Crooks relation and forward–reverse protocols

Let the forward protocol take the control from A to B over a specified duration. Its reverse follows the reversed control schedule from B to A with the same duration. Each ensemble starts in canonical equilibrium for its own initial Hamiltonian at the same temperature. For underdamped systems, time reversal changes momenta as well; time-odd external parameters must be transformed consistently. The needed microscopic-reversibility condition relates probabilities of paired paths under these two experiments. It is stronger than observing a stationary histogram in either experiment.

Work is positive on the system in each direction. If a forward path has work W, its appropriately reversed partner has reverse work minus W. Consequently the denominator in the Crooks work ratio is $P_R(-W)$, not $P_R(W)$. These quantities are probability densities for continuous work or probability masses for discrete work, used consistently. The free-energy change is always $F_B-F_A$ in the displayed forward convention, whereas the reverse experiment has the opposite equilibrium free-energy change.

A forward work value above the free-energy difference is more probable than its sign-reversed reverse counterpart by the stated exponential factor. At W equal to the free-energy difference the two probability curves coincide, provided they have overlapping support there. The crossing locates the equilibrium difference without equating the forward and reverse mean works. Integrating the Crooks ratio after rearrangement also gives the Jarzynski equality by normalization of the reverse distribution. This is a consistency check linking the two results, not a replacement for verifying their experimental assumptions.

For compatible Gaussian work distributions with equal variance, the ratio's logarithm is linear in work. Matching its slope and intercept imposes relations among the two means, variance, and free-energy difference. Gaussianity is useful in linear driven models or as an empirical approximation, but it is not required by the theorem. Strongly non-Gaussian distributions may satisfy the exact ratio. Conversely, visually smooth Gaussian-looking histograms can violate it if their preparation, work calibration, or assigned reverse protocol is inconsistent.

A finite-time forward drive typically ends in a nonequilibrium distribution. Starting the reverse schedule immediately from that unrelaxed distribution generally does not meet the canonical initial condition of the standard Crooks work theorem. One must independently equilibrate at B or justify a more general relation with the required distribution boundary term. This difference is easy to miss because the control value B is correct in both preparations, while the state probabilities can be different. Control settings alone do not specify a statistical ensemble.

In measurements, forward and reverse distributions should be compared using matched work conventions and comparable density estimates. Unequal sample counts do not directly equal probability-density ratios without normalization. Sparse tails or disjoint practical samples make crossing estimates unreliable, even though the ideal ensemble relation remains well defined. Protocol changes in the interactive labs recompute their toy model at the retained clock value; they do not create a time-reversed thermodynamic protocol. Resetting and setting controls can compare fresh mathematical distributions, but no work history is thereby accumulated.