Frictional Mpemba Effect Dynamics

• Frictional Mpemba effect is a nonequilibrium anomaly where hot systems relax faster to a target state due to friction-induced dynamics.
• The effect arises from the coupling of temperature with hidden variables like non-Gaussian velocity distributions and excess kurtosis.
• Experimental studies on dry-friction active matter reveal that threshold activation and temperature overshooting can lead to anomalous relaxation times.

The frictional Mpemba effect is a nonequilibrium relaxation anomaly in which a system governed by friction or friction-like dissipation reaches a colder or target steady state faster when prepared hotter than when prepared at a merely intermediate temperature. In the most explicit recent formulation, the effect is predicted for a single macroscopic active body moving on a surface under Coulomb dry friction, where a hotter initial state relaxes faster after a quench because an intermediate initial state undergoes a large temperature overshoot below the bath value and then requires delayed reheating (Antonov et al., 30 Jul 2025). Closely related Mpemba-like anomalies have also been identified in molecular gases with nonlinear drag, driven granular gases and Maxwell models, driven binary mixtures, and active particles in traps, where the common structure is multi-mode dissipative relaxation controlled by additional state variables beyond temperature alone (Santos et al., 2020, Lasanta et al., 2016, Biswas et al., 2020, Biswas et al., 2021, González et al., 2020, Biswas et al., 2024).

1. Definition and diagnostic criteria

In frictional settings, the effect is usually defined through either a crossing of relaxation curves or a reversal of relaxation times. For the dry-friction active-matter model, the protocol fixes a bath noise parameter $\epsilon_{\rm bath}$, prepares the initial state as the steady velocity distribution $P_{\rm st}(v;\epsilon_{\rm init})$ with $\epsilon_{\rm init}>\epsilon_{\rm bath}$, then quenches to $\epsilon=\epsilon_{\rm bath}$ and follows the relaxation of the full velocity distribution $P(v,t)$ toward $P_{\rm st}(v;\epsilon_{\rm bath})$ (Antonov et al., 30 Jul 2025). The distance to the bath state is taken to be

$$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$

and the relaxation time is the first $t_{\rm relax}$ such that $\mathcal{D}(t_{\rm relax})=2\times 10^{-3}$. A Mpemba effect is present when two initial temperatures satisfy

$$T_{\rm hot}>T_{\rm warm}>T_{\rm bath},\qquad t_{\rm relax}(T_{\rm hot})<t_{\rm relax}(T_{\rm warm}),$$

despite

$P_{\rm st}(v;\epsilon_{\rm init})$0

Granular and kinetic-theory studies use an equivalent criterion in terms of crossing temperature curves or shorter equilibration times. For granular fluids, the hotter sample can become colder than the initially cooler one after a crossover time, or equivalently have a shorter relaxation time to a prescribed tolerance (Lasanta et al., 2016). In driven Maxwell gases and driven binary mixtures, the comparison is made through the total granular energy or temperature $P_{\rm st}(v;\epsilon_{\rm init})$1 and its crossing under a common post-quench dynamics (Biswas et al., 2020, González et al., 2020). These formulations all encode the same ordering anomaly: initial distance from the final state does not determine the late-time approach rate.

A persistent theme across these definitions is that temperature is not the only relevant state variable. In dry-friction active matter the full velocity distribution matters; in granular and molecular gases the excess kurtosis or other moments matter; in mixtures the energy partition between components matters (Antonov et al., 30 Jul 2025, Lasanta et al., 2016, Santos et al., 2020, González et al., 2020). This already distinguishes frictional Mpemba phenomena from single-parameter exponential relaxation.

2. Dry-friction active matter as the canonical frictional realization

The most direct “frictional Mpemba effect” in the narrow sense is the dry-friction model of an active macroscopic body moving along a line on a solid surface (Antonov et al., 30 Jul 2025). In dimensional form its velocity obeys

$P_{\rm st}(v;\epsilon_{\rm init})$2

with Coulomb friction

$P_{\rm st}(v;\epsilon_{\rm init})$3

Gaussian white noise $P_{\rm st}(v;\epsilon_{\rm init})$4 of strength $P_{\rm st}(v;\epsilon_{\rm init})$5, active force amplitude $P_{\rm st}(v;\epsilon_{\rm init})$6, and an Ornstein–Uhlenbeck active drive $P_{\rm st}(v;\epsilon_{\rm init})$7. After nondimensionalization the dynamics become

$P_{\rm st}(v;\epsilon_{\rm init})$8

$P_{\rm st}(v;\epsilon_{\rm init})$9

with reduced activity amplitude $\epsilon_{\rm init}>\epsilon_{\rm bath}$0 and noise parameter

$\epsilon_{\rm init}>\epsilon_{\rm bath}$1

The study fixes $\epsilon_{\rm init}>\epsilon_{\rm bath}$2 and varies $\epsilon_{\rm init}>\epsilon_{\rm bath}$3.

