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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}

Why this formula appears here

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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e−βWe^{-\beta W}

Symbol e^-β W

e−e^-β W is part of the quantity the equation computes from the expression on the right.

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e−βΔFe^{-\beta\Delta F}

Symbol e^-βΔ F

e−e^-βΔ F is one of the signed contributions combined to compute the quantity on the left.

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kBk_{\mathrm B}

Symbol k_mathrm B

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

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TT

Symbol T

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

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kBTk_{\mathrm B}T

Denominator: k_mathrm BT

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

⟨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}.

Equation 13 · Physics

The Statistical Mechanics of Irreversibility at Molecular Scale

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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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