Fixing the flux: A dual approach to computing transport coefficients

We present a method to compute transport coefficients in molecular dynamics. Transport coefficients quantify the linear dependencies of fluxes in non-equilibrium systems subject to small external forcings. Whereas standard non-equilibrium approaches fix the forcing and measure the average flux induced in the system driven out of equilibrium, a dual philosophy consists in fixing the value of the flux, and measuring the average magnitude of the forcing needed to induce it. A deterministic version of this approach, named Norton dynamics, was studied in the 1980s by Evans and Morris. In this work, we introduce a stochastic version of this method, first developing a general formal theory for a broad class of diffusion processes, and then specializing it to underdamped Langevin dynamics, which are commonly used for molecular dynamics simulations. We provide numerical evidence that the stochastic Norton method provides an equivalent measure of the linear response, and in fact demonstrate that this equivalence extends well beyond the linear response regime. This work raises many intriguing questions, both from the theoretical and the numerical perspectives.