Entropy production in the nonreciprocal Cahn-Hilliard model

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Entropy production in the nonreciprocal Cahn-Hilliard model

Thomas Suchanek, Klaus Kroy, and Sarah A. M. Loos

Phys. Rev. E 108, 064610 – Published 18 December 2023

Abstract

We study the nonreciprocal Cahn-Hilliard model with thermal noise as a prototypical example of a generic class of non-Hermitian stochastic field theories, analyzed in two companion papers [Suchanek, Kroy, and Loos, Phys. Rev. Lett. 131, 258302 (2023); Phys. Rev. E 108, 064123 (2023)]. Due to the nonreciprocal coupling between two field components, the model is inherently out of equilibrium and can be regarded as an active field theory. Beyond the conventional homogeneous and static-demixed phases, it exhibits a traveling-wave phase, which can be entered via either an oscillatory instability or a critical exceptional point. By means of a Fourier decomposition of the entropy production rate, we quantify the associated scale-resolved time-reversal symmetry breaking, in all phases and across the transitions, in the low-noise regime. Our perturbative calculation reveals its dependence on the strength of the nonreciprocal coupling. Surging entropy production near the static-dynamic transitions can be attributed to entropy-generating fluctuations in the longest wavelength Fourier mode and heralds the emerging traveling wave. Its translational dynamics can be mapped on the dissipative ballistic motion of an active (quasi)particle.

Received 11 May 2023
Accepted 13 November 2023

DOI:https://doi.org/10.1103/PhysRevE.108.064610

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Entropy production Nonequilibrium & irreversible thermodynamics Pattern formation Phase transitions Stochastic thermodynamics

Living matter & active matter Non-Hermitian systems Nonequilibrium systems

PT-symmetry

Statistical Physics & ThermodynamicsFluid DynamicsPolymers & Soft MatterPhysics of Living Systems

Authors & Affiliations

Thomas Suchanek¹, Klaus Kroy¹, and Sarah A. M. Loos ^2,*

¹Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
²DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

^*sl2127@cam.ac.uk

Time-reversal and parity-time symmetry breaking in non-Hermitian field theories

Thomas Suchanek, Klaus Kroy, and Sarah A. M. Loos

Phys. Rev. E 108, 064123 (2023)

Irreversible Mesoscale Fluctuations Herald the Emergence of Dynamical Phases

Thomas Suchanek, Klaus Kroy, and Sarah A. M. Loos

Phys. Rev. Lett. 131, 258302 (2023)

Irreversibility across a Nonreciprocal $P T$ -Symmetry-Breaking Phase Transition

Henry Alston, Luca Cocconi, and Thibault Bertrand

Phys. Rev. Lett. 131, 258301 (2023)

