Equation 7 · 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 Δ s_tot
Δ ot is part of the quantity the equation computes from the expression on the right.
Symbol Gamma
Gamma is part of the quantity the equation computes from the expression on the right.
Symbol k_mathrm B
athrm B is an input to the expression that computes the quantity on the left.
Symbol P_mathrm F
athrm F occurs above the fraction bar. The numerator is divided by the entire denominator below it.
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.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →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.
Numerator: mathcal P_mathrm F[Gamma]
The complete quantity above the fraction bar.
Denominator: mathcal P_mathrm R[Gamma^dagger]
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
Under broad conditions, total stochastic entropy production can be represented as a log-likelihood ratio, . This equation supplies a precise meaning for the arrow of time. If a path is far more probable under the forward process than its conjugate is under the reverse process, observing it provides evidence about temporal direction. If the probabilities are equal, the path itself carries no arrow-of-time information. Irreversibility is therefore related to statistical distinguishability, not to an absolute prohibition on reverse-looking motion.
For background, read the article’s source list.
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