Thermodynamically consistent Langevin dynamics with spatially correlated noise predicting frictionless regime and transient attraction effect

While the origins of temporal correlations in Langevin dynamics have been thoroughly researched, the understanding of spatially correlated noise (SCN) is rather incomplete. In particular, very little is known about the relation between friction and SCN. In this article, starting from the microscopic, deterministic model, we derive the analytical formula for the spatial correlation function in the particle-bath interactions. This expression shows that SCN is the inherent component of binary mixtures, originating from the effective (entropic) interactions. Further, employing this spatial correlation function, we postulate the thermodynamically consistent Langevin equation driven by the Gaussian SCN and calculate the adequate fluctuation-dissipation relation. The thermodynamical consistency is achieved by introducing the spatially variant friction coefficient, which can be also derived analytically. This coefficient exhibits a number of intriguing properties, e.g., the singular behavior for certain types of interactions. Eventually, we apply this new theory to the system of two charged particles in the presence of counter-ions. Such particles interact via the screened-charge Yukawa potential and the inclusion of SCN leads to the emergence of the anomalous frictionless regime. In this regime the particles can experience active propulsion leading to the transient attraction effect. This effect suggests a nonequilibrium mechanism facilitating the molecular binding of the like-charged particles.

Figure 1

The density plot of $K\left(r\right)/\gamma$ for the DLVO-based model as a function of tracer-tracer distance $r$ and the counter-ion concentration $n_2$. The region of inertial motion $\left[K\right(r)\simeq 0]$ can be clearly distinguished from the region of ordinary overdamped motion where $K\left(r\right)\simeq \gamma$. The two areas are separated by the narrow region of divergent behavior, $K\left(r\right)\to \pm \infty$ (see Fig. 2).

Figure 2

The exemplary plots of $K\left(r\right)/\gamma$ in the DLVO-based model for the three values of $n_2$. $K\left(r\right)$ becomes divergent as it passes from $K\left(r\right)\simeq 0$ to $K\left(r\right)\simeq \gamma$.

Figure 3

The mean numerical solution $〈r\left(t\right)〉$ of equation (47) for $n_2={10}^{-8}\mathrm{mol}/{\mathrm{dm}}^3$ ($r_c=12.84d$, dashed line) in $T=300\text{K}$ and for several initial positions $r_0$. Despite the highly repulsive potential $U_Y\left(r\right)$, for $r_0\lesssim r_c$ the system exhibits systematic attraction in the initial phase of motion (especially $r_0=0.98r_c$, black solid line), marking the transient attraction effect. Inset: For $r_0=0.98r_c$ the trajectories are either of the escape type (54% of cases) or the entrant type (46%). ${〈r\left(t\right)〉}_{ent}$ is calculated over the entrant trajectories and indicates the depth of penetration in the attraction effect.