Statistical Mechanics

Fundamental Principles

Statistical Ensembles

Classical Statistical Mechanics

Quantum Statistical Mechanics

Ideal Gases

Classical Ideal Gas

Partition function: \(Z = \frac{V^N}{N!\lambda^{3N}}\)

Where $\lambda = \sqrt{\frac{2\pi\hbar^2}{mk_BT}}$ is the thermal de Broglie wavelength.

Equation of state: $PV = Nk_BT$

Quantum Ideal Gases

Fermi Gas

At low temperature, forms a Fermi sphere in momentum space.

Fermi energy: $E_F = \frac{\hbar^2}{2m}(3\pi^2n)^{2/3}$

Specific heat at low T: $C_V \propto T$

Bose Gas

Below critical temperature: \(T_c = \frac{2\pi\hbar^2}{mk_B}\left(\frac{n}{2.612}\right)^{2/3}\)

Bose-Einstein condensation occurs.

Interacting Systems

Virial Expansion

For weakly interacting gas: \(\frac{PV}{Nk_BT} = 1 + B_2(T)n + B_3(T)n^2 + ...\)

Second virial coefficient: \(B_2(T) = -\frac{1}{2V}\int (e^{-\beta u(r)} - 1)d^3r\)

Mean Field Theory

Approximate interactions by average field.

Example - Ising model magnetization: \(m = \tanh\left(\frac{m z J}{k_B T}\right)\)

Critical temperature: $T_c = \frac{zJ}{k_B}$

Correlation Functions

Two-point correlation: \(G(r) = \langle s_i s_j \rangle - \langle s_i \rangle\langle s_j \rangle\)

Near critical point: $G(r) \sim \frac{e^{-r/\xi}}{r^{d-2+\eta}}$

Phase Transitions

Fluctuations

Gaussian Fluctuations

For energy fluctuations: \(\langle (\Delta E)^2 \rangle = k_B T^2 C_V\)

For particle number: \(\langle (\Delta N)^2 \rangle = k_B T \left(\frac{\partial N}{\partial \mu}\right)_{T,V}\)

Fluctuation-Dissipation Theorem

Connects response functions to equilibrium fluctuations: \(\chi(\omega) = \frac{1}{k_B T} \int_0^{\infty} \langle A(t)A(0) \rangle e^{i\omega t} dt\)

Einstein’s Relations

• Diffusion: $D = \mu k_B T$ (mobility $\mu$)
• Conductivity: $\sigma = ne^2\tau/m$ (Drude model)

Non-equilibrium Statistical Mechanics

Boltzmann Equation

Evolution of distribution function: \(\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_r f + \frac{\mathbf{F}}{m} \cdot \nabla_v f = \left(\frac{\partial f}{\partial t}\right)_{coll}\)

H-theorem

Boltzmann’s H-function decreases: \(H = \int f \ln f d^3v\) \(\frac{dH}{dt} \leq 0\)

Linear Response Theory

For small perturbation $F(t)$: \(\langle A(t) \rangle = \langle A \rangle_0 + \int_{-\infty}^t \chi(t-t') F(t') dt'\)

Kubo formula for conductivity: \(\sigma = \lim_{\omega \to 0} \frac{1}{\omega} \int_0^{\infty} dt e^{i\omega t} \langle J(t)J(0) \rangle\)

Applications

Condensed Matter Physics

• Electronic properties of solids
• Superconductivity (BCS theory)
• Quantum Hall effects
• Topological phases

Soft Matter

• Polymer physics
• Liquid crystals
• Colloids
• Biological membranes

Cosmology

• Early universe thermodynamics
• Dark matter freeze-out
• Cosmic microwave background

Quantum Information

• Thermal states in quantum computing
• Entanglement entropy
• Quantum thermodynamics

Computational Methods

Monte Carlo

• Metropolis algorithm
• Cluster algorithms (Wolff, Swendsen-Wang)
• Quantum Monte Carlo

Molecular Dynamics

• Verlet algorithm
• Nosé-Hoover thermostat
• Parrinello-Rahman barostat

Density Functional Theory

Hohenberg-Kohn theorem: Ground state density determines all properties.

