Fluctuations of Non-Conservative Systems

When a non-conservative system fluctuates around its steady configuration, in general, neither equipartition nor the fluctuation–dissipation theorem are satisfied. Using a path integral approach, we show that in this case the probability distribution is determined in terms of the energy dissipated along the minimum path. The latter is the path of minimum energy dissipation of a fictitious, unit mass particle, moving with constant energy under the influence of an electric and a magnetic field. In addition, the instantaneous speed of this particle equals the mean backward velocity of the Brownian particle. At the end, a Boltzmann-like probability distribution is obtained, which allows us to define an effective temperature kernel. In particular, when the forces applied to the particle are linearly dependent on the distance from the origin, the effective temperature turns out to be the sum between an isotropic and an antisymmetric tensor, which allows us to generalize the fluctuation–dissipation theorem.