A Cold Tracer in a Hot Bath: In and Out of Equilibrium

TL;DR

This work addresses how a zero-temperature tracer in a hotter Brownian bath transitions between nonequilibrium activity and effective equilibrium as bath density grows. Using a fully-connected linear-coupling model, the bath can be integrated out to yield a non-Markovian tracer dynamics that reduces to an equilibrium Langevin process at the bath temperature $T$ in the limit $N\to\infty$ (or $\rho\to\infty$), with mobility $N^{-1}$. A systematic $1/N$ perturbation theory reveals departures from Boltzmann statistics at order $1/N$ and an entropy production rate scaling as $\sigma \sim \mathcal{O}(N^{-3})$, implying an intermediate time-reversible regime before true irreversibility emerges. When bath connectivity is finite and arranged as a lattice (a gel), the cold tracer drives the bath out of equilibrium, producing long-range suppression of fluctuations that decays as $r^{-2d}$, a striking nonequilibrium effect with potential experimental implications for active enzymes and soft active solids.

Abstract

We study the dynamics of a zero-temperature overdamped tracer in a bath of Brownian particles. As the bath density is increased, numerical simulations show the tracer to transition from an active dynamics, characterized by boundary accumulation and ratchet currents, to an effective equilibrium regime. To account for this analytically, we eliminate the bath degrees of freedom under the assumption of linear coupling to the tracer and show convergence, in the large density limit, to an equilibrium dynamics at the bath temperature. We then develop a perturbation theory to characterize the tracer's departure from equilibrium at large but finite bath densities, revealing an intermediate time-reversible yet non-Boltzmann regime, followed by a fully irreversible one. Finally, we show that when the bath particles are connected as a lattice, mimicking a gel or a soft active solid, the cold tracer drives the entire bath out of equilibrium, leading to a long-ranged suppression of bath fluctuations.

Figures (4)

• Figure 1: (a) A zero-temperature tracer in a bath of $N$ Brownian particles at $T>0$, with short-ranged repulsive interactions between the tracer and bath particles. (b) Tracer position in a ratchet potential $U(x)$ of period $L$ in units of tracer radius. (c) The mean position $\langle x(t)\rangle$ scales as $t \langle \dot x \rangle$, demonstrating the existence of a ratchet current $\langle \dot x \rangle$ that decreases with the bath density $\rho=N/L^d$. (d) An external potential $U(x)$ models both confining walls at the system's ends and an asymmetric obstacle in the bulk. The stationary probability density $p(x)$ of the tracer shows that the rectification and boundary accumulation observed for finite bath densities disappear as $\rho\to\infty$.
• Figure 2: (a) Fully-connected model: The zero-temperature tracer is coupled by springs of stiffness $k$ to $N$ interacting particles. (b) Tracer probability density $p(x)$ in a quartic potential $U(x) = x^4/4$ (red dashed line). Left panel is the short-ranged model of Fig. \ref{['fig:fig1']}; right panel is the fully-connected model of Eq. \ref{['eq:fullyconnectedmodel']}. The colored curves are numerical results for different values of $\rho$ in the short-ranged model and $N$ in the fully-connected model. The black dashed lines correspond to the Boltzmann distribution $\propto e^{-U(x)}$.
• Figure 3: Departure from the large-$N$ equilibrium limit. (a) Stationary probability density $p(x)$ in a potential $U(x) = x^4/4$ (dotted black line). Blue markers are simulation results ($N=10$); solid lines are successive orders in perturbation theory; dotted magenta line is the effective potential at $\mathcal{O}(N^3)$. (b) Steady-state current, $\langle \dot{x} \rangle$, simulated in a potential $U(x) = \sin(\pi x/2) + \sin(\pi x)$ (inset). The magenta line is Eq. \ref{['eq:current']}, showing the $N^{-4}$ scaling. (c) Successive regimes of tracer dynamics as $N$ is increased.
• Figure 4: (a) Loop model: A zero-temperature particle is inserted within a loop of particles at temperature $T$ and subjected to a potential $U$. (b) Generalization to higher-dimensional lattices. (c) Effective temperature suppression of a particle $m$ sites away from the tracer. Blue markers are Eq. \ref{['eq:1dTeff']}$(n=d=1)$. Red markers are exact numerical results for $N=400^2$ particles with $n=1, d=2$. Black lines correspond to the field-theoretic prediction Eq. \ref{['eq:tefffieldtheory']}. $T=k=1$.