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Equation 18 · Part 6 · The Statistical Mechanics of Irreversibility at Molecular Scale

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

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

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

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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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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