entropy example 1

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< Entropy
Tomasz Downarowicz (2007), Scholarpedia, 2(11):3901. doi:10.4249/scholarpedia.5166 revision #25680 [link to/cite this article]

Curator: Tomasz Downarowicz, Institute of Mathematics Computer Science, Wroclaw University of Technology, Poland

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.


Invited by: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Action editor: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
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