Time Reversal Breaking and Entropy Production in Non-Equilibrium Systems: Insights from Mean Back Relaxation
by Gabriel Johann Werner Knotz
Date of Examination:2025-11-10
Date of issue:2025-11-20
Advisor:Prof. Dr. Matthias Krüger
Referee:Prof. Dr. Matthias Krüger
Referee:Prof. Dr. Peter Sollich
Persistent Address:
http://dx.doi.org/10.53846/goediss-11645
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Abstract
English
The distinction between equilibrium thermal fluctuations, exemplified by Brownian motion, and active non-equilibrium processes is an essential problem in various fields of non-equilibrium statistical physics and biophysics. While Brownian motion is created by the random collision of a colloidal particle with much smaller surrounding particles, many biological and artificial systems exhibit externally and self-driven motion, for example, induced by external energy sources and molecular motors. This thesis addresses the problem of differentiating these motion types by focusing on the breaking of time reversal symmetry, a hallmark of non-equilibrium dynamics. We investigate a novel quantity, the mean back relaxation (MBR), to detect said time reversal symmetry breaking. The mean back relaxation is a three point correlation function and is defined as the negative ratio of the displacement between $[0,t]$ to the displacement from $[-\tau,0]$. We find that the long time value of MBR approaches $\frac{1}{2}$ if later displacements get statistically independent from earlier ones, and if time reversal symmetry is valid. We demonstrate that MBR shows time reversal breakage for an active Brownian particle in a harmonic potential. We derive a solution for the finite time MBR for a Gaussian process in terms of the mean squared displacement (MSD) with a path integral formalism. This allows the analytic investigation of the time dependence of MBR curves in a Gaussian model. Also, we introduce a new density based MBR generalizing MBR to other types of observables. By analyzing the variance of the back relaxation (VBR), we characterize MBRs statistical properties and give estimates on how to select MBR parameters for practical applications. Connecting MBR to fundamental concepts, we derive a new bound for entropy production, a central quantity in non-equilibrium dynamics. The bound is valid for time antisymmetric observables whose absolute value is bounded, and we find that the tightest observable is the sign of entropy production. This leads to an interesting coarse graining scheme that leaves the bound intact by grouping paths according to their stochastic entropy production. We test the entropy bound for a time discrete ring and compare it to other known relations. We apply MBR to experimental data obtained from colloidal particles in living cells. We can qualitatively reconstruct the curves found from cell data with a Random Horse and Cart model. Both in the model and cell data for different cell types, the long time value of MBR is linearly related to effective energy amplitude, which quantifies the violations of the fluctuation dissipation theorem (FDT), a common indirect marker of breaking of time reversal symmetry. Finally, we test the time reversal properties of MBR for cell data. We show that for colloidal particles in cells, MBR detects breaking of time reversal symmetry. This is one of the first direct empirical evidences for observable time reversal breaking of colloidal particles in living cells. The time reversal breaking is greatly reduced by depolymerization of microtubules, which hints that the effect is linked to cytoskeletal activity. We also apply the entropy production bound by inserting the anti-symmetric part of MBR and find that the lower bound given by MBR is correlated to FDT violations for different cell types, quantified by the effective energy amplitude.
Keywords: non-equilibrium; time reversal symmetry; cells; entropy production; Brownian motion; Langevin equations; path integral; stochastic dynamics; statistical physics
