Thermal Interaction

Consider a purely thermal interaction between two systems, $A$ and $A'$, that contain a large number of particles. Suppose that the internal energies of these two systems are $U$ and $U'$, respectively. The external parameters are held fixed, so that systems $A$ and $A'$ cannot do work on one another. However, we shall assume that the systems are free to exchange heat energy (i.e., they are in thermal contact). It is convenient to divide the energy scale into small subdivisions of width $\delta U$. The number of accessible states of $A$ (i.e., states in which the internal energy of the whole system lies between $U$ and $U+\delta U$) is denoted ${\mit\Omega}(U)$. Likewise, the number of accessible states of $A'$ is denoted ${\mit\Omega}'(U')$.

The combined system $A^{(0)} = A + A'$ is assumed to be isolated (i.e., it neither does work on, nor exchanges heat with, its surroundings). It follows the total internal energy, $U^{(0)}$, is constant. When speaking of thermal contact between two distinct systems, we usually assume that the mutual interaction is sufficiently weak for the internal energies to be additive. Thus,

$$U + U' \simeq U^{(0)} = {\rm constant}.$$ (5.302)

According to Equation (5.302), if the internal energy of $A$ lies in the range $U$ to $U+\delta U$ then the internal energy of $A'$ must lie between $U^{(0)}- U -\delta U$ and $U^{(0)}- U$. Thus, the number of accessible states for each system is given by ${\mit\Omega}(U)$ and ${\mit\Omega}'(U^{(0)}-U)$, respectively. Because every possible state of $A$ can be combined with every possible state of $A'$ to form a distinct state, the total number of distinct states accessible to $A^{(0)}$ when the energy of $A$ lies in the range $U$ to $U+\delta U$ is

$${\mit\Omega}^{(0)}(U) = {\mit\Omega}(U)\, {\mit\Omega}'(U^{(0)} - U).$$ (5.303)

Consider an ensemble of pairs of thermally interacting systems, $A$ and $A'$, that are left undisturbed, so that they can attain thermal equilibrium. The principle of equal a priori probabilities is applicable to this situation. (See Section 5.4.2.) According to this principle, the probability of occurrence of a given macroscopic state is proportional to the number of accessible microscopic states, because all microscopic states are equally likely. Thus, the probability that the system $A$ has an energy lying in the range $U$ to $U+\delta U$ can be written

$$P(U) = C\, {\mit\Omega}(U) \,{\mit\Omega}'(U^{(0)} - U),$$ (5.304)

where $C$ is a constant that is independent of $U$.

We know, from Section 5.4.4, that the typical variation of the number of accessible states with energy is of the form

$${\mit\Omega} \propto U^{3N/2},$$ (5.305)

where $N$ is the number of molecules. For a macroscopic system, $N$ is an exceedingly large number. It follows that the probability, $P(U)$, in Equation (5.304) is the product of an extremely rapidly increasing function of $U$, and an extremely rapidly decreasing function of $U$. Hence, we would expect the probability to exhibit a very pronounced maximum at some particular value of the energy, $U_f$.