Temperature and the Canonical Ensemble

Let us consider the more realistic case in which our system is in thermal contact with the environment, allowing energy to be exchanged in the form of heat. This situation corresponds to a system in a heat reservoir or heat bath. Now consider that the total energy of the environment and the system is fixed and constant, but the energy of the system can vary. Imagine a large number of copies of the composite system+reservoir, and we consider that this composite supersystem is isolated and can be described by the microcanonical ensemble. However, since we are only interested in the physical properties of the system without the reservoir, we need to know the probability $P_s$ of finding the laboratory system in the state $s$ with energy $E_s$. The ensemble which describes the probability distribution of a system in thermal equilibrium with a heat bath is known as the ``canonical ensemble''.

An example of such a system is our demon, immersed in the heat bath of the other $N$ particles. The microstate of the demon is defined by its energy $E_d$.

Our strategy for finding the form of the probability distribution of the canonical ensemble is to do a computer simulation of a demon which can exchange energy with an ideal gas of $N$ particles. The ideal gas will serve as the heat bath and we will determine the probability $P(E_d)$ that the demon has energy $E_d$. The exact form of this probability distribution is given by

$$P(E_d)=\frac{1}{Z}e^{-E_d/k_BT}$$ (112)

where $Z$ is the normalization constant such that the sum over all states of the demon is unity. The parameter $T$ in (113) is called the ``absolute temperature''. If $T$ is measured in Kelvin, $k_B=1.38\times 10^-{16} erg deg^{-1}$. The probability distribution (113) is called the ``Boltzmann'' or ``canonical'' distribution.

Note that the Boltzmann distribution is specified by the temperature. If we assume that the form (113) is applicable to any system in thermal equilibrium with a heat bath, we see that the macrostate in the canonical ensemble is specified by $T$, $N$ and $V$. In contrast, a macrostate in the microcanonical ensemble is specified by $E$, $N$ and $V$.

The form (113) of the Boltzmann distribution provides a simple way of computing $T$ from the mean energy $\langle E_d \rangle$ of the demon. Since $\langle E_d \rangle$ is given by

$$\langle E_d \rangle \frac{\int dE E e^{-E/k_BT}}{\int dE e^{-E/k_BT}}=k_BT,$$ (113)

we see that $T$ is simply the average demon energy divided by $k_B$. Note that the result $\langle E_d \rangle = k_BT$ holds only if the energy of the demon can take continuous values.