Statistical Mechanics

The statistical mechanics domain provides tools for calculating thermodynamic properties from microscopic physics.

Overview 

The statistical mechanics module implements:

Boltzmann distribution for thermal equilibrium
Ideal gas calculations
Ising model for magnetic systems
Fermi-Dirac distribution for fermions
Bose-Einstein distribution for bosons

Quick Start 

from mechanics_dsl.domains.statistical import (
    BoltzmannDistribution, IdealGas, IsingModel,
    FermiDirac, BoseEinstein
)

# Boltzmann distribution
boltz = BoltzmannDistribution(temperature=300)
v_rms = boltz.rms_speed(mass=4.65e-26)  # N2 molecules

# Ideal gas
gas = IdealGas(n_moles=1.0, temperature=273.15, volume=0.0224)
P = gas.pressure()  # ~1 atm

# Ising model
ising = IsingModel(L=20, dimension=2, temperature=2.5)
ising.initialize_random()
ising.monte_carlo_sweep()
M = ising.magnetization_density()

Classes 

BoltzmannDistribution 

Thermal equilibrium probability distribution:

\[P(E) \propto g(E) \exp\left(-\frac{E}{k_B T}\right)\]

Key methods:

boltzmann_factor(E): exp(-βE)
maxwell_speed_distribution(v, m): Speed distribution
most_probable_speed(m): v_p = √(2kT/m)
rms_speed(m): v_rms = √(3kT/m)

IdealGas 

Ideal gas equation of state: PV = NkT

Methods:

pressure(): Calculate from T, V, N
internal_energy(f): U = (f/2)NkT
heat_capacity_V(): C_V = (f/2)Nk
entropy(): Sackur-Tetrode formula

IsingModel 

Ising model Hamiltonian:

\[H = -J \sum_{\langle i,j \rangle} s_i s_j - h \sum_i s_i\]

Features:

1D, 2D, 3D lattices
Metropolis Monte Carlo dynamics
Critical temperature calculation (2D exact)

FermiDirac & BoseEinstein 

Quantum statistics:

\[f_{FD}(E) = \frac{1}{e^{(E-\mu)/(k_B T)} + 1}\]

\[n_{BE}(E) = \frac{1}{e^{(E-\mu)/(k_B T)} - 1}\]

Physical Constants 

from mechanics_dsl.domains.statistical import (
    BOLTZMANN_CONSTANT,  # 1.38e-23 J/K
    AVOGADRO_NUMBER,     # 6.02e23 /mol
    GAS_CONSTANT,        # 8.314 J/(mol·K)
    PLANCK_CONSTANT      # 6.626e-34 J·s
)