Hello, this is the 17 th class in the physics
of material course, and today we are look at the Fermi-Dirac statistics. So, the Fermi-Dirac statistics - the Fermi-Dirac
statistics describes to as the distribution of Fermions in a system of thermal equilibrium;
distribution of Fermions across energy levels for a system
in thermal equilibrium. Fermi-Dirac statistics describe the distribution of Fermions across
energy levels for a system that is in thermal equilibrium. For our purposes this is of relevant,
because the electrons that we are trying to examine electrons with solute that we are
trying to examine, electrons qualifier Fermions, they have all characteristics that that going
to this terminology called the Fermions. So, they have half into get a spin they obey policy
exclusion principle and and therefore, they have qualifiers Fermions.
So, we would like to see this, this of course is useful to as because there it tells as
something about the behavior of those electrons, what they can do, what they cannot do, what
they will attempt to do, this is the basic idea that we wish to examine in detail, right.
So, this is Fermi-Dirac statistics so what are the conditions we would just take them
very briefly because we have discuss them in detail earlier. First of all it says that
those the Fermions fall obey. So the particles of the Paulies exclusion
principle and therefore, this this discussion applies for them. The particle are identical
and indistinguishable. So, this simply means that when you have energy levels and you have
certain number of particles in one energy level and another number of particles in another
energy level, simply swapping particle between two energy levels is not is not considered
as a new arrangement. So, if you have N 1 at energy level m E 1 and N 2 at an energy
level E 2, if you simply take one particle from E 1 and exchange it with a particle at
E 2 that is not considered as a new arrangement, so that is the important thing you will through
it in to the, we will incorporated into the mathematics as we go along.
So, these are the two major thing points. The third thing of course, is that it is in
thermal equilibrium plus this is a system that is a that has a fixed extend so in other
words the number of particles in it it is a constant. So, therefore, the total number
of particles which we have call N total is the total number of particle is constant is
fixed it is an thermal equilibrium therefore, the total energy of the system e total is
also fixed. So, N total is fixed, E total is fixed, which these are this is the framework
that we are operating in and this is the framework with with in which Fermi-Dirac statistics
is use to describe the distribution of particles. So, N total is fixed, E total is fixed and
we are basically looking; we are looking for this information, what is the probability
that an energy level that state with energy level with energy E i is occupied. So, we
wish to find out what is the probability that is state with energy E i is occupied. This
is the piece of information that we wish to find out, this the framework within which
we are going do this calculation, right. So, this how we are draw about it so as you proceed
you will see how this comes about. I will point out that we would put will put down
some equations and proceed step by step and each each put down please carefully, note
down what are the conditions under which we are claiming that that equation is operating.
You can always re examine it when you when you done with the class, just to see that
you understand what it means when you put down the kind of an equation. Because at each
stage we may make some simplification, we will sometimes eliminate some conditions;
sometimes focus on a particular condition and so on. So, it is not that as you all the
equations are to be just remembered as it is. Please, understand what are the conditions
under which we are putting down those equations. So, the system that we are dealing with has
some parameters associated with it. So, we will just list those down we will just
say that it has energy levels E 0, E 1, E 2, E 3 and so on, E r. At each energy level
this is the system we are we are basically saying Paulies exclusion principle applies
and and for it to have any meaning they has to be a certain fixed number of states. If
they are infinite state there is no meaning in saying Paulies; even then you would say
you know if Paulies exclusion principle applies into electrons for not have exactly the same
state, will not occupy exactly the same state, where the state incorporate for of the quantum
number information. So, we basically say that these are the energy
levels. At each energy level there is a certain finite number of states so you just call that
S 0, this is the number of state available at energy level E 0, S 1 number of energy
level available at with energy E 1, S 2, S 3 and S r, which is the number of states available
at the energy level r. so, we will just say that this is the steam that we have. Now,
what we; so, if you take our system given that is in a thermal equilibrium and given
that other constraint to the system around fixed, the energy levels it is a fixed volume
system it is a closed system so the energy levels are all fixed.
So, in our calculation, in all the calculation that we will do, up front these numbers are
all fixed. We are not specifically giving a energy at value in terms of joule or in
terms of joule specifically we are not explicitly saying so but regardless the point is for
our calculation purposes this are certain fixed numbers, we have to be we have to recognize
that. So, this is not a variable in our calculation it is fixed all of this numbers are fixed.