Kohn-Sham equations: \(\left[-\frac{\hbar^2}{2m}\nabla^2 + v_{\text{eff}}\left[n\right](r)\right]\psi_i(r) = \epsilon_i\psi_i(r)\)

Advanced Topics

Renormalization Group

• Block spin transformations
• Fixed points and universality
• Epsilon expansion
• Functional renormalization

Conformal Field Theory

At critical points, systems exhibit conformal symmetry.

Central charge characterizes universality class.

AdS/CFT Correspondence

Connects strongly coupled field theories to weakly coupled gravity.

Applications to quark-gluon plasma and condensed matter.

Graduate-Level Mathematical Formalism

Information Theory and Statistical Mechanics

Shannon entropy: \(S = -k_B \sum_i p_i \ln p_i\)

Maximum entropy principle: The equilibrium distribution maximizes entropy subject to constraints.

Canonical ensemble from MaxEnt: Maximize S subject to:

• Normalization: $\sum_i p_i = 1$
• Energy constraint: $\sum_i p_i E_i = \langle E \rangle$

Using Lagrange multipliers: \(p_i = \frac{e^{-\beta E_i}}{Z}\)

Jaynes’ principle: Statistical mechanics as inference theory

Relative entropy (Kullback-Leibler divergence): \(D_{KL}(p||q) = \sum_i p_i \ln\left(\frac{p_i}{q_i}\right) \geq 0\)

Advanced Ensemble Theory

Generalized Ensembles

Tsallis statistics: \(S_q = \frac{k_B (1 - \sum_i p_i^q)}{q - 1}\)

Pressure ensemble: (NPT) \(\Delta(N,P,T) = \int_0^{\infty} dV \, e^{-\beta PV} Z(N,V,T)\)

Isothermal-isobaric partition function: \(\Delta = \frac{k_B T}{P} Z(N,\langle V \rangle,T) e^{\beta P \langle V \rangle}\)

Jarzynski Equality and Fluctuation Theorems

Jarzynski equality: \(\langle e^{-\beta W} \rangle = e^{-\beta \Delta F}\)

Crooks fluctuation theorem: \(\frac{P_F(W)}{P_R(-W)} = e^{\beta(W - \Delta F)}\)

Work distribution: Gaussian near equilibrium \(P(W) \approx (2\pi\sigma^2)^{-1/2} \exp\left[-\frac{(W - \langle W \rangle)^2}{2\sigma^2}\right]\)

Path Integral Formulation

Quantum partition function: \(Z = \text{Tr}(e^{-\beta H}) = \int \mathcal{D}[q] \exp(-S_E[q]/\hbar)\)

Euclidean action: \(S_E = \int_0^{\beta\hbar} d\tau \left[\frac{m\dot{x}^2}{2} + V(q)\right]\)

Feynman-Kac formula: Connection to diffusion \(\langle q_f|e^{-\beta H}|q_i\rangle = \int_{q(0)=q_i}^{q(\beta\hbar)=q_f} \mathcal{D}[q] \, e^{-S_E[q]/\hbar}\)

Effective action at finite temperature: \(\Gamma[q_c] = -k_B T \ln Z[J] + \int d\tau \, J(\tau)q_c(\tau)\)

Field Theoretic Methods

Hubbard-Stratonovich Transformation

For interaction term: \(\exp\left[\frac{\beta}{2} \sum_{ij} J_{ij}s_i s_j\right] = \int \mathcal{D}[\phi] \exp\left[-\frac{\beta}{2} \sum_{ij} \phi_i(J^{-1})_{ij}\phi_j + \beta\sum_i \phi_i s_i\right]\)

Replica Method

For disordered systems: \(\langle \ln Z \rangle = \lim_{n\to 0} \frac{\langle Z^n \rangle - 1}{n}\)

