 Microscopic reversibility

In an equilibrium stochastic process with detailed balance, a sample trajectory and its physical time reversal have equal full probabilities when their equilibrium starting probabilities are included. If $P[\gamma \mid i]$ is the conditional weight for a path from $i$ to $j$, thenFor coarse equilibrium states with free energies $F_i$ and $p_i^{\mathrm{eq}}\propto e^{-F_i/(k_{B}T)}$, this givesThese are conditional path weights, rather than probabilities of a path after conditioning additionally on both endpoint events. Time reversal also reverses time-odd quantities such as momentum; a time-even order parameter is only read in reverse order. Magnetic fields and prescribed driving must be reversed in the comparison when required by the physical dynamics.^[ref]

Microscopic reversibility is a principle in statistical mechanics and thermodynamics that states that the underlying microscopic processes of a system can occur in either direction, and the statistical behavior of the system remains invariant when those processes are reversed. This idea is rooted in the concept that at the molecular or atomic level, the laws of physics—particularly the laws of motion—are time-invariant, meaning they don't change if time is reversed.