Microscopic modifications of equilibrium probabilities due to non-conservative perturbations with applications to anharmonic systems

Abstract

The standard relationships of statistical mechanics are upended my the presence of active forces. In particular, it is no longer usually possible to simply write down what the stationary probability of a state of such a system will be, as can be done in ordinary statistical mechanics. While exact expressions are possible for harmonic systems, anharmonic systems tend to be much more difficult to treat. In this manuscript, we investigate how the microscopic probability for anharmonic systems is modified in the presence of non-conservative forces. We recount how non-conservative corrections to microscopic probabilities in generic systems can be represented as integrals over Green's function kernels, and that these Green's functions take the form of path integrals. We show how using analytically tractable form of these functions allows us to calculate corrections to the microscopic probability for a generic anharmonic system. These results are compared to active Brownian particles inside a harmonic or anharmonic well. From this it can be shown that accumulation of probability away from its equilibrium minima is a natural effect arising in anharmonic but not harmonic systems. Finally, we extend the microscopic results to study small active polymers with anharmonic backbone potentials. The interplay of the active driving with the anharmonic backbone potential can be used to understand how the end-to-end distance and mode space distributions of anharmonic active polymers scales with active driving. In particular, the polymer modes with the smallest eigenvalues will be affected by non-conservative forces the most.