Temperature overshooting in the Mpemba effect of frictional active matter

TL;DR

The work investigates Mpemba-like relaxation in active matter on a Coulomb friction surface. It introduces a minimal one-dimensional inertial active-particle model with dry friction and active Ornstein–Uhlenbeck forcing, analyzed via a distance measure $\mathcal{D}(t)$ and a kinetic-temperature metric $T(t)$. It finds a pronounced entropic Mpemba effect (EME) where hotter initial states relax faster toward the bath distribution, and a concurrent thermal Mpemba effect (TME) reflected in the early-time decay $T(t) \approx (\epsilon^2/k_B) e^{-t/\epsilon}$ that overshoots the bath. The results highlight how nonlinear friction combined with activity yields Mpemba-like relaxation and suggest potential applications in guiding macroscopic active systems such as vibrobots and granular robots.

Abstract

The traditional Mpemba effect refers to an anomalous cooling phenomenon when an initial hotter system cools down faster than an initial warm system. Such counterintuitive behavior has been confirmed and explored across phase transitions in condensed matter systems and also for colloidal particles exposed to a double-well potential. Here we predict a frictional Mpemba effect for a macroscopic body moving actively on a surface governed by Coulomb (dry) friction. For an initial high temperature, relaxation towards a cold state occurs much faster than that for an intermediate initial temperature, due to a large temperature overshooting in the latter case. This frictional Mpemba effect can be exploited to steer the motion of robots and granules.

Figures (4)

• Figure 1: (a) An illustrative example of active granular particle (vibrating robot), subjected to dry friction due to the solid-solid contact between the surface and its legs. (b)-(d) Trajectories sketched to illustrate different temperatures. The motion occurs due to an interplay between active force and dry friction force, while the temperature regulates the impact of the fluctuations, resulting in the gradual trajectories for cold (b) temperatures and in fluctuating trajectories when increasing the temperature for warm (c) and hot (d). (e) Dependence of the system temperature $T$ on the noise parameter $\epsilon$. Colored rectangle projections mark the cold, warm (intermediate), and hot states, highlighting the associated ranges of temperature $T$ and noise strength $\epsilon$.
• Figure 2: (a) Steady-state velocity probability distribution $P_{\rm st}(v)$ for various noise parameters $\epsilon$ and fixed activity amplitude $f_0 = 0.5$. (b) Kurtosis of the steady-state distribution as a function of noise parameter $\epsilon$ at a fixed activity amplitude $f_0 = 0.5$. Color phases correspond to those depicted in Fig. \ref{['fig:illustration']}(e), and the dashed vertical lines correspond to the noise parameters of density profiles shown in panel (a).
• Figure 3: (a) Relaxation dynamics of the distance measure $\mathcal{D}(t)$. (b) Relaxation time in the temperature range $T_{\rm init}/T_{\rm bath} \in [1, 900]$. The blue rectangle indicates the temperature range where the entropic Mpemba effect (a reduction in the distance relaxation time) is observed. (c) Monotonic dependence of $\mathcal{D}_{\rm init}$ on $T_{\rm init}$ for the parameters employed in the numerical simulations.
• Figure 4: (a) Time evolution of system temperatures for warm and hot initial temperatures. (b) Relaxation time determined from the temperature evolution \ref{['eq:temp-relax']}. The dark-blue rectangle (Mpemba effect) indicates the temperature range where both the thermal Mpemba effect (a reduction in the temperature relaxation time) and entropic Mpemba effects are observed; the light-blue rectangle (TME) indicates the temperature range where only the thermal Mpemba effect occurs, while the gray rectangle (EME) marks the range where only the entropic Mpemba effect is present.