Math 2.0

Statistical mechanics, structure of matter Nonequilibrium statistical mechanics Theorem

Jarzynski equality

Builds on Helmholtz free energyThermodynamic workExpected valueJensen's inequality

Statement

Let a classical system have Hamiltonian HλH_\lambda depending on a control parameter λ\lambda. Let it be coupled to a heat bath at temperature TT, and let β=1/(kBT)\beta = 1/(k_B T), where kBk_B is the Boltzmann constant. Let Fλ=−kBTln⁡ZλF_\lambda = -k_B T \ln Z_\lambda be the Helmholtz free energy, where ZλZ_\lambda is the partition function of HλH_\lambda. Let the system start in canonical equilibrium at λ=A\lambda = A. Change λ\lambda along a fixed protocol λ(t)\lambda(t), 0≤t≤τ0 \le t \le \tau, from λ(0)=A\lambda(0) = A to λ(τ)=B\lambda(\tau) = B. Assume the dynamics is either Hamiltonian, or Markovian with the canonical distribution of HλH_\lambda stationary for each fixed λ\lambda. Let WW be the work done on the system in one realisation, W=∫0τλ˙ ∂λHλ(x(t)) dtW = \int_0^\tau \dot\lambda\, \partial_\lambda H_\lambda(x(t))\, dt. Then 𝔼[e−βW]=e−β(FB−FA),\mathbb{E}\left[e^{-\beta W}\right] = e^{-\beta (F_B - F_A)}, where 𝔼\mathbb{E} is the expected value over realisations.

Remarks

For Hamiltonian dynamics the proof is a change of variables: W=HB(x(τ))−HA(x(0))W = H_B(x(\tau)) - H_A(x(0)), and Liouville's theorem turns 𝔼[e−βW]\mathbb{E}[e^{-\beta W}] into ZB/ZAZ_B / Z_A. The final state need not be in equilibrium, and the protocol can be arbitrarily fast. By Jensen's inequality, e−β𝔼[W]≤𝔼[e−βW]e^{-\beta \mathbb{E}[W]} \le \mathbb{E}[e^{-\beta W}], so 𝔼[W]≥FB−FA\mathbb{E}[W] \ge F_B - F_A, which is the second law for this process. Example: a quasistatic process has W=FB−FAW = F_B - F_A in every realisation, and the equality is trivial. Without the equilibrium initial state the result fails; it is an essential hypothesis.

Going deeper

For a student

The average is dominated by rare realisations with small work, so many repetitions are needed when the process is far from quasistatic. The equality was found by Christopher Jarzynski in 1997. It is one of the fluctuation relations of stochastic thermodynamics, and it follows by integration from the Crooks fluctuation theorem.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Convex function, definition, no evidence levelConvex functiondefinitionExpected value, concept, no evidence levelExpected valueconceptHelmholtz free energy, concept, no evidence levelHelmholtz free energyconceptThermodynamic work, concept, no evidence levelThermodynamic workconceptJensen's inequality, statement, no evidence levelJensen's inequalitystatementJarzynski equality, statement, no evidence levelJarzynski equalitystatement
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. Expected value concept
  3. Helmholtz free energy concept
  4. Thermodynamic work concept
  5. Jensen's inequality statement
  6. Jarzynski equality statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Jarzynski_equality
About this page
Tag1VPYJ7R7DT
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Kindstatement, theorem
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MSC82C05, 82C35, 82B30
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AboutHelmholtz free energy
UsesHelmholtz free energy, Thermodynamic work, Expected value, Jensen's inequality
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Changed3 November 2026

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