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Equation 6 · Part 1 · The Arrow of Time and the Engine of Evolution

Symbol e^-β W

⟨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}
e−βWe^{-\beta W}

What this part means

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

Its job in the formula

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

The passage around this formula

Christopher Jarzynski’s 1997 equality was the opening result. For a system beginning in thermal equilibrium and then driven away from it by an arbitrarily fast, arbitrarily far-from-equilibrium protocol, the exponential average of the work W performed on it recovers the equilibrium free-energy difference Δ\Delta F between the protocol’s start and end points exactly: ⟨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}. [ 5 ] . The result does not say average work equals Δ\Delta F — dissipation makes the ordinary average of W larger than Δ\Delta F whenever the protocol is not quasistatic — it says a specific nonlinear average of a driven, dissipative, irreversible process reproduces an equilibrium quantity exactly, which is a…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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