Hypotheses for microscopic fluctuation claims

Scope.

The adopted Macroscopic second law for isolated compounds compares the total macroscopic entropies of specified equilibrium endpoints of an admitted isolated process record. It assigns no entropy function to nonequilibrium intermediate configurations and does not assert that every microscopic realization has nonnegative entropy change. The macrostate and finite-model quantities remain distinct as defined in Macroscopic and microscopic entropy quantities.

A stochastic fluctuation claim needs its own state space or path space, probability law, preparation, driving protocol, time interval, measured or modelled random variable, and averaging convention. A trajectory observable has the units stated by its model (for physical entropy production, $\mathrm{J}{\mathrm{K}}^{-1}$; for dimensionless entropy production, no units). If a trajectory variable is called entropy production, define it and state the hypotheses of the relation being used, including the forward and reverse protocols and the support and integrability conditions needed for the stated averages. Positivity of an ensemble mean does not imply positivity in every realization.

For a precise model, let $X$ be a specified set of states and let $\tau \in (0,\mathrm{\infty })\subseteq \mathbb{R}$ be the dimensionless numerical duration in seconds. The physical duration is $(1\mathrm{s})\tau$. The numerical time domain $[0,\tau ]$ is the closed real interval of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length; its coordinate $t$ represents physical elapsed time $(1\mathrm{s})t$. Let $\mathrm{\Gamma }$ be a specified set of allowed sample paths $\gamma :[0,\tau ]\to X$, where each $\gamma$ is a function in the sense of A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain. Any further path regularity or admissibility conditions are part of that model; none is imposed universally here. Equip $\mathrm{\Gamma }$ with an event $\sigma$-algebra $\mathcal{F}$ and choose a probability measure $P$ on $(\mathrm{\Gamma },\mathcal{F})$. A trajectory observable is a measurable map $\mathrm{\Sigma }:(\mathrm{\Gamma },\mathcal{F})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$, where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra; if it represents physical entropy production, the model assigns its values units $\mathrm{J}{\mathrm{K}}^{-1}$, while a dimensionless entropy production has no units. When $\mathrm{\Sigma }$ is $P$-integrable, meaning ${\int }_{\mathrm{\Gamma }}\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }dP<\mathrm{\infty }$, its expectation is the integral ${\mathbb{E}}_P[\mathrm{\Sigma }]:={\int }_{\mathrm{\Gamma }}\mathrm{\Sigma }dP$ by Expectation of a nonnegative or integrable random variable. A driving protocol is a declared control map $\lambda :[0,\tau ]\to \mathrm{\Lambda }$, with its parameter space $\mathrm{\Lambda }$, regularity and endpoint controls specified by the model; $\lambda$ is also a function as in A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain. Any reverse-protocol comparison needs its own path measure and support conditions.

The fluctuation–dissipation theorem discussed in the cited treatments relates equilibrium fluctuations to a specified linear response of a coupled model. It is distinct from a trajectory fluctuation relation and does not by itself prove the macroscopic second-law postulate. No universal microscopic fluctuation formula or trajectory-wise entropy inequality is asserted here.