Math 2.0

Statistical mechanics, structure of matter Nonequilibrium statistical mechanics Theorem

Crooks fluctuation theorem

Builds on Jarzynski equalityFluctuation theoremHelmholtz free energyThermodynamic workProbability distribution

Statement

Let a system with energy function HλH_\lambda evolve by Markovian dynamics coupled to a heat bath at temperature TT, with β=1/(kBT)\beta = 1/(k_B T). Assume that for each fixed λ\lambda the dynamics satisfies detailed balance with respect to the canonical distribution of HλH_\lambda. In the forward process, the system starts in canonical equilibrium at λ=A\lambda = A, and λ\lambda follows a protocol λ(t)\lambda(t), 0≤t≤τ0 \le t \le \tau, from AA to BB. In the reverse process, the system starts in canonical equilibrium at λ=B\lambda = B, and λ\lambda follows the time-reversed protocol λ(τ−t)\lambda(\tau - t). Let PFP_F and PRP_R be the probability densities of the work WW done on the system in the two processes. Let ΔF=FB−FA\Delta F = F_B - F_A be the difference of the Helmholtz free energies. Then, for every real WW, PF(W)PR(−W)=eβ(W−ΔF).\frac{P_F(W)}{P_R(-W)} = e^{\beta (W - \Delta F)}.

Remarks

The proof compares the probability of each forward trajectory with that of its time reverse in the reverse process. Detailed balance makes their ratio eβQe^{\beta Q}, where QQ is the heat given to the bath, and the equilibrium initial states add the factor eβ(ΔE−ΔF)e^{\beta(\Delta E - \Delta F)}; with W=Q+ΔEW = Q + \Delta E this gives eβ(W−ΔF)e^{\beta(W - \Delta F)}. Multiplying by e−βWPR(−W)e^{-\beta W} P_R(-W) and integrating over WW gives the Jarzynski equality. The two distributions cross at W=ΔFW = \Delta F, which gives a way to measure free energy differences in experiments. It is a fluctuation theorem for the work, in the form for driven systems that start in equilibrium.

Going deeper

For a student

The theorem refines the Jarzynski equality: it gives a relation between whole distributions, not only one average. It is used with optical tweezers in single-molecule biophysics, where it has been checked in experiments on RNA hairpins. Its hypotheses are the same as for detailed balance: an equilibrium bath and microscopically reversible dynamics.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Convex function, definition, no evidence levelConvex functiondefinitionEntropy production, concept, no evidence levelEntropy productionconceptExpected value, concept, no evidence levelExpected valueconceptHelmholtz free energy, concept, no evidence levelHelmholtz free energyconceptProbability distribution, concept, no evidence levelProbability distribut…conceptThermodynamic work, concept, no evidence levelThermodynamic workconceptJensen's inequality, statement, no evidence levelJensen's inequalitystatementFluctuation theorem, statement, no evidence levelFluctuation theoremstatementJarzynski equality, statement, no evidence levelJarzynski equalitystatementCrooks fluctuation theorem, statement, no evidence levelCrooks fluctuation th…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Convex function definition
  2. Entropy production concept
  3. Expected value concept
  4. Helmholtz free energy concept
  5. Probability distribution concept
  6. Thermodynamic work concept
  7. Jensen's inequality statement
  8. Fluctuation theorem statement
  9. Jarzynski equality statement
  10. Crooks fluctuation theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Crooks_fluctuation_theorem
About this page
TagK3DP51N0M7
URIm2://K3DP51N0M7
Kindstatement, theorem
WritingW0 stub
MSC82C05, 82C35, 82C31
StatusThis copy of the site has no build data. Run m2 status K3DP51N0M7.
AboutHelmholtz free energy
UsesJarzynski equality, Fluctuation theorem, Helmholtz free energy, Thermodynamic work, Probability distribution
Same as
Changed3 November 2026

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