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For the flux $W_{ij}=-M{\partial }_i{\mu }_j+N_{ij}$ of a vector order parameter, microscopic reversibility requires independent Gaussian white noise components with covarianceThe current changes sign under time reversal. The resulting path-probability ratio contributes $-(k_{B}T{)}^{-1}\int W_{ij}{\partial }_i{\mu }_j$ to its logarithm; integration by parts converts this to the free-energy loss from the conserved dynamics.

Put $u_j={\dot{p}}_j+{\partial }_i{W}_{ij}$. The additive independent noises give the forward actionFor a time-even order parameter and time-odd flux, the backward action replaces $u,W$ by $-u,-W$ while keeping $\mu$ fixed on the corresponding configurations. Common normalization and time-reversal invariance of a path Jacobian then giveAt the two thermal noise strengths, periodic boundary conditions cancel the two spatial terms by integration by parts, leaving $-(F_2-F_1)/(k_{B}T)$. Thus the joint dynamics satisfies microscopic reversibility.

For nonconserved order-parameter dynamics, the positive kinetic coefficient $\Gamma$ converts the functional derivative of the free energy into a relaxation rate through $\dot{p}=-\Gamma \delta F/\delta p$. Microscopic reversibility fixes the accompanying Gaussian white noise variance by the Model A fluctuation-dissipation relation.

With constant positive kinetic coefficient $\Gamma$ and mobility $M$, two independent relaxation channels giveThe deterministic mobility operator is $\Gamma -M{\nabla }^2$. Each channel separately satisfies microscopic reversibility when its noise obeys the corresponding Model A fluctuation-dissipation relation or fluctuation-dissipation relation for a conserved flux.

For nonconserved order-parameter dynamics, microscopic reversibility with equilibrium weight proportional to $e^{-F/(k_{B}T)}$ requiresIndeed the Onsager--Machlup path probability gives $\log (P_F/P_B)=-2\Gamma \int \dot{p}\delta F/\delta pdt/{\sigma }^2=-2\Gamma (F_2-F_1)/{\sigma }^2$. Equating this to the detailed balance value proves the relation. The functional chain rule is understood with the same midpoint convention as the path action.

At thermal equilibrium, microscopic reversibility equates the probabilities of a path and its reversed path when both include their Boltzmann distribution initial weights. Denote their endpoint states by $z_1,z_2$, including velocity if needed. Since the Hamiltonian is even under time reversal in classical mechanics,This detailed balance condition and the energy balance for an autonomous Lagrangian yieldThis is the fluctuation-dissipation relation for a Langevin particle: the strength of Gaussian white noise is fixed by the damping and temperature, with $k_B$ the Boltzmann constant.

For an equilibrium coarse-grained variable, the unresolved microscopic states contribute entropy; their statistical weight is encoded in the Helmholtz free energy, rather than in a single microscopic energy. Relative to the same reference measure, ${\rho }_{\mathrm{eq}}(x)\propto e^{-\beta F(x)}$, so microscopic reversibility becomesThis extension assumes an equilibrium coarse-grained description with reversible path statistics; externally driven dynamics need not obey this relation.

The difference between the two squared actions is $|\dot{p}+\Gamma \mu {|}^2-|-\dot{p}+\Gamma \mu {|}^2=4\Gamma \dot{p}\cdot \mu$. The common factor cancels, givingThe last equality uses the functional chain rule with the same midpoint convention as the Onsager–Machlup functional.

At equilibrium, microscopic reversibility requires $e^{-F_1/(k_{B}T)}{P}_F=e^{-F_2/(k_{B}T)}{P}_B$, equivalently $P_F/P_B=e^{-(F_2-F_1)/(k_{B}T)}$. Comparing the coefficients of an arbitrary free-energy change gives the model A fluctuation-dissipation relation

Set the noise strengths to the model A fluctuation-dissipation relation and the fluctuation-dissipation relation for a conserved flux:The preceding logarithmic ratio becomesThe last two terms combine as ${\partial }_i({\mu }_j{W}_{ij})$. They integrate to zero under the periodic boundary conditions, by integration by parts. The remaining term is $F_2-F_1$ by the functional chain rule, henceAfter including the equilibrium initial weights, the forward and reversed histories have equal probability. Thus microscopic reversibility is satisfied by the two channels together, with each channel's own thermal noise strength.

Time reversal in classical mechanics reverses the order of a trajectory's states and the sign of its velocities. A coordinate is time-even, while its velocity is time-odd. Microscopic reversibility compares a path with this physical reversal, including the correct equilibrium initial weights.