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Equation 18 · The Statistical Mechanics of Irreversibility at Molecular Scale

What does this equation mean?

ssys(t)=−kBln⁡p(xt,t).s_{\mathrm{sys}}(t) = -k_{\mathrm B}\ln p(x_t,t).

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Inputs and operations-k_mathrm Bln p(x_t,t)
Result or conditions_sys(t)
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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.

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ssyss_{\mathrm{sys}}

Symbol s_sys

sss_sys is part of the quantity the equation computes from the expression on the right.

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tt

Symbol t

t is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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kBk_{\mathrm B}

Symbol k_mathrm B

kmk_mathrm B is one of the signed contributions combined to compute the quantity on the left.

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pp

Symbol p

p is one of the signed contributions combined to compute the quantity on the left.

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xtx_t

Symbol x_t

xtx_t is one of the signed contributions combined to compute the quantity on the left.

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=

=

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

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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

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What the article says around this equation

The system’s stochastic entropy is defined from the ensemble density evaluated at the realized state, ssys(t)=−kBln⁡p(xt,t)s_{\mathrm{sys}}(t) = -k_{\mathrm B}\ln p(x_t,t). For a single thermal bath, the medium entropy change is related to dissipated heat, subject to sign convention, and total entropy combines system and medium contributions. This allows every observed path to receive a thermodynamic accounting while preserving the distinction between a random trajectory-level quantity and its ensemble expectation [ 4 ] .

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