Article Text

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References

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Issue

Vol. 108, Iss. 6 — December 2023

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Images

Figure 1
(a) Phase diagram of the stationary solutions of Eq. (1) as established in Ref. [26] for $κ = 0.01, γ = 0.015$ and $β = 0.05$ . The mixed phase is marked in yellow, the static-demixed phase in blue and the traveling-wave phase in red. The transition from the static-demixed phase to the traveling wave phase is through a line of exceptional points, which is here marked by a vertical blue line. The transition from the homogeneous phase to the traveling wave phase is through a line of oscillatory instabilities, here marked by a horizontal red line. The dotted gray lines ( $c_{1} - c_{6}$ ) mark paths along which the behavior of the entropy production is displayed in Figs. 2, 4, and 5; (b)–(d) snapshots of the stationary solutions for $ε = 2.5 \times 10^{- 5}$ . Unlike to the static-demixed phase, the maxima and minima of the profiles of the fields $ϕ_{A}$ (gray) and $ϕ_{B}$ (black) in the traveling-wave phase have a characteristic mean phase shift $〈 Δ θ^{π} 〉 > 0,$ with $Δ θ^{π} \equiv θ_{A} - (θ_{B} - π)$ , which is aligned with the direction of propagation. This coupling of the propagation direction to the relative phase shift (which determines the parity of the profile) entails the $PT$ symmetry breaking in the nonreciprocal Cahn-Hilliard model.
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Figure 2
Entropy production rate $S$ and the first-mode contribution $σ^{1}$ [defined in Eqs. (14), (15)] as a function of the “demixing” parameter $α$ , inside the homogeneous phase in the vicinity of the phase boundaries (at $α_{c}$ , indicated by dashed vertical lines): (a) when approaching a conventional critical point transition to the static-demixed phase along path $c_{1}$ from Fig. 1 (here, $δ = 0.8 δ_{c}$ ); (b) approaching an oscillatory instability to the traveling-wave phase along path $c_{2}$ from Fig. 1 (here, $δ = 1.2 δ_{c}$ ). The analytical prediction of Eq. (19) is shown by red solid lines. Symbols depict simulation results, dashed lines serve as guides to the eye. Insets depict zooms into a smaller $y$ -axis range. We observe very distinct behavior near the transitions. Toward the static-static transition [in panel (a)], the entropy production rate only mildly increases but stays regular. Toward the static-dynamic transition [in panel (b)], the first-mode contribution $σ^{1}$ , and hence also $S$ , increase steeply, and formally diverges in the zero-noise limit, even after UV regularization. Other parameters are $κ = 0.01, γ = 0.015, β = 0.05, ε = 10^{- 10}$ .
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Figure 3
Scaling of the entropy production rate $S$ and the first-mode contribution $σ^{1}$ [defined in Eqs. (14), (15), shown here in panel (a)], $S$ and $σ_{θ}^{1}$ [defined in Eqs. (B13) and (B8), and shown in panels (b,c)] with the noise intensity $ε$ , for $ε \to 0$ within the different phases: (a) Mixed phase ( $δ = 0.03, α = - 0.02$ ); (b) static-demixed phase ( $δ = 0.045, α = - 0.07$ ); (c) traveling-wave phase ( $δ = 0.06, α = - 0.07$ ). The analytical results for the low-noise limits given in Eqs. (19), (30), and (32) are shown by solid lines. The numerical results confirm the analytics, including that $S \sim ε^{0}$ in the static phases, and $S \sim ε^{- 1}$ in the traveling-wave phase. Other parameters are $κ = 0.01, γ = 0.015, β = 0.05$ .
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Figure 4
Total entropy production rate $S$ from Eq. (B13), and its divergent contribution $σ_{θ}^{1}$ due to the first Fourier mode from Eqs. (B8), as a function of the control parameters $α$ and $δ$ , inside the static-demixed phase: (a) beyond the conventional critical point at $α_{c}$ , along path $c_{3}$ from Fig. 1 (here, $δ = 0.8 δ_{c}$ ), and (b) when approaching a CEP where the system undergoes a transition to the traveling-wave phase along path $c_{4}$ from Fig. 1 (here, $α = - 0.07$ ). The analytical prediction Eq. (30) is shown by red solid lines. Symbols depict simulation results, dashed lines are guides to the eye. Insets show magnifications. We observe that $S$ decreases away from the phase boundaries. For large $| α |$ , the data indicates saturation to a finite value. Similarly to the homogeneous phase (Fig. 2), $S$ remains regular toward to static-static transition [panel (a)], while it surges toward the static-dynamic transition [panel (b)]. The decomposition of $S$ clearly shows that the contribution $σ_{θ}^{1}$ from the fluctuations of the first Fourier mode, $θ^{1} = {(θ_{A}^{1}, {\tilde{θ}}_{B}^{1})}^{T}$ , is responsible for the surge in entropy production. It can exclusively be attributed to excitations of the Goldstone mode [32]. Other parameters are $κ = 0.01, γ = 0.015, β = 0.05, ε = 10^{- 10}$ .
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Figure 5
Total entropy production $S$ from Eq. (B13) and its first-mode's contribution $σ^{1}$ from Eq. (B12) within the traveling-wave phase in the vicinity of the phase boundaries: (a) beyond the CEP transition to the static-demixed phase along path $c_{5}$ from Fig. 2 (here, $α = - 0.07$ ), and (b) beyond the oscillatory instability, along path $c_{6}$ from Fig. 1 (here, $δ = 1.2 δ_{c}$ ). The analytical result for $σ^{1}$ in Eq. (32) is shown by solid red lines, symbols depict corresponding simulation results. As one moves deeper into the traveling-wave phase, $S$ generally increases, i.e., it increases with increasing amplitude of the traveling-wave ${(A_{A, B}^{1, *})}^{2} \propto | α - α_{c} |$ [26], as well as with an increase of the deterministic speed of the traveling waves $v^{2} \sim | δ - δ_{c} |$ [panel (a)].
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Figure 6
Scaled variance of the phase fluctuations near the transition as a function of $δ$ for $α = - 0.07, β = 0.05, γ = 0.015, κ = 0.01, ε = 10^{- 10}$ from the numerical simulation of Eq. (1). The solid line shows the analytical result Eq. (C3) which holds within the one mode approximation. The inset shows the same results in a log-scale.
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