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Equation 12 · Part 15 · The Arrow of Time and the Engine of Evolution

Denominator: k_B T

PF(+W)PR(−W)=exp⁡ ⁣(W−ΔFkBT)\frac{P_{\mathrm F}(+W)}{P_{\mathrm R}(-W)} = \exp\!\left(\frac{W - \Delta F}{k_{\mathrm{B}} T}\right)
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

Gavin Crooks sharpened this two years later into a statement about entire probability distributions rather than a single average. If PF(W)P_{\mathrm F}(W) is the distribution of work values measured over many repetitions of a forward protocol, and PR(W)P_{\mathrm R}(W) is the distribution measured over the time-reversed protocol run from its own correctly defined reverse-equilibrium starting ensemble, the two distributions are related by PF(+W)PR(−W)=exp⁡ ⁣(W−ΔFkBT)\frac{P_{\mathrm F}(+W)}{P_{\mathrm R}(-W)} = \exp\!\left(\frac{W - \Delta F}{k_{\mathrm{B}} T}\right). [ 6 ] . Where W exactly equals Δ\Delta F , the ratio is one and the forward distribution crosses the reflected reverse distribution; away from that point, the ratio grows exponentially in how far W departs from Δ\Delta F , which is the precise,…

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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.