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Equation 13 · Part 5 · The Statistical Mechanics of Irreversibility at Molecular Scale

Symbol T

⟨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}.
TT

What this part means

T occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Its job in the formula

T occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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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A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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