multipliers. What is that method of undetermined multiplier for that you need to have constraints one constraint is the total number of phase points has to be same that means all changes in the number of phase point d n i's should add up to 0. Second summation e i d n i should be 0 these are same thing which we discussed in Maxwell Boltzmann distribution there we multiplied by alpha and minus beta here we will multiply by log beta and beta of course, if you go back we are adding here and you are adding minus log beta minus beta i i and now you treat this as 0. Once you treat this as 0 you rearrange you get log g plus n i naught by n i naught is equal to log b plus beta e i and n i naught upon g is 1 over b into exponential b e i minus 1 this is Bose-Einstein distribution function. Remember Bose-Einstein distribution function you have n i naught by g g is the number of compartments and b exponential beta e i minus 1 sometimes this 1 can also be conditionally neglected. Now if you want to further get into this theory of Bose-Einstein distribution function you can find out the expression for b you can also find out the expression for beta which will turn out to be 1 over k d. So, the distribution function that we just derived this is similar to Maxwell Boltzmann distribution function except that on the left hand side we have number of phase points per compartment and on the right side we have this factor which includes b beta and energy levels. This has the resemblance with Maxwell Boltzmann function, but you see in the Bose-Einstein distribution function you have additional consideration for the phase points per compartment. That means you now start considering the number of compartments in a given cell. So, when you compare with Maxwell Boltzmann distribution Bose-Einstein distribution offers some different features and Maxwell Boltzmann distribution offers some features which were not considered in the Bose-Einstein distribution function. So, as I said that in this discussion in this treatment we have not applied poly exclusion principle. If this theory is applied to electrons then poly exclusion principle has to be adhered to. How that is done that we will discuss in the next lecture. Thank you very much.