Enumeration

Consider a model in which the particles are distinguishable, non-interacting, and have only two possible velocities $v_0$ and $-v_0$. Since the particles are non-interacting, the size of the system and the positions of the particles are irrelevant. Consider a system of $N=4$ of such particles and fixed energy $E=2v_0^2$. (the mass of the particles is taken to be unity) The number of possible microstates with these constraints is $2^N=16$. The enumeration of these configuration allows us to calculate the ensemble averages for the physical quantities of the system. The enumeration procedure is equivalent to the one used to enumerate 1D random walks. If we fix the number $n_R$ of particles moving to the right, the number of configurations for a given $n$ are 1, 4, 6, 4, 1 for $N=$ 0, 1, 2, 3, 4, respectively, and the corresponding probabilities $P_n$ are 1/16, 4/16, 6/16, 4/16, 1/16. Hence, the mean number of particles moving to the right is

$$\langle n \rangle = \sum{nP_n}=(0\times 1 + 1\times 4 + 2 \times 6 + 3 \times 4 + 4\times 1)/16=2.$$