Statistical mechanics of a cold tracer in a hot bath

TL;DR

The paper develops a general framework to describe a zero-temperature tracer linearly coupled to a hot Brownian bath via a generalized Langevin equation, enabling exact elimination of bath degrees of freedom for arbitrary tracer potentials. It analyzes two canonical bath topologies—the fully-connected model and the loop model—showing that the tracer equilibrates at large bath density in the former but not in the latter, which exhibits persistent nonequilibrium behavior. A perturbative expansion for large but finite bath size reveals departures from Boltzmann statistics, ratchet currents, and entropy production, establishing a link to active matter. The work also provides exact solutions for quadratic external potentials, showing bath fluctuations are suppressed by the cold tracer in a long-range, power-law manner, and extends these results to elastic field theories, where the suppression scales as $r^{-2d}$ in $d$ dimensions. Together, these results illuminate nonequilibrium steady states, irreversible currents, and the impact of cold inclusions in hot baths and elastic media, with potential relevance to active enzymes and driven gels.

Abstract

We study the dynamics of a zero-temperature particle interacting linearly with a bath of hot Brownian particles. Starting with the most general model of a linearly-coupled bath, we eliminate the bath degrees of freedom exactly to map the tracer dynamics onto a generalized Langevin equation, allowing for an arbitrary external potential on the tracer. We apply this result to determine the fate of a tracer connected by springs to $N$ identical bath particles or inserted within a harmonic chain of hot particles. In the former "fully-connected" case, we find the tracer to transition between an effective equilibrium regime at large $N$ and an FDT-violating regime at finite $N$, while in the latter "loop" model the tracer never satisfies an FDT. We then study the fully-connected model perturbatively for large but finite $N$, demonstrating signatures of irreversibility such as ratchet currents, non-Boltzmann statistics, and positive entropy production. Finally, we specialize to harmonic external potentials on the tracer, allowing us to exactly solve the dynamics of both the tracer and the bath for an arbitrary linear model. We apply our findings to show that a cold tracer in a hot lattice suppresses the fluctuations of the lattice in a long-ranged manner, and we generalize this result to linear elastic field theories.

Figures (3)

• Figure 1: A zero-temperature tracer in an external potential $U(x)$ coupled to two models of a linearly-interacting hot bath. (a) Fully-connected model: The tracer is connected by linear springs to all bath particles. (b) Loop model: The tracer is inserted within a harmonic chain of hot particles. These are two instances of the more general coupling described by Eq. \ref{['eq:genlanham']} and studied throughout the article.
• Figure 2: Steady-state current, $\langle \dot{x} \rangle$, in an asymmetric periodic potential $U(x) = \sin(\pi x/2) + \sin(\pi x)$ (inset), for two variants of the fully-connected model: first, a harmonically-coupled model where the tracer temperature is $T_0=T/2$ rather than $T_0 = 0$, and, second, a model in which the harmonic springs coupling tracer and bath particles are replaced by quartic springs with interaction potential $({\bf r} - {\bf r}_i)^4/4$. The spatial dimension $d$ is indicated. Both models show an $N^{-4}$ decay, consistent with our perturbative results on the harmonic fully-connected model with a zero-temperature tracer.
• Figure 3: A zero-temperature tracer in a bath of Brownian colloids at temperature $T=0$, with short-ranged soft repulsive interactions between the tracer and bath particles. A periodic potential $U(x) = \sin(\pi x/2) + \sin(\pi x)$ is applied to the tracer. (a) Stationary probability density of the tracer $P(x)$ for different bath densities $\rho$, showing convergence to the Boltzmann weight at large $\rho$. (b) The deviation from the Boltzmann distribution $\left[P(x) - P_\mathrm{eq}(x)\right]$ scales as $1/\rho$, and fits the fully-connected theory of Eq. \ref{['eq:statMeasure']} with an effective $k$ fitted to $k=6.5$. (c) The steady current decays as $1/\rho^4$, with a prefactor comparable to the fully-connected prediction of Eq. \ref{['eq:current']}, with the value $k=6.5$ fitted from panel (b). The simulation domain is periodic with size $L=4$ and the interaction potential is $V(r) = 5\Theta(1-r) \times (r-1)^2/2$.