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"""
example.py — Usage demonstrations for hv_fluctuation_os.
NumPy only.
"""
import numpy as np
from hv_fluctuation_os import (
MovingHarmonicTrap, TunableQuadratic, DoubleWellTilted,
LinearSchedule, sample_equilibrium, sample_forward,
sample_forward_both, jarzynski_estimate, crooks_test,
)
def demo_jarzynski_quadratic():
print("=" * 72)
print("Demo 1 — Jarzynski on the tunable quadratic")
print("=" * 72)
print()
system = TunableQuadratic()
schedule = LinearSchedule(lam_start=1.0, lam_end=4.0, T_total=5.0)
beta = 1.0
dt = 1e-3
n_steps = int(schedule.T_total / dt)
exact_df = (system.free_energy(schedule.lam_end, beta)
- system.free_energy(schedule.lam_start, beta))
print(f" System: {system.name}")
print(f" λ: {schedule.lam_start} → {schedule.lam_end}")
print(f" T_total: {schedule.T_total}")
print(f" β: {beta}")
print(f" Exact ΔF: {exact_df:.6f}")
print()
rng = np.random.default_rng(0)
x0 = sample_equilibrium(system, schedule.lam_start, beta,
n_traj=10000, rng=rng)
W = sample_forward(system, schedule, beta, x0, dt, n_steps, rng)
j = jarzynski_estimate(W, beta)
print(f" n_traj: {j['n']}")
print(f" <W>: {W.mean():.4f}")
print(f" σ_W: {W.std():.4f}")
print(f" <exp(-βW)>: {j['mean_exp']:.6f}")
print(f" exp(-βΔF): {np.exp(-beta * exact_df):.6f}")
print(f" βΔF_naive: {beta * j['delta_F_naive']:.4f}")
print(f" βΔF_bc: {beta * j['delta_F_bc']:.4f}")
print(f" Exact βΔF: {beta * exact_df:.4f}")
print(f" Error (naive): "
f"{abs(j['delta_F_naive'] - exact_df) / exact_df:.2%}")
print()
def demo_crooks():
print("=" * 72)
print("Demo 2 — Crooks fluctuation theorem")
print("=" * 72)
print()
system = TunableQuadratic()
schedule = LinearSchedule(lam_start=1.0, lam_end=4.0, T_total=5.0)
beta = 1.0
dt = 1e-3
n_steps = int(schedule.T_total / dt)
rng = np.random.default_rng(0)
W_f, W_r = sample_forward_both(system, schedule, beta, n_traj=5000,
dt=dt, n_steps=n_steps,
n_equil=0, rng=rng)
exact_df = (system.free_energy(schedule.lam_end, beta)
- system.free_energy(schedule.lam_start, beta))
crooks = crooks_test(W_f, W_r, beta, n_bins=25)
print(f" <W_F>: {W_f.mean():+.4f} σ: {W_f.std():.4f}")
print(f" <W_R>: {W_r.mean():+.4f} σ: {W_r.std():.4f}")
print()
print(f" Crooks slope: {crooks['slope']:+.4f} (expected β={beta})")
print(f" Crooks intercept:{crooks['intercept']:+.4f}")
print(f" ΔF from fit: {crooks['delta_F']:+.4f} "
f"(exact {exact_df:+.4f})")
print(f" Abs error: {abs(crooks['delta_F'] - exact_df):.4f}")
print(f" Bins used: {crooks['bins_used']}")
print()
def demo_convergence():
print("=" * 72)
print("Demo 3 — 1/√N convergence of the Jarzynski estimator")
print("=" * 72)
print()
system = TunableQuadratic()
schedule = LinearSchedule(1.0, 4.0, 5.0)
beta = 1.0
dt = 1e-3
n_steps = int(schedule.T_total / dt)
exact_df = (system.free_energy(schedule.lam_end, beta)
- system.free_energy(schedule.lam_start, beta))
print(f" Exact βΔF = {beta * exact_df:.4f}")
print()
print(f" {'n_traj':>7s} {'mean βΔF':>10s} {'std over seeds':>15s}")
print(" " + "-" * 36)
for n_traj in [100, 500, 2000, 10000]:
dfs = []
for seed in range(10):
rng = np.random.default_rng(seed * 1000 + n_traj)
x0 = sample_equilibrium(system, schedule.lam_start, beta,
n_traj, rng)
W = sample_forward(system, schedule, beta, x0, dt, n_steps, rng)
j = jarzynski_estimate(W, beta)
dfs.append(j['delta_F_naive'])
dfs = np.array(dfs)
print(f" {n_traj:>7d} {dfs.mean():>+10.4f} {dfs.std():>15.4f}")
print()
def demo_double_well():
print("=" * 72)
print("Demo 4 — double-well tilt (barrier crossing)")
