Einstein Solid Entropy

The Einsein model consists of $N$ oscillators and $E$ packets of energy which can be distributed among the oscillators in $\Omega$ ways, where $$\Omega = \frac{(E+N-1)!}{E! (N-1)!}.$$ Each way is a microstate.

Program EinsteinSolidEntropy considers two systems which can share energy; the number of oscillators in each system is fixed. The key idea is to explore the total entropy $S = k\ln{ \Omega}$ and the inverse temperature $\partial S/\partial E$ for each system at equilibrium. In the program $k = 1$.

Problem: Two Einstein solids in thermal contact

The entropies of two Einstein solids in thermal contact are computed and plotted in Program EinsteinSolidEntropy.

  1. Consider $N_A = 50$, $N_B = 30$, and total energy $E_{\rm tot} = 200$. Discuss the qualitative dependence of $S_A$, $S_B$, and $S_{\rm tot}$ on the energy $E_A$. Why is $S_A$ is an increasing function of $E_A$ and $S_B$ a decreasing function of $E_A$? Given this dependence of $S_A$ and $S_B$ on $E_A$, why does $S_{\rm tot}$ have a maximum at a particular value of $E_A$?
  2. The values of the inverse of $\partial S_A/\partial E_A$ and $\partial S_B/\partial E_B$ are shown in the lower right corner by clicking on the corresponding plots. What is the relation of these values of the inverse of the slopes to the temperature? Determine the value of $E_A$ at which the slopes are equal. What can you say about the total entropy at this value of $E_A$? Consider several values of $N_A$, $N_B \neq N_A$ and $E_{\rm tot}$.

Resource

Problem 4.6 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).

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