A minimal model with stochastically broken reciprocity

TL;DR

This work studies a minimal two-body system in which Newton's third law is stochastically broken, while the average reciprocity is preserved. The authors derive stochastic equations for the center-of-mass and relative motions, develop corresponding Fokker-Planck descriptions for the COM, relative, and joint PDFs, and quantify how reciprocity-breaking fluctuations generate an energy input and nonzero entropy production. They show explicit results: the COM velocity variance grows as $\langle V^2\rangle = \langle V_0^2\rangle + g_{11}t$ and the COM MSD as $\langle (\Delta X)^2\rangle = \langle V_0^2\rangle t^2 + (g_{11}/3) t^3$, while the relative motion under a harmonic potential develops a time-dependent covariance from $g_{22}$. In a thermal bath, the relative motion obeys a Smoluchowski equation with an effective temperature $T_e = T + \tfrac{\nu}{2}(g_A/\gamma_A^2 + g_B/\gamma_B^2)$, yielding a Boltzmann steady state $P(x) \propto e^{-\varphi(x)/T_e}$, and an extremely minimal model decouples the relative motion to classical mechanics, leaving only the COM fluctuations. Overall, the framework provides a minimal, thermodynamically consistent way to study nonreciprocal fluctuations, their energetic and entropic implications, and potential experimental signatures and extensions to many-body systems.

Abstract

We introduce a minimal model consisting of a two-body system with stochastically broken reciprocity (i.e., random violation of Newton's third law) and then investigate its statistical behaviors, including fluctuations of velocity and position, time evolution of probability distribution functions, energy gain, and entropy production. The effective temperature of this two-body system immersed in a thermal bath is also derived. Furthermore, we heuristically present an extremely minimal model where the relative motion adheres to the same rules as in classical mechanics, while the effect of stochastically broken reciprocity only manifests in the fluctuating motion of the center of mass.

Figures (2)

• Figure 1: Two-body system. Two particles are labeled with A and B, respectively. $\mathbf{F}_\mathrm{BA}$ represents the action force exerted by A on B, while $\mathbf{F}_\mathrm{AB}$ represents the reaction force exerted by B on A.
• Figure 2: Time dependence of reduced mean square displacements (rMSD). The solid, dashed and dotted lines correspond to $\left\langle (\Delta X)^2\right\rangle/D\tau_c$, $\left\langle (\Delta X)^2\right\rangle_\mathrm{CM}/D\tau_c$, and $\left\langle (\Delta X)^2\right\rangle_\mathrm{BM}/D\tau_c$, respectively. For the plotting of these curves, the parameters employed are $\left\langle V_0^2\right\rangle=D/\tau_c$ and $g_{11}=D/10\tau_c^2$.