print("=" * 72)
print()
system = DoubleWellTilted(a=4.0)
schedule = LinearSchedule(lam_start=0.0, lam_end=1.5, T_total=20.0)
beta = 1.0
dt = 5e-4
n_steps = int(schedule.T_total / dt)
exact_df = (system.free_energy(schedule.lam_end, beta)
- system.free_energy(schedule.lam_start, beta))
print(f" System: {system.name}")
print(f" λ: {schedule.lam_start} → {schedule.lam_end}")
print(f" Exact ΔF: {exact_df:.6f}")
print()
rng = np.random.default_rng(0)
x0 = sample_equilibrium(system, schedule.lam_start, beta,
n_traj=2000, rng=rng)
W = sample_forward(system, schedule, beta, x0, dt, n_steps, rng)
j = jarzynski_estimate(W, beta)
print(f" <W>: {W.mean():+.4f}")
print(f" σ_W: {W.std():.4f}")
print(f" βΔF_naive: {beta * j['delta_F_naive']:+.4f}")
print(f" Exact βΔF: {beta * exact_df:+.4f}")
print(f" 2nd law: <W> = {W.mean():.4f} ≥ ΔF = "
f"{exact_df:.4f}: {W.mean() >= exact_df}")
print()
def demo_quasi_static():
print("=" * 72)
print("Demo 5 — quasi-static limit")
print("=" * 72)
print()
system = TunableQuadratic()
beta = 1.0
dt = 1e-3
exact_df = (system.free_energy(4.0, beta)
- system.free_energy(1.0, beta))
print(f" Exact ΔF = {exact_df:.4f}")
print()
print(f" {'T_total':>8s} {'mean W':>10s} {'std W':>10s} "
f"{'|W - ΔF|':>10s}")
print(" " + "-" * 44)
for T in [1.0, 2.0, 5.0, 20.0, 50.0]:
schedule = LinearSchedule(1.0, 4.0, T)
n_steps = int(T / dt)
rng = np.random.default_rng(42)
x0 = sample_equilibrium(system, 1.0, beta, 500, rng)
W = sample_forward(system, schedule, beta, x0, dt, n_steps, rng)
print(f" {T:>8.1f} {W.mean():>+10.4f} {W.std():>10.4f} "
f"{abs(W.mean() - exact_df):>10.4f}")
print()
def demo_custom_system():
print("=" * 72)
print("Demo 6 — custom system (quartic confinement)")
print("=" * 72)
print()
class QuarticTrap:
name = "quartic_trap"
def potential(self, x, lam):
return 0.25 * lam * x ** 4
def force(self, x, lam):
return -lam * x ** 3
def dU_dlam(self, x, lam):
return 0.25 * x ** 4
def free_energy(self, lam, beta):
# Z = ∫ exp(-β λ x^4/4) dx
# Sub u = (β λ / 4)^{1/4} x, Z = (4/(β λ))^{1/4} · Γ(5/4) · 2
# F = -(1/β) ln Z
Z = (4.0 / (beta * lam)) ** 0.25 * 2 * 0.906402477055477 # Γ(5/4)
return -np.log(Z) / beta
def equilibrium_std(self, lam, beta):
# Numerical: compute std via quadrature
xs = np.linspace(-10, 10, 8001)
U = self.potential(xs, lam)
p = np.exp(-beta * U)
p /= p.sum()
mean = float(np.sum(xs * p))
var = float(np.sum((xs - mean) ** 2 * p))
return var ** 0.5
system = QuarticTrap()
schedule = LinearSchedule(lam_start=0.5, lam_end=2.0, T_total=10.0)
beta = 1.0
dt = 5e-4
n_steps = int(schedule.T_total / dt)
exact_df = (system.free_energy(schedule.lam_end, beta)
- system.free_energy(schedule.lam_start, beta))
rng = np.random.default_rng(0)
x0 = sample_equilibrium(system, schedule.lam_start, beta,
n_traj=2000, rng=rng)
W = sample_forward(system, schedule, beta, x0, dt, n_steps, rng)
j = jarzynski_estimate(W, beta)
print(f" System: {system.name}")
print(f" U(x, λ) = (λ/4) x^4")
print(f" λ: {schedule.lam_start} → {schedule.lam_end}")
print()
print(f" Exact ΔF: {exact_df:+.4f}")
print(f" <W>: {W.mean():+.4f}")
print(f" βΔF_naive: {beta * j['delta_F_naive']:+.4f}")
print(f" Error: {abs(beta * j['delta_F_naive'] - beta * exact_df):.4f}")
print()
print(" Any system implementing potential / force / dU_dlam /")
print(" free_energy can be plugged into the engine.")
print()
def main():
demo_jarzynski_quadratic()
demo_crooks()
demo_convergence()
demo_double_well()
demo_quasi_static()
demo_custom_system()
if __name__ == "__main__":
main() |