Statistical Ensembles Questions and Answers

Q: Consider a closed system is subdivided into two subsystems 1 and 2, which are connected such that internal energy and particles may be exchanged and volume of the two remain constant. Under this condition the minimum value of the quantity \( \left(\frac{dU_{1}}{dN_{1}}\right) \) is given by (here μ's and T's are the respective chemical potential and temperature) :

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.

Q: If a thermodynamic system has N particles at temperature T, then the fluctuation in average energy is proportional to :

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 \(N\).

Q: If the system under consideration is in contact with both a heat reservoir and a particle reservoir, then the ensemble is :

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.

Let \(\rho(\vec{p}, \vec{q}, t)\) be the phase space density of an ensemble of a system. The Hamiltonian of the system is \(H(\vec{p}, \vec{q})\). If \(\{A, B\}\) denotes the Poisson bracket of \(A\) and \(B\), then

\(\dfrac{d\rho}{dt} = 0\)

implies

A.\(\dfrac{\partial\rho}{\partial t} = 0\)
B.\(\dfrac{\partial\rho}{\partial t} \propto \{\rho, H\}\)
C.\(\frac{\partial\rho}{\partial t}\propto\{\rho,\frac{\vec{p}\cdot\vec{q}}{2}\}\)
D.\(\frac{\partial\rho}{\partial t}\propto\{\rho,\frac{\vec{q}\cdot\vec{q}}{2}\}\)

Exam Year(s)

GATE Physics - 2024,

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.

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Consider a closed system is subdivided into two subsystems 1 and 2, which are connected such that internal energy and particles may be exchanged and volume of the two remain constant. Under this condition the minimum value of the quantity \( \left(\frac{dU_{1}}{dN_{1}}\right) \) is given by (here μ's and T's are the respective chemical potential and temperature) :

A.\( \frac{\mu_{1}T_{2} - \mu_{2}T_{1}}{T_{2} - T_{1}} \)
B.\( \frac{\mu_{1}T_{2} + \mu_{2}T_{1}}{T_{2} - T_{1}} \)
C.\( \frac{\mu_{1}T_{2} - \mu_{2}T_{1}}{T_{2} + T_{1}} \)
D.\( \frac{\mu_{1}T_{2} + \mu_{2}T_{1}}{T_{1} + T_{2}} \)

Exam Year(s)

MH SET - 2018, SET Exam - 2018,

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.

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If a thermodynamic system has N particles at temperature T, then the fluctuation in average energy is proportional to :

A.1/N
B.\(1/\sqrt{N}\)
C.\(1/(3\sqrt{N})\)
D.N

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

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 \(N\).

Discuss in Forum

If the system under consideration is in contact with both a heat reservoir and a particle reservoir, then the ensemble is :

A.Canonical
B.Microcanonical
C.Grandcanonical
D.Isobaric-isothermal

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

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.

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