Thermodynamics of fluctuations based on time-and-space averages

We develop nonequilibrium theory by using averages in time and space as a generalized way to upscale thermodynamics in nonergodic systems. The approach offers a classical perspective on the energy dynamics in fluctuating systems. The rate of entropy production is shown to be explicitly scale dependent when considered in this context. We show that while any stationary process can be represented as having zero entropy production, second law constraints due to the Clausius theorem are preserved due to the fact that heat and work are related based on conservation of energy. As a demonstration, we consider the energy dynamics for the Carnot cycle and for Maxwell's demon. We then consider nonstationary processes, applying time-and-space averages to characterize nonergodic effects in heterogeneous systems where energy barriers such as compositional gradients are present. We show that the derived theory can be used to understand the origins of anomalous diffusion phenomena in systems where Fick's law applies at small length scales, but not at large length scales. We further characterize fluctuations in capillary-dominated systems, which are nonstationary due to the irreversibility of cooperative events.

Figure 1

The length scale for ergodicity can be estimated based on $\sqrt{D{\tau }_m}$, the mean distance for diffusion over timescale ${\tau }_m$. We consider heterogeneous systems where the ergodic hypothesis holds at the scale of $\mathcal{V}$, but not at the larger scale of $\mathrm{\Omega }$.

Figure 2

Carnot cycle as a fluctuation: time averaging around the entire cycle results in constant average values of the entropy $\overline{S}$ and temperature $\overline{T}$; volume $\overline{V}$ and pressure $\overline{p}$. For any cyclic process, a stationary time series will be obtained for all thermodynamic quantities. Using a time average, the energy dynamics can be embedded within fluctuation terms that capture the net energy contribution over the cycle.

Figure 3

“Maxwell's demon” considers a thought experiment in which a demon selectively permits the migration of hot molecules from one chamber to the other. Due to the heat flux carried by the transmitted molecules, the temperature of the hot chamber increases, leading to a thermal gradient between the chambers. The demon's actions are constrained by a symmetry law relating thermal fluctuations.

Figure 4

Fluctuations due to operation of a conservative Maxwell's demon: the demon is activated at $t=0$ second and deactivated at time $t=1.0$ second to obtain a stationary time series. Fluctuations to the intensive properties are observed based on the redistribution of mass and energy within the system.

Figure 5

Droplet coalescence leads to fluctuations in the pressure field (color) due to the rapid change in capillary forces. Scaling for the coalescence event is limited by viscous forces during the initial bridge formation, and is dominated by inertial forces at late times.

Figure 6

Pressure fluctuation due to the coalescence of two fluid droplets. (a) Rate of energy change associated with pressure fluctuation relative to the final surface energy during droplet coalescence event. (b) Noise spectrum associated with capillary fluctuations can be predicted by ${log}_{10}{S}_P\sim \alpha F^p$.

Figure 7

Capillary fluctuations during displacement in porous media are linked with spontaneous pore-scale events that occur due to capillary forces within the solid microstructure. Haines jumps occur spontaneously during immiscible displacement as the system “jumps” from (a) one energy minimum to (b) the subsequent quasistatic configuration; (c) fluid pressure fluctuations arise due to these dynamics; and (d) scaling behavior for the power spectrum is independent of the time averaging interval.