Physical systems

Let us now consider a physical system, for example a gas of $N$ molecules inside a box of volume $V$. For simplicity, let us assume that the molecules of gas do not have internal degrees of freedom and that do not interact with each other. In fact, we need to be slightly careful here. If we consider truly non interacting particles, and we also isolate the system from the environment, then we condemn it to remain in the same microstate forever. We shall therefore partially lift the non-interaction assumption and allow the particles to exchange energy by occasional collisions. Since we are not interested in any time dependent description, we can make these occasional collisions as rare as we want, so that they do not affect appreciably the physical properties of the system at any one time and have the only purpose to allow the system to visit all its accessible microstates.

We shall also treat the system as having a discrete set of energy levels. This would correspond to the real physical situation, in which energy levels are quantised, and we will return as appropriately on the separation of these energy levels and what that means for the behaviour of the particles. Each particle will have possible energies $\epsilon_1 < \epsilon_2 < \epsilon_3 \dots < \epsilon_r \dots$, and the total energy of the system for a particular distribution of these single particles energies is $E = \sum_{i=1}^N \epsilon_{j_i}$, where $\epsilon_{j_i}$ is the energy of particle $i$. Here we have assumed that the single particle energies are non degenerate, for simplicity.

The number of ways a particular value of the total energy $E$ can be realised depends on its value, just as we discussed for the variable $f$ in the previous section. For example, the value $E_1 = N \epsilon_1$ can only be realised with all particles having energy $\epsilon_1$, and so there is a single microstate available for such value of $E_1$. The next possible energy level is obtained by having all particles but one in the ground state, $E_2 = \epsilon_2 + (N-1) \epsilon_1$, which can be realised in $N$ different ways ^3.2. The next energy level $E_3$ is realised by promoting two particles from $\epsilon_1$ to $\epsilon_2$, and we have $E_3 = (N-2)\epsilon_1 + 2 \epsilon_2$. There are $N (N-1)/2$ ways of choosing two of the $N$ particles to have energy $\epsilon_2$, and so we see that the degeneracy of $E$ quickly increases with its value. If there is a maximum value $\epsilon_{max}$, then we are in a similar situation of that for $f$ in the previous section, and $E$ can assume a maximum value $E_{max} = N \epsilon_{max}$, which can also only be realised by a single microstate. In this case the degeneracy of $E$ increases at first, reaches a maximum, and then decreases back to 1.

In the next section we will consider the situation in which the system is isolated from the environment, and so its total energy is fixed to some particular value $E$.