Mpemba Effect in Many-Body Systems Near Equilibrium

Abstract

The Mpemba effect, in which a system initially farther from equilibrium relaxes faster than a closer one, is often associated with nonlinear or far-from-equilibrium dynamics. We show that this effect can arise entirely within the linear-response regime of many-body systems. In reciprocal systems, a uniform Mpemba effect emerges for three or more degrees of freedom via spectral separation of fast and slow modes. Breaking reciprocity renders the relaxation operator non-normal, enabling a strict componentwise Mpemba effect, with the hotter state relaxing faster even in every individual degree of freedom.

Figures (3)

• Figure 1: (a) Mechanical analogy of the Mpemba effect: relaxation corresponds to overdamped motion in an effective potential $V_{\mathrm{eff}}(\Theta)=\tfrac{1}{2} \Theta^T M \Theta$, with the bottom representing equilibrium. Two initial states are shown: a hotter state (red) aligned with steep directions, and a colder state (green) along shallow directions. (b) Non-uniform Mpemba effect with a many-body system: the hotter state relaxes faster globally without being componentwise larger. (c) True Mpemba effect with a many-body system: the hotter state is strictly larger in all components and still relaxes faster.
• Figure 2: Mpemba effect in a many-body system (a) made with three SiC nanoparticles (radius $r = 50~\mathrm{nm}$) arranged in a scalene triangle which exhange heat between them and with an external bath by radiation in near-field and far-field, respectively. Double arrows indicate the strength of radiative coupling $G_{ij}$, the shortest separation $d_{12}=3r$ corresponding to the strongest interaction. All particles are coupled to the bath (wavy arrows) at $T_b = 300~\mathrm{K}$. (b) Distance to the equilibrium $\mathcal{D}(t)=\|\Theta^{(h,c)}(t)\|$ for a "hot" initial state $\Theta^{(h)}(0)=(25,-23,-1.5)$ (red) and a "cold" initial state $\Theta^{(c)}(0)=(4,4,4)$ (blue). Despite starting farther from equilibrium, the hot state relaxes faster at short times due to suppressed projection onto the slowest mode, producing a clear crossing, showing a non-uniform Mpemba effect.
• Figure 3: True Mpemba effect in an active many-body system. (a) The system is an electronic circuit composed of three aligned nodes whose voltages represent the dynamical variables. Each node is connected to ground by a capacitor and a resistor in parallel, setting the local relaxation rates $M_{11}=0.1$, $M_{22}=1.0$, and $M_{33}=4.0$. Nodes $1$ and $2$ are coupled by two operational amplifiers arranged in parallel and in opposite directions; each op-amp injects a current through a series resistor before reaching the receiving node, thereby implementing directional couplings. The same architecture connects nodes $2$ and $3$. This design realizes asymmetric interactions with strengths $M_{12}=3.5$, $M_{21}=0.02$, $M_{23}=2.5$, $M_{32}=0.03$, while $M_{13}=M_{31}=0$. (b) Time evolution of the distance $\mathcal{D}$ for the three coupled nodes, starting from two initial states in which the hot state is strictly larger componentwise than the cold state. The asymmetry of the couplings aligns the hot state with faster decaying modes, leading to a finite-time crossing of the Euclidean norms and providing evidence of a genuine componentwise Mpemba effect.