Replica symmetry breaking: Order parameter $q_{ab}$

Functional Integral Representation

Grand canonical ensemble: \(\Xi = \int \mathcal{D}[\psi^*, \psi] \exp(-S[\psi^*, \psi])\)

Action for bosons: \(S = \int_0^{\beta} d\tau \int d^dr \left[\psi^*(\partial_\tau - \mu)\psi + \frac{\hbar^2}{2m}|\nabla\psi|^2 + U(\psi^*\psi)\right]\)

Critical Phenomena: Advanced Treatment

Scaling Theory

Scaling hypothesis: Near $T_c$, singular part of free energy: \(f_s(t, h) = b^{-d}f_s(b^{y_t}t, b^{y_h}h)\)

Where $y_t = 1/\nu$, $y_h = d - \beta/\nu$

Scaling relations derivation:

• From $f_s$: $\alpha = 2 - d\nu$
• From $m = -\partial f/\partial h$: $\beta = (d - y_h)\nu$
• From $\chi = \partial^2f/\partial h^2$: $\gamma = (2y_h - d)\nu$

Renormalization Group: Field Theory

$\phi^4$ theory action: \(S = \int d^dx \left[\frac{1}{2}(\nabla\phi)^2 + \frac{r}{2}\phi^2 + \frac{u}{4!}\phi^4\right]\)

RG flow equations (one-loop): \(\frac{dr}{dl} = (2 - \eta)r + Au \frac{r^2}{1 + r}\) \(\frac{du}{dl} = \varepsilon u - Bu^2 + \frac{Cu^3}{(1 + r)^2}\)

Fixed points:

• Gaussian: $(r^, u^) = (0, 0)$
• Wilson-Fisher: $(r^, u^) = (-\varepsilon/A, \varepsilon/B)$

Critical exponents ($\varepsilon$-expansion): \(\nu = \frac{1}{2} + \frac{\varepsilon}{12} + O(\varepsilon^2)\) \(\eta = \frac{\varepsilon^2}{54} + O(\varepsilon^3)\)

Conformal Field Theory at Criticality

Conformal algebra in 2D: Virasoro algebra \([L_m, L_n] = (m - n)L_{m+n} + \frac{c}{12} m(m^2 - 1)\delta_{m+n,0}\)

Central charge: Characterizes universality class

• Ising: $c = 1/2$
• XY model: $c = 1$
• Potts model (q states): $c = 1 - 6/[q(q+1)]$

Operator product expansion: \(\phi_i(z)\phi_j(0) = \sum_k C_{ijk}z^{h_k-h_i-h_j}\phi_k(0)\)

Exact Solutions

2D Ising Model (Onsager Solution)

Transfer matrix method: \(Z = \text{Tr}(T^N)\)

Critical temperature: \(\sinh\left(\frac{2J}{k_B T_c}\right) = 1\)

Free energy per site: \(f = -k_B T \ln(2\cosh(2\beta J)) - \frac{k_B T}{2\pi} \int_0^\pi d\theta \, \ln\left[1 + \sqrt{1 - \kappa^2\sin^2\theta}\right]\)

Where $\kappa = 2\sinh(2\beta J)/\cosh^2(2\beta J)$

Magnetization ($T < T_c$): \(m = \left[1 - \sinh^{-4}(2\beta J)\right]^{1/8}\)

Bethe Ansatz

1D Heisenberg chain: \(H = J\sum_i \boldsymbol{\sigma}_i \cdot \boldsymbol{\sigma}_{i+1}\)

Bethe equations: \(k_j L = 2\pi I_j - \sum_{k\neq j} \theta(k_j - k_k)\)

Ground state energy: \(\frac{E_0}{N} = -J \ln 2 + \frac{J}{4}\)

Non-equilibrium Field Theory

Keldysh Formalism

Contour ordering: Forward and backward branches

Green’s functions: \(G^{++}(t,t') = -i\langle T\phi(t)\phi(t')\rangle\) \(G^{--}(t,t') = -i\langle \tilde{T}\phi(t)\phi(t')\rangle\) \(G^{+-}(t,t') = -i\langle \phi(t')\phi(t)\rangle\) \(G^{-+}(t,t') = -i\langle \phi(t)\phi(t')\rangle\)

