Statistical Physics and Kinetic Theory: /20 Summary

Context and scope

the supplied physics reference, Applications of Classical Physics — Chapters 3–6 Kinetic Theory · Statistical Mechanics · Statistical Thermodynamics · Random Processes

/20 Summary

• Central object: the distribution function $\mathcal{N}(\mathbf{x},\mathbf{p},t)$ in 6-D (or 7-D relativistic) phase space — number of particles per unit volume per unit momentum.
• Three pillars of statistical physics:
  1. Kinetic theory (Ch. 3): non-equilibrium evolution via Liouville / Boltzmann equations.
  2. Statistical mechanics (Ch. 4–5): equilibrium ensembles connect microstates ↔︎ thermodynamics.
  3. Random processes (Ch. 6): time-series fluctuations, noise, Brownian motion, FDT.
• Bedrock principle: maximum entropy under given constraints ⇒ all equilibrium distributions (Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac, Planck).
• The triple connection between (i) microscopic dynamics, (ii) equilibrium thermodynamics, and (iii) fluctuations is the fluctuation-dissipation theorem — the signature result of this Part.
• Three ensembles as Legendre transforms:
  • Microcanonical: fixed $(E, V, N)$ → entropy $S$
  • Canonical: fixed $(T, V, N)$ → Helmholtz $F$
  • Grand canonical: fixed $(T, V, \mu)$ → grand potential $\Omega$
• Practical payoff: every later Part (fluids, plasma, optics, GR) imports equations of state, transport coefficients, and noise spectra from here.

Master Map

mindmap
  root((Stat Physics))
    Ch.3 Kinetic Theory
      Phase space
      Distribution function N
      Liouville theorem
      Quantum statistics
        Maxwell-Boltzmann
        Bose-Einstein
        Fermi-Dirac
        Planck
      Equations of state
      Mean free path
      Boltzmann eqn
    Ch.4 Stat Mechanics
      Microcanonical
      Canonical
      Grand canonical
      Partition function Z
      Entropy
      Density of states
      Bose condensation
      Blackbody radiation
    Ch.5 Stat Thermo
      U F G H Omega
      Maxwell relations
      Phase transitions
      Saha equation
      Chemical potential
      Fluctuations
    Ch.6 Random Processes
      Markov chains
      Stationary ensembles
      Wiener-Khinchin
      Langevin
      Fokker-Planck
      Brownian motion
      Fluctuation-Dissipation
      Shot noise
      1/f noise

Kinetic Theory

The Distribution Function

The fundamental object of kinetic theory is the number density in phase space:

$$\mathcal{N}(\mathbf{x}, \mathbf{p}, t)\, d^3x\, d^3p = \text{number of particles in } d^3x\, d^3p$$

Properties:

• Units: (length)⁻³ (momentum)⁻³
• Always $\ge 0$
• $\mathcal{N}$ is a Lorentz invariant (in B&T's convention) — the volume element $d^3x\, d^3p/E$ is the invariant measure, but conventions vary. Always verify.

Reduced/integrated quantities:

Mean values (single-particle averages over $\mathcal{N}$): $$\langle Q\rangle = \frac{\int Q\, \mathcal{N}\, d^3p}{\int \mathcal{N}\, d^3p}$$

Liouville's Theorem

For a Hamiltonian system, phase-space volume is preserved along the flow:

$$\boxed{\frac{D\mathcal{N}}{Dt} \equiv \frac{\partial \mathcal{N}}{\partial t} + \dot{x}^i\frac{\partial \mathcal{N}}{\partial x^i} + \dot{p}^i\frac{\partial \mathcal{N}}{\partial p^i} = 0 \quad \text{(collisionless)}}$$

Equivalently (with Hamilton's equations $\dot{x}^i = \partial H/\partial p_i$, $\dot{p}_i = -\partial H/\partial x^i$):

$$\frac{\partial \mathcal{N}}{\partial t} + \{H, \mathcal{N}\} = 0$$

where $\{\cdot,\cdot\}$ is the Poisson bracket.

Physical picture: the distribution function is constant along particle trajectories in phase space. It's the analog of incompressibility for the "fluid" of particles in $(\mathbf{x},\mathbf{p})$ space.

