Фізика статистичної механіки

📊 Ансамблі

Microcanonical

E, V, N fixed. Equal а priori probability: microstates. Entropy S = k_B ln(Ω), де Ω = number of states.

Canonical

T, V, N fixed. Partition function Z = Σ exp(-βE_i), β = 1/(k_B T). F = -k_B T ln(Z).

Grand canonical

T, V, μ fixed. Ξ = Σ exp(-β(E - μN)). Chemical potential μ. Open systems.

Equivalence

Thermodynamic limit N → ∞: ансамблі equivalent. Fluctuations ~ 1/√N. Central limit theorem.

🔬 Больцмана розподіл

Canonical ensemble

P(E_i) = exp(-βE_i)/Z. Boltzmann factor. Partition function: Z = Σ exp(-βE_i).

Maxwell-Boltzmann

f(v) = 4πn(m/(2πk_B T))^(3/2)·v²·exp(-mv²/(2k_B T)). Velocity distribution. Ideal gas.

Ideal gas

P = nk_B T. U = 3/2 Nk_B T. Heat capacities C_V, C_P. Diatomic: rotational, vibrational modes.

Applications

Reaction rates, atmospheric physics. Kinetic theory. Transport phenomena.

⚛️ Фермі-Дірака

Distribution

f(E) = 1/(exp((E-μ)/(k_B T))+1). Pauli exclusion. Fermi energy μ ≈ E_F at T = 0.

Degenerate regime

T < T_F: Fermi degenerate. Step function f(E) at T = 0. Filled up to E_F.

Fermi gas

Metals, white dwarfs, neutron stars. T_F = ℏ²(6π²n)^(2/3)/(2mk_B). High density.

Applications

Electrons in solids, degenerate matter. White dwarfs: electron degeneracy. Chandrasekhar limit.

🌊 Бозе-Ейнштейна

Distribution

n(E) = 1/(exp((E-μ)/(k_B T))-1). Photons, phonons. No exclusion principle.

Condensation

BEC: T < T_c. Macroscopic population lowest state. ¹⁶⁷Rb, ²³Na. 1995 experiments.

Photons

μ = 0 для photons. Planck distribution: n(ν) = 1/(exp(hν/(k_B T))-1). Blackbody radiation.

Phonons

Lattice vibrations. Debye model: C_V ~ T³ для T < Θ_D. Heat capacity, thermal conductivity.

🔄 Phase transitions

Critical phenomena

Order parameter η ~ (T_c - T)^β. Correlation length ξ ~ |T-T_c|^(-ν). Scale invariance.

Symmetry breaking

Spontaneous: ferromagnetic, superconducting. Landau theory: F(η) = aη² + bη⁴.

Renormalization Group

RG flow: coarse-graining. Fixed points: critical, stable. Universality classes.

Applications

Magnetism, superconductivity, liquid-gas. Ising model, XY model, Heisenberg model.

🌊 Transport phenomena

Diffusion

Fick's law: J = -D∇n. D ~ v·l (mean free path l). Random walk: ⟨r²⟩ = 6Dt.

Viscosity

Newtonian: τ = η·(∂v/∂x). η ~ ρvl. Poiseuille flow. Reynolds number Re.

Thermal conduction

Fourier: q = -κ∇T. κ ~ cvl. Wiedemann-Franz: κ/σ ~ T. Metals.

Fluctuation-dissipation

Einstein relation: D = k_B T/(mγ). Langevin equation. Brownian motion.

📊 Графіки та діаграми

Maxwell-Boltzmann

Velocity distribution:

Ideal gas: MB distribution. Most probable, average, RMS speeds. Temperature dependent.

Fermi-Dirac

Occupancy:

Degenerate: T < T_F. Step function at T = 0. Smearing у k_B T at finite T.

f(v) = 4πn(m/(2πk_B T))^(3/2)·v²·exp(-mv²/(2k_B T)) Peak: v_mp = √(2k_B T/m) Average: v_avg = √(8k_B T/(πm)) RMS: v_rms = √(3k_B T/m)

🧪 Практичні приклади

Приклад 1: Ideal gas

Air: P = nk_B T, C_V = 3/2 Nk_B. Maxwell-Boltzmann distribution. Transport coefficients.

Приклад 2: Electron gas

Metals: Fermi energy E_F ~ 1-10 eV. Degenerate T < T_F. Heat capacity C ~ T.

Приклад 3: BEC

¹⁶⁷Rb: T_c ~ 100 nK. Macroscopic condensate. Superfluidity, coherence.

Приклад 4: Ferromagnetic transition

Ising: T_c, β ≈ 1/3. Order parameter, correlations. Critical exponents.

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