entropy example 1 - davidar/scholarpedia GitHub Wiki

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

The change of entropy as the system passes from state <math>A</math> to <math>B</math> equals

<math>

\Delta S = \int_{Q_1}^{Q_0} \frac cQ\,dQ + \int_{Q_2}^{Q_0} \frac cQ\,dQ. </math>

By an elementary calculus,

<math>

\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)]. </math>

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