Statistical Ensembles Questions and Answers

Answer: B

The condition implies \(\dfrac{\partial\rho}{\partial t} \propto \{\rho, H\}\).

The total time derivative of a phase-space function along the motion is its explicit time derivative plus its Poisson bracket with the Hamiltonian:

\[ \begin{aligned} \frac{d\rho}{dt} &= \frac{\partial\rho}{\partial t} + \{\rho, H\} = 0 \\ \frac{\partial\rho}{\partial t} &= -\{\rho, H\} \end{aligned} \]

This is Liouville's theorem: the phase-space density is constant along trajectories, and its local rate of change is fixed by the bracket with H. ∂ρ/∂t = 0 holds only for a stationary ensemble, which dρ/dt = 0 alone does not guarantee. Brackets with p·q/2 or q·q/2 involve functions that are not the Hamiltonian of the system and do not generate its motion.

Answer: A

The minimum value is \( \frac{\mu_{1}T_{2} - \mu_{2}T_{1}}{T_{2} - T_{1}} \).

The whole system is closed, so \( dU_{2} = -dU_{1} \) and \( dN_{2} = -dN_{1} \), with volumes fixed. The total entropy change is

\[ \begin{aligned} dS = &\left(\frac{1}{T_{1}} - \frac{1}{T_{2}}\right) dU_{1} \\ - &\left(\frac{\mu_{1}}{T_{1}} - \frac{\mu_{2}}{T_{2}}\right) dN_{1} \end{aligned} \]

The second law requires \( dS \ge 0 \). Taking \( T_{2} \gt T_{1} \) and particles flowing into subsystem 1 (\( dN_{1} \gt 0 \)), divide through and multiply the fraction by \( T_{1}T_{2} \):

\[ \frac{dU_1}{dN_1}\ge\frac{\mu_1T_2-\mu_2T_1}{T_2-T_1} \]

The other three options change a sign in the numerator or the denominator, which no step of this derivation produces.

Answer: B

The fluctuation is proportional to \(1/\sqrt{N}\).

In the canonical ensemble, the mean square energy fluctuation is \(\langle(\Delta E)^{2}\rangle = kT^{2}C_{V}\). Both \(C_{V}\) and \(\langle E \rangle\) are proportional to \(N\), so the relative fluctuation is:

\[ \begin{aligned} \frac{\Delta E}{\langle E \rangle} &= \frac{\sqrt{kT^{2}C_{V}}}{\langle E \rangle} \propto \frac{\sqrt{N}}{N} \\ &= \frac{1}{\sqrt{N}} \end{aligned} \]

This is why the energy of a macroscopic system is sharply defined. 1/N is too fast a fall-off, and \(1/(3\sqrt{N})\) adds an arbitrary factor. The absolute fluctuation grows like \(\sqrt{N}\), not \(N\).

Answer: C

The ensemble is Grandcanonical.

Exchange of energy with a heat reservoir fixes the temperature, and exchange of particles with a particle reservoir fixes the chemical potential. A system with variable energy and particle number at fixed T, V and \(\mu\) is described by the grand canonical ensemble.

The canonical ensemble has a fixed particle number (heat reservoir only); the microcanonical ensemble is isolated, with fixed energy and particle number; the isobaric-isothermal ensemble fixes pressure instead of volume, not chemical potential.