Equation 14 · The Statistical Mechanics of Irreversibility at Molecular Scale
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol P_mathrm F
athrm F occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol W
W is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol P_mathrm R
athrm R occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol β
β is one of the signed contributions combined to compute the quantity on the left.
Symbol Δ F
Δ F is one of the signed contributions combined to compute the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subtraction
Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.
change
Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
Denominator: P_mathrm R(-W)
The complete quantity below the fraction bar; it must be nonzero for this division.
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.
What the article says around this equation
Crooks sharpened the connection by comparing complete forward and reverse work distributions [ 3 ] : . At W= F , the ratio equals one, so the forward distribution and the reflected reverse distribution cross. This provides both a geometric interpretation and an experimental route to free-energy differences.
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.
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