How Statistical Forces Depend on the Thermodynamics and Kinetics of Driven Media

We study the statistical force of a nonequilibrium environment on a quasistatic probe. In the linear regime, the isothermal work on the probe equals the excess work for the medium to relax to its new steady condition with a displaced probe. Also, the relative importance of reaction paths can be measured via statistical forces, and from second order onwards the force on the probe reveals information about nonequilibrium changes in the reactivity of the medium. We also show that statistical forces for nonequilibrium media are generally nonadditive, in contrast with the equilibrium situation. Both the presence of nonthermodynamic corrections to the forces and their nonadditivity put serious constraints on any formulation of nonequilibrium steady state thermodynamics.

Figure 1

(a) A slow probe (the light gray disk) is immersed in a nonequilibrium medium (the dark green circles), in contact with an equilibrium reservoir (the small blue circles). (b) A dilute colloid solution diffusing on a ring, with each colloid particle being coupled to an internal degree of freedom which is driven out of equilibrium.

Figure 2

(a) The equilibrium statistical force $f_{\mathrm{eq}}(x)$ and the nonequilibrium correction to it, $g(x)$, for field $ϵ=2.0$. Here ${\varphi }_0=1,a=2$. The inset shows the rotational part of the force $f_{\mathrm{rot}}$ as a function of the field $ϵ$. (b) The excess work $\mathrm{đ}{W}^{\mathrm{ex}}(x)$ is the extra dissipated work for relaxing to the new steady nonequilibrium (at a different power $\dot{W}$) as the probe position $x\to x+dx$ has changed.

Figure 3

(a) The statistical force $g(x=\pi /2)$ as a function of $ϵ$. The initial slope depends on the relative equilibrium reactivity ${\varphi }_0$ (here ${\varphi }_0=0.5$). For larger drive $ϵ$, the dependence on the reactivity parameter $a$ becomes visible. (b) Linear $g_1(x)$ and second order correction $g_2(x)$ as a function of the probe position for $a=1,2$ and ${\varphi }_0=1$. The linear order is independent of $a$, while the second order depends explicitly on it.

Figure 4

(a) Coupling of two internal degrees of freedom to the probe. The $A$ reservoir is driven and will also create a current in the $B$ reservoir, making the forces nonadditive. (b) Nonadditivity in the rotational component of the total force $\mathrm{\Delta }{f}_{\mathrm{rot}}$ versus $ϵ$ for $\gamma =1.0,5.0$ and $a=2,{\varphi }_0=1$. (Inset) The deviation from additivity $\mathrm{\Delta }f(x)$ versus the probe position $x$ at $ϵ=2.0$.