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Published equation contexts

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)

Why this formula appears here

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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PFP_{\mathrm F}

Symbol P_mathrm F

PmP_mathrm F occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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PRP_{\mathrm R}

Symbol P_mathrm R

the distribution measured over the time-reversed protocol run from its own correctly defined reverse-equilibrium starting ensemble, the two distributions are related by [displayed formula].

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kBk_{\mathrm{B}}

Symbol k_B

kBk_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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PR(−W)P_{\mathrm R}(-W)

Denominator: P_mathrm R(-W)

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

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kBTk_{\mathrm{B}} T

Denominator: k_B T

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.

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)

Equation 12 · Evolutionary Physics

The Arrow of Time and the Engine of Evolution

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

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,…

Meanings in this article

  • PRP_{\mathrm R}: the distribution measured over the time-reversed protocol run from its own correctly defined reverse-equilibrium starting ensemble, the two distributions are related by [displayed formula].
  • ΔF\Delta F: the w exactly equals.
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