NeuralBlitz Statistical Mechanics
What I Do
I provide comprehensive statistical mechanics tools including ensemble theory, partition functions, phase transitions, Monte Carlo simulations, and non-equilibrium dynamics for physics and materials science applications.
When to Use Me
- Thermodynamic property prediction
- Phase transition analysis
- Monte Carlo simulations
- Critical phenomena
- Non-equilibrium dynamics
- Materials modeling
Core Concepts
- Ensembles: Microcanonical, canonical, grand canonical
- Partition Functions: Translational, rotational, vibrational
- Phase Transitions: First/second order, critical phenomena
- Monte Carlo: Metropolis, importance sampling
- Mean Field Theory: Bragg-Williams, Landau theory
- Fluctuations: Variance, correlation functions
- Non-Equilibrium: Langevin, Fokker-Planck
- Renormalization: Wilson RG, scaling
Code Examples
Ensemble Calculations
import numpy as np
def canonical_partition_function(T, N, V, E_levels):
beta = 1 / (8.617e-5 * T) # eV/K
Z = sum(np.exp(-beta * E) for E in E_levels)
return Z
def free_energy_canonical(T, Z):
kB = 8.617e-5 # eV/K
return -kB * T * np.log(Z)
def entropy_canonical(T, Z, E_avg):
kB = 8.617e-5
return (E_avg - free_energy_canonical(T, Z)) / T
def heat_capacity(E2, E1, T2, T1):
return (E2 - E1) / (T2 - T1)
T = 300
E_levels = np.array([0, 0.01, 0.02, 0.05, 0.1])
Z = canonical_partition_function(T, 1, 1, E_levels)
F = free_energy_canonical(T, Z)
print(f"Partition function: {Z:.4f}")
print(f"Free energy: {F:.4f} eV")
Ising Model Simulation
def ising_energy(config, J, h):
E = 0
n = len(config)
for i in range(n):
for j in range(n):
E -= J * config[i,j] * config[(i+1)%n, j]
E -= J * config[i,j] * config[i, (j+1)%n]
E -= h * config[i,j]
return E
def metropolis_ising(config, T, J=1, h=0, n_steps=10000):
N = len(config)
E = ising_energy(config, J, h)
for _ in range(n_steps):
i, j = np.random.randint(0, N, 2)
delta_E = 2 * config[i,j] * (J * (config[(i-1)%N,j] + config[(i+1)%N,j] +
config[i,(j-1)%N] + config[i,(j+1)%N]) + h)
if delta_E <= 0 or np.random.random() < np.exp(-delta_E / T):
config[i,j] *= -1
E += delta_E
return config, E
config = np.random.choice([-1, 1], (10, 10))
config_final, E_final = metropolis_ising(config, 2.5)
print(f"Final energy: {E_final:.2f}")
print(f"Magnetization: {np.sum(config_final)}")
Critical Phenomena
def susceptibility(chi, beta, gamma, h):
return chi * np.abs(h)**(-gamma / beta) if h != 0 else np.inf
def correlation_length(xi, nu, T, Tc):
if T > Tc:
return xi * (T - Tc)**(-nu)
return np.inf
def order_parameter(T, Tc, beta):
if T < Tc:
return (1 - T/Tc)**beta
return 0
def scaling_relation(delta, eta, gamma, nu):
return delta = (d + 2 - eta) / (d - 2 + eta)
Tc_ising = 2.269 # 2D Ising critical temperature
beta = 1/8
print(f"Order parameter at T=2.0: {order_parameter(2.0, Tc_ising, beta):.4f}")
Langevin Dynamics
def langevin_step(x, v, m, gamma, T, dt, force_func):
kB = 1.38e-23
sigma = np.sqrt(2 * gamma * kB * T / dt)
noise = np.random.normal(0, sigma)
deterministic = force_func(x) - gamma * v
v_new = v + (deterministic / m) * dt + noise / m
x_new = x + v_new * dt
return x_new, v_new
def langevin_equation(m, gamma, k, T):
kBT = 1.38e-23 * T
relaxation_time = m / gamma
x0 = 1.0
x_final = x0 * np.exp(-gamma / m * t)
return x_final
def friction_dissipation(gamma, v):
return -gamma * v
def noise_correlation(gamma, kB, T):
return 2 * gamma * kB * T
Fokker-Planck Equation
def fokker_planck_coefficients(drift, diffusion, x, t):
A = drift(x, t)
B = diffusion(x, t)
dB_dx = np.gradient(B, x)
return A, dB_dx
def probability_current(J, x):
return J
def solve_fokker_planck(x0, x_max, t_max, drift, diffusion, n_points=100):
dx = (x_max - x0) / n_points
dt = 0.01
x = np.linspace(x0, x_max, n_points)
P = np.zeros_like(x)
P[np.argmin(np.abs(x - x0))] = 1 / dx
for t in np.arange(0, t_max, dt):
A, dB_dx = fokker_planck_coefficients(drift, diffusion, x, t)
P_new = P - dt * np.gradient(A * P, dx) + dt * np.gradient(dB_dx * np.gradient(P, dx), dx)
P = np.maximum(P_new, 0)
return x, P
Best Practices
- Equilibration: Allow sufficient equilibration time
- Autocorrelation: Account for correlated samples
- Finite Size: Account for finite-size effects near Tc
- Error Estimation: Use block averaging
- Boundary Conditions: Periodic vs open boundaries
Common Patterns
# Wang-Landau algorithm
def wang_landau_simulation():
pass
# Transfer matrix method
def transfer_matrix_isizing(J, h, N):
pass
Core Competencies
- Ensemble theory
- Monte Carlo simulation
- Phase transition analysis
- Non-equilibrium dynamics
- Critical phenomena