Part IV: Advanced Topics | Chapter 4

Fluctuation Theorems

Jarzynski equality, Crooks fluctuation theorem, stochastic thermodynamics, and entropy production in non-equilibrium processes

Historical Context

The fluctuation theorems, developed in the 1990s and 2000s, represent a revolution in our understanding of the second law of thermodynamics. Christopher Jarzynski (1997) showed that the equilibrium free energy difference can be extracted from non-equilibrium work measurements. Gavin Crooks (1999) generalized this to a detailed fluctuation theorem relating forward and reverse process probabilities.

These results extend thermodynamics to the microscopic scale, where fluctuations are large and the second law is statistical rather than absolute. The field of stochastic thermodynamics, developed by Udo Seifert and others, provides a complete thermodynamic framework for individual trajectories of small systems -- molecular motors, colloidal particles, RNA hairpins, and single molecules.

1. The Jarzynski Equality

Derivation 1: Free Energy from Non-Equilibrium Work

Consider a system initially in thermal equilibrium at temperature \(T\) with Hamiltonian \(H_A\). An external parameter (e.g., the extension of a molecule) is varied according to a protocol, changing the Hamiltonian from \(H_A\) to\(H_B\). The work done on the system along a particular trajectory is:

\[W = \int_0^{\tau}\frac{\partial H(\lambda(t))}{\partial\lambda}\dot{\lambda}\,dt\]

Jarzynski proved that:

\[\boxed{\langle e^{-\beta W}\rangle = e^{-\beta\Delta F}}\]

where \(\Delta F = F_B - F_A\) is the equilibrium free energy difference and the average is over all realizations of the non-equilibrium process (different initial conditions and noise histories). This is exact for any protocol speed -- fast or slow.

Proof sketch: Start from \(P_A(\Gamma_0) = e^{-\beta H_A(\Gamma_0)}/Z_A\). For Hamiltonian dynamics (or Langevin with heat bath), the work satisfies\(W = H_B(\Gamma_\tau) - H_A(\Gamma_0) + Q\) where \(Q\) is heat dissipated. Using the Liouville theorem and integrating:

\[\langle e^{-\beta W}\rangle = \frac{1}{Z_A}\int d\Gamma_0\,e^{-\beta H_B(\Gamma_\tau)} = \frac{Z_B}{Z_A} = e^{-\beta\Delta F}\]

The Second Law as a Consequence

By Jensen's inequality (\(\langle e^x\rangle \ge e^{\langle x\rangle}\)):

\[e^{-\beta\langle W\rangle} \le \langle e^{-\beta W}\rangle = e^{-\beta\Delta F}\]

Therefore \(\langle W\rangle \ge \Delta F\): on average, the work done exceeds the free energy change. The excess \(W_{\text{diss}} = \langle W\rangle - \Delta F \ge 0\)is the dissipated work. This is the second law. But individual trajectories can violate it: \(W < \Delta F\) is possible for small systems over short times.

2. The Crooks Fluctuation Theorem

Derivation 2: Forward and Reverse Process Relation

Crooks' theorem is a more detailed statement. Consider the forward process (A to B) and the reverse process (B to A, with time-reversed protocol). The probability distributions of work satisfy:

\[\boxed{\frac{P_F(W)}{P_R(-W)} = e^{\beta(W - \Delta F)}}\]

where \(P_F(W)\) is the work distribution in the forward process and\(P_R(-W)\) is the distribution of the negative work in the reverse process.

This implies the Jarzynski equality (integrate both sides over \(W\)) and also:

The forward and reverse work distributions cross at \(W = \Delta F\)
The probability of observing a “second-law violating” trajectory (\(W < \Delta F\)) is exponentially suppressed: \(P(W < \Delta F)/P(W > \Delta F) = e^{-\beta W_{\text{diss}}}\)

Experimentally, one can determine \(\Delta F\) by finding where the forward and reverse work distributions intersect -- a powerful method for measuring free energies of biomolecular processes.

