Entropy/entropy example 1

Consider a system consisting of two identical and homogeneous solid bodies, of temperatures $T_1$ and $T_2$, respectively (state $A$). For our purposes, we take the states to be parameterized completely by $T_1$ and $T_2$; thus, the state space is two-dimensional. Assuming that temperature depends linearly on the heat content, the heat contained in the solids amounts to $Q_1=cT_1$ and $Q_2=cT_2$, respectively. All states with $Q_1+Q_2 = {const}$ have the same energy. Let $B$ denote the state where both solids contain the same amount of heat, $Q_0 = \frac {Q_1+Q_2}2$.

The change of entropy as the system passes from state $A$ to $B$ equals

$$\Delta S = \int_{Q_1}^{Q_0} \frac cQ\,dQ + \int_{Q_2}^{Q_0} \frac cQ\,dQ.$$

By an elementary calculus,

$$\Delta S = c(\log Q_0 - \log Q_1) + c(\log Q_0 - \log Q_2) = 2c\left[\log\left(\frac{Q_1+Q_2}2\right) - \frac{\log Q_1+\log Q_2}2\right].$$

Since the logarithmic function is strictly concave, this expression is positive, which means that the state $B$ has entropy larger then $A$. Thus $B$ has the largest entropy among all states with the same level of energy and so it is the equilibrium state.