Nonequilibrium statistical mechanics of mixtures of particles in contact with different thermostats

We introduce a novel type of locally driven systems made of two types of particles (or a polymer with two types of monomers) subject to a chaotic drive with approximately white noise spectrum, but different intensity; in other words, particles of different types are in contact with thermostats at different temperatures. We present complete systematic statistical mechanics treatment starting from first principles. Although we consider only corrections to the dilute limit due to pairwise collisions between particles, meaning we study a nonequilibrium analog of the second virial approximation, we find that the system exhibits a surprisingly rich behavior. In particular, pair correlation function of particles has an unusual quasi-Boltzmann structure governed by an effective temperature distinct from that of any of the two thermostats. We also show that at sufficiently strong drive the uniformly mixed system becomes unstable with respect to steady states consisting of phases enriched with different types of particles. In the second virial approximation, we define nonequilibrium “chemical potentials” whose gradients govern diffusion fluxes and a nonequilibrium “osmotic pressure,” which governs the mechanical stability of the interface.

Figure 1

Field of currents. For illustration purposes, the interaction potential is chosen in the form $u\left(r\right)=1/(r^2+1)$, and the temperatures are $T_{\mathcal{A}}=1$ and $T_{\mathcal{B}}=0.5$. In the main figure, the arrows are the current vectors with components $J_{\mathcal{A}}=\left(-1+\frac{T_{\mathcal{A}}}{\overline{T}}\right)e^{-u\left(r\right)/\overline{T}}{\partial }_{r}u/\zeta$ and $J_{\mathcal{B}}=\left(1-\frac{T_{\mathcal{B}}}{\overline{T}}\right)e^{-u\left(r\right)/\overline{T}}{\partial }_{r}u/\zeta$ [assuming for illustration ${\zeta }_{\mathcal{A}}={\zeta }_{\mathcal{B}}$; see Eq. (4)]. In the enlarged window, the drift current (upward and to the left arrows, corresponding to the first terms in the expressions for $J_{\mathcal{A}}$ and $J_{\mathcal{B}}$) and the diffusion current (downward and to the right arrows, corresponding to the second terms in the expressions of the currents) are shown separately; unlike for equilibrium systems, they are not collinear, indicating a detailed balance violation. Their vector sum gives rise to the nonpotential current field.

Figure 2

Main figure: Triangular phase diagram, for a three-component system of $\mathcal{A}+\mathcal{B}+\mathrm{solvent}$; for every point in the triangle (exemplified by a star), the volume fractions of $\mathcal{A}$ and $\mathcal{B}$ particles are given by the distances to the triangle sides, while the volume fraction of the solvent $1-{\varphi }_{\mathcal{A}}-{\varphi }_{\mathcal{B}}$ is the distance to the bottom of the triangle. The symmetric line is the “spinodal” Eq. (13); below this line, the uniformly mixed state is unstable. For this figure, we used the parameters $\beta =B_{\mathcal{AB}}/\sqrt{B_{\mathcal{A}}{B}_{\mathcal{B}}}=2.95$ (an instability in an equilibrium system requires $\beta >3$), $T_{\mathcal{A}}/T_{\mathcal{B}}=3$ (which is sufficient to open an instability region). The asymmetric curve is a constant osmotic pressure line (14) $p{B}_{\mathcal{A}}{B}_{\mathcal{B}}/B_{\mathcal{AB}}\overline{T}=0.493$, assuming $B_{\mathcal{A}}=B_{\mathcal{B}}$; this line crosses the instability region thus indicating the possibility of two coexisting phases. Left inset: Lines of constant pressure (14). Right inset: Level lines of the left-hand side of formula (13).

Figure 3

Spinodal line (C4), shown for one particular choice of parameters ($T_{\mathcal{A}}/T_{\mathcal{B}},{\zeta }_{\mathcal{A}}/{\zeta }_{\mathcal{B}},B_{\mathcal{A}}{B}_{\mathcal{B}}/B_{\mathcal{A}B}^2$). Physical meaning has only region ${\varphi }_{\mathcal{A}}+{\varphi }_{\mathcal{B}}<1$, the instability region within this region is shaded.

Figure 4

Instability exists above this surface; see condition (D4).

Figure 5

For every $\kappa$, instability exists above the line, according to Eqs. (D4).

Figure 6

Ramp potential used to calculate pressure.