Irreversibility across a Nonreciprocal $\mathcal{P}\mathcal{T}$-Symmetry-Breaking Phase Transition

Nonreciprocal interactions are commonplace in continuum-level descriptions of both biological and synthetic active matter, yet studies addressing their implications for time reversibility have so far been limited to microscopic models. Here, we derive a general expression for the average rate of informational entropy production in the most generic mixture of conserved phase fields with nonreciprocal couplings and additive conservative noise. For the particular case of a binary system with Cahn-Hilliard dynamics augmented by nonreciprocal cross-diffusion terms, we observe a nontrivial scaling of the entropy production rate across a parity-time symmetry breaking phase transition. We derive a closed-form analytic expression in the weak-noise regime for the entropy production rate due to the emergence of a macroscopic dynamic phase, showing it can be written in terms of the global polar order parameter, a measure of parity-time symmetry breaking.

Figure 1

Nontrivial scaling of $\dot{\mathcal{S}}$ across a $\mathcal{P}\mathcal{T}$-breaking phase transition in a nonreciprocal system. (a) For $D={10}^{-7}$, we identify a non-trivial scaling in the entropy production rate in the binary system governed by Eq. (8), with a discontinuous derivative (inset) at a critical value of the nonreciprocal coupling $\alpha ={\alpha }_c$. [Numerical simulations: symbols; analytic results from Eqs. (13) and (15): solid lines.] (b) Discontinuity in the scaling of $\dot{\mathcal{S}}$ coinciding with the breaking of $\mathcal{P}\mathcal{T}$ symmetry, characterized by the nonzero value of the polar order parameter ${\mathcal{J}}^{(0)}$ for the deterministic equations [i.e., $D=0$ in Eq. (8)]. The critical nonreciprocal coupling corresponds to a transition between (c) static states and (d) dynamic states (traveling waves). The simulations were run for parameter values $L=2\pi$, ${\gamma }_1=0.04$, ${\gamma }_2=0$, ${\chi }_1=-0.05$, ${\chi }_2=0.005$, and $\kappa =0.005$.

Figure 2

Evaluation of contributions to entropy production rate $\dot{\mathcal{S}}$. (a) Transition to motion for nonreciprocal coupling strengths exceeding critical value, ${\alpha }_c=\sqrt{{\kappa }^2+{\chi }_2^2}$. (b) Numerically evaluated contribution ${\dot{\mathcal{S}}}_A$ shown to scale quadratically with the nonreciprocal parameter $\alpha$. (c) A second contribution ${\dot{\mathcal{S}}}_B$ to the entropy production rate stems directly from the emergence of macroscopic dynamics as exhibited in (a).

Figure 3

Scaling of ${\dot{\mathcal{S}}}_A$ and ${\dot{\mathcal{S}}}_B$ in the weak-noise regime. Given in the regime $\alpha >{\alpha }_c$ to ensure that both contributions to $\dot{\mathcal{S}}$ are nonzero, we confirm through the numerical simulations (symbols) the analytic results obtained through the one-mode approximation (solid lines): ${\dot{\mathcal{S}}}_A\propto D^0$ [Eq. (13)] (prefactor determined by fitting numerical data) and ${\dot{\mathcal{S}}}_B\propto D^{-1}$ [Eq. (15)].