Crooks equation for steered molecular dynamics using a Nosé-Hoover thermostat

The Crooks equation [Eq. (10) in J. Stat. Phys. 90, 1481 (1998)], originally derived for microscopically reversible Markovian systems, relates the work done on a system during an irreversible transformation to the free energy difference between the final and the initial state of the transformation. In the present work we provide a theoretical proof of the Crooks equation in the context of constant volume, constant temperature steered molecular dynamics simulations of systems thermostated by means of the Nosé-Hoover method (and its variant using a chain of thermostats). As a numerical test we use the folding and unfolding processes of decaalanine in vacuo at finite temperature. We show that the distribution of the irreversible work for the folding process is markedly non-Gaussian thereby implying, according to Crooks equation, that also the work distribution of the unfolding process must be inherently non-Gaussian. The clearly asymmetric behavior of the forward and backward irreversible work distributions is a signature of a non-Markovian regime for the folding/unfolding of decaalanine.