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Wyatt's Notes — University

Irreversible Thermodynamics and Fluctuations

19.1 Fluctuation-Dissipation in the Time Domain

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

χ(t)=1kBTddt⟨A(t)A(0)⟩\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:

⟨v(t)v(0)⟩=kBTme−t/τ\langle v(t)v(0)\rangle = \frac{k_BT}{m}e^{-t/\tau}

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

19.2 Johnson—Nyquist Noise Spectrum

The voltage noise spectrum across a resistor RR at temperature TT:

SV(f)=4kBTRS_V(f) = 4k_BTR

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

The voltage fluctuation in bandwidth Δf\Delta f:

⟨V2⟩=4kBTR Δf\langle V^2 \rangle = 4k_BTR\,\Delta f

19.3 Jarzynski Equality

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

⟨e−βW⟩=e−βΔF\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 AA to equilibrium state BB in time τ\tau.

Consequences:

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

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

19.4 Crooks Fluctuation Theorem

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

PF(W)PR(−W)=eβ(W−ΔF)\frac{P_F(W)}{P_R(-W)} = e^{\beta(W - \Delta F)}

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

This implies the Jarzynski equality as a special case:

∫PF(W) e−βW dW=∫PR(−W) e−βΔF dW=e−βΔF\int P_F(W)\,e^{-\beta W}\,dW = \int P_R(-W)\,e^{-\beta\Delta F}\,dW = e^{-\beta\Delta F}

Worked Example 19.1: Jarzynski Equality for a Two-Level System

Consider a two-level system with ϵ1=0\epsilon_1 = 0 and ϵ2=ϵ\epsilon_2 = \epsilonInitially in equilibrium at inverse temperature β\beta.

The free energy: F=−kBTln⁡Z=−kBTln⁡(1+e−βϵ)F = -k_BT\ln Z = -k_BT\ln(1 + e^{-\beta\epsilon}).

Now the energy gap is suddenly changed from ϵ\epsilon to ϵ"\epsilon". The work done is:

W={0withprob.p1=1/Zϵ′−ϵwithprob.p2=e−βϵ/ZW = \begin{cases} 0 & \text{with} prob. p_1 = 1/Z \\ \epsilon' - \epsilon & \text{with} prob. p_2 = e^{-\beta\epsilon}/Z \end{cases}

The Jarzynski average:

⟨e−βW⟩=p1⋅e0+p2⋅e−β(ϵ′−ϵ)=1Z+e−βϵ′Z=1+e−βϵ′Z\langle e^{-\beta W}\rangle = p_1 \cdot e^0 + p_2 \cdot e^{-\beta(\epsilon' - \epsilon)} = \frac{1}{Z} + \frac{e^{-\beta\epsilon'}}{Z} = \frac{1 + e^{-\beta\epsilon'}}{Z}

The new free energy: F′=−kBTln⁡(1+e−βϵ′)F' = -k_BT\ln(1 + e^{-\beta\epsilon'}).

e−βΔF=e−β(F′−F)=e−βF′eβF=(1+e−βϵ′)1Z=⟨e−βW⟩✓e^{-\beta\Delta F} = e^{-\beta(F' - F)} = e^{-\beta F'}e^{\beta F} = (1 + e^{-\beta\epsilon'})\frac{1}{Z} = \langle e^{-\beta W}\rangle \quad \checkmark

The Jarzynski equality is verified exactly for this two-level system, even though the process is far from equilibrium (sudden quench).