Onsager symmetry from mesoscopic time reversibility and the hydrodynamic dispersion tensor for coarse-grained systems

Onsager reciprocity relations derive from the fundamental time reversibility of the underlying microscopic equations of motion. This gives rise to a large set of symmetric cross-coupling phenomena. We here demonstrate that different reciprocity relations may arise from the notion of mesoscopic time reversibility, i.e., reversibility of intrinsically coarse-grained equations of motion. We use Brownian dynamics as an example of such a dynamical description and show how it gives rise to reciprocity in the hydrodynamic dispersion tensor as long as the background flow velocity is reversed as well.

Figure 1

The effect of time reversal with the implied $\mathbf{u}\to -\mathbf{u}$ inversion in a porous medium (shaded). The forward time path in (a) is inverted in (b).

Figure 2

Illustration of the Brownian dynamics simulation setup. The background velocity $\mathbf{u}\left(\mathbf{x}\right)$ vanishes outside the central region, and the particle number $N_1$ outside in the reservoir region is recorded. The subcells have volume $\mathrm{\Delta }V$.

Figure 3

The distributions $P\left(N\right)$ for different flow conditions in the central cell illustrated in Fig. 2. The circles, squares, and triangles show measurements, and the straight lines show the theoretical expectation of Eq. (25). The red circles show the distribution that results from $\mathbf{u}=0$, the green squares show $u_x=u_{0}sin(2\pi y/L)$, and the blue triangles show $u_x=u_0$, all with a velocity of $u_0=1$. The Péclet number is $\mathrm{Pe}=u_{0}L/D=50$ in all cases. The total particle number is 1000, and the average is taken over 400 000 time steps.

Figure 4

The porous medium connected to four reservoirs of given constant temperatures and concentration deviations $\mathrm{\Delta }{C}_i$. The reservoirs are assumed to be in equilibrium, the central cell is in the steady state, and the concentration deviations are symmetric in the sense that $\mathrm{\Delta }{C}_{i+2}=-\mathrm{\Delta }{C}_i$.