Because the system is active and out of equilibrium, temperature is defined kinetically:

$\epsilon_{\rm init}>\epsilon_{\rm bath}$4

The bath “temperature” is therefore controlled indirectly through $\epsilon_{\rm init}>\epsilon_{\rm bath}$5, which determines the steady-state value of $\epsilon_{\rm init}>\epsilon_{\rm bath}$6. The stationary velocity distribution $\epsilon_{\rm init}>\epsilon_{\rm bath}$7 is strongly non-Gaussian. At large $\epsilon_{\rm init}>\epsilon_{\rm bath}$8 it is Brownian-like with Laplace-like tails characteristic of dry-friction Brownian motion; at small $\epsilon_{\rm init}>\epsilon_{\rm bath}$9 it exhibits a sharp peak at $\epsilon=\epsilon_{\rm bath}$0 and heavy tails associated with intermittent active bursts. The stationary distribution of the OU variable is Gaussian and independent of $\epsilon=\epsilon_{\rm bath}$1; $\epsilon=\epsilon_{\rm bath}$2 changes its timescale rather than its variance (Antonov et al., 30 Jul 2025).

These non-Gaussian stationary states are not incidental. The reported Mpemba anomaly occurs in a regime where the cold bath has weak white noise, rare activation events, and strong sticking near $\epsilon=\epsilon_{\rm bath}$3. Under those conditions the state space explored by the dynamics is organized by threshold crossing rather than by linear damping, which makes the relaxation strongly history-dependent.

3. Temperature overshooting and the threshold mechanism

The central mechanism in the dry-friction system is temperature overshooting. For certain intermediate initial temperatures, the kinetic temperature $\epsilon=\epsilon_{\rm bath}$4 does not relax monotonically to $\epsilon=\epsilon_{\rm bath}$5 after the quench; instead it dips below the bath value,

$\epsilon=\epsilon_{\rm bath}$6

for a finite interval and only later returns upward to the final steady state (Antonov et al., 30 Jul 2025). By contrast, hotter initial states show weaker overshooting and approach the bath more directly.

The dynamics separate into two stages. At early times, Coulomb friction dominates. Because the friction force is velocity-independent and depends only on $\epsilon=\epsilon_{\rm bath}$7, it removes motion without preferentially damping faster states more strongly. The white-noise component is weak in the cold-bath regime, and active motion requires threshold crossing by the OU force. The relevant activation probability is the probability that the active force exceeds the friction threshold,

$\epsilon=\epsilon_{\rm bath}$8

which is small but nonzero for $\epsilon=\epsilon_{\rm bath}$9. The corresponding mean waiting time is estimated as

$P(v,t)$0

Before $P(v,t)$1, dry friction mainly cools the system; after $P(v,t)$2, intermittent active bursts re-inject energy (Antonov et al., 30 Jul 2025).

This produces an asymmetry between “warm” and “hot” initial states. For a moderately warm initial distribution, dry friction efficiently arrests many modest-velocity realizations, and the scarcity of threshold-crossing events creates a prolonged low-energy interval. The result is an over-cooled state with $P(v,t)$3, followed by a slow reheating stage. For a very hot initial state, heavy tails in the velocity distribution and the eventual active bursts reduce the depth of the overshoot, so the trajectory in distribution space reaches the bath distribution sooner. The hotter state is therefore initially farther from the target in $P(v,t)$4 but closer in the dynamically relevant sense that controls $P(v,t)$5 (Antonov et al., 30 Jul 2025).

The same paper explicitly contrasts Coulomb and viscous friction. Under viscous drag $P(v,t)$6, faster states are cooled more strongly, and the specific threshold-sticking mechanism responsible for the overshoot is absent. The reported anomaly is therefore tied to dry friction as a physical mechanism, not merely to generic activity.