Similarly, the number of states that are available at each energy level are is also fixed. So,
what they are fixed at is is another calculation which we will do later on so that is not something
that immediately concerns us. For our system this is fixed, this is fixed we do not have
any choice, it is fixed. What is not fixed, what is variable is the
actual number of particle number of electrons in this case that are happen to be sitting
at E 0, E 1, E 2, E 3 and E 4 and so on. So, those are N 0, N 1, N 2, N 3, N r for the
state r and so on. So, this is not something that is fixed, at is as it start out the calculation,
as we start out the calculation we do not know what this values are. In other words,
we can consider a verity of situations in each of the situation the difference will
be the actual value of N 0 was is the actual value of N 1, N 2, N 3 and so on, those numbers
can change. You could add some particles here, you could remove some particles here you could
add some more particles here all of this possibilities are allowed.
Where as you cannot do that with the the number states available at the energy levels or the
energy level themselves, this is all fixed. You cannot just increase E 0 you cannot increase
S 0 and so on. When you say E 0 it is fixed E 0 there is no other value you can have,
S 0is also the number states that you cannot change that anywhere. Whereas, the number
of particles that are sitting there, subject to the fact that it is not greater than S
0 you can actually increase the number of particles or decrease the number of particles
so this is the system that we have. Now, we wish to know; so let us what we will do, we
will we will starting some limited fashion then we will in our in our calculation if
narrow down to a certain starting point from there we will start.
We will then write a more general equation then we will find so this is the layout of
what we are going to do. We are going to start with narrow our focus of one particular situation,
then we will write a more general equation which covers the entire arrangement that we
can think of. And then we were find that even though there is a larger equation that is
of interest in that larger equation there is only one term that is of real interest.
If you focus on that one term we will get the information that we want. So, we will,
so even though you are write a much more elaborate equation, most of that equation we can set
aside. We can say that most of that equation is not is not necessary for as to solve or
analyze that entire equation, that one term in the equation is good enough.
It it is the most important term that we need to analyze so we were focus on that one term
and our calculations will be based on that term, so that is what we will do. So, what
is this narrow starting point that we will start with? So, our narrow starting point
is that we will first say that I told you that these numbers can change, right, N 0,
N 1, N 2, N 3, N 5 and so on can change. Now, I will start with a situation where and we
were call this a distribution. So, this this entire information that we have here I will
call this N bar. So, as a particular N bar or I call this n bar I for example, so our
particular N bar implies that a particular set of N 0, N 1, N 2, N 3, N 4 and a particular
arrangement has been chosen. A particular combination of N 0, N 1 a set
a particular set of N 0, N 1, N 2, N 3 has been chosen as N i. If you change this it
will be some other N, N bar so that is that is what we are referring to. Now, I put something
down here and then you will, I will explain what it means this is the probability this
f (E) is the is the probability function which tells as the probability of that an f (E i)
let say, this is the probability that the energy of the this is the probability that
state at energy level E i is occupied. So, this is some E i is occupied so that is the
probability that it is occupied. So, this is so this is for a particular n bar will
will remove the side for it does not confusing at the movement we leave it like that n bar
is a particular set that we having. So, this is the probability that are state
at energy level E i is occupied, now what are it say is this is about general probability
we will narrow down and I put the equation down here then then we will discuss it. What
I mean is this is called the conditional probability that state at energy level E i is occupied
so conditional probability. So, this is the conditional probability that
state at energy level E i is occupied. So, what this means is, you have various different
N bars available to you, I will picking one particular N bar. So, one particular arrangement
of N bar I am picking, which means a particular set of N 0, N 1, N 2, N 3, N 4 and so on.
So, at this point I have frozen all three numbers, right.
All three numbers are anyway these two are frozen I also frozen this number for this
particular N bar because I happen took deliberately choose to do so. Given that I have frozen
it, what is the probability that are state at energy level E 3 is occupied, it is imply
the ratio of the number of particles occupying those state to the number of state that are
available. So, there are S 3 states available, there are N 3 particles occupying those states.
So, any one of the states if you take, in if there are let us have we are 10 states
here and there are 4 particles here so the chance that a particular state at S 3 is occupied
is simply 40 percent. Because there are 4 particles occupying one
of ten states, I mean 4 of 10 states so that is what you have is 40 percent. So, but that
is for a particular N bar, if I change that N bar that this number would change at at
the next N bar that I select, this N 3 might change to instead of 4 it might become 7,
in which case for that new N bar the probability of occupancy is 70 percent, right. So, this
when when I say N i by S i that is for a particular n bar so this is called conditional probability.
