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Equation 6 · Part 12 · The Arrow of Time and the Engine of Evolution

Denominator: k_B 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}
kBTk_{\mathrm{B}} T

What this part means

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

Its job in the formula

kBk_B 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

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

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.