3. Stochastic Thermodynamics

Derivation 3: Trajectory-Level First and Second Laws

For a single overdamped Langevin trajectory \(x(t)\) in a potential\(V(x, \lambda(t))\), define the stochastic work and heat:

\[dW = \frac{\partial V}{\partial\lambda}\dot{\lambda}\,dt, \qquad dQ = \left(\frac{\partial V}{\partial x}\right)\circ dx\]

The first law holds for each trajectory: \(dU = dW - dQ\) where\(U = V(x(t), \lambda(t))\) is the internal energy. The stochastic entropy production along a trajectory is:

\[\boxed{\Delta S_{\text{tot}} = \Delta S_{\text{sys}} + \Delta S_{\text{med}} = -\ln\frac{P(x_\tau, \tau)}{P(x_0, 0)} + \frac{Q}{k_BT}}\]

where \(\Delta S_{\text{med}} = Q/(k_BT)\) is the entropy change of the medium (heat bath). The total entropy production satisfies the integral fluctuation theorem:

\[\langle e^{-\Delta S_{\text{tot}}/k_B}\rangle = 1\]

which implies \(\langle\Delta S_{\text{tot}}\rangle \ge 0\): the second law holds on average. Individual trajectories can have \(\Delta S_{\text{tot}} < 0\)(transient second-law violations).

4. Entropy Production Rate

Derivation 4: Steady-State Entropy Production

In a non-equilibrium steady state (NESS), such as a system driven by a constant force\(f\) or a temperature gradient, the entropy production rate is:

\[\dot{S}_{\text{tot}} = k_B\int dx\,\frac{j^2(x)}{D\cdot P^{ss}(x)}\]

where \(j(x)\) is the probability current and \(P^{ss}(x)\) is the steady-state distribution. At equilibrium, \(j = 0\) everywhere (detailed balance) and \(\dot{S}_{\text{tot}} = 0\). In NESS, persistent currents drive entropy production.

For a particle dragged at velocity \(v\) through a fluid at temperature \(T\):

\[\dot{S}_{\text{tot}} = \frac{\gamma v^2}{T} = \frac{fv}{T}\]

This is the rate at which the dragging force does work, converted to heat and dissipated into the bath. The efficiency of molecular machines is bounded by the ratio of useful work to total entropy production.

5. Experimental Verification

Derivation 5: Free Energy of RNA Hairpin Unfolding

The most celebrated experimental test uses optical tweezers to unfold single RNA molecules. Liphardt et al. (2002) measured the work distribution for unfolding an RNA hairpin at various pulling speeds and verified the Jarzynski equality:

\[\Delta G = -k_BT\ln\langle e^{-\beta W}\rangle\]

They found \(\Delta G \approx 110\,k_BT\) for a P5abc RNA hairpin, consistent with the equilibrium (reversible) measurement. The key challenge is sampling: the exponential average is dominated by rare trajectories with low work, requiring careful statistical analysis.

Key Experimental Systems

RNA/DNA hairpins: Optical tweezers measure work distributions for folding/unfolding. Crooks theorem determines \(\Delta G\) from the crossing of forward/reverse work distributions.

Colloidal particles: Dragged through fluid by optical traps, directly measuring entropy production and verifying transient fluctuation theorems.

Molecular motors: F1-ATPase, kinesin, and other molecular machines operate in the stochastic thermodynamics regime, with single-molecule experiments measuring individual steps.

Electronic circuits: Small tunnel junctions at low temperatures exhibit measurable charge fluctuations that satisfy fluctuation theorems.

6. Applications and Extensions

Modern Developments

Thermodynamic uncertainty relations: Bounds on the precision of currents in terms of entropy production:\(\text{Var}(J)/\langle J\rangle^2 \ge 2k_B/\langle\dot{S}_{\text{tot}}\rangle\tau\). Higher precision requires more dissipation.

Information thermodynamics: Maxwell's demon and feedback control are described by generalized fluctuation theorems that include information terms. The Sagawa-Ueda equality: \(\langle e^{-\beta(W - \Delta F) + I}\rangle = 1\).

Optimal transport: Minimizing entropy production for finite-time processes (thermodynamic geometry, optimal protocols).

Active matter: Self-propelled particles violate detailed balance at the microscopic level, producing entropy that fuels their motion.

7. Computational Exploration

This simulation demonstrates the Jarzynski equality, Crooks theorem, work distributions for fast and slow processes, and entropy production statistics.