Boltzmann equation (add collisions): $$\frac{D\mathcal{N}}{Dt} = \left(\frac{\partial \mathcal{N}}{\partial t}\right)_{\text{coll}}$$

The collision integral (binary collisions, classical): $$\left(\frac{\partial f}{\partial t}\right)_{\text{coll}} = \int d^3p_1\, d\Omega\, |\mathbf{v}-\mathbf{v}_1|\,\sigma(\Omega)\, [f' f'_1 - f f_1]$$

Equilibrium Distributions — Master Form

Any equilibrium distribution in B&T takes the form:

$$\boxed{\eta(E) = \frac{1}{e^{(E-\mu)/k_B T} \pm 1 \text{ or } 0}}$$

where $\eta$ is the mean occupation number per quantum state.

Then $\mathcal{N} = (g_s/h^3)\,\eta(E)$ where $g_s$ counts spin/polarization states.

Specific Cases — Number, Energy, Pressure

For an ideal classical gas (M-B distribution, non-relativistic):

$$\mathcal{N}(\mathbf{p}) = n\left(\frac{1}{2\pi m k_B T}\right)^{3/2} e^{-p^2/(2 m k_B T)}$$

• $n = N/V$
• Equipartition: $\langle E\rangle = \tfrac{3}{2} k_B T$ per particle (translation)
• $P = n k_B T$ (ideal gas law — emerges from $\int p_i v_j \mathcal{N} d^3 p$)
• $c_V = \tfrac{3}{2} k_B$ per particle (monatomic)

Speed distribution (Maxwell): $$f(v)\, dv = 4\pi n \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-mv^2/(2 k_B T)}\, dv$$

Equation of State — Polytropic Index

For an ideal classical gas: $$P = (\gamma-1)\, u \qquad u = \text{internal energy density}$$

Adiabatic index $\gamma = c_P/c_V$:

Universal relation (from kinetic theory): $P = \tfrac{1}{3}\int p v\, \mathcal{N}\, d^3 p$ holds for any isotropic distribution.

Degenerate Fermi Gas (Cold Limit)

When $k_B T \ll E_F$ (Fermi energy), $\eta \to \Theta(E_F - E)$ (step function).

Non-relativistic degenerate:

$$E_F = \frac{\hbar^2}{2m}(3\pi^2 n)^{2/3}, \qquad P = \tfrac{2}{5} n E_F \propto n^{5/3}$$

→ supports white dwarfs (Chandrasekhar)

Ultra-relativistic degenerate:

$$E_F = \hbar c (3\pi^2 n)^{1/3}, \qquad P = \tfrac{1}{4} n E_F \propto n^{4/3}$$

→ Chandrasekhar mass limit $M_{\text{Ch}} \approx 1.4 M_\odot$ (no stable equilibrium above this for cold electron-degenerate stars)

Photon Gas (Planck Distribution)

$$\mathcal{N}_\gamma(\omega)\,d\omega = \frac{1}{\pi^2 c^3} \frac{\omega^2}{e^{\hbar\omega/k_BT} - 1} d\omega$$

Integrated quantities:

Adiabatic index: $\gamma_{\text{rad}} = 4/3$.

Mean Free Path & Transport (Heuristic)

Mean free path: $\ell = 1/(n\sigma)$ where $\sigma$ is cross section.

Collision time: $\tau = \ell/\bar v$.

Order-of-magnitude transport coefficients (random-walk argument):

Wiedemann-Franz law (metals): $\kappa/(\sigma_e T) = L$, with Lorenz number $L = \pi^2 k_B^2/(3 e^2)$ — both transport coefficients carried by same electrons.

Statistical Mechanics

Microstates, Ensembles, Ergodicity

Microstate: complete specification of quantum state of the system (a point in phase space classically, or a basis state quantum-mechanically).

Macrostate: specified by macroscopic parameters $(E, V, N, T, P, \mu, \ldots)$ — many microstates per macrostate.

Ergodic hypothesis: time average along a trajectory = ensemble average over all accessible microstates with equal a priori probability (subject to conservation laws).

Statistical equilibrium: the ensemble distribution is stationary under microscopic evolution.

