# Statistics and thermodynamics

## The fundamental postulate of statistical mechanics[edit | edit source]

We still have not established a link between the ensemble formalism that we are developing and the thermodynamics of a system. This is what we are now going to do.

From thermodynamics we know (see Entropy and Thermodynamic potentials) that all the properties of a system can be obtained from its entropy through appropriate differentiations; we would therefore like to define the entropy of a system within the microcanonical ensemble.

However, we have no clues on what we can do in order to define it; of course we must link with some property of the microcanonical ensemble of our system, but we have nothing that can suggest us what we might use.
After all the only really new concept that we have introduced with the ensemble formalism is the phase space volume , so we can think that this is what must be related to the entropy.

We can understand that since it looks like we are at a dead end we have to "artificially" introduce something, i.e. we *must* do some assumptions in order to proceed.
The assumption we make is the *fundamental postulate of statistical mechanics*:

^{[1]}. Unfortunately there is no intuitive way to justify this postulate (in general we could have supposed that was proportional to some power of , or some other function different from the logarithm), it something we have to take as it is. However this doesn't mean that we must "religiously" belive in it; it is true that being a postulate we can't really "prove" , but we certainly can try to see if it is "reasonable". In other words what we would like to do now is to verify if as defined in the fundamental postulate of statistical mechanics is

*really*the entropy of a system; this means that we want to see, even if only qualitatively, if the consequences of the fundamental postulate of statistical mechanics do agree with what we know about the thermodynamics of a system or not. In particular we have seen in Entropy that in order to obtain all the thermodynamic information we need about a macroscopic system we just need to take appropriate derivatives of the entropy:

Therefore what we are going to do in the following is to see if the relations we have previously written follow from the fundamental postulate of statistical mechanics.

Before proceeding let us make an observation that will be useful for the computations.
Since and is extensive, the quantity is itself extensive and thus we can always write it as^{[2]}:

## Temperature and statistical mechanics[edit | edit source]

Let us now see how the temperature of a system comes into play within the microcanonical ensemble. Consider two systems, which we call 1 and 2, of volume , energy , at temperature and each composed of particles, with , separated by a fixed wall that allows only the exchange of energy between the two systems. The situation is as shown in the following figure:

We call the total energy and the total number of particles, and we know from thermodynamics that if initially then after some time the two systems reach an equilibrium and have the same temperature .
Now, it is intuitively clear^{[3]} that the phase space volume of the whole system is:

*any*of the possible microstates of the system 1 the system 2 can be in

*any*of its possible states; therefore the total number of the possible states of the whole system can be obtained integrating over all the possible values of . Now, since is extensive and can be written as in , we have (neglecting the dependence of on the volume, since it is fixed):

*saddle point*approximation (see the appendix The saddle point approximation). The result is

^{[4]}:

^{[5]}. Now, in order to explicitly see that the main contribution to comes from the configuration where , let us expand in a Taylor series the integrand around (and we write explicitly and because the second derivatives of and are negative in and , since they are maxima):

## Pressure and statistical mechanics[edit | edit source]

We now want to see, just like we have done with the temperature, what role does the pressure of a system play in the microcanonical ensemble. Let us therefore consider a system on which we can act using a piston of cross section :

If the pressure of the gas is , in order to maintain the system in equilibrium we must exert a force on the piston. If the gas inside the box is ideal the Hamiltonian of the system can be written as:

Now, by definition:

## Chemical potential and statistical mechanics[edit | edit source]

We conclude with the chemical potential. Let us therefore consider a system divided into two subsystems 1 and 2 connected by a hole that allows the exchange of particles, like the one represented here:

Its Hamiltonian will be:

We have already previously encountered the first equation, which has led us to . Focusing now on the second one, this leads to:

^{[6]}.

## Conclusions[edit | edit source]

Therefore with these qualitative reasonings we have shown that the fundamental postulate of statistical mechanics, leads to what we expect about the thermodynamics of a system. We can therefore conclude that it is indeed a reasonable assumption.

- ↑ In general, we should have used a generic constant , but at the end of the computations that allow to rederive the thermodynamics of the system we would find out that is precisely , so we use from the beginning just for the sake of simplicity.
- ↑ In fact, since is some extensive function then factorizing an we have:
- ↑ If this is not the case we can anyway easily obtain this result in a more "formal" way. In fact, we have: where we have integrated over in the last step. Now, using a "trick":
- ↑ The derivatives are intended to be taken with respect to the argument of the function:
- ↑ Alternatively we could have used a different approach to come to the same result. In fact from our analysis we have that two systems at thermal equilibrium in the microcanonical ensemble are such that has the same value for both, and from thermodynamics we know that two systems at thermal equilibrium share the same temperature; thus must be related to the temperature of a system, and since it has the dimensions of the inverse of a temperature we can
*define*the temperature in the microcanonical ensemble as:Also in this case we get to the result . - ↑ Remember that this, in the end, is only a qualitative way to show that the fundamental postulate of statistical mechanics is reasonable and leads to results compatible to what we know about the thermodynamics of a system.