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Equation 13 · Part 6 · The Statistical Mechanics of Irreversibility at Molecular Scale

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⟨e−βW⟩=e−βΔF,β=1kBT.\left\langle e^{-\beta W}\right\rangle = e^{-\beta\Delta F}, \qquad \beta = \frac{1}{k_{\mathrm B}T}.
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What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

In equilibrium statistical mechanics, the free-energy difference Δ\Delta F between two control-parameter values is a state function. A rapid drive between them generally performs work W>Δ\Delta F on average because dissipation is positive. It would seem that estimating Δ\Delta F requires an impractically slow, quasistatic protocol. The Jarzynski equality demonstrates otherwise: ⟨e−βW⟩=e−βΔF,β=1kBT\left\langle e^{-\beta W}\right\rangle = e^{-\beta\Delta F}, \qquad \beta = \frac{1}{k_{\mathrm B}T}. The equality holds for an ensemble of realizations beginning in equilibrium, even when each realization is driven far from equilibrium [ 2 ] . It does not say that the arithmetic mean work equals the free-energy difference. The exponential average disproportionately weights rare, low-work…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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