Inferring entropy production in anharmonic Brownian gyrators

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Inferring entropy production in anharmonic Brownian gyrators

Biswajit Das, Sreekanth K. Manikandan, and Ayan Banerjee

Phys. Rev. Research 4, 043080 – Published 7 November 2022

Abstract

A nonvanishing entropy production rate is one of the defining characteristics of any nonequilibrium system, and several techniques exist to determine this quantity directly from experimental data. The short-time inference scheme, derived from the thermodynamic uncertainty relation, is a recent addition to the list of these techniques. Here we apply this scheme to quantify the entropy production rate in a class of microscopic heat engine models called Brownian gyrators. In particular, we consider models with anharmonic confining potentials. In these cases, the dynamical equations are indelibly nonlinear, and the exact dependencies of the entropy production rate on the model parameters are unknown. Our results demonstrate that the short-time inference scheme can efficiently determine these dependencies from a moderate amount of trajectory data. Furthermore, the results show that the nonequilibrium properties of the gyrator model with anharmonic confining potentials are considerably different from its harmonic counterpart; especially in setups leading to a nonequilibrium dynamics and the resulting gyration patterns.

Received 7 May 2022
Accepted 27 September 2022

DOI:https://doi.org/10.1103/PhysRevResearch.4.043080

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)

Brownian motion Irreversible processes Nonequilibrium statistical mechanics Stochastic inference

Statistical Physics & Thermodynamics

Authors & Affiliations

Biswajit Das ^1,*, Sreekanth K. Manikandan ^2,†, and Ayan Banerjee ^1,‡

¹Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Mohanpur, West Bengal 741246, India
²Nordita Stockholm University and KTH Royal Institute of Technology, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

^*bd18ip005@iiserkol.ac.in
^†sreekanth.km@fysik.su.se
^‡ayan@iiserkol.ac.in

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Issue

Vol. 4, Iss. 4 — November - December 2022

Subject Areas

Statistical Physics

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Images

Figure 1
Brownian gyrator with complex potential: (a) Brownian gyrator model: A Brownian particle is confined in a generic potential in the presence of two heat baths of different temperatures ( $T_{1}$ and $T_{2})$ along two axes. In some cases, the potential needs to be rotated by an angle $θ$ along with $T_{1} \neq T_{2}$ to obtain the gyration effect of the trapped particle. Shapes of the generic potentials of our study: (b) double-well potential and (c) quartic potential. (d) Short-time inference technique is used in this study to characterize the nonequilibrium features of the system through the estimation of entropy generation rate and the thermodynamic force field from the trajectory of the trapped particle.
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Figure 2
(a) Velocity field and (b) thermodynamic force field of Brownian gyrator with harmonic confining potential are plotted (streamlines) on top of the steady-state probability (color maps) of the position of the particle. Parameters used: $k_{1} = 1$ , $k_{2} = 5$ , and $θ = 45^{\circ}$ . Ratio of the temperatures along the orthogonal axis is fixed at $α = \frac{T_{2}}{T_{1}} = 0.1$ . The closed loops denote the equipotential contours.
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Figure 3
Quantitative analysis of $σ$ corresponding to the Brownian gyrator model with the bistable potential: (a) Particle trajectories in the 2D coordinate space of the bistable potential when the potential is rotated by an angle $θ = 45^{\circ}$ . Parameter values: $b = 1, k = 2, α = \frac{T_{2}}{T_{1}} = 0.1$ . The displacement of the particle shows jumps along any direction due to the presence of the finite barrier height of the potential (inset). (b) Entropy production rate ( $σ$ ) of the system is quantified as a function of the ratio of temperatures ( $α = T_{2} / T_{1}$ ) along the two axes in different conditions of the gyrator indicated by the different values of the parameters mentioned in the legend. $σ$ of each set is normalized by dividing with the $σ$ value corresponding to $α = 0.1$ of that particular set. (c) Absolute value of $σ$ is plotted as a function of the angle of rotation ( $θ$ ) and the stiffness constant ( $k$ ) of the harmonic part of the potential while keeping the value of $α = 0.1$ . (Inset: $σ$ is plotted with $θ$ at fixed $k$ , and shows $\sim {sin}^{2} (2 θ)$ behavior.) (d) $σ$ is quantified as a function of the parameters “ $k$ ” and “ $b$ ” with $α = 0.1$ and $θ = 45^{\circ}$ . (e) Thermodynamic force field and (f) the phase-space velocity field plotted as streamlines. Parameter values: $b = 1, k = 2, α = 0.1, θ = 45^{\circ}$ . In (e) and (f), the color maps represent the steady-state probability distribution of the system, while the black loops denote the equipotential contours. Error bars are given as the standard deviation over ten independent measurements of the entropy production rate $σ$ for a fixed choice of model parameters.
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Figure 4
Quantitative analysis of $σ$ corresponding to the Brownian gyrator model with the quartic potential: (a) Brownian trajectories of the trapped particle in the isotropic quartic potential with $k_{1} = k_{2} = 1$ and $α = 0.1$ . (b) Stochastic trajectories of the trapped particle in the anisotropic quartic potential with $k_{1} = 1, k_{2} = 2$ , and $α = 0.1$ . (c) Entropy production rate ( $σ$ ) of the system with the anisotropic quartic potential is quantified as a function of the ratio of temperatures ( $α = T_{2} / T_{1}$ ) along two axes in different conditions of the gyrator indicated by the different values of the parameters mentioned in the legend. $σ$ of each set is normalised by dividing with the $σ$ value corresponding to $α = 0.1$ of that particular set. (d) $σ$ is plotted as a function of $θ$ for the anisotropic potential with $k_{1} = 1, k_{2} = 2$ . We observe deviation from $\sim {sin}^{2} (2 θ)$ as $σ \neq 0$ for $θ = 0$ . (e) Absolute value of $σ$ shows a linear nature with the parameter $k$ for the gyrator system with isotropic quartic potential. (f) The thermodynamic force field of the system with the isotropic quartic potential ( $k_{1} = k_{2} = 1, α = 0.1$ ). (g) The thermodynamic force field of the system in the anisotropic quartic potential ( $k_{1} = 1, k_{2} = 2, α = 0.1, θ = 45^{\circ}$ ). (h) Velocity field for the system with isotropic quartic potential ( $k_{1} = k_{2} = 1, α = 0.1$ ). (i) Velocity field for the system with the anisotropic quartic potential ( $k_{1} = 1, k_{2} = 2, α = 0.1, θ = 45^{\circ}$ ). In (f)–(i) color maps represent the steady-state probability distribution of the system while the black loops denote the equipotential contours. Error bars are given as the standard deviation over ten independent measurements of the entropy production rate $σ$ for a fixed choice of model parameters.
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Figure 5
Effect of hyperparameters on the inference of entropy production rate ( $σ$ ): $σ$ is plotted as a function of (a) length of the trajectory, (b) sampling interval, and (c) order of the polynomial basis for both double-well and quartic confining potential. Error bars are given as the standard deviation over ten independent measurements of entropy production rate $σ$ for a fixed choices of model parameters. Parameters: double-well potential: $b = 1, k = 2, θ = 45^{\circ}, α = 0.1$ ; quartic well potential: $k_{1} = 1, k_{2} = 2, θ = 45^{\circ}, α = 0.1$ .
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Figure 6
Kramers rates for double-well potential: The quantity $1 / 2 - p_{+} (t)$ [30] is plotted for all configurations of the double-well potential we study.
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