Accumulation of Particles and Formation of a Dissipative Structure in a Nonequilibrium Bath

Abstract

The standard textbooks contain good explanations of how and why equilibrium thermodynamics emerges in a reservoir with particles that are subjected to Gaussian noise. However, in systems that convert or transport energy, the noise is often not Gaussian. Instead, displacements exhibit an α-stable distribution. Such noise is commonly called Lévy noise. With such noise, we see a thermodynamics that deviates from what traditional equilibrium theory stipulates. In addition, with particles that can propel themselves, so-called active particles, we find that the rules of equilibrium thermodynamics no longer apply. No general nonequilibrium thermodynamic theory is available and understanding is often ad hoc. We study a system with overdamped particles that are subjected to Lévy noise. We pick a system with a geometry that leads to concise formulae to describe the accumulation of particles in a cavity. The nonhomogeneous distribution of particles can be seen as a dissipative structure, i.e., a lower-entropy steady state that allows for throughput of energy and concurrent production of entropy. After the mechanism that maintains nonequilibrium is switched off, the relaxation back to homogeneity represents an increase in entropy and a decrease of free energy. For our setup we can analytically connect the nonequilibrium noise and active particle behavior to entropy decrease and energy buildup with simple and intuitive formulae.

Variance of fluctuations from Noether invariance

“…In future work it it would be highly interesting to explore connections of our results to statistical thermodynamics [13], to the study of liquids under shear [30], to the large fluctuation functional [31], as well as to recent progress in systematically incorporating two-body correlations into classical density functional theory [32,33]. Investigating the implications of our variance results for Levy-noise [34] is interesting. Future use of the sum rules can be manifold, ranging from the construction and testing of new theories, such as approximate free energy functionals within the classical density functional framework, to validation of simulation data and numerical theoretical results.…”

Variance of fluctuations from Noether invariance

“…In future work it it would be highly interesting to explore connections of our results to statistical thermodynamics 13 , to the study of liquids under shear 32 , to the large fluctuation functional 33 , as well as to recent progress in systematically incorporating twobody correlations into classical density functional theory 34,35 . Investigating the implications of our variance results for Levynoise 36 is interesting. As the displacement vector ϵ is arbitrary both in its orientation and its magnitude our reasoning does not stop at second order in the Taylor expansion, see Eqs.…”