The Three Ensembles

flowchart LR
    A[Microcanonical<br/>E, V, N fixed<br/>S = k ln W]
    B[Canonical<br/>T, V, N fixed<br/>F = -kT ln Z]
    C[Grand Canonical<br/>T, V, μ fixed<br/>Ω = -kT ln Ξ]
    A -->|exchange E w/ bath| B
    B -->|exchange N w/ bath| C

Microcanonical Ensemble

System isolated. All accessible microstates equally probable.

$$P_n = \frac{1}{W(E,V,N)}, \qquad W = \#\text{microstates with energy } E$$

Boltzmann's relation: $$\boxed{S(E,V,N) = k_B \ln W(E,V,N)}$$

— inscribed on Boltzmann's tombstone, the foundation of statistical thermodynamics.

Canonical Ensemble

System exchanges energy with reservoir at temperature $T$. Probability of microstate $n$ with energy $E_n$:

$$P_n = \frac{e^{-\beta E_n}}{Z}, \qquad \beta = \frac{1}{k_B T}$$

Partition function:

$$\boxed{Z(T,V,N) = \sum_n e^{-\beta E_n}}$$

Everything derives from $Z$:

Last identity = fluctuation-response relation: energy fluctuations ↔︎ heat capacity.

Grand Canonical Ensemble

System exchanges both energy and particles with reservoir.

$$P_{n,N} = \frac{e^{-\beta(E_{n,N} - \mu N)}}{\Xi}$$

Grand partition function:

$$\Xi(T,V,\mu) = \sum_N e^{\beta\mu N} Z(T,V,N) = \sum_{n,N} e^{-\beta(E_{n,N} - \mu N)}$$

Grand potential: $\Omega = -k_B T \ln\Xi = U - TS - \mu N = -PV$.

Density of States $g(E)$

Continuum limit of $W(E)$. Number of states with energy in $(E, E+dE)$.

Examples:

Use: convert $\sum_n \to \int g(E)\, dE$. All ensemble integrals become 1-D.

Bose-Einstein Condensation

For bosons with conserved number ($\mu < 0$ always for free Bose gas):

• $\mu \to 0^-$ at finite density ⇒ ground state acquires macroscopic occupation.
• Critical temperature:

$$k_B T_c = \frac{2\pi\hbar^2}{m}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}, \quad \zeta(3/2) \approx 2.612$$

• Below $T_c$: condensate fraction $N_0/N = 1 - (T/T_c)^{3/2}$.
• $P$ at $T < T_c$: depends only on $T$, not on $n$ (the condensate has zero pressure).

Observed in: ultra-cold alkali vapors (Cornell, Wieman, Ketterle, Nobel 2001), liquid ⁴He (superfluid), exciton-polariton systems.

Black-Body Radiation (Detail)

Apply grand canonical to photons (massless, $\mu=0$, spin-1 with 2 polarizations):

$$u(\omega)\,d\omega = \frac{\hbar\omega^3}{\pi^2 c^3} \frac{d\omega}{e^{\hbar\omega/k_BT} - 1}$$

Limits:

• Rayleigh-Jeans ($\hbar\omega \ll k_BT$): $u(\omega) \approx \omega^2 k_BT/(\pi^2 c^3)$
• Wien ($\hbar\omega \gg k_BT$): $u(\omega) \approx (\hbar\omega^3/\pi^2 c^3)\,e^{-\hbar\omega/k_BT}$

Wien's displacement law: $\hbar\omega_{\text{peak}} \approx 2.82 k_B T$, $\lambda_{\text{peak}} T \approx 2.898\times10^{-3}$ m·K.

Total flux: $F = \sigma T^4$, Stefan-Boltzmann constant $\sigma = 5.67\times10^{-8}$ W/(m²·K⁴).

Specific Heat of Solids

Debye $T^3$ law (low T):

$$C_V \approx \frac{12\pi^4}{5} N k_B \left(\frac{T}{\Theta_D}\right)^3$$

with Debye temperature $\Theta_D = \hbar\omega_D/k_B$, typically 100–500 K.