This is just a general probability that is state at energy level E i is occupied and
we are going to write a expression for that. This is the conditional probability that the
particular state is occupy and the condition is that this the distribution that currently
exist so that is the condition we have placed that is why it is called the conditional probability.
So, this conditional is have to do with this choice of this N bar, fine. So, this is the
conditional probability that it exist state of at energy level E i is occupied. So, what
do we now need? So, if you want to look at if you you want now, what we are going to
do now is to find what we need to do to this term, so that we can end up on this term so
that is the more general statement that we wish to make. What are we have to do using
this term to come up with this term. So, what is it that we have to do? We have
to see what is the probability that; now, we have different variety of arrangements
possible each each arrangements corresponds to a set like this so we need one information
which is what is that probability that we are at this particular N bar, right. So, I
have 1000 particles out of which I have put 10 here and I have put 100 here, I have put
something here, something here, something here. In another arrangement, I have put 500
here I have put 50 here and so on. So, all this possible arrangement might exist each
of which corresponds to an N bar. So, for every N bar there is a there is a probability
you will be at that N bar I mean as suppose to any of the number of other N bar set are
possible. So, you will write an another equation and then we will see what it means; again
I will put down the equation then I describe it to you. We will say first of all that p of N bar equals
W N bar by W total. What does this what does we, what do we mean by this? This is the probability
that the system actually has that specific distribution N bar that specific distribution
N bar. What is that probability it is the number of ways in which you can create that
N bar divided by the total number of ways in which you can create any N bar. Or in other
words this is simply W N bar by a sum of all the N bars if the let say if you put a j across
all j. So, N bar so, you have N 1 N 2 N 3 and so on, you have you can come up with N
bar j you can say, we call this j for example, and this is j.
So, if you sum it across all the N bars then whatever you will get is called total so,
we will we are just interested in one them so, we just say N bar by W total. And this
is the total number of ways in which you can arrange the particles across all possible
combinations, all possible N bars that you can get. So, all specific ways in which you
can specify N 1 N 2 N 3 N 4 and so on you take all of that together you can come up
with one W total of which for one of them whatever you get is appear. So, this is what
it is, this is the probability that the system is at particular state N bar. So, therefore,
f of E is now simply I will write it down. So, what it means is, the general probability
that an state at energy level E i is occupied, which is this term that is there, is this
is the term here which is the conditional probability that that particular state is
occupied. The condition begin that the state is at the overall set is at N bar times the
probability that it is at N bar right and summed over all possible N bars. So, this
states in to account all possibilities so, we will I will so I just write it down and
then we will examine this little so that so, what we mean is so, this is f (E i), so we
have all the information here, I will just restate what I have just put together here,
so that you follow what is being done right. This is the probability that an energy level
at that state at energy level E i is occupy so, this is the probability right so this
is the term that we are interested in. So, this is an in general regardless of all the
conditions in all conditions taken into account in all possible ways that the system could
exist all information put together this is the probability that state at energy level
at energy level E i is occupied right. Now, if you go back to what we put down here we
had this three possibilities our energy levels are fixed the number of states at those energy
levels are fixed. But we have freedom in selecting how many particles happen to be sitting at
that energy level. We had this freedom, we can keep we do have
this freedom we can select different number of particles and then distribute them at all
this energy levels. So, all these values can change so, we since they can change and we
have many many possibilities of those changes, we started out with one such possibility which
we we have designated as N bar. And when it we say N bar that means, for that instant
we have froze on the values of N 0 N 1 N 2 N 3 N 4 and so on. So, if you want to consider
the entire gamete of possibilities available to you, you have to look at all possible N
bar arrangements right N bar meaning this collection. So, I this one specification of
N bar another way you could specify it you will called it another N bar and so on.
So, you have lot of possibilities of N bar of which only one is put down on the board
here right. So, what we are basically saying is that the probability that state is occupied
here of energy level some energy level is occupied, is it is equal to the probability
that it is occupied, when you have specified a particular N bar. Times that total number
of the probability that it is the; it is at that N bar and you have to sum that up right.
