GATE Physics Thermodynamics and Statistical Physics Syllabus

State the equipartition theorem.

In classical equilibrium, each quadratic degree of freedom in the Hamiltonian contributes an average energy of $\frac{1}{2}k_{B}T$ to the internal energy.

Write the Maxwell–Boltzmann speed distribution $f(v)$.

$$f(v) = 4\pi n \left(\frac{m}{2\pi k_{B}T}\right)^{3/2} v^{2}\, e^{-\tfrac{mv^{2}}{2k_{B}T}}$$

Give the most probable speed, mean speed, and rms speed of a Maxwell–Boltzmann gas.

$$v_{p}=\sqrt{\frac{2k_{B}T}{m}},\quad \langle v\rangle=\sqrt{\frac{8k_{B}T}{\pi m}},\quad v_{rms}=\sqrt{\frac{3k_{B}T}{m}}$$

Write the Fermi–Dirac distribution function.

$$\langle n_{i}\rangle = \frac{1}{e^{\beta(\varepsilon_{i}-\mu)}+1}$$

Write the Bose–Einstein distribution function.

$$\langle n_{i}\rangle = \frac{1}{e^{\beta(\varepsilon_{i}-\mu)}-1}$$

Define the Fermi energy $E_{F}$ of a degenerate Fermi gas.

The Fermi energy is the chemical potential at $T=0$: it is the energy of the highest occupied single-particle state, below which all states are filled and above which all are empty at absolute zero.

Give the Fermi energy of a 3D non-relativistic electron gas in terms of number density $n$.

$$E_{F} = \frac{\hbar^{2}}{2m}\left(3\pi^{2} n\right)^{2/3}$$

What is the average energy per particle of a degenerate Fermi gas at $T=0$?

$$\langle E \rangle = \frac{3}{5}E_{F}, \qquad U = \frac{3}{5}N E_{F}$$

What is the form of the electronic heat capacity of a degenerate Fermi gas at low temperature?

It is linear in temperature: $$C_{V} = \frac{\pi^{2}}{2}N k_{B}\frac{T}{T_{F}} \propto T$$ where $T_{F}=E_{F}/k_{B}$ is the Fermi temperature.

What is the degeneracy pressure of a Fermi gas at $T=0$?

A non-zero pressure arising purely from the Pauli exclusion principle: $$P = \frac{2}{5}n E_{F} = \frac{2U}{5V}$$ It exists even at absolute zero and supports white dwarfs and neutron stars.

State the density of states $g(\varepsilon)$ for a 3D free non-relativistic gas (including spin factor $g_{s}$).

$$g(\varepsilon) = \frac{g_{s} V}{4\pi^{2}}\left(\frac{2m}{\hbar^{2}}\right)^{3/2}\sqrt{\varepsilon}$$

State Planck's law for the spectral energy density of black-body radiation (per unit frequency).

$$u(\nu)\,d\nu = \frac{8\pi h \nu^{3}}{c^{3}}\,\frac{1}{e^{h\nu/k_{B}T}-1}\,d\nu$$

What is the average number of photons in a mode of frequency $\nu$ (Planck distribution), and why is $\mu=0$?

$$\langle n\rangle = \frac{1}{e^{h\nu/k_{B}T}-1}$$ The chemical potential $\mu=0$ because photon number is not conserved (photons are freely created and destroyed).

State the Stefan–Boltzmann law for the total energy density and emitted power of black-body radiation.

Energy density $u \propto T^{4}$, and radiated power per unit area $$j = \sigma T^{4}, \qquad \sigma = \frac{2\pi^{5}k_{B}^{4}}{15 h^{3}c^{2}}$$

State Wien's displacement law.

$$\lambda_{\max} T = b, \qquad b \approx 2.898\times 10^{-3}\ \text{m·K}$$ The peak wavelength of black-body radiation is inversely proportional to temperature.

What limit of Planck's law gives the Rayleigh–Jeans law, and what failure does it have?

In the low-frequency / high-temperature limit $h\nu \ll k_{B}T$, Planck's law reduces to $u(\nu)=\frac{8\pi\nu^{2}}{c^{3}}k_{B}T$. Extended to all $\nu$ it diverges as $\nu\to\infty$ — the ultraviolet catastrophe.