Fluctuation Theorems: Jarzynski Equality, Crooks Theorem, and Entropy Production

Python

script.py189 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Fluctuation Theorems: Jarzynski and Crooks
# ============================================================
np.random.seed(42)

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Fluctuation Theorems', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#1a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('#fca5a5')
    for spine in ax.spines.values():
        spine.set_color('#333')

# System: harmonic oscillator with shifting center
# V(x, lambda) = k/2 * (x - lambda)^2
# lambda goes from 0 to L
# Delta F = 0 (same potential, just shifted)

k_spring = 4.0
kBT = 1.0
beta = 1.0 / kBT
gamma = 1.0
L = 3.0  # total shift
dt = 0.005

def simulate_protocol(n_steps, n_traj, direction='forward'):
    """Simulate pulling protocol."""
    work_vals = []
    for _ in range(n_traj):
        x = np.random.randn() * np.sqrt(kBT / k_spring)  # equilibrium start
        if direction == 'reverse':
            x += L  # start at the end
        W = 0.0
        for step in range(n_steps):
            if direction == 'forward':
                lam = L * step / n_steps
                lam_next = L * (step + 1) / n_steps
            else:
                lam = L * (1 - step / n_steps)
                lam_next = L * (1 - (step + 1) / n_steps)

# Langevin step
            force = -k_spring * (x - lam)
            noise = np.sqrt(2 * gamma * kBT * dt) * np.random.randn()
            x += (force / gamma) * dt + noise

# Work = partial V / partial lambda * d_lambda
            dW = k_spring * (x - lam) * (lam_next - lam)
            W += dW

work_vals.append(W)
    return np.array(work_vals)

# --- Panel 1: Work distributions for different speeds ---
ax1 = axes[0, 0]
Delta_F = 0.0  # free energy difference is 0

speeds = [
    (200, '#f87171', 'Fast (200 steps)'),
    (1000, '#fbbf24', 'Medium (1000 steps)'),
    (5000, '#34d399', 'Slow (5000 steps)'),
]

n_traj = 3000

for n_steps, color, label in speeds:
    W = simulate_protocol(n_steps, n_traj, 'forward')
    bins = np.linspace(-5, 15, 60)
    ax1.hist(W, bins=bins, density=True, alpha=0.4, color=color, label=label)
    jarzynski_F = -kBT * np.log(np.mean(np.exp(-beta * W)))
    ax1.axvline(x=np.mean(W), color=color, linestyle='--', alpha=0.7)

ax1.axvline(x=Delta_F, color='white', linewidth=2, linestyle=':', label='Delta F = 0')
ax1.set_xlabel('Work W')
ax1.set_ylabel('P(W)')
ax1.set_title('Work Distributions (Forward)')
ax1.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 2: Jarzynski equality verification ---
ax2 = axes[0, 1]
step_vals = [50, 100, 200, 500, 1000, 2000, 5000]
n_traj_j = 5000

F_jarzynski = []
W_avg_vals = []
for ns in step_vals:
    W = simulate_protocol(ns, n_traj_j, 'forward')
    F_j = -kBT * np.log(np.mean(np.exp(-beta * W)))
    F_jarzynski.append(F_j)
    W_avg_vals.append(np.mean(W))

ax2.semilogx(step_vals, F_jarzynski, 'o-', color='#f87171', linewidth=2, markersize=6, label='Jarzynski Delta F')
ax2.semilogx(step_vals, W_avg_vals, 's--', color='#60a5fa', linewidth=2, markersize=6, label='<W> (average work)')
ax2.axhline(y=Delta_F, color='#fbbf24', linewidth=2, linestyle=':', label='Exact Delta F = 0')
ax2.set_xlabel('Number of steps (slower ->)')
ax2.set_ylabel('Energy (kBT units)')
ax2.set_title('Jarzynski Equality: <exp(-beta*W)> = exp(-beta*Delta_F)')
ax2.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 3: Crooks theorem - forward vs reverse ---
ax3 = axes[1, 0]
n_steps_c = 500
n_traj_c = 5000