To stated again N bar is this collection here and since you have many such N bars, there
is even associated with N bar; there is a probability that you are there at that N bar
right. Because you have you know several several
possibilities of N that you can several changes, you can make here so, for every given change
there is a new N bar so, you have a huge number of N bars available. Given how easy or how
many times you can make it that way you can have a certain probability that you are at
that particular N bar, when you put down the terms you will see that. So, all all I am
saying is that you have a variety of values available for N bar and if you look at those
values you will find that given that the system. This hole analysis is also based on the assumption
that all the possibilities that we can think of or all equally or individually equally
probable all right. So, in other words if I come with a particular
arrangement of the system it is just as as probable as another arrangement right. So,
what we are interested in is therefore, to see that if you take a particular arrangement,
in how many ways you can attain that arrangement. Each of those rows is equally probable and
therefore, if one arrangement can occur in ten in ten different ways another arrangement
can occur in 100 different ways. Individually the first arrangement is equally probable
as this each of those ten arrangements is as equally probable as each of those 100 arrangements.
But because one can occur in ten different ways, the other can occur in 100 different
ways; the second arrangement is ten times more probable than the first arrangement this
is the basic idea alright. So, you have ten ways to ten ways to attain
N bar in a particular manner and 100 ways to attain another another N bar, each of those
ten ways is just as probable as each of those 100 ways. And therefore, if you look at it
taken together this 100 ways will occur ten times more likely than this ten ways will
occur so, this is the basic idea. So, so, we have here the probability that are state
at energy level E i is occupied, that is equal to the conditional probability that state
E i is occupied the condition being that it is at N bar. Times the number of ways in which,
you can accomplish this N bar by the total number of ways in which you can accomplish
everything. So, so this is the information now what we
will find is, which is the basic statistical mechanics tool that we employ; is that, if
you look at the number of ways in which you can accomplish N bar this term right. You
have for every arrangement there is certain number ways in which you can accomplish it,
if you look at if you focus on this term you will find that for every arrangement you can
come up with a value of W N bar. And you will find one arrangement, one particular N bar
which corresponds to a particular value of N 0 N 1 N 2 N 3 N 4 and so on. That particular
arrangement will be the most probable arrangement, meaning the number of ways in which, you can
obtain that arrangement will be huge. And statistical mechanics which operates for
these kinds of system basically says that, when you identify that particular arrangement,
which has the maximum number of ways, in which you can accomplish it. You will find that
it is way more probable, significantly more probable than any other arrangement possible
with any other N bar that is possible right. Therefore, at that point what you will find
is that we will put that down here, I will call that most probable arrangement as N bar
zero. So, N bar zero is the most probable arrangement that you can get for which implies N bar zero has maximized this W N bar zero
alright. When W N bar zero is a maximum you will find
that its value will be much higher than all the other N bars that are present in the system,
all the W N bars that are present in the system, which will imply W N bar will be approximately
equal to one, when this happens and we say that in our system that given the kind of
system that we deal with, the most probable arrangement will will be will be much more
probable than even the next most probable arrangement. So, it will be not just much
more; we are talking of you know we did this example right at the beginning, we did when
we consider the maximum Bolts man statistics, we did this example where we found that as
you increase the number of particles in the system.
The most probable state in the system starts becoming more and more probable relative to
all the other possible arrangements in the system. The most probable micro state becomes
more and more probable than the other micro states combined and we did that for you know
two three and four particles and I basically highlighted the idea. That when you go to
larger and larger systems and we know when you talking of one mole of a solid of a material
which has ten power 23 particles and not just two three four particles, ten power twenty
three particles. When you go to that kind of a system the statistical mechanics indicates
that in the; in that kind of a system that most probable arrangement will essentially
be 99.999 percent probable, something like almost like 100 percent probable.
All the other arrangements combine will constitute a very mini squall number of arrangements
related to the most probable arrangement. So, therefore, statistical mechanics basically
operates on the idea that the most probable arrangement has a probability of occurrence
of almost equal to one. So, when we do a statistical mechanical approach, what we are essentially
doing is; even though the system has all the possible arrangements available to it and
this case we need all the possible N bars available to it. Because each N bar is one
arrangement, one micro state, so; one micro state is one N bar, even though you have so
many different possible micro states available. The most probable micro state the most probable
micro state which is N bar zero that micro state will occur in so many ways, that it
will it will be way more likely to occur, than all the other micro states in that system
combined. Not just the second one, not just the third one, second plus, third plus, four
plus whatever, whatever it is all the possible micro states that you can think of of the
system. You you add them all up the number of ways in which they can occur as a sum will
be a very tiny fraction of the number of ways in which the most probable micro state can
occur. So, we have not put down the equations for the most for the manner in which we will
calculate this number of ways in which the most probable micro state will occur, but
this is the idea all right. So, therefore, if you keep that in mind this
is when I say N i N i N bar zero, we are talking of the most occurring micro state. So, therefore
further most occurring micro state that is the idea that I just mention, the probability
that it exist will be almost equal to one. So, the number of ways in which you will accomplish
the most probable micro state, which is N bar zero divided by the total number of ways
in which you can attain any micro state. This this term in the denominator also includes
W N bar zero which includes W N bar zero, which is the most probable micro state plus
second most probable micro state plus the third most probable micro state and so on.
