Question

Energy distributions in equivalent ensembles (Thermodynamics)

a) Why is there a maximum in the energy distribution of the Boltzmann distribution although the probability to find the system in a specific microstate n with energy E_n decreases monotonously like exp(-bE_n)/Z with E_n?

b) A given system can be described by a micro canonical, canonical or grand canonical ensemble. Sketch the energy distribution w(e) for each of these ensembles. Which properties are equal and which are different? Consider the variance of these distributions and their dependence of the systems volume. How do particle number fluctuations effect the grand canonical ensemble?

c) A closed system is divided into two subsystems of volume V1and V2 with equal particle densities. The subsystems are in thermal contact. When and under which conditions for V1 and V2

i) does an equilibrium require equal temperatures in the subsystems

ii) exists a maximum in the energy distribution w_1(e) of subsystem 1

iii) can subsystem 1 be described by a Boltzmann distribution?

Sketch the according zones in a (ln(V1), ln(V2)) diagram.

Answer 1

(a) There is no need of any confusion as Boltzmann distribution says that high energy microstates are less probable. This is because Boltzmann distribution applies only to small part of the system. If the small system is in a microstate of high energy then rest of the system has slightly lower energy. For the large system , we expect number of microstates to be rapidly increasing function of energy, so the probability is the product of rapidly increasing function of e and another rapidly decreasing function i.e. Boltzmann factor. This gives the sharp maxima of P(E) at some particular value of energy.

(c) Considering the two subsystems be filled with ideal gas the ideal gas equation can be applied for finding the condition between two volumes for the system to be in thermal equilibrium.

Next, for the energy to be maximum in the subsystem 1 I think, Volumes of two subsystems should be equal.