Keldysh rotation: \(G^R = G^{++} - G^{+-}\) \(G^A = G^{++} - G^{-+}\) \(G^K = G^{++} + G^{--} - G^{+-} - G^{-+}\)

Langevin Dynamics

Stochastic equation: \(\partial_t\phi = -\Gamma\frac{\delta F}{\delta\phi} + \eta\)

Noise correlations: \(\langle \eta(x,t)\eta(x',t') \rangle = 2\Gamma k_B T\delta(x-x')\delta(t-t')\)

Martin-Siggia-Rose formalism: Path integral with response field \(Z = \int \mathcal{D}[\phi, \tilde{\phi}] \exp(iS[\phi, \tilde{\phi}])\)

Quantum Many-Body Systems

Fermi Liquid Theory

Quasiparticle concept: Landau parameters $f^s$, $f^a$

Effective mass: \(\frac{m^*}{m} = 1 + \frac{F_1^s}{3}\)

Compressibility: \(\frac{\kappa}{\kappa_0} = (1 + F_0^s)^{-1}\)

Collective modes: Zero sound velocity \(s = v_F\sqrt{1 + \frac{F_0^s}{3}}\)

BCS Theory of Superconductivity

BCS Hamiltonian: \(H = \sum_k \varepsilon_k c^\dagger_{k\sigma}c_{k\sigma} - g\sum_{kk'} c^\dagger_{k\uparrow}c^\dagger_{-k\downarrow}c_{-k'\downarrow}c_{k'\uparrow}\)

Gap equation: \(\Delta = g\sum_k \frac{\Delta}{2E_k} \tanh(\beta E_k/2)\)

Critical temperature: \(k_B T_c = 1.14\hbar\omega_D \exp(-1/N(0)g)\)

Luttinger Liquids (1D)

Bosonization: Fermion operators to Boson fields \(\psi(x) \sim \exp[i\phi(x)]\)

Luttinger parameter: $K < 1$ repulsive, $K > 1$ attractive

Power-law correlations: \(\langle \psi^\dagger(x)\psi(0) \rangle \sim x^{-1/(2K)}\)

Modern Developments

Tensor Network Methods

Matrix Product States (MPS): \(|\psi\rangle = \sum_{s_1...s_N} \text{Tr}(A^{s_1}...A^{s_N})|s_1...s_N\rangle\)

DMRG algorithm: Variational optimization of MPS

Area law entanglement: $S \sim L^{d-1}$ for ground states

Machine Learning in Statistical Mechanics

Neural network representation of states: \(\psi(s) = \exp\left[\sum_i a_i s_i + \sum_{ij} W_{ij}h_i(s)s_j + ...\right]\)

Variational Monte Carlo with NNs: \(E = \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle}\)

Unsupervised learning of phases:

• Principal component analysis
• Autoencoders
• Diffusion maps

Quantum Thermalization

Eigenstate Thermalization Hypothesis (ETH): \(\langle E_n|O|E_m\rangle = O(E)\delta_{nm} + e^{-S(E)/2}f_O(E,\omega)R_{nm}\)

Many-body localization: Failure of thermalization

Floquet systems: Time-periodic Hamiltonians

Stochastic Processes and Field Theory

Doi-Peliti Formalism

Creation/annihilation operators for classical particles: \(a^\dagger|n\rangle = |n+1\rangle\) \(a|n\rangle = n|n-1\rangle\)

Master equation to “Schrodinger” equation: \(\partial_t|\psi\rangle = H|\psi\rangle\)

Coherent state path integral: \(P(n,t) = \int \mathcal{D}[\phi^*,\phi] \exp(-S[\phi^*,\phi])\)