It is just that from the second the term if you set aside this term all the other terms
are a very negligible contribution to this term. So, the denominator is actually the
same as the numerator plus a very very very tiny at extra term. So, all so, the denominator
w total is n bar zero plus a very very tiny amount over and above that so therefore, this
ratio works out to be one in the most probable micro state. So, we are saying that if you
have manage to identify that particular micro state which which is the most probable micro
state then the of course, that it it directly means; that the number of ways in which you
can accomplish it is a maximum. So, therefore, we are basically saying that when you have
done that. So, this is the frame work so this is the
equation that that is that helps us narrow down the calculation that we wish to do, what
which is the basically directly take taking this equation which is the general case. The
general case is that the probability of occupancy at energy level E I, in general; is the probability
of occupancy at energy E i given a particular arrangement, times the probability that that
arrangements exists. When it becomes the most probable arrangement this ratio becomes equal
to one, in the most probable arrangement we are designate the designating this as N bar
zero and therefore, this is also N bar zero. When this ratio becomes equal to one essentially
we are basically saying that the probability of occupancy is simply the occupancy at at
the at this at this particular value N bar zero. So, where you have already reach the
maximum so that then the sum becomes meaningless so, then you will basically just directly
become this probability of occupancy at E i is the probability of occupancy at E i times
the at this particular condition. In principle the sum would include other terms, but all
those other terms are approaching zero, relative to this term. So therefore, the conditional
probability that a particular energy level E I, a particular a particular state at energy
level E i is occupied. When you have attained the most probable distribution
is therefore, the actual probability that state is occupy regardless of all condition.
So, this is the simplification we wish to them or we find is valid for the system right.
Now, our problem now changes to like I said we started with a narrow condition we widened
it, our narrow was that we only looked at in general; what is the conditional probability
that at particular state at energy level E i is occupied given a particular distribution
N bar. That is the that is the narrow point or the focus point where we started our calculation,
we widen that to a more general equation here, which is the, which gives as a more complete
picture of a what is that probability that state at energy level E i is occupied.
Then we find that even though with this is the even though this complete term is the
more more complete whole picture of the system. Most of the term in it are negligible so,
we can eliminate in fact all of those other terms, it is enough if you find out, what
is the probability that are state at energy level E i is occupy in the most probable distribution
that exist in the system. If you if you find that out you already automatically found this
out so, we have started some of focused point, come up with large larger expression out here
and from that again we find that it is enough if you if you just focus on a particular term.
What is this? This is simply if you compare with what we have here, it is simply N i by
S i subject to at N bar N i. So, or I put it over here it is simply so,
this is the final generalization that we make or final simplification that we make. So,
when the distribution is that most probable distribution N i N bar naught, for that distribution
when if you know the ratio N i by S i that is equal to the probability that state is
occupied so, that is the direct result of this equation here right. So, what I have
written here, when distribution is N i N bar naught N i by S i that is simply this term
here so, this term is equal to this term which is what which is what has out here so that
is. So, for the most probable distribution if you know N i by S i you automatically know
the probability of occupancy of an state at energy level E i subject to this conditions
that we have spoken of. So therefore, what we know come down to is
we need to find out N i by S i for N i naught for this particular distribution N bar naught.
So, if you we only need to focus on this distribution N bar naught and in that distribution if you
know N i by S i we have got the answer that we are looking for. So, to do that what we
are going to do is, we are going to look at the number of ways in which you can accomplish
any one given N bar any one given N bar and then maximize it, to see in under what conditions
you get the maximum value for this. In the calculation that we do, you will at some point
you will get a value for N i by S i and that is that is basically all the information that
we will get all right. So, this is the approach we are going to take all right. So, we are looking for an expression for W
N bar naught So, this is the number of ways in which the
most probable micro state N bar naught can be attained this is what we are looking for
and that is what that is the expression that we are interested in. So, what do we have?