What limit of Planck's law gives Wien's distribution law?

In the high-frequency limit $h\nu \gg k_{B}T$, where $$u(\nu) \approx \frac{8\pi h\nu^{3}}{c^{3}}\,e^{-h\nu/k_{B}T}$$

How does the heat capacity of a photon gas (black-body radiation) depend on temperature?

$$C_{V} \propto T^{3}$$ since the total energy $U \propto T^{4}$.

What is Bose–Einstein condensation (BEC)?

Below a critical temperature, a macroscopic fraction of bosons in an ideal gas occupies the single lowest-energy (ground) quantum state, forming a Bose–Einstein condensate.

Give the condition (in terms of phase-space density / chemical potential) for the onset of BEC.

BEC sets in when the chemical potential $\mu \to \varepsilon_{0}$ (the ground-state energy, taken as $0$), equivalently when $n\lambda^{3} = \zeta(3/2)\approx 2.612$, the degeneracy condition.

Write the BEC critical temperature $T_{c}$ for an ideal Bose gas.

$$T_{c} = \frac{2\pi\hbar^{2}}{m k_{B}}\left(\frac{n}{g_{s}\,\zeta(3/2)}\right)^{2/3}$$

Below $T_{c}$, what fraction of bosons is in the condensate (ground state)?

$$\frac{N_{0}}{N} = 1 - \left(\frac{T}{T_{c}}\right)^{3/2}$$

Why do photons and phonons not undergo Bose–Einstein condensation?

Their number is not conserved (chemical potential $\mu=0$ always), so there is no constraint forcing macroscopic ground-state occupation as $T$ decreases; instead their number simply vanishes as $T\to 0$.

State the Gibbs phase rule.

$$F = C - P + 2$$ where $F$ is the number of degrees of freedom, $C$ the number of components, and $P$ the number of coexisting phases.

What is the condition for two phases to be in equilibrium (phase coexistence)?

At coexistence the two phases have equal temperature, equal pressure, and equal chemical potential (molar Gibbs free energy): $$T_{1}=T_{2},\quad P_{1}=P_{2},\quad \mu_{1}=\mu_{2}$$

State the Clausius–Clapeyron equation for the slope of a phase-coexistence curve.

$$\frac{dP}{dT} = \frac{L}{T\,\Delta V} = \frac{\Delta S}{\Delta V}$$ where $L$ is the latent heat and $\Delta V$ the volume change of the transition.

Classify phase transitions by Ehrenfest: what distinguishes first-order from second-order (continuous) transitions?

First-order: a discontinuity in the first derivatives of $G$ (entropy/volume), so there is latent heat. Second-order (continuous): first derivatives are continuous but second derivatives (e.g. $C_{P}$, compressibility) are discontinuous or divergent, with no latent heat.

What is a triple point?

The unique combination of temperature and pressure at which three phases (e.g. solid, liquid, vapour) of a single-component substance coexist in equilibrium; by the phase rule it has zero degrees of freedom ($F=0$).

What is the critical point of a liquid–gas system?

The end point of the liquid–vapour coexistence curve, at temperature $T_{c}$ and pressure $P_{c}$, beyond which the distinction between liquid and gas disappears; latent heat and density difference vanish there.

What conditions on the $P$–$V$ isotherm define the critical point in the van der Waals model?

At the critical point the isotherm has an inflection: $$\left(\frac{\partial P}{\partial V}\right)_{T_{c}} = 0, \qquad \left(\frac{\partial^{2} P}{\partial V^{2}}\right)_{T_{c}} = 0$$

Give the critical constants $T_{c}$, $V_{c}$, $P_{c}$ for the van der Waals equation in terms of $a$ and $b$.

$$V_{c}=3b,\quad T_{c}=\frac{8a}{27\,b\,R},\quad P_{c}=\frac{a}{27 b^{2}}$$ giving the universal ratio $\frac{P_{c}V_{c}}{R T_{c}}=\frac{3}{8}$.