Einstein Solid Microstates

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

Program EinsteinSolidMicrostates considers two systems which can share energy; the number of oscillators in each system is fixed. The idea is to explore the probability of various distributions of energy between the two systems.

Problem: Two Einstein solids in thermal contact

Consider two Einstein solids with $N_A = 4$ and $E_A = 10$ and $N_B = 4$ and $E_B = 2$. The two subsystems are thermally isolated from one another and from their surroundings. Verify that the number of microstates accessible to subsystem $A$ is 286 and the number of microstates accessible to subsystem $B$ is 10. What is the initial number of accessible microstates for the composite system? Program EinsteinSolidMicrostates determines the number of accessible microstates of an Einstein solid using (\ref{1}) and will help you answer the following questions.

  1. The internal constraint is removed so that the two subsystems may exchange energy. Determine the probability $P_A(E_A)$ that system $A$ has energy $E_A$. Plot $P_A$ versus $E_A$ and discuss its qualitative energy dependence.
  2. What is the probability that energy is transferred from system $A$ to system $B$ and from system $B$ to system $A$? Which system would you identify as the colder system?
  3. Determine the mean energy $\overline{E}_A$, the most probable energy $\skew3\tilde E_A$, the standard deviations $\sigma_A$ and $\sigma_B$, and the relative fluctuations $\sigma_A/\overline{E}_A$.
  4. What is the number of accessible microstates of the composite system after the internal constraint has been relaxed? What is the change in $S$ of the system (choose units such that $k=1$)?
  5. The quantity $S$ of the composite system when each subsystem is in its most probable macrostate is $\ln \Omega_A(\skew3\tilde E_A) \Omega_B(E_{\rm tot} - \skew3\tilde E_A)$. Compare the value of this contribution to the value of $S_{\rm tot} = \ln \sum_{E_A} \Omega_A(E_A) \Omega_B(E_{\rm tot}- E_A)$.
  6. Consider successively larger systems and describe the qualitative behavior of the various quantities.

Resources

Problems 4.5 and 4.7 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).

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