We have E 0 energy level, S 0 is the number of state available at that energy level and
N 0 is the number of particles in that particular state, then we have E 1 S 1 N 1 E 2 S 2 N
2 and so on so, we just say E r S r N r some r we have all of this. Now, this is not some
arbitrary number, we have some at this point we do not know what these numbers are, we
want to see the condition under which the probability, the number of ways in which this
arrangement can be attain it is a maximum. So, we will start with so, what we are saying
in general is; we will start like this we will say energy level E i has S i states and
in this S i states there are N i particles. We want an expression now to see the number
of ways in which we can accomplish this. So, number of ways in which we can do this arrangement
even that we have an particular collection of N 1 N 0 N 1 N 2 N 3 N 4 and so on. So,
to do that first we will take any one particular energy value S i for example here, we will
see the number of ways in which this can be accomplished right, times the number of ways
in which this can be accomplished. The total number of ways in which this this
arrangement can be accomplished, the total number of ways in which this arrangement that
have been put on the board can be accomplished is is the product of the number of ways in
which you can attain this, times the number of ways in which you can attain this, times
the number of ways in which you can attain the other one and so on right. So, the product
of all the ways in which you can attain each of these combinations, given that you have
selected a combination so this is the way we have go about it. So, we will just take
a particular case E i energy level E i which has S i states and N i particles available,
the number of ways in which you can do this is simply S i factorial by N i factorial times
S i minus N i factorial. Why I said this? We it is essentially the
same as aim that you have N i particles and S i minus N i empty spaces. Because you have
S i states right you have some number of states here state state state state you have, some
number of particles you have, some particle here, some particle here it is same as saying
you have two particles, and two empty states. In how many ways can you jumble them up, that
the number of ways in which you can jumble them jumble them up at the number of unique
ways in which you can attain them, and therefore that represents the number of ways in which
you can accomplish that condition right. So, that is same as so, you have this is just
a particular example S i is the number of states we have so, the total number of ways.
In which you can arrange those S i states is S i factorial or select those states, in
which N i factorial is the number of particles that happens to be there or or comes from
the number of particles there and the number of vacant states that are available. So, if
you do this that gives you the total number of ways in which you can attain a particular
a combination of N i particles sitting in S i states right. And I said that the for
the entire so, this we will call a small w so, small w for a particular N i is this one.
So, the total so, this is for any one of this combinations right which are put down here
any one of those combinations can be attained for once you specify the value of N i you
can attain it in these many number of ways. So, for the overall system given that you
have N 0 at E 0 N 1 at E 1 N 2 at E 2 and so on. The total number of ways in which you
can accomplish this is simply the product of the number of ways you can accomplish each
one of them. So therefore, W of N 0 bar is simply the product S 0 factorial by N i N
0 factorial times S i sorry S 0 minus N 0 factorial product times S 1 factorial by N
1 factorial times S 1 minus N 1 factorial and so on. So, for the for energy level E
0 this is the number of ways you can accomplish the arrangement, energy level E 1 this is
the number of ways you can accomplish this arrangement, product similar term will come
for energy level E 2 E 3 E 4 E 5 and so on so, this is a product, a product of terms
that put like this right. So, the way of writing it is just the way
we and we describe discuss this earlier, just the way we write sigma for a sum, we write
pi for a product so, we write W N bar is pi over i of. So, W N bar naught is the total number of
ways in which you can attain this particular arrangement, all things considered so, all
things consider meaning; we have already said that this is one particular arrangement. So,
what is a total number of ways in which, we should not designate this yet as the N N bar
naught, we we do not yet know that this is the N bar naught, we are going to find the
condition under which this becomes N bar naught so, this is N bar in general. So, up given
N bar is there a given arrangement is there a given set of N 0 N 1 N 2 N 3 N 4 and so
on up to N r is there. So, given arrangement exist for which this is the number of ways
in which you can attain that arrangement. So, number of ways in which you can attain
the arrangement is the number of ways in which you can attain the arrangement for energy
level E 0 times the number of ways you can attain it for E 1 and so on. And so that is
pi over i this, when you maximize this, when this becomes maximum it becomes N bar 0 all
right. So, our calculation is in fact our interest in fact is to maximize this so, when
you maximize this you will get N bar zero all right so, that is what we are going to
do. So, we just rewrite that here this is pi over
i pi over i of S i factorial by N i factorial
minus times S i minus N i factorial. So, this is the total number of ways. This is the total number of ways in which
the micro state N bar can be attained all right so, this is the this is what we are
looking at and. We wish to We wish to identify that particular N bar
