CSIR NET Physical Sciences Thermodynamic and Statistical Physics Syllabus

What is the exchange interaction and why is it essential for ferromagnetism?

A quantum-mechanical interaction arising from the Pauli exclusion principle and Coulomb repulsion, favoring parallel (or antiparallel) spin alignment. It is far stronger than magnetic dipole interactions and is what produces ordering at room-temperature scales.

Write the Hamiltonian of the Ising model.

$$H = -J\sum_{\langle i,j\rangle} s_i s_j - h\sum_i s_i, \qquad s_i = \pm 1$$ where $J$ is the coupling (exchange) constant, the first sum is over nearest-neighbor pairs, and $h$ is the external field.

Does the 1D Ising model exhibit a phase transition at finite temperature?

No. The one-dimensional Ising model has no spontaneous magnetization at any $T > 0$; its critical temperature is $T_c = 0$. Domain-wall (entropy) arguments show order is destroyed at any finite temperature.

What is the exact critical temperature of the 2D square-lattice Ising model (Onsager)?

$$\frac{k_B T_c}{J} = \frac{2}{\ln(1+\sqrt{2})} \approx 2.269$$ Onsager's exact solution gives a continuous transition with a logarithmically diverging specific heat ($\alpha = 0$) and $\beta = 1/8$.

What are the mean-field (Bragg-Williams) critical exponents of the Ising model?

Mean-field theory gives $\alpha = 0$ (jump), $\beta = \frac{1}{2}$, $\gamma = 1$, and $\delta = 3$. These are exact above the upper critical dimension $d = 4$.

Solve for the transition temperature of the Ising model in mean-field approximation.

With $z$ nearest neighbors, self-consistency $m = \tanh\!\left(\frac{zJm + h}{k_B T}\right)$ gives a nonzero spontaneous $m$ below $$k_B T_c = zJ.$$

What is Bose-Einstein condensation (BEC)?

A macroscopic occupation of the single-particle ground state by a finite fraction of bosons below a critical temperature $T_c$, occurring because the chemical potential approaches the ground-state energy. It is a phase transition driven purely by quantum statistics.

Give the BEC critical temperature for an ideal Bose gas.

$$T_c = \frac{2\pi\hbar^{2}}{m k_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}$$ where $n = N/V$ is the number density and $\zeta(3/2) \approx 2.612$ is the Riemann zeta function.

What is the condensate fraction of an ideal Bose gas below $T_c$?

$$\frac{N_0}{N} = 1 - \left(\frac{T}{T_c}\right)^{3/2}$$ for $T < T_c$. At $T = 0$ all particles occupy the ground state; $N_0 = 0$ for $T \geq T_c$.

What is the Bose-Einstein distribution function?

$$\langle n_\varepsilon\rangle = \frac{1}{e^{(\varepsilon - \mu)/k_B T} - 1}$$ giving the mean occupation of a single-particle state of energy $\varepsilon$. It requires $\mu \leq \varepsilon_{min}$.

What condition on the thermal de Broglie wavelength signals the onset of BEC?

BEC occurs when the phase-space density satisfies $n\lambda^{3} \approx \zeta(3/2) \approx 2.612$, where $\lambda = \frac{h}{\sqrt{2\pi m k_B T}}$. Physically, when interparticle spacing becomes comparable to $\lambda$, wavefunctions overlap.

Write the one-dimensional diffusion (heat) equation.

$$\frac{\partial \phi}{\partial t} = D\frac{\partial^{2}\phi}{\partial x^{2}}$$ where $\phi(x,t)$ is the concentration (or temperature) and $D$ is the diffusion constant.

State Fick's first law of diffusion.

$$\vec{J} = -D\,\nabla \phi$$ The diffusive flux $\vec{J}$ is proportional to and directed down the concentration gradient, with $D$ the diffusion coefficient.

How does Fick's second law (diffusion equation) follow from Fick's first law?

Combining the continuity equation $\frac{\partial \phi}{\partial t} + \nabla\cdot\vec{J} = 0$ with $\vec{J} = -D\nabla\phi$ gives $$\frac{\partial \phi}{\partial t} = D\nabla^{2}\phi.$$

What is the fundamental (Green's function) solution of the 1D diffusion equation for a point source?

$$\phi(x,t) = \frac{1}{\sqrt{4\pi D t}}\exp\!\left(-\frac{x^{2}}{4Dt}\right)$$ a spreading Gaussian whose variance grows as $\langle x^{2}\rangle = 2Dt$.

What is a simple random walk and how does its mean displacement behave?

