Evaluation of observables

In the previous exercises we have used the number of particles in the left side of the box, and the measure of the temperature $T$ to characterize the equilibrium of the system. Other parameter that can be used are the mean pressure $P$ and as well as the total energy $E$.

Other thermal quantity is the heat capacity $C_{V}=(\partial E/\partial T)_{V}$. $C_{V}$ is a measure of the amount of heat needed to produce a change in the temperature. Since this is and extensive quantity (depends on the size of the system) it is convenient to use the specific heat $% c_{V}=C_{V}/N$ instead. The easiest way to obtain it is to determine the mean potential energies at neighboring temperatures $T$ and $T+\Delta T$.

In order to determine the mean pressure, suppouse for a moment that the container has rigid walls. We know that the collisions of particles with the walls will cause a mean net force to be exerted on each element of the wall. The mean force per unit of area is the pressure $P$ of the gas. The force can be found by relating the force in the horizontal direction to the rate of change of the linear component of the linear momentum of the particles hitting the wall.

We can use a similar argument to calculate the pressure in the absence of walls. Since the pressure is uniform at equilibrium, we can relate the pressure to the transfer of momentum across an element of area anywhere in the system. Consider an element of area $dA$ and let $\mathbf{K}_{+}$ be the mean momentum crossing the surface per unit of time from left to right, and let $\mathbf{K}_{-}$ be the mean momentum crossing the surface per unit of time from right to left. The the mean force $\mathbf{F}$ is

$$\mathbf{F=K}_{+}\mathbf{-K}_{-}$$

and the mean pressure is given by

$$P=\frac{dF_{\alpha }}{dA}$$

where $F_{\alpha }$ is the component of the force normal to the element of area. (For two dimensions, the pressure is defined per unit of length)

An alternative way of calculating the pressure is from the virial theorem:

$$PV=NkT+\frac{1}{d}\langle \sum_i^N \mathbf{r}_i \cdot \mathbf{F}_i \rangle$$