Numerical Test of the Onsager Relations in a Driven System

The Onsager reciprocity relations were formulated in the context of irreversible thermodynamics, but they are based on assumptions that have a wider applicability. Here, we present simulations testing the Onsager relations between surface-coupled diffusive and bulk fluxes in a system prepared in a nonequilibrium steady state. The system consists of a mixture of two identical species maintained at different temperatures inside a channel. In order to tune the friction of the two species with the walls independently, while keeping the particle-wall interaction potentials the same, we allow the kinematics of particle-wall collisions to be different: “bounce-back” ($B$) or “specular” ($S$). In the $BB$ case, diffusio-capillary transport can only take place if the two species have different temperatures. We find that the Onsager reciprocity relations are obeyed in the linear regime, even in the $BB$ case where all fluxes are the result of perturbing the system from a nonequilibrium steady state in a way that does not satisfy time-reversal symmetry. Our Letter provides a direct, numerical illustration of the validity of the Onsager relations outside their original range of application, and suggests their relevance for transport in driven or active systems.

Figure 1

Schematic illustration of the system: the fluid of hot and cold particles is confined between two parallel flat square walls with size $L\times L$ fixed at a separation $W$ in the $z$ direction. The system is periodically extended in the $x$ and $y$ directions and all the forces are applied in the $x$ direction. The drawing represents an $xz$ cross section of the system. It illustrates the specular and bounce-back collision rules. Dashed lines represent infinitely thin specular walls implemented to determine the fluxes in the “bulk” of the channel (see text). Particles on either side of such a wall can interact with each other. However, they cannot cross these walls: rather they undergo specular reflection.

Figure 2

Relation between the fluxes and the force $F_C$ acting on cold particles only ($F_H=0$) for the system with $BB$ boundary conditions and $\mathrm{\Delta }T/T=0$. We plot the ratio $J/F_C$ in order to visualize the range of validity of the linear response. The symbols represent the “cold” particles $J_C/F_C$ (blue squares) and “hot” particles $J_H/F_C$ (red triangles). In the inset, $J/F_C$ is plotted against $F_C^2$ to quantify the lowest-order nonlinear term. All simulations in the main text were carried out with forces of magnitude 0.01, which is clearly deep within the validity of the linear response. A systematic overview of this relation for all parameters, can be found in SM (Figs. S2–S5) [9].

Figure 3

Ratio of the off-diagonal elements of the transport matrix relating the flux of cold ($C$) and hot ($H$) particles subject to color forces $F_C$ and $F_H$. The figure shows the results for weak (${\mathrm{\Gamma }}_t={10}^{-3}$) and strong thermal driving (${\mathrm{\Gamma }}_t={10}^{-2}$). (a) Results for the case where the applied forces and measured fluxes are confined to the bulk of the system. (b) Results for simulations in the entire channel (without the additional specular walls) with surfaces-induced coupling between diffusion and flow. Note that the $SB$ and $BS$ cases are distinct: one is obtained from the other by permuting $S$ and $B$, and replacing $\mathrm{\Delta }T$ with $-\mathrm{\Delta }T$. In contrast, the $BB$ case is even in $\mathrm{\Delta }T$. In spite of the linear dependence of the $SB$ and $BS$ transport coefficients on $\mathrm{\Delta }T$, the Onsager relations still seem to be satisfied. The error bars (about the size of the symbols) correspond to one standard deviation.

Figure 4

Ratio of the off-diagonal elements of the transport matrix, describing the coupling between diffusion and flow for three versions of the boundary conditions: $BB$ (a), $BS$ (b), and $SB$ (c). The figure shows the normalized off-diagonal transport coefficients for weak (${\mathrm{\Gamma }}_t={10}^{-3}$) and strong thermal driving (${\mathrm{\Gamma }}_t={10}^{-2}$). We have plotted the normalized matrix elements ${\mathrm{\Lambda }}_{TD}^{\prime }\equiv {\mathcal{L}}_{TD}^{\prime }/\sqrt{{\mathcal{L}}_{TT}^{\prime }{\mathcal{L}}_{DD}^{\prime }}$ (blue) and ${\mathrm{\Lambda }}_{DT}^{\prime }\equiv {\mathcal{L}}_{DT}^{\prime }/\sqrt{{\mathcal{L}}_{TT}^{\prime }{\mathcal{L}}_{DD}^{\prime }}$ (red). Note that the values of the elements of the transport matrix depend on the strength of the thermal driving. However, the coincidence of red and blue lines shows that Onsager symmetry seems to be conserved. The unnormalized matrix ${\mathcal{L}}^{\prime }$ is given in SM [9].