Temperature

The parameter $T \equiv 1/k \beta = \frac{1}{\gamma}$ has been defined so far as nothing but the inverse of the Lagrange Multiplier $\gamma$. Note that according to Eq. (36), however, $T$ can be defined as follows:

$$\frac{1}{T}= \frac{\partial S}{\partial E} \Bigg )_N.$$ (39)

The goal of this section is to show that $T$ can be identified with the temperature of the system because it is the same through out the system whenever the system is at thermal equilibrium. Consider a system at equilibrium, with ensemble average internal energy $E$, in the state of maximum entropy at fixed N. Consider a distribution of S, T, and E in compartments (1) and (2) as specified by the following diagram:

Consider a small displacement of heat $\delta E$ from compartment (1) to compartment (2):

$$\delta E^{(1)}= -\delta E, \qquad \delta E^{(2)}= \delta E.$$ (40)

Since the system was originally at the state of maximum entropy, such a displacement would produce a change of entropy

$$\delta S)_{E,N} \leq 0,$$ (41)

where

$$\delta S = \delta S^{(1)} + \delta S^{(2)} = \frac{\partial S^{(1... ...\delta E^{(2)} =\left ( \frac{1}{T_1} - \frac{1}{T_2} \right ) \delta E \leq 0.$$ (42)

Since the inequality introduced by Eq. (42) has to be valid for any positve or negative $\delta E$, then $\boxed{T_1 = T_2}$.