Statistical Thermodynamics

The Four Laws

Thermodynamic Potentials — Legendre Transforms

flowchart TD
    U["U(S,V,N)\nInternal Energy"] -->|−PV| H["H = U + PV\nEnthalpy\nH(S,P,N)"]
    U -->|−TS| F["F = U − TS\nHelmholtz\nF(T,V,N)"]
    F -->|−PV| G["G = F + PV\nGibbs\nG(T,P,N)"]
    F -->|−μN| O["Ω = F − μN\nGrand Potential\nΩ(T,V,μ)"]

Differential forms:

$$dU = TdS - PdV + \mu dN$$ $$dH = TdS + VdP + \mu dN$$ $$dF = -SdT - PdV + \mu dN$$ $$dG = -SdT + VdP + \mu dN$$ $$d\Omega = -SdT - PdV - Nd\mu$$

Natural variables determine which potential is minimized:

Maxwell Relations

From equality of mixed partials of potentials (e.g., $\partial^2 F/\partial T\partial V = \partial^2 F/\partial V\partial T$):

$$\left(\frac{\partial T}{\partial V}\right)_{S,N} = -\left(\frac{\partial P}{\partial S}\right)_{V,N} \quad (\text{from } U)$$ $$\left(\frac{\partial T}{\partial P}\right)_{S,N} = +\left(\frac{\partial V}{\partial S}\right)_{P,N} \quad (\text{from } H)$$ $$\left(\frac{\partial S}{\partial V}\right)_{T,N} = +\left(\frac{\partial P}{\partial T}\right)_{V,N} \quad (\text{from } F)$$ $$\left(\frac{\partial S}{\partial P}\right)_{T,N} = -\left(\frac{\partial V}{\partial T}\right)_{P,N} \quad (\text{from } G)$$

Mnemonic "thermodynamic square":

   V — F — T
   |       |
   U       G
   |       |
   S — H — P

Read each side: $dU = TdS - PdV$ etc. Maxwell relations follow from rotating the square.

Heat Capacities & Identities

$$C_V = \left(\frac{\partial U}{\partial T}\right)_{V,N}, \qquad C_P = \left(\frac{\partial H}{\partial T}\right)_{P,N}$$

General relation (always true):

$$C_P - C_V = T V \frac{\alpha^2}{\kappa_T}$$

with $\alpha = (1/V)(\partial V/\partial T)_P$ (thermal expansion), $\kappa_T = -(1/V)(\partial V/\partial P)_T$ (isothermal compressibility).

For ideal gas: $C_P - C_V = N k_B$, $\gamma = C_P/C_V$.

Sound speed:

$$c_s^2 = \left(\frac{\partial P}{\partial \rho}\right)_S = \gamma \left(\frac{\partial P}{\partial \rho}\right)_T = \frac{\gamma P}{\rho}$$

Chemical Potential — The "Energy Cost of Adding a Particle"

$$\mu = \left(\frac{\partial U}{\partial N}\right)_{S,V} = \left(\frac{\partial F}{\partial N}\right)_{T,V} = \left(\frac{\partial G}{\partial N}\right)_{T,P}$$

Ideal classical gas:

$$\mu = k_B T \ln\!\left(\frac{n \lambda_{\text{th}}^3}{g_s}\right), \quad \lambda_{\text{th}} = \sqrt{\frac{2\pi\hbar^2}{m k_B T}}$$

$\lambda_{\text{th}}$ is the thermal de Broglie wavelength. Classical limit valid when $n\lambda_{\text{th}}^3 \ll 1$.

Equilibrium conditions:

• Two phases coexist: $\mu_1 = \mu_2$ at fixed $T,P$.
• Reaction $A + B \rightleftharpoons C$: $\mu_A + \mu_B = \mu_C$.

Saha Equation (Ionization Equilibrium)

For $A \rightleftharpoons A^+ + e^-$ with ionization energy $\chi$:

$$\boxed{\frac{n_+ n_e}{n_0} = \left(\frac{m_e k_B T}{2\pi\hbar^2}\right)^{3/2} \frac{2 Z_+(T)}{Z_0(T)} e^{-\chi/k_B T}}$$

where $Z_+, Z_0$ are internal partition functions.

