The arrow of time encoded in work distributions P_F(W) and P_R(−W)
Work distributions — forward (F) & reverse (R) protocols
Log-ratio ln[P_F(W)/P_R(−W)] vs W — slope = β = 1/kT
⟨W_F⟩ = ? | ΔF (Jarzynski) = ? | ⟨σ⟩ = ⟨W−ΔF⟩/kT = ?
For a system driven from equilibrium state A to state B by a protocol λ(t), with time-reverse protocol giving B→A, the Crooks Fluctuation Theorem (1999) states:
P_F(W) / P_R(−W) = exp[(W − ΔF) / kT]
where W is the work done, ΔF = F_B − F_A is the free energy difference, and the ratio compares the probability of work W in the forward process to the probability of work −W in the reverse process.
Key consequences: (1) The two distributions cross at W = ΔF — this provides a way to measure free energy differences from nonequilibrium pulling experiments. (2) Integrating gives the Jarzynski equality: ⟨e^{−βW}⟩_F = e^{−βΔF}. (3) The mean entropy production ⟨σ⟩ = β⟨W−ΔF⟩ ≥ 0 — the second law follows as an average of the fluctuation theorem.
Applications: measuring folding free energies of single molecules (RNA hairpin experiments, Liphardt 2002), protein unfolding (Bustamante), and testing non-equilibrium statistical mechanics.