Question

Consider a system of identical but distinguishable particles, each of which has two states, with energies ε and −ε. a) calculate the entropy in the microcanonical ensemble. Use Stirling approx- imation to simplify factorials and express Smc in terms of x = E/(N ε) and N only. b) Find the entropy Sc in the canonical ensemble and express it in terms of x = ⟨E⟩c/(N ε) and N only. Under which condition Smc ≃ Sc? c) Finally, calculate the entropy SGC in the grand-canonical ensemble and express it in terms of x = ⟨E⟩GC /(NGC ε) and NGC only. In order to do so express ZGC in terms of NGC and then invert the expression for NGC to get βμ in terms of NGC , x. How does SGC compare to Sc, Smc? Under which condition they will be equivalent?

Answer 1

To calculate the entropy in different ensembles for a system of identical but distinguishable particles, we use different expressions. The entropy in the microcanonical ensemble is given by Smc = Nk(ln 2). In the canonical ensemble, the entropy is Sc = Nk[(x + 1)ln(x + 1) - xln x], where x is the average energy per particle. In the grand-canonical ensemble, the entropy is SGC = -k[(1+x)ln(1+x) - xln x] - Nkln(N), where x is the average energy per particle and NGC is the particle number.

Step-by-step explanation:

To calculate the entropy in the microcanonical ensemble for a system of identical but distinguishable particles, we first need to determine the number of microstates or configurations. In this case, each particle can be in one of two states, with energies ε and -ε, resulting in 2^N possible configurations. The entropy, Smc, is then given by Smc = k ln(2^N), where k is Boltzmann's constant. Using Stirling approximation, we can simplify the expression to Smc = Nk(ln 2)

In the canonical ensemble, the entropy, Sc, is given by Sc = Nk[(x + 1)ln(x + 1) - xln x], where x = ⟨E⟩c/(N ε) is the average energy per particle. Smc will be approximately equal to Sc when x is much larger than 1 or much smaller than -1, which means the average energy per particle is significantly higher or lower than the energy states ε and -ε.

In the grand-canonical ensemble, the entropy, SGC, is given by SGC = -k[(1+x)ln(1+x) - xln x] - Nkln(N), where x = ⟨E⟩GC /(NGC ε) is the average energy per particle and NGC is the particle number. The entropy in the grand-canonical ensemble will typically be higher or lower than Sc and Smc, depending on the values of x and NGC. When x is much larger than 1 or much smaller than -1 and NGC is large, SGC will be approximately equal to Sc and Smc.