Configurational temperature in active matter. II. Quantifying the deviation from thermal equilibrium

Shibu Saw, Lorenzo Costigliola, and Jeppe C. Dyre

Phys. Rev. E 107, 024610 – Published 17 February 2023

Abstract

This paper proposes using the configurational temperature $T_{conf}$ for quantifying how far an active-matter system is from thermal equilibrium. We measure this “distance” by the ratio of the systemic temperature $T_{s}$ to $T_{conf}$ , where $T_{s}$ is the canonical-ensemble temperature for which the average potential energy is equal to that of the active-matter system. $T_{conf}$ is “local” in the sense that it is the average of a function, which depends only on how the potential energy varies in the vicinity of a given configuration. In contrast, $T_{s}$ is a global quantity. The quantity $T_{s} / T_{conf}$ is straightforward to evaluate in a computer simulation; equilibrium simulations in conjunction with a single steady-state active-matter configuration are enough to determine $T_{s} / T_{conf}$ . We validate the suggestion that $T_{s} / T_{conf}$ quantifies the deviation from thermal equilibrium by data for the radial distribution function of the 3D Kob-Andersen and 2D Yukawa active-matter models with active Ornstein-Uhlenbeck and active Brownian Particle dynamics. Moreover, we show that $T_{s} / T_{conf}$ , structure, and dynamics of the homogeneous phase are all approximately invariant along the motility-induced phase separation boundary in the phase diagram of the 2D Yukawa model. The measure $T_{s} / T_{conf}$ is not limited to active matter and can be used for quantifying how far any system involving a potential-energy function, e.g., a driven Hamiltonian system, is from thermal equilibrium.

Received 17 December 2022
Accepted 23 January 2023

DOI:https://doi.org/10.1103/PhysRevE.107.024610

Physics Subject Headings (PhySH)

Collective behavior

Physics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

Shibu Saw ^*, Lorenzo Costigliola^†, and Jeppe C. Dyre ^‡

Glass and Time, IMFUFA, Department of Science and Environment, Roskilde University, P.O. Box 260, DK-4000 Roskilde, Denmark

^*shibus@ruc.dk
^†lorenzo.costigliola@gmail.com
^‡dyre@ruc.dk

Configurational temperature in active matter. I. Lines of invariant physics in the phase diagram of the Ornstein-Uhlenbeck model

Shibu Saw, Lorenzo Costigliola, and Jeppe C. Dyre

Phys. Rev. E 107, 024609 (2023)

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Vol. 107, Iss. 2 — February 2023

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Images

Figure 1
Structure and dynamics of the Yukawa ABP model in two dimensions. (a) The left panel shows the RDF as a function of the pair distance $r$ along the active-matter isomorph, the middle panel shows the same data in reduced units, and the right panel shows the reduced RDF for the same parameters (Table 1) at the reference state-point density $ρ = 1.0$ . (b) The left panel shows the MSD as a function of time $t$ along the active-matter isomorph, the middle panel shows the same data in reduced units where the dashed line marks slope unity, i.e., ordinary diffusion; the right panel shows the reduced MSD for the same parameters at the reference state-point density $ρ = 1.0$ .
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Figure 2
Determination of the ratio of systemic to configurational temperature, $T_{s} / T_{conf}$ , quantifying how far an active-matter system is from thermal equilibrium. (a) Data for $T_{s}$ and $T_{conf}$ for the 3D Kob-Andersen AOUP model (Paper I, [6]) as functions of $τ$ with the remaining model parameters kept fixed. (b) $T_{s} / T_{conf}$ for the same data. For $τ$ values around $10^{- 4}$ the system begins to move away from equilibrium, and for $τ > 10^{- 3}$ significant deviations from equilibrium are predicted. (c) Data for $T_{s}$ and $T_{conf}$ for the 2D Yukawa AOUP model as functions of $τ$ with the remaining model parameters kept fixed. (d) $T_{s} / T_{conf}$ for the same data. For $τ$ values above $10^{- 4}$ the system starts to deviate from equilibrium. (e) Data for $T_{s}$ and $T_{conf}$ for the 2D Yukawa ABP model as functions of $v_{0}$ with the remaining model parameters kept fixed. (f) $T_{s} / T_{conf}$ for the same data. For $v_{0}$ values around 10 the system begins to move away from equilibrium.
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Figure 3
RDFs of active-matter states predicted to be close to and not close to thermal equilibrium (left and right column, respectively). The red curves are the active-matter data and the black dashed lines are the RDFs of the corresponding equilibrium system for $T = T_{s}$ . (a–d) Results for the AA and BB RDFs of the Kob-Andersen AOUP model for $τ = 10^{- 4}$ and $τ = 4 \times 10^{- 2}$ (red curves) corresponding to $T_{s} / T_{conf} = 1.13$ and $T_{s} / T_{conf} = 6.59$ . (e, f) Results for the 2D Yukawa AOUP model for $τ = 10^{- 4}$ and $τ = 8 \times 10^{- 3}$ corresponding to $T_{s} / T_{conf} = 1.09$ and $T_{s} / T_{conf} = 2.59$ . (g, h) Results for the 2D Yukawa ABP model for $v_{0} = 10$ and $v_{0} = 50$ corresponding to $T_{s} / T_{conf} = 1.13$ and $T_{s} / T_{conf} = 2.18$ .
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Figure 4
Using the ratio of systemic to configurational temperature to quantify how far the 2D Yukawa ABP system is from thermal equilibrium [corresponding to $v_{0} = 0$ in Eq. (3)]; the parameters kept fixed here are $ρ = 1, D_{r} = 3$ , and $D_{t} = 2$ . (a) shows how the dissipation (“Power”) varies with $v_{0}$ (MD units). From Fig. 2 we see that when $v_{0} \to 0$ , the two temperatures become identical (equal to 2 because $D_{t} = 2$ corresponds to that thermal equilibrium temperature); at the same time the dissipation goes to zero. (b, c) Power as a function of $T_{s} / T_{conf}$ (black points). The quantity $T_{s} / T_{conf}$ goes to unity as thermal equilibrium is approached, which presents an advantage compared to using the dissipated power for quantifying deviations from thermal equilibrium. The red points in (b) give the power in reduced units.
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Figure 5
$(ρ, D_{t})$ phase diagrams showing MIPS state points as red stars and homogeneous state points as black squares (the green circles are gaslike states of minor relevance here). The MIPS phase consists of coexisting phases that differ in density, the denser phase being a “solid” phase of hexagonal crystal structure. The reference state point $(ρ, D_{r}, D_{t}, v_{0}) = (1.01, 3, 1, 367)$ is located in the homogeneous (solid) phase close to the phase boundary. From this an active-matter isomorph was traced out using Eq. (11) (black line). The figure gives data in the $(ρ, D_{t})$ phase diagram with $D_{r}$ and $v_{0}$ given by Eq. (11) at density $ρ$ . The blue dashed lines mark $\pm 5$ % variations in density. We see that the phase-transition line is an approximate active-matter isomorph, which is consistent with the expectation that the degree of deviation from thermal equilibrium is constant along this line.
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Figure 6
Structure and dynamics probed along the active-matter isomorph approximately delimiting the MIPS phase boundary of the 2D ABP Yukawa system, slightly into the homogeneous phase (Fig. 5). Panels (a) and (b) show log-log plots of the RDF and MSD, respectively; panels (c) and (d) show the same data in reduced units.
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