4. Related frictional and dissipative realizations

The broader literature shows that “frictional Mpemba effect” is not confined to Coulomb friction. Several dissipative mechanisms play an analogous role by coupling temperature to hidden variables that evolve on comparable timescales.

System class	Dissipative mechanism	Auxiliary variable or structure
Dry-friction active matter	Coulomb friction plus intermittent active drive	Heavy-tailed $P(v,t)$7, threshold activation, overshooting
Molecular gas under nonlinear drag	Nonlinear viscous drag $P(v,t)$8	Excess kurtosis $P(v,t)$9
Granular fluids and Maxwell gases	Inelastic collisions plus driving	Excess kurtosis, correlations, $P_{\rm st}(v;\epsilon_{\rm bath})$0, anisotropy
Driven binary mixtures	Species-dependent Stokes drag plus noise	Partial temperatures and $P_{\rm st}(v;\epsilon_{\rm bath})$1
Active Brownian particle in a trap	Overdamped viscous damping plus activity	Mode coefficient $P_{\rm st}(v;\epsilon_{\rm bath})$2 and phase-space shift

In a molecular gas coupled to a thermal bath through nonlinear drag,

$P_{\rm st}(v;\epsilon_{\rm bath})$3

the Enskog–Fokker–Planck description yields coupled evolution of temperature and excess kurtosis. The Mpemba effect is absent for linear drag $P_{\rm st}(v;\epsilon_{\rm bath})$4 and appears only when nonlinear friction makes the cooling rate depend on the non-Gaussianity $P_{\rm st}(v;\epsilon_{\rm bath})$5 of the velocity distribution (Santos et al., 2020). In uniformly heated and freely cooling granular fluids, the relevant coupling is between granular temperature and excess kurtosis, again producing multi-exponential relaxation and explicit crossing criteria (Lasanta et al., 2016).

Exact Maxwell-gas analyses make the hidden variables particularly transparent. In the mono-disperse driven Maxwell gas, the effect requires non-stationary initial states with nonzero inter-particle velocity correlations $P_{\rm st}(v;\epsilon_{\rm bath})$6, because steady states have $P_{\rm st}(v;\epsilon_{\rm bath})$7 and therefore lack the extra mode needed for crossings (Biswas et al., 2020). In bi-disperse Maxwell gases, and in anisotropically driven inelastic Maxwell gases, the internal variable can instead be an energy difference $P_{\rm st}(v;\epsilon_{\rm bath})$8 or an anisotropy, so Mpemba, inverse Mpemba, and strong Mpemba effects can occur even when the initial states are non-equilibrium steady states (Biswas et al., 2020, Biswas et al., 2021). Driven molecular binary mixtures under viscous drag and Langevin forcing show the same structure: the bath couples differently to the two species through $P_{\rm st}(v;\epsilon_{\rm bath})$9 and $$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$0, and the global temperature relaxes through coupled partial temperatures $$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$1 and $$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$2 (González et al., 2020).

Active-particle models without metastable states extend this picture to overdamped traps. For an active Brownian particle in a single-well potential, the asymptotic criterion for a Mpemba effect is the non-monotonic dependence of the slowest-mode amplitude $$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$3 on the initial temperature. Activity can induce or suppress the effect by producing an effective translational shift in the phase space of steady states and eigenmodes (Biswas et al., 2024). Taken together, these studies suggest that the essential ingredient is not a specific microscopic friction law but dissipative dynamics in which temperature is slaved to additional slow degrees of freedom.

5. Spectral and dynamical interpretations

Several papers formulate the anomaly in spectral language. In the active Brownian-particle study, the post-quench probability distribution is expanded in right eigenfunctions of the Fokker–Planck operator,

$$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$4

so the late-time ordering is governed by the coefficient of the slowest nonzero mode, $$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$5 (Biswas et al., 2024). Photonic diffusion under dephasing offers a closely analogous Liouvillian picture: a far-from-equilibrium state relaxes faster when its overlap with the slowest mode is suppressed, while a seemingly closer state retains a larger slow tail (Longhi, 2024). Random-circuit work in the quantum setting reaches a related conclusion in terms of overlaps with conserved and nonconserved operator sectors (Turkeshi et al., 2024).

