Contrary to what Crooks asserts, the self-consistency requirement put forward in J. Chem. Phys.129, 091101 (2008) is necessary for the Jarzynski equality (JE) and the Crooks fluctuation theorem (CFT) to be valid regardless which definition of work one uses, the force-displacement definition or the thermodynamic definition. The self-consistency check limits the applicability of the JE/CFT to linear systems and reversible processes in nonlinear systems. The fluctuation-dissipation theorem of J. Chem. Phys.129, 144113 (2008) is valid where the JE/CFT is applicable and where the JE/CFT is inapplicable.
In the comment1 on my recent communication,2 Crooks asserts that the self-consistency check of the Jarzynski equality (JE) (Ref. 3) and the Crooks fluctuation theorem (CFT) (Ref. 4) are invalid because the force-displacement definition of work is used rather than the thermodynamic definition of work. I disagree. First, the self-consistency check put forward in Ref. 2 is valid regardless which definition of work one uses. For the JE/CFT to be valid, the self-consistency factor has to be equal to one, namely, . Here , in which is the Boltzmann constant and is the absolute temperature. is the work along a forward transition path from state A to state B or that along a reverse path from state B to state A. The brackets and stand for, respectively, the statistical averages over all forward and reverse paths. Second, the two conclusions suggested in Ref. 2 do stand in spite of the fact that the example used to illustrate the points turned out to be ineffective: (1) The CFT does not have a wider range of applicability than the JE because CFT leads to the JE and many different equalities that have to be equivalent to JE within its range of validity. (2) The JE/CFT is only valid within the linear-response regime. The toy model system [a particle (mass ) in one dimension attached to a spring (elastic constant ) and pulled with a constant force for time ] turns out to be an ineffective example because it does satisfy the self-consistency requirement when one uses the thermodynamic definition of work. This, however, does not mean that the JE/CFT is applicable beyond the linear-response regime because the toy model is a linear system. It takes a nonlinear system to illustrate the points.5
In the same comment, Crooks also objects to my derivation of a nonequilibrium fluctuation-dissipation theorem within the context of Brownian dynamics6 for the force-displacement “expressions for work fails a basic symmetry requirement; the work is not odd under a time reversal.” This objection is invalid because a system of Brownian particles is subject to frictional and stochastic forces. The work done to the system along a forward transition path, , and the work done to it along a reverse path that is the exact reverse of the forward path, , are not symmetrical under time reversal. Namely, , where is the dissipative work. The proof of CFT (Ref. 4) relies on such a time-reversal symmetry, , in terms of the thermodynamic definition of work. Reference 6 finds the lack of such a time-reversal symmetry and derives a new fluctuation theorem in terms of the force-displacement definition of work, , for the free energy difference between states A and B. Reference 5, Fig. 1, illustrates the accuracy and efficiency of this fluctuation theorem and also shows that the JE/CFT is inapplicable for irreversible processes.
In summary, in spite of the trivial toy model failing as an effective illustration, the self-consistency requirement put forward in Ref. 2 is necessary for the JE and the CFT to be valid regardless which definition of work one uses. It limits the applicability of the JE/CFT to linear systems and reversible processes in nonlinear systems. The new fluctuation theorem of Ref. 6 is valid where the JE/CFT is applicable and where the JE/CFT is inapplicable.