Fluctuation-dissipation relation and stationary distribution of an exactly solvable many-particle model for active biomatter far from equilibrium

Abstract

An exactly solvable, Hamiltonian-based model of many massive particles that are coupled by harmonic potentials and driven by stochastic non-equilibrium forces is introduced. The stationary distribution and the fluctuation-dissipation relation are derived in closed form for the general non-equilibrium case. Deviations from equilibrium are on one hand characterized by the difference of the obtained stationary distribution from the Boltzmann distribution; this is possible because the model derives from a particle Hamiltonian. On the other hand, the difference between the obtained non-equilibrium fluctuation-dissipation relation and the standard equilibrium fluctuation-dissipation theorem allows us to quantify non-equilibrium in an alternative fashion. Both indicators of non-equilibrium behavior, i.e., deviations from the Boltzmann distribution and deviations from the equilibrium fluctuation-dissipation theorem, can be expressed in terms of a single non-equilibrium parameter α that involves the ratio of friction coefficients and random force strengths. The concept of a non-equilibrium effective temperature, which can be defined by the relation between fluctuations and the dissipation, is by comparison with the exactly derived stationary distribution shown not to hold, even if the effective temperature is made frequency dependent. The analysis is not confined to close-to-equilibrium situations but rather is exact and thus holds for arbitrarily large deviations from equilibrium. Also, the suggested harmonic model can be obtained from non-linear mechanical network systems by an expansion in terms of suitably chosen deviatory coordinates; the obtained results should thus be quite general. This is demonstrated by comparison of the derived non-equilibrium fluctuation dissipation relation with experimental data on actin networks that are driven out of equilibrium by energy-consuming protein motors. The comparison is excellent and allows us to extract the non-equilibrium parameter α from experimental spectral response and fluctuation data.

Approach to equilibrium and nonequilibrium stationary distributions of interacting many-particle systems that are coupled to different heat baths

“…Two prominent features of NEQ systems are departures from the canonical Boltzmann distribution and violation of the fluctuation-dissipation theorem, which were recently demonstrated to go hand in hand for a harmonically coupled particle system [19], the fundamental hallmark of NEQ systems is the violation of the detailed balance condition [5,8]. A particularly striking example * rnetz@physik.fu-berlin.de for the breakdown of equilibrium statistical mechanics is the occurrence of phase transitions induced by NEQ driving.…”

“…To rescue the effective temperature picture one would have to ascribe different temperatures to each covariance matrix element, which clearly is far from the usefulness of the temperature definition at equilibrium. When basing the effective temperature definition on the fluctuation-dissipation relation, as an additional effect a frequency dependence appears [19,42,43,92], which is not reflected in the effective temperatures one would obtain based on the covariance matrix elements.…”

Negative friction memory induces persistent motion

“…Examples include the diffusion of a tracer bead in viscoleastic [3][4][5][6] and heterogeneous [7,8] media, polymer dynamics [9][10][11][12][13] and dynamics in rough energy landscapes [14][15][16]. Furthermore, many systems far from equilibrium, such as self-propelled particles [17][18][19] or passive tracer particles in active media [20][21][22], exhibit non-Markovian dynamics.…”

Effective Temperature in Active Brownian Particles

“…Much attention has been recently payed to analyze whether e®ective thermodynamic concepts could be applied to describe active matter. Most studies in this direction focus on the de¯nition of e®ective temperatures [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], active pressure [16,17], and activity-induced interactions [18].…”

“…Finally, Netz [12] asked whether deviations from FDT and the Boltzmann distributions are necessarily coupled. He considered a solvable Hamiltonian-based model of massive particles harmonically coupled to a central one and driven by a stochastic force out of equilibrium.…”