Mathematical relations concerning particle systems require knowledge of the applicability conditions to become physically relevant and not merely formal. We illustrate this fact through the analysis of the Jarzynski equality (JE), whose derivation for Hamiltonian systems suggests that the equilibrium free-energy variations can be computational or experimentally determined in almost any kind of non-equilibrium processes. This apparent generality is surprising in a mechanical theory. Analytically, we show that the quantity called “work” in the Hamiltonian derivation of the JE is neither a thermodynamic quantity nor mechanical work, except in special circumstances to be singularly assessed. Through molecular dynamics simulations of elastic and plastic deformations induced via nano-indentation of crystalline surfaces that fall within the formal framework of the JE, we illustrate that the JE cannot be verified and that the results of this verification are process dependent.
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From this point of view, protein stretching experiments are noteworthy, since they are dissipative and look rather different from low dimensional Langevin processes.29
Ref. 9 states: “Since such values of the work represent statistically very rare events, it would require an unreasonably large number of measurements of W to determine with accuracy.” “This condition pretty much rules out macroscopic systems of interest.” Here, β = 1/kBT and W is the quantity identified as work in Ref. 9.
Note that the equilibria at which Eq. (11) holds are not realized during the JE process, except at time t = 0.
The fact that U∗ (τ, ω) depends on τ is not per se a problem; it is a τ-dependent fα-equilibrium quantity at fixed ω.
This case is analogous to those investigated in Refs. 22 and 39, in which the initial state cannot be restored after a perturbation. In that highly idealized case, explicit calculations are possible and show that the JE must be modified. Then, the verification of the JE is not merely statistically difficult, as this equality is fundamentally inaccurate.
Similar drops are observed in DNA pulling experiments Ref. 29.