The dry-friction active-matter paper does not present a full eigenmode analysis of its Fokker–Planck operator, but it explicitly states that the behavior can be interpreted in that spirit: threshold behavior and fat tails imply that relaxation is not dominated by a simple exponential mode, and different initial conditions project differently onto slow directions of evolution (Antonov et al., 30 Jul 2025). This aligns with the granular and Maxwell-gas literature, where temperature relaxation is a sum of at least two exponentials and strong Mpemba effects correspond to initial conditions with vanishing projection on the slowest mode (Lasanta et al., 2016, Biswas et al., 2020, Biswas et al., 2021).

A more geometric variant appears in the conservative mechanical analogue of the Mpemba effect. There the system is frictionless, so it is not itself a frictional realization, but it isolates a useful structural motif: non-monotonic relaxation times arise when state space contains fast and slow regions and the initial condition carries dynamical memory between them (Finkelstein, 28 Mar 2025). Landau theory for second-order phase transitions expresses the same logic in overdamped form: the effective restoring coefficient in the order-parameter direction changes sign along the trajectory, and the time to reach that stability boundary can be shorter for hotter initial conditions (Holtzman et al., 2022). These works suggest that frictional Mpemba effects are best understood as state-dependent alignment with fast and slow relaxation channels rather than as anomalies of temperature alone.

6. Scope, experimental relevance, and open issues

The dry-friction active-matter realization is explicitly motivated by macroscopic active systems such as vibrobots and robotic granules. Because the required ingredients are dry friction, stochastic driving, and tunable noise amplitude, the effect is presented as experimentally accessible. The observable set is likewise straightforward: particle velocities can be tracked, kinetic temperature can be inferred from $$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$6, and overshooting of $$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$7 below $$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$8 can be measured directly (Antonov et al., 30 Jul 2025). The same paper emphasizes a control perspective: one may steer relaxation by choosing a sufficiently high initial activity or noise level, or avoid parameter regimes in which overshooting produces long transients.

Several misconceptions are explicitly contradicted by the literature. First, the effect does not require water, freezing, or any phase transition. Granular fluids remain homogeneous and phase-transition free, active dry-friction matter has no latent heat, and active particles in single-well traps can show the effect without metastable states (Lasanta et al., 2016, Antonov et al., 30 Jul 2025, Biswas et al., 2024). Second, it is not a violation of thermodynamic principles; even the mechanical analogue presents the ordering anomaly as a dynamical consequence of geometry and inertia rather than any conflict with fundamental laws (Finkelstein, 28 Mar 2025). Third, it is not generic under linear homogeneous relaxation. In the nonlinear-drag molecular gas, the effect disappears for linear drag; in the one-parameter Landau setting, the phase-transition time is $$\mathcal{D}(t)=\int_{-\infty}^{\infty} dv\,\left|P(v,t)-P_{\rm st}(v;\epsilon_{\rm bath})\right|,$$9 for all hot initial states, so no Mpemba ordering remains (Santos et al., 2020, Holtzman et al., 2022).

The current formulations also impose sharp modeling assumptions. The dry-friction study is one-dimensional, single-particle, and based on ideal Coulomb friction with Gaussian white noise and an OU active force (Antonov et al., 30 Jul 2025). Molecular-gas and granular analyses often rely on first Sonine closures or Maxwell simplifications, while driven-mixture studies use multitemperature Maxwellians and homogeneous states (Santos et al., 2020, Biswas et al., 2020, González et al., 2020). A plausible implication is that the detailed location and strength of Mpemba regimes are model-dependent even when the qualitative mechanism is robust.

Open questions recur across the papers. Full spectral analysis of the dry-friction active model remains outstanding (Antonov et al., 30 Jul 2025). Extensions to dense granular flows, active particles in disordered environments, more realistic friction laws with stick–slip or hysteresis, and controlled quench protocols are explicitly suggested (Antonov et al., 30 Jul 2025). Broader work on dissipative and even conservative analogues indicates that the decisive problem is to identify the slow variables that control the projection onto the slowest relaxation sector. In that sense, the frictional Mpemba effect has evolved from a narrow paradoxical statement about “hotter cools faster” into a technically precise question about how friction, non-Gaussianity, and hidden modes organize nonequilibrium relaxation.