Applications: stellar atmospheres, primordial recombination (z≈1100, CMB decoupling), HII regions, ionosphere.

Phase Transitions

First-order: discontinuity in first derivatives of $G$ (entropy, volume) → latent heat. Examples: water-ice, condensation.

Clausius-Clapeyron equation:

$$\frac{dP}{dT}\bigg|_{\text{coex}} = \frac{L}{T\Delta V}$$

where $L = T\Delta S$ is latent heat per particle.

Second-order (continuous): $G$ derivatives continuous, but second derivatives diverge. Examples: ferromagnet at $T_C$, superfluid $\lambda$-transition, critical point CO₂.

Critical exponents (universal, depend on dimension + symmetry, not on microscopic details):

with $t = (T - T_c)/T_c$.

Fluctuations of Thermodynamic Variables

In the canonical ensemble:

$$\langle (\Delta E)^2\rangle = k_B T^2 C_V$$ $$\langle (\Delta V)^2\rangle = k_B T V \kappa_T$$ $$\langle (\Delta N)^2\rangle = k_B T (\partial N/\partial \mu)_{T,V}$$

Key insight: fluctuations scale as $\sqrt{N}$, so relative fluctuations $\sim 1/\sqrt{N}$ — explains why thermodynamics works (huge $N$).

Connection to response functions: $\langle(\Delta E)^2\rangle / k_B T^2 = C_V$ — variance equals the response to a thermodynamic perturbation. This is the seed of the fluctuation-dissipation theorem.

Random Processes

Random Variables & Distributions

Random variable $X$: maps outcomes to numbers.

Probability density $p(x)$: $\Pr\{x < X < x+dx\} = p(x)\,dx$.

Moments: $\langle X^n\rangle = \int x^n p(x)\, dx$.

Cumulants: $\kappa_1 = \langle X\rangle$, $\kappa_2 = \text{Var}$, $\kappa_3 = $ skewness × $\sigma^3$, etc. Generated by $\ln M(t) = \sum \kappa_n t^n/n!$ with $M(t) = \langle e^{tX}\rangle$.

Common Distributions

Central Limit Theorem: sum of $N$ iid finite-variance RVs $\to$ Gaussian as $N\to\infty$, scaled as $\sqrt{N}\sigma$. Foundation of why Gaussians dominate.

Random Processes

A random process $Y(t)$ is a family of random variables indexed by $t$. Realization (sample path): one outcome over time.

Ensemble average: $\langle Y(t)\rangle = $ average over realizations at fixed $t$.

Time average: $\bar Y = \lim_{T\to\infty} \tfrac{1}{T}\int_0^T Y(t)\, dt$.

Ergodic theorem: for stationary processes, ensemble average = time average.

Stationarity, Correlation, Spectral Density

Stationary: statistical properties invariant under time translation. Then:

$$\langle Y(t)\rangle = \mu_Y = \text{const}, \quad \langle Y(t)Y(t')\rangle = C_Y(t-t')$$

Autocorrelation function $C_Y(\tau)$:

• $C_Y(0) = \langle Y^2\rangle$ (mean square)
• $C_Y(\infty) = \mu_Y^2$ (uncorrelated)
• Decay scale = correlation time $\tau_c$

Spectral density $S_Y(f)$ (one-sided, $f\ge 0$):

$$\boxed{S_Y(f) = 4\int_0^\infty C_Y(\tau)\cos(2\pi f\tau)\, d\tau \quad \text{(Wiener-Khinchin)}}$$

Inverse: $C_Y(\tau) = \int_0^\infty S_Y(f)\cos(2\pi f \tau)\, df$.

Parseval / variance: $\langle Y^2\rangle - \mu_Y^2 = \int_0^\infty S_Y(f)\, df$.

Units: $[S_Y] = [Y]^2 / $ Hz.