A process where a particle takes successive steps of fixed length $a$ in random directions. For an unbiased walk, the mean displacement is zero, $\langle x\rangle = 0$, since forward and backward steps are equally likely.

What is the mean square displacement of a 1D random walk after $N$ steps?

$$\langle x^{2}\rangle = N a^{2}$$ so the root-mean-square displacement grows as $x_{rms} = a\sqrt{N}$, characteristic of diffusive (sub-ballistic) spreading.

How does a random walk connect to diffusion in the continuum limit?

Taking many small steps, the discrete random walk yields the diffusion equation with $$D = \frac{a^{2}}{2\tau}$$ in 1D, where $a$ is the step length and $\tau$ the time per step. The probability distribution becomes Gaussian.

What is the probability distribution of displacement for a 1D random walk after many steps?

By the central limit theorem it approaches a Gaussian: $$P(x, N) = \frac{1}{\sqrt{2\pi N a^{2}}}\exp\!\left(-\frac{x^{2}}{2Na^{2}}\right)$$ with mean $0$ and variance $Na^{2}$.

What is Brownian motion?

The random, erratic motion of a microscopic particle suspended in a fluid, caused by incessant collisions with thermally agitated fluid molecules. It is the physical realization of a continuous random walk and provided evidence for the atomic theory of matter.

State Einstein's relation for the diffusion coefficient of a Brownian particle.

$$D = \frac{k_B T}{\gamma} = \mu k_B T$$ where $\gamma$ is the friction coefficient and $\mu = 1/\gamma$ the mobility. This is a form of the fluctuation-dissipation theorem.

Give the Stokes-Einstein relation for a spherical Brownian particle.

$$D = \frac{k_B T}{6\pi\eta r}$$ where $\eta$ is the fluid viscosity and $r$ the particle radius, using Stokes' drag $\gamma = 6\pi\eta r$.

What is Einstein's result for the mean square displacement of a Brownian particle?

$$\langle x^{2}\rangle = 2Dt$$ in one dimension (and $\langle r^{2}\rangle = 6Dt$ in three dimensions). The displacement grows as $\sqrt{t}$, not linearly in $t$.

Write the Langevin equation for a Brownian particle.

$$m\frac{dv}{dt} = -\gamma v + \eta(t)$$ where $-\gamma v$ is the systematic drag and $\eta(t)$ is a random (fluctuating) force with $\langle \eta(t)\rangle = 0$ and $\langle \eta(t)\eta(t')\rangle = 2\gamma k_B T\,\delta(t - t')$.

What does the fluctuation-dissipation theorem state in the context of Brownian motion?

It relates the dissipative friction $\gamma$ to the fluctuating random force: the strength of thermal fluctuations and the dissipation arise from the same molecular collisions. Quantitatively, $\langle \eta(t)\eta(t')\rangle = 2\gamma k_B T\,\delta(t - t')$, ensuring equilibrium at temperature $T$.

What is the velocity autocorrelation function for a free Brownian particle from the Langevin equation?

$$\langle v(0)v(t)\rangle = \frac{k_B T}{m}\, e^{-\gamma t/m}$$ It decays exponentially on the relaxation time $\tau = m/\gamma$, with $\langle v^{2}\rangle = k_B T/m$ at $t = 0$ by equipartition.

What is the Fokker-Planck equation and what does it describe?

An equation for the time evolution of the probability density $P(x,t)$ of a stochastic process: $$\frac{\partial P}{\partial t} = -\frac{\partial}{\partial x}\big(A(x)P\big) + \frac{\partial^{2}}{\partial x^{2}}\big(B(x)P\big)$$ where $A$ is drift and $B$ is diffusion. It is the macroscopic counterpart of the Langevin equation.

What distinguishes nonequilibrium processes from equilibrium ones?

Nonequilibrium processes involve net fluxes (of heat, particles, charge) and entropy production, with thermodynamic forces (gradients) driving currents. Unlike equilibrium states, they are not described by a single set of state variables and require transport/kinetic theory.

State Onsager's reciprocal relations for linear nonequilibrium transport.

For coupled fluxes $J_i = \sum_j L_{ij} X_j$ driven by forces $X_j$, the kinetic coefficients are symmetric: $$L_{ij} = L_{ji}$$ (in the absence of magnetic fields), a consequence of microscopic time-reversal symmetry.

What is the principle of detailed balance in a system at equilibrium?

At equilibrium, each elementary process is balanced by its exact reverse, so every microscopic transition rate satisfies $$P_i W_{i\to j} = P_j W_{j\to i}$$ giving zero net flux between any pair of states. Its violation signals a nonequilibrium (driven) steady state.