Jarzynski equality in driven models

Use work W positive on the system and $\beta=1/(k_BT)$. The standard isothermal relation assumes initial canonical equilibrium at control value A, a fixed externally prescribed protocol ending at B, and thermal dynamics consistent with microscopic reversibility or an equivalent equilibrium-preserving framework. For the usual weak-coupling description, $\Delta F$ refers to the system's equilibrium Helmholtz free energies at the same bath temperature. Strong coupling, feedback, or noncanonical preparation needs a separately justified extension rather than direct reuse of this formula.

Each realization begins with a new draw from the A equilibrium distribution and follows the same protocol timing. Inclusive work is the energy supplied by changing the Hamiltonian's control parameter along that trajectory. The average is over the resulting full ensemble of work values. The system may finish far from equilibrium; the final free energy in the equality still refers to the equilibrium state associated with the final Hamiltonian. Adding a control-dependent constant to all energy levels shifts both work and free-energy difference, leaving the equality consistent.

During an instantaneous quench the microscopic state does not change, but its assigned energy can. The trajectory work equals the final energy minus the initial energy evaluated at that same state. Multiplying its exponential by the initial canonical probability cancels the initial energy factor. Summing over states then produces the final partition function divided by the initial one. This algebra proves the equality for the quench without any relaxation assumption afterward. It also shows why sampling initial states uniformly is valid only when their initial energies make the canonical distribution uniform.

The exponential average is not the exponential of average work. Convexity gives $\langle W\rangle\ge\Delta F$, with equality in an appropriate reversible limit, while individual work values can lie below the free-energy difference. Low-work trajectories receive disproportionately large exponential weights and are essential to the equality. The inequality does not mean every realization dissipates a positive amount of work. Stochastic violations of that trajectory-level claim are part of the mechanism by which the exact ensemble identity holds.

For independent correctly sampled trajectories, the finite exponential sample mean is an unbiased estimator of the ensemble exponential average. Taking its logarithm introduces finite-sample bias in the free-energy estimator; under suitable integrability its expected estimate is no smaller than the true free-energy difference. A particular dataset can deviate either way. Rare low-work trajectories may dominate the exact average but be absent from a practical sample. Faster driving can make this overlap problem much worse even though the mathematical equality remains exact.

Useful analysis therefore reports preparation, work calibration, protocol repeatability, statistical dependence, and evidence for tail coverage. A narrow-looking work histogram alone cannot prove that an unobserved tail contributes negligibly to an exponential average. Reverse-process data can help, motivating the Crooks relation next. Instrument parameter changes that recompute a toy distribution are also not a physical driving experiment: a work protocol requires a time-ordered energy change experienced along each realized trajectory, with its history included in the work calculation.