Active Matter

Toner-Tu equations: Flocking \(\partial_t\rho + \nabla\cdot(\rho\mathbf{v}) = 0\) \(\partial_t\mathbf{v} + \lambda(\mathbf{v}\cdot\nabla)\mathbf{v} = \alpha\mathbf{v} - \beta|\mathbf{v}|^2\mathbf{v} - \nabla P + \nu\nabla^2\mathbf{v} + \mathbf{f}\)

Motility-induced phase separation: \(\partial_t\rho = \nabla\cdot[(D(\rho) + D_t)\nabla\rho]\)

Advanced Computational Methods

Quantum Monte Carlo

Path integral Monte Carlo: \(\rho(R,R';\beta) = (2\pi\lambda\beta)^{-3N/2}\sum_P (\pm)^P \exp\left[-\beta\sum_i V(R_i)\right]\)

Sign problem: Fermionic systems, frustrated magnets

Continuous-time algorithms: Worm algorithm, CT-QMC

Machine Learning Acceleration

import torch
import torch.nn as nn

class VariationalWavefunction(nn.Module):
    def __init__(self, L, hidden_dim=100):
        super().__init__()
        self.L = L
        self.net = nn.Sequential(
            nn.Linear(L, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, 2)  # Real and imaginary parts
        )
    
    def forward(self, states):
        """states: (batch_size, L) binary spin configurations"""
        out = self.net(states.float())
        log_amp = out[:, 0]
        phase = out[:, 1]
        return log_amp, phase
    
    def sample(self, n_samples):
        """Metropolis sampling from |psi|^2"""
        states = torch.randint(0, 2, (n_samples, self.L))
        # Implement Metropolis-Hastings...
        return states

Research Frontiers

Quantum Information and Statistical Mechanics

Entanglement entropy scaling:

• Volume law: S ∼ L^d (thermal, excited states)
• Area law: S ∼ L^{d-1} (ground states)
• Logarithmic: S ∼ log L (1D critical)

Tensor network representations:

• MPS, PEPS, MERA
• Entanglement renormalization

Non-equilibrium Quantum Systems

Prethermalization: Quasi-stationary states

Dynamical phase transitions: Non-analytic behavior in Loschmidt echo

Floquet engineering: Designer Hamiltonians

Machine Learning and Physics

Reverse engineering Hamiltonians: Learning from data

Accelerating simulations: Neural network quantum states

Discovering order parameters: Unsupervised learning

Topological Phases

Symmetry-protected topological phases:

• Classification by cohomology
• Edge states

Topological order:

• Anyonic excitations
• Topological entanglement entropy

Many-Body Localization

Phenomenology:

• Area law entanglement
• Emergent integrability
• l-bits (localized integrals of motion)

Transitions:

• Thermal to MBL
• MBL to ergodic

References and Further Reading

Classic Textbooks

1. Pathria & Beale - Statistical Mechanics
2. Kardar - Statistical Physics of Particles & Fields
3. Landau & Lifshitz - Statistical Physics (Parts 1 & 2)
4. Huang - Statistical Mechanics

Advanced Monographs

1. Altland & Simons - Condensed Matter Field Theory
2. Sachdev - Quantum Phase Transitions
3. Nishimori & Ortiz - Elements of Phase Transitions and Critical Phenomena
4. Täuber - Critical Dynamics

Specialized Topics

1. Gogolin, Nersesyan & Tsvelik - Bosonization and Strongly Correlated Systems
2. Schollwöck - The density-matrix renormalization group in the age of matrix product states
3. Eisert, Cramer & Plenio - Colloquium: Area laws for the entanglement entropy
4. Carleo & Troyer - Solving the quantum many-body problem with artificial neural networks

Recent Reviews

1. Nandkishore & Huse - Many-body localization and thermalization (2015)
2. Calabrese, Cardy & Doyon - Special issue on quantum integrability in out of equilibrium systems (2016)
3. Abanin et al. - Colloquium: Many-body localization, thermalization, and entanglement (2019)
4. Carrasquilla - Machine learning for quantum matter (2020)

See Also