Irreversible Thermodynamics and Fluctuations

The classical fluctuation-dissipation theorem relates the autocorrelation function of a fluctuating variable to the linear response function:

$\chi(t) = \frac{1}{k_BT}\frac{d}{dt}\langle A(t)A(0)\rangle$

For example, the velocity autocorrelation function of a Brownian particle:

$\langle v(t)v(0)\rangle = \frac{k_BT}{m}e^{-t/\tau}$

Gives the mobility $\mu = e\tau/m$ (Einstein relation).

The voltage noise spectrum across a resistor $R$ at temperature $T$:

$S_V(f) = 4k_BTR$

This is white noise (frequency-independent up to $f \sim k_BT/h$).

The voltage fluctuation in bandwidth $\Delta f$:

$\langle V^2 \rangle = 4k_BTR\,\Delta f$

The Jarzynski equality (1997) connects non-equilibrium work to equilibrium free energy differences:

$\langle e^{-\beta W}\rangle = e^{-\beta\Delta F}$

Where the average is over many realisations of a process that drives the system from equilibrium state $A$ to equilibrium state $B$ in time $\tau$.

Consequences:

• By Jensen”s inequality: $\langle W \rangle \geq \Delta F$ (the average work is never less than the free energy change).
• For quasi-static processes: $\langle W \rangle = \Delta F$ and the distribution of $W$ is a delta function.
• For fast (far-from-equilibrium) processes: $\langle W \rangle > \Delta F$But the exponential average still equals $e^{-\beta\Delta F}$.

This remarkable result has been verified experimentally in single-molecule pulling experiments (RNA, DNA hairpins) using optical tweezers.

The Crooks theorem (1999) relates the work distributions for forward and reverse processes:

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

Where $P_F(W)$ is the probability distribution of work for the forward process and $P_R(W)$ for the reverse process.

This implies the Jarzynski equality as a special case:

$\int P_F(W)\,e^{-\beta W}\,dW = \int P_R(-W)\,e^{-\beta\Delta F}\,dW = e^{-\beta\Delta F}$

The fluctuation-dissipation theorem unifies these: the Einstein relation and Johnson-Nyquist formula are special cases of the FDT applied to Brownian motion and electrical circuits, respectively.

• Applying the FDT only to equilibrium systems. The standard FDT assumes the system is in thermal equilibrium. Fix: For non-equilibrium steady states, use generalised fluctuation-dissipation relations that include additional correlation terms.
• Confusing white noise with infinite power. Johnson-Nyquist noise is white only up to $f \sim k_BT/h \approx 6$ THz at 300 K; above this, quantum effects cut off the spectrum. Fix: Use $S_V(f) = 4k_BTR \cdot [\hbar f/(k_BT)]/[\exp(\hbar f/k_BT) - 1]$ for the quantum-corrected spectrum.
• Assuming Jarzynski equality only applies to slow processes. The equality holds for arbitrarily fast (even instantaneous) processes. Fix: The work distribution for a fast process has large tails, but the exponential average still equals $e^{-\beta\Delta F}$ — verify with the two-level example.
• Forgetting to take the exponential average in experiments. The average $\langle e^{-\beta W}\rangle$ is dominated by rare trajectories with negative work, requiring many samples to converge. Fix: Use Bennett’s acceptance ratio or Hummer-Szabo estimator for better convergence.

• Single-molecule biophysics: Optical tweezers measure the work needed to unfold RNA/DNA hairpins; the Jarzynski equality extracts the folding free energy without requiring reversible pulling.
• Nanoscale heat transfer: The FDT predicts thermal noise in nanomechanical resonators (cantilevers, membranes), limiting force sensitivity in AFM and gravitational-wave detectors.
• Circuit design: Johnson-Nyquist noise sets the fundamental noise floor in amplifiers and receivers; cryogenic cooling reduces $S_V$ linearly with $T$.
• Molecular dynamics: The Crooks theorem is used to compute free energy differences from non-equilibrium pulling simulations, avoiding expensive equilibrium sampling.
• Brownian ratchets: Fluctuation theorems constrain the efficiency of molecular motors and information-driven devices (Maxwell’s demon).

Problem. A $10$ k$\Omega$ resistor at $T = 300$ K is connected to a $1$ nF capacitor. Compute the RMS voltage fluctuation across the capacitor and the noise power in a $1$ MHz bandwidth.

Solution. The voltage noise spectral density is $S_V(f) = 4k_BTR = 4(1.38\times10^{-23})(300)(10^4) \approx 1.66\times10^{-16}$ V$^2$/Hz. In a $\Delta f = 1$ MHz bandwidth:

$\langle V^2\rangle = S_V\,\Delta f \approx 1.66\times10^{-16} \times 10^6 = 1.66\times10^{-10}\;\mathrm{V}^2$

$V_{\rm rms} = \sqrt{1.66\times10^{-10}} \approx 1.29\times10^{-5}\;\mathrm{V} = 12.9\;\mu\mathrm{V}$

The RC low-pass filter ($f_c = 1/(2\pi RC) \approx 16$ kHz) limits the effective bandwidth if the capacitor is considered, but at $f \ll f_c$ the full $1$ MHz bandwidth applies. This illustrates why sensitive electronics are cryogenically cooled: reducing $T$ from 300 K to 4 K reduces $V_{\rm rms}$ by $\sqrt{300/4} \approx 8.7\times$.

$\blacksquare$