Temperature overshooting in the Mpemba effect of frictional active matter

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.

💡 Research Summary

In this paper the authors investigate a novel manifestation of the Mpemba effect in a macroscopic active particle subject to dry (Coulomb) friction. The system is described by a one‑dimensional Langevin equation that includes three forces: a velocity‑independent friction term σ(v)=ΔC sign(v), a Gaussian white‑noise term with strength K, and an active force n(t) f where n(t) follows an Ornstein‑Uhlenbeck process with persistence time τ. After nondimensionalisation the dynamics depend only on two parameters: the reduced activity amplitude f₀=f/ΔC (fixed at 0.5 in the study) and a composite noise parameter ϵ that controls both the thermal white noise and the active Ornstein‑Uhlenbeck noise.

When ϵ is large the dynamics are noise‑dominated; the active term can be neglected and the steady‑state velocity distribution becomes a Laplacian (V‑shaped) P_st(v;ϵ)=½ϵ exp(−|v|/ϵ). This is the classic distribution for a particle under dry friction driven by thermal fluctuations. For small ϵ the active noise dominates, producing a distribution with a sharp peak at zero velocity and heavy tails, reflecting rare but long‑lived acceleration events.

The authors define the kinetic temperature as T=½m⟨v²⟩/k_B and study cooling from an initial temperature T_init (set by ϵ_init) to a colder bath temperature T_bath (set by ϵ_bath). Two complementary measures of relaxation are used: (i) an entropic distance D(t)=∫|P(v,t)−P_st(v;ϵ_bath)| dv, and (ii) the relative deviation of the kinetic temperature from the bath value. The “Entropic Mpemba Effect” (EME) and the “Thermal Mpemba Effect” (TME) are identified when the hotter initial condition reaches the steady state faster than a colder one, according to D(t) and T(t), respectively.

Simulation results (Euler‑Maruyama integration, 10⁸ realizations) reveal a pronounced Mpemba effect in a broad range of ϵ (approximately 0.1–5). The key mechanism is a temperature‑overshoot phenomenon: for intermediate initial temperatures the kinetic temperature initially drops below T_bath before rising again, which lengthens the overall relaxation time. In contrast, a very hot initial state possesses fat tails in its velocity distribution, allowing a direct exponential decay of temperature, T(t)≈T_init exp(−t/ϵ), without overshoot. Consequently, the hot system cools faster than the intermediate one, producing the Mpemba effect.

Analytical arguments support these findings. In the early stage, before noise becomes important, the deterministic part of the Langevin equation yields v(t)=v_init−t (for v_init>t) and thus T(t) decays exponentially with a rate set by ϵ. Because dry friction is velocity‑independent, the decay rate depends on the initial temperature, unlike viscous (Stokes) friction where the rate is fixed by the viscosity. This leads to faster cooling for higher initial temperatures and to the observed overshoot for moderate temperatures.

The paper emphasizes that dry friction’s constant resistance makes the effect fundamentally different from previously reported Mpemba phenomena in systems with velocity‑dependent (Stokes) drag. Moreover, the authors show that both entropic and thermal measures give consistent Mpemba windows, indicating that the interplay of non‑linear friction and active fluctuations governs the anomalous relaxation.

Finally, the authors discuss practical implications. Since many robotic platforms, vibrating granular media, and other macroscopic active matter systems experience dry friction, the temperature‑overshoot mechanism can be exploited to steer motion, design faster stopping protocols, or create controlled delays by tuning the noise (or activity) level. The study thus extends the Mpemba concept to non‑equilibrium, friction‑dominated systems and provides a theoretical foundation for engineering applications where rapid thermal or kinetic relaxation is desired.