Computation of free energy differences through nonequilibrium stochastic dynamics: The reaction coordinate case

The computation of free energy differences through an exponential weighting of out-of-equilibrium paths (known as the Jarzynski equality [C. Jarzynski, Equilibrium free energy differences from nonequilibrium measurements: a master equation approach, Phys. Rev. E 56 (5) (1997) 5018-5035, C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78 (14) (1997) 2690-2693]) is often used for transitions between states described by an external parameter in the Hamiltonian. An extension to transitions between states defined by different values of some reaction coordinate is presented here, using a projected Brownian dynamics. In contrast with other approaches (see e.g. [S. Park, F. Khalili-Araghi, E. Tajkhorshid, K. Schulten, Free energy calculation from steered molecular dynamics simulations using Jarzynski's equality, J. Chem. Phys. 119 (6) (2003) 3559-3566]), a projection is used rather than a constraining potential to let the constraints associated with the reaction coordinate evolve. It is shown how to use the Lagrange multipliers associated with these constraints to compute the work associated with a given trajectory. Appropriate discretizations are proposed. Some numerical results demonstrate the applicability of the method for the computation of free energy difference profiles.