which maximize the value of this term here so, that is all we wish to do. And when we
do that the we will designate that which we will designate
So, we will designate that as N bar and W N bar zero respectively so, that is what we
will do so, our calculation will proceed on this phases. And in fact I will also tell
you that as we precede through the calculation our purpose even though this is the framework
within which we are actually operating. And this is this is the important idea that we
are following; our intent actually is in this process of identifying this N bar zero and
therefore, W N bar 0. In this process at an appropriate step we will find the ratio N
i by S i and we will corresponding to this W to this particular N bar zero, corresponding
to this particular arrangement. So, we will designate that as N i zero and
S i zero so, this is the probability that as energy level that is state at energy level
E i is occupied in the most probable distribution that exist within the system. So, that is
important then this is the probability that an state at energy level E i is occupy in
the most probable N the distribution that is available in the system. Then we do this
we have got the this is equal to f (E), since it is most probable distribution all the other
terms from the calculations that we have done so far can be neglected and we will be able
to say that this is the f (E), so this is all that we have all interested in.
Now, in our system so, this is what we are looking at there are two more pieces of information
that are relevant in our system which we will use for our calculation. The first is that
we said right at the beginning that the system is at thermal equilibrium which means the
total energy available to the system is fixed. So, E total is fixed. So, what is E total
this is simply implies the sum of the sum of all the the number of particles times the
energy level that they are at sum over i so, you have N i particles sitting at energy level
E i so, N i E i is the total energy of those particles at that energy, because of that
energy level, because of those particles at that energy level.
So, if you sum it across all energy levels that is the total energy of the system right.
You have different number of particles at each energy level at each of that different
energy levels so, within that energy level, the energy available is the number of particles
times that energy level. You sum it across all energy levels that is a total energy available
to the system, since this system is at equilibrium that total energy is fixed, we do not change
the total energy. So therefore, this term is a constant all right, so that is how this
is going to be and we also said the total number of particles in the system is fixed.
N total which is the total number of particles in the available in the system it is also
fixed. So, this is what is it, in terms of what we have designated within the system
in terms of the number of, in terms of the way we have distributed the particles across
the system and so on or the state of our system. It this simply means the sum over i of N i
the two constrains that we have here is that the total energy of the system is fixed. Therefore,
the sum of the number of particle at each energy level times the energy level; times
that energy level that sum is a constant, because that sum is equal to the total energy
of the system. So, that is fixed so that is a constant so, this is one constraint that
we are placing on the system, the other constraint we are placing on the system is; that the
total number of particles in the system is fixed. So, any any time you do an rearrangement
you you have to remove some particles from some state to put it in to another state right.
So, you cannot just on arbitrarily add on, you cannot arbitrarily increase the number
of particles in any given energy level. They have to come from one of the other energy
levels or some of those other energy levels so therefore, the total number of particles
is fixed N total is fixed. That means; the sum of the number of particles at all the
energy levels which then therefore, totals to the total number of particles present in
the system that is fixed that is a constant. So, our job in fact our immediate task for
example, is basically to see how we can maximize this term, if you maximize this term we are
able to identify the most probable state of the system.
Our task is to maximize this term here subject to the constraints that the energy is fixed
therefore, this term is fixed and this number of particles is fixed therefore, this term
is fixed. So, we have two sums here and you have product here, so we want to maximize
this subject to these two constraints. When we do that, we will get under the maximum
conditions, under the conditions that give as the maximum number of ways we can attain
a particular micro state. We will automatically find that the corresponding N i naught by
S i naught is the info piece of information that we are looking for which is simply at
the end of it all is the probability that has state at energy level E i is occupy.
So, that is all finally, it comes down to this we want that one particular probability
we have a lot of equations available to us from which we narrow down to this three equations.
If you look at focus at these three equations we will get this term, when we have got this
term, we have got the answer we are looking for and at that point you will look at the
implications of that term. So, with this we will halt this in this class we will continue
with this derivation in the next class, where we will start from this point here and we
will move forward with our calculations and see how we can arrive at this by focusing
on this equation here subject to these two concepts. So, we will see that in the next
class. Thank you.
