Finite trajectory entropy production ($\Sigma =\Delta S_{\mathrm{tot}}/k_B$)

Consider a finite-duration Markov process with discrete, time-even states coupled to one thermal reservoir at positive temperature $T$. Let $P_F[\gamma ]$ include the initial probability ${\rho }_0(x_0)$ of the sample trajectory $\gamma$. Define $P_R$ by reversing the control protocol and initializing it with the actual forward final probability distribution ${\rho }_{\tau }$. Write $\Theta \gamma$ for the same states visited in reverse time order. Require normalized path laws with matching probability support: a path has positive forward weight exactly when its reverse has positive reverse weight. This excludes irreversible forbidden transitions and missing initial states.

Under local detailed balance, with positive heat $Q_{\mathrm{bath}}$ delivered to the thermal reservoir, the full path-weight logarithm isThe endpoint term is the change in stochastic system entropy; the conditional path ratio alone accounts for the thermal reservoir. Here $k_B$ is the Boltzmann constant. Both terms matter even for a short observation. ^[ref]

Define the conjugate reverse functional by ${\Sigma }_R[\eta ]=\ln (P_R[\eta ]/P_F[\Theta \eta ])$, so ${\Sigma }_R[\Theta \gamma ]=-{\Sigma }_F[\gamma ]$. Summing the path identity over paths with a given value gives the exact finite fluctuation theoremfor nonzero denominator, using probability masses or corresponding probability density functions. This ${\Sigma }_R$ is defined by the paired laws: it need not be the total entropy production obtained by independently restarting the thermodynamic construction for the reverse experiment. A generic driven experiment therefore does not imply $p_F(s)/p_F(-s)=e^s$.

Normalization of the reverse law gives an exact finite identity:The sum notation describes discrete paths; the corresponding path integral has the same change-of-variable argument. Jensen's inequality for the exponential function then gives $e^{-⟨{\Sigma }_F{⟩}_F}\le 1$, hence $⟨{\Sigma }_F{⟩}_F\ge 0$. Equivalently this mean is the Kullback-Leibler divergence between the forward path law and the reverse path law pulled back by $\Theta$. Equality requires those two laws to agree. ^[ref]

Rare negative entropy production contributes disproportionately to this expectation through $e^{-{\Sigma }_F}$. Missing such events in a finite sample can spoil an estimate of the identity. Matching probability support is also essential: reverse paths absent from the forward law cannot be recovered by cancelling forward weights. This Second law of thermodynamics statement concerns an ensemble average; an individual negative value is compatible with it.