W_forward = simulate_protocol(n_steps_c, n_traj_c, 'forward')
W_reverse = simulate_protocol(n_steps_c, n_traj_c, 'reverse')

bins_c = np.linspace(-8, 15, 80)
ax3.hist(W_forward, bins=bins_c, density=True, alpha=0.4, color='#f87171', label='P_F(W) forward')
ax3.hist(-W_reverse, bins=bins_c, density=True, alpha=0.4, color='#60a5fa', label='P_R(-W) reverse')
ax3.axvline(x=Delta_F, color='#fbbf24', linewidth=2, linestyle=':', label='Delta F = 0 (crossing)')
ax3.set_xlabel('Work W')
ax3.set_ylabel('Probability density')
ax3.set_title('Crooks Theorem: P_F(W)/P_R(-W) = exp(beta(W-Delta_F))')
ax3.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Entropy production distribution ---
ax4 = axes[1, 1]

# Entropy production = beta * W_diss = beta * (W - Delta_F)
S_prod = beta * (W_forward - Delta_F)  # Delta F = 0
bins_s = np.linspace(-5, 20, 80)

ax4.hist(S_prod, bins=bins_s, density=True, alpha=0.5, color='#a78bfa', label='P(Delta S_tot / kB)')

# Verify <exp(-Delta_S_tot/kB)> = 1
exp_neg_S = np.mean(np.exp(-S_prod))

ax4.axvline(x=0, color='#fbbf24', linewidth=2, linestyle=':', label='Delta S = 0')
ax4.axvline(x=np.mean(S_prod), color='#f87171', linewidth=2, linestyle='--',
            label='<Delta S/kB> = ' + str(round(np.mean(S_prod), 2)))
ax4.set_xlabel('Delta S_tot / k_B')
ax4.set_ylabel('Probability density')
ax4.set_title('Entropy Production Distribution')
ax4.legend(fontsize=7, facecolor='#1a0a0a', edgecolor='#333', labelcolor='white')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

# Numerical results
print("\n" + "=" * 60)
print("FLUCTUATION THEOREMS: NUMERICAL RESULTS")
print("=" * 60)

print("\n--- Jarzynski Equality Verification ---")
print("  Exact Delta F = " + str(Delta_F))
print("  Protocol: harmonic trap shifted by L = " + str(L))
for ns, fj, wav in zip(step_vals, F_jarzynski, W_avg_vals):
    print("  n_steps = " + str(ns).ljust(6) + ": Jarzynski F = " + str(round(fj, 4)).ljust(10) +
          "<W> = " + str(round(wav, 4)).ljust(10) +
          "W_diss = " + str(round(wav - Delta_F, 4)))

print("\n--- Crooks Theorem ---")
print("  Forward: <W> = " + str(round(np.mean(W_forward), 4)))
print("  Reverse: <W> = " + str(round(np.mean(W_reverse), 4)))
print("  Distributions should cross at W = Delta F = 0")

print("\n--- Entropy Production ---")
print("  <Delta S_tot / kB> = " + str(round(np.mean(S_prod), 4)) + " (should be >= 0)")
print("  <exp(-Delta S_tot / kB)> = " + str(round(exp_neg_S, 4)) + " (should = 1)")
frac_negative = np.mean(S_prod < 0)
print("  Fraction with Delta S < 0: " + str(round(frac_negative, 4)))
print("  (transient second-law violations)")

print("\n--- Second Law from Jensen's Inequality ---")
print("  <W> >= Delta F always holds on average")
print("  Individual trajectories can have W < Delta F")
frac_violation = np.mean(W_forward < Delta_F)
print("  Fraction with W < Delta F: " + str(round(frac_violation, 4)))

print("\n--- Key Formulas ---")
print("  Jarzynski: <exp(-beta*W)> = exp(-beta*Delta_F)")
print("  Crooks: P_F(W)/P_R(-W) = exp(beta*(W - Delta_F))")
print("  Integral FT: <exp(-Delta_S_tot/kB)> = 1")
print("  Second law: <W> >= Delta_F, <Delta_S_tot> >= 0")
print("\n" + "=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

8. Summary and Key Results

Core Formulas

Jarzynski: \(\langle e^{-\beta W}\rangle = e^{-\beta\Delta F}\)
Crooks: \(P_F(W)/P_R(-W) = e^{\beta(W - \Delta F)}\)
Integral FT: \(\langle e^{-\Delta S_{\text{tot}}/k_B}\rangle = 1\)
Second law: \(\langle W\rangle \ge \Delta F\)

Physical Insights

Free energy from non-equilibrium measurements
Second law is statistical, not absolute, for small systems
Transient violations probability decreases exponentially
Stochastic thermodynamics: complete framework for trajectories

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