Standard Noise Spectra

Markov Processes & Chapman-Kolmogorov

Markov property: future depends only on present, not on history:

$$p(y_{n+1} | y_n, y_{n-1}, \ldots, y_1) = p(y_{n+1} | y_n)$$

Chapman-Kolmogorov equation:

$$p(y_3, t_3 | y_1, t_1) = \int dy_2\, p(y_3, t_3 | y_2, t_2)\, p(y_2, t_2 | y_1, t_1)$$

Master equation (continuous-time Markov, discrete states):

$$\frac{dP_n}{dt} = \sum_m \left[W_{n\leftarrow m} P_m - W_{m\leftarrow n} P_n\right]$$

Detailed balance (in equilibrium): $W_{n\leftarrow m} P_m^{eq} = W_{m\leftarrow n} P_n^{eq}$.

Brownian Motion & Langevin Equation

Langevin equation (1-D):

$$m \dot v = -\gamma v + F(t)$$

where $F(t)$ is white-noise force with:

• $\langle F(t)\rangle = 0$
• $\langle F(t) F(t')\rangle = 2 D_F \delta(t-t')$

Solution moments (relaxation):

$$\langle v(t)^2\rangle = v_0^2 e^{-2t/\tau_v} + \frac{D_F}{m\gamma}(1 - e^{-2t/\tau_v}), \quad \tau_v = m/\gamma$$

Equipartition at long times: $\langle v^2\rangle \to k_B T/m$. Matching:

$$\boxed{D_F = \gamma k_B T \quad \text{(Einstein/FDT)}}$$

Position diffusion (long-time):

$$\langle (\Delta x)^2\rangle = 2 D t, \qquad D = \frac{k_B T}{\gamma} = \mu k_B T \quad \text{(Einstein relation)}$$

with mobility $\mu = 1/\gamma$. For Stokes drag on sphere of radius $a$ in fluid of viscosity $\eta$:

$$\gamma = 6\pi \eta a \;\Rightarrow\; D = \frac{k_B T}{6\pi\eta a}$$

Fluctuation-Dissipation Theorem (FDT)

The unifying theorem of stat. physics:

For a system in equilibrium at $T$, the dissipation (imaginary part of response) is determined by the fluctuations (correlation/spectrum) of the conjugate variable:

$$\boxed{S_X(f) = \frac{4 k_B T}{\omega} \chi''(\omega), \quad \omega = 2\pi f}$$

with $\chi(\omega) = \chi'(\omega) + i\chi''(\omega)$ the linear response function: $\hat X(\omega) = \chi(\omega)\hat F(\omega)$.

Concrete realizations:

Fokker-Planck Equation

When jumps are small + Markov → continuous-state evolution governed by:

$$\boxed{\frac{\partial p}{\partial t} = -\partial_x\!\left[A(x) p\right] + \tfrac{1}{2}\partial_x^2\!\left[B(x) p\right]}$$

with drift $A(x) = \langle \Delta x\rangle/\Delta t$ and diffusion $B(x) = \langle(\Delta x)^2\rangle/\Delta t$ (from Kramers-Moyal expansion).

Special case — Brownian motion in velocity (Ornstein-Uhlenbeck):

$$\frac{\partial p(v,t)}{\partial t} = \frac{\gamma}{m}\partial_v(v p) + \frac{k_B T \gamma}{m^2}\partial_v^2 p$$

Stationary solution: Maxwellian $p_{\text{eq}}(v) \propto e^{-mv^2/(2 k_B T)}$.

Special case — diffusion (zero drift, constant $B = 2D$):

$$\partial_t p = D\partial_x^2 p \quad\Rightarrow\quad p(x,t) = \frac{1}{\sqrt{4\pi Dt}} e^{-x^2/(4Dt)}$$

→ $\langle x^2\rangle = 2 D t$.

Boltzmann form (general): equilibrium $p_{\text{eq}}(x) \propto e^{-U(x)/k_BT}$ where $U$ is a potential. The FP equation can be cast as

$$\partial_t p = \partial_x \left[D\,e^{-\beta U}\partial_x(e^{\beta U} p)\right]$$

making detailed balance manifest.

Kramers' Escape Rate (a key application of FP)

Particle in potential well, escape over barrier of height $\Delta U$:

$$r_{\text{escape}} = \frac{\omega_a \omega_b}{2\pi\gamma} e^{-\Delta U / k_B T} \quad \text{(high friction)}$$

with $\omega_a$ = well frequency, $\omega_b$ = barrier curvature. Arrhenius form — exponential temperature dependence governs chemical reaction rates, vacancy diffusion, nucleation, etc.

