4. Statistical Physics of Non-isolated Systems

In principle, with the tools of the last section we could tackle all the problems we want now. But it turns out to be hard to calculate the entropy of any isolated system more complicated than an ideal paramagnet. This is because in an isolated system the energy is fixed, and it becomes complicated to work out all the possible ways the total energy can be split between all the atoms of the system: we can't treat each atom as independent of all the others, even if they are non-interacting.

We don't have to consider isolated systems though. In this section we will consider systems in contact with a heat bath, so that their temperature, rather than their energy, is constant. This has the advantage that if the atoms of a system don't interact with one another, they can be treated independently.

For a macroscopic system, there is very little difference in the results from the two approaches. If the temperature is held constant the energy will fluctuate, but the fractional size of the fluctuations decreases as $1/\sqrt{N}$ and so, from a macroscopic point of view, the energy does not appear to vary and it makes little difference whether the heat bath is there or not. So lots of results we obtain in this section are also applicable to isolated, macroscopic systems.

We will introduce something call the partition function from which we can calculate the energy, pressure etc. The heart of the partition function is the Boltzmann distribution, already met last year, which gives the probability that a particle in contact with a heat bath will have a given energy.