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Equation 6 · Part 8 · The Arrow of Time and the Engine of Evolution

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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

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

Its job in the formula

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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

Addition combines quantities; subtraction measures the signed difference between them. Parentheses show what is combined before the rest of the expression is evaluated.

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Sources cited in the surrounding passage

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