Shot Noise

For Poisson process with rate $\lambda$: counts in window $T$ have $\langle N\rangle = \lambda T$, $\langle(\Delta N)^2\rangle = \lambda T$.

Current $I = e\lambda$, two-sided spectrum $S_I = e I$, one-sided $S_I = 2 e I$ (Schottky formula).

Comparison with thermal: thermal noise $\propto T$ (vanishes at $T=0$); shot noise $\propto I$ (survives at $T=0$). Crossover at $eV \sim k_BT$.

Workflow / Process — How to attack a stat-phys problem

flowchart TD
    A["What is the question?"] --> B["Equilibrium or non-equilibrium?"]
    B -->|Equilibrium| C["Which constraints?"]
    B -->|Non-equilibrium| D["Kinetic / random process"]

    C -->|"E, V, N fixed"| E["Microcanonical:\nS = k ln W"]
    C -->|"T, V, N fixed"| F["Canonical:\nF = -kT ln Z"]
    C -->|"T, V, μ fixed"| G["Grand canonical:\nΩ = -kT ln Ξ"]

    D --> H["Closed-form distribution f?"]
    H -->|Yes| I["Boltzmann equation"]
    H -->|No| J["Markov?"]

    J -->|"Yes, small jumps"| K["Fokker–Planck"]
    J -->|"Yes, discrete"| L["Master equation"]
    J -->|"Linear w/ noise"| M["Langevin → FDT"]

    E --> N["Compute g(E), derive thermo"]
    F --> N
    G --> N

    I --> O["Transport coefficients"]
    K --> O
    L --> O
    M --> O

Comparison Tables

Quantum Statistics

Ensemble Comparison

Noise & Their Origins

Adiabatic Indices

Common Mistakes

• ❌ Conflating microcanonical and canonical. They give same thermodynamics in thermodynamic limit, but very different fluctuations (microcanonical: $\Delta E=0$; canonical: $\Delta E = \sqrt{k_BT^2 C_V}$).
• ❌ Using classical statistics when $n\lambda_{\text{th}}^3 \gtrsim 1$. Must use B-E or F-D. Common in electrons in metals, photons, ultra-cold atoms, neutron star matter.
• ❌ Forgetting spin degeneracy $g_s$. Electrons: $g_s = 2$. Photons: $g_s = 2$ (transverse polarizations only).
• ❌ Confusing mean and most-probable speeds. Maxwell distribution: $\bar v \ne v_p \ne v_{rms}$. Use the right one for the right physical quantity.
• ❌ Applying ideal-gas thermodynamics to interacting/degenerate systems. Virial corrections, exchange, screening all matter.
• ❌ Ignoring $\mu \ne 0$ for bosons. Only photons have $\mu = 0$. Atomic bosons have $\mu < 0$ approaching zero at BEC.
• ❌ Treating heat as a state variable. $\delta Q$ is inexact (depends on path); $S$, $U$ are state functions.
• ❌ Confusing reversible and quasi-static. Quasi-static needn't be reversible (e.g., slow Joule expansion).
• ❌ Wrong sign on chemical potential. $\mu$ for ideal gas at standard conditions is negative (large positive $\ln(n\lambda^3/g_s)$ argument is < 1).
• ❌ Time-averaging non-stationary process. Wiener-Khinchin and FDT require stationarity. For non-stationary, use windowed analysis or two-time correlations.
• ❌ Forgetting factor of 2 (one-sided vs two-sided spectra). $S_{\text{two-sided}}(-f) = S_{\text{two-sided}}(f)$, so $S_{\text{one-sided}} = 2 S_{\text{two-sided}}$ for $f>0$.
• ❌ Misapplying central limit theorem. Fails for heavy-tailed (Lévy) distributions and for strongly correlated variables.
• ❌ Confusing Fokker-Planck and Boltzmann. FP: small jumps (continuous), Boltzmann: arbitrary collisions (kernel).
• ❌ Using mean-field exponents in 2-D or 3-D Ising. Mean-field is exact only above upper critical dimension ($d_c = 4$ for Ising-like).