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Equation 13 · Part 7 · The Statistical Mechanics of Irreversibility at Molecular Scale

fraction

⟨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}.
fraction

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

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

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.