Which of the systems listed below is the largest, having $N_{A}=N_{B}$ and $U=6 N_{A} \varepsilon$ for which there is a better than 1 in a billion chance of seeing one or the other solid having zero energy? (Hint: Using StatMech to check cases is probably the easiest approach.)
A) $N_{A}=6$
B) $N_{A}=8$
C) $N_{A}=10$
D) $N_{A}=12$
E) $N_{A}=15$

Question

Quizplus · Accepted Answer

The correct answer is based on statistical mechanics principles, specifically the Boltzmann distribution and the concept of microstates. For a system with $N_{A}=N_{B}$ and $U=6N_{A}\varepsilon$, the probability of one solid having zero energy while the other has all the energy can be calculated using the number of microstates ($\Omega$) corresponding to these conditions. The probability of observing one solid at zero energy is significantly higher for smaller systems due to the exponential dependence on the system size in the Boltzmann factor. However, as $N_{A}$ increases, the number of microstates for distributing energy increases, making it increasingly unlikely to find all energy in one solid. The hint suggests using statistical mechanics to evaluate the likelihood for different $N_{A}$ values, and among the given options, $N_{A}=8$ is the largest system size for which there is still a better than 1 in a billion chance of observing one of the solids at zero energy, based on the calculation of probabilities from the distribution of energy states.

Which of the Systems Listed Below Is the Largest, Having NA=NBN_{A}=N_{B}NA​=NB​

Which of the Systems Listed Below Is the Largest, Having $N_{A}=N_{B}$