Non-equilibrium theory for non-ergodic systems based on time-and-space averaging

In an ergodic system spatial, temporal and ensemble averages are equivalent. Many theoretical approaches explicitly require ergodicity, and few general strategies have been advanced to treat non-equilibrium thermodynamic behavior associated with non-ergodic systems. We develop a non-equilibrium theory using time-and-space averaging, assuming that ergodic conditions hold only at very small length scales. Since the timescale for mixing is fast at small length scales, many non-ergodic systems can be described based on this approach. We show that fluctuations in these systems are constrained by the internal energy dynamics, deriving quasi-ergodic requirements that must hold for any stationary process due to conservation of energy. Since these requirements are formulated in terms of observable quantities, they can be used to explicitly identify the timescale where valid transport coefficients can be obtained. This result is significant because it provides a straightforward way to homogenize the dynamics of systems that do not obey equipartition of energy, particularly systems with slow fluctuations. We apply our theory to derive transport coefficients for immiscible fluid flow through porous media, demonstrating that pressure fluctuations observed in experiments can be non-Gaussian due to cooperative effects that are caused by capillary events. We show that the macroscopic dynamics can still be homogenized as long as the timescale for averaging is chosen such that these fluctuations perform no net work on the system. We further demonstrate that changes to fluid topology are responsible for non-ergodic effects, and that time-and-space averages provide a natural mechanism to account for discrete changes based on the topological residence time associated with particular micro-states of the system.