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Equation 8 · The Statistical Mechanics of Irreversibility at Molecular Scale

What does this equation mean?

⟨exp⁡(−ΔstotkB)⟩=1.\left\langle \exp\left(-\frac{\Delta s_{\mathrm{tot}}}{k_{\mathrm B}}\right) \right\rangle = 1.

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.

Inputs and operations1
Result or conditionlangle exp(-fracΔ s_totk_mathrm B) rangle
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

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

Δstot\Delta s_{\mathrm{tot}}

Symbol Δ s_tot

Δ sts_tot occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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kBk_{\mathrm B}

Symbol k_mathrm B

kmk_mathrm 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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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

Divide the expression above the line by the one below it.

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change

change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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subscript

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.

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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.

What the article says around this equation

Exponentiating and averaging over forward paths produces the integral fluctuation theorem, ⟨exp⁡(−ΔstotkB)⟩=1\left\langle \exp\left(-\frac{\Delta s_{\mathrm{tot}}}{k_{\mathrm B}}\right) \right\rangle = 1. Jensen’s inequality then gives

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