Manifestations of Projection-Induced Memory: General Theory and the Tilted Single File

Abstract

Over the years the field of non-Markovian stochastic processes and anomalous diffusion evolved from a specialized topic to mainstream theory, which transgressed the realms of physics to chemistry, biology and ecology. Numerous phenomenological approaches emerged, which can more or less successfully reproduce or account for experimental observations in condensed matter, biological and/or single-particle systems. However, as far as their predictions are concerned these approaches are not unique, often build on conceptually orthogonal ideas, and are typically employed on an ad hoc basis. It therefore seems timely and desirable to establish a systematic, mathematically unifying and clean approach starting from more finegrained principles. Here we analyze projection-induced ergodic non-Markovian dynamics, both reversible as well as irreversible, using spectral theory. We investigate dynamical correlations between histories of projected and latent observables that give rise to memory in projected dynamics, and rigorously establish conditions under which projected dynamics is Markovian or renewal. A systematic metric is proposed for quantifying the degree of non-Markovianity. As a simple, illustrative but non-trivial example we study single file diffusion in a tilted box, which, for the first time, we solve exactly using the coordinate Bethe ansatz. Our results provide a solid foundation for a deeper and more systematic analysis of projection-induced non-Markovian dynamics and anomalous diffusion.

Faster Uphill Relaxation in Thermodynamically Equidistant Temperature Quenches

“…Tilted single file.-In the context of tracer-particle dynamics, we consider N hard-core Brownian point particles with positions fx i ðtÞg (the extension to a finite diameter is straightforward [5,6]) diffusing in a box of unit length in the presence of a linear potential (e.g., the gravitational field), Uðfx i gÞ ¼ P N i¼1 gx i . The probability density of fx i ðtÞg upon a quench fromT, PTðfx i ðtÞg; tÞ, evolves according toL 1 ¼ P N i¼1 ð∂ 2 x i þ g∂ x i Þ under noncrossing conditions [7,8]. In [60] we solve the problem exactly via the coordinate Bethe ansatz [7,8], both for the Markovian complete single file and the non-Markovian probability density of a tagged particle PTðz; tÞ (i.e., q ≡ x T ¼ z).…”

“…Relaxation processes are a paradigm for condensed matter [1,2], single-molecule experiments [3] and tracer-particle transport in complex media [4][5][6][7][8]. Relaxation close to equilibrium was described by the mechanical Onsager-Casimir [9,10] and thermal Kubo-Yokota-Nakajima [11] linear laws.…”

“…1) remains elusive. Moreover, as a result of the projection to a lower-dimensional subspace, it is expected that observables in many experiments, in particular those tracking individual particles [4] and single molecules [46,47], relax in a manner that is not Markovian [8].…”

“…This always holds for single-well potentials and occurs in near degenerate multiwell potentials with high energy barriers under Markovian dynamics, as well as for a class of non-Markovian observables probed by single-molecule and particle-tracking experiments. We demonstrate the asymmetry on the hand of the Gaussian polymer chain [56], single-file diffusion in a tilted box [8], and for diffusion in nearly degenerate multiwell potentials. For relaxation near a stable minimum and thus for all reversible Ornstein-Uhlenbeck processes, we prove that the asymmetry, albeit counterintuitive, is general.…”

“…If (and only if) latent degrees of freedom (i.e., those integrated out) relax much faster than qðtÞ, Eqs. (4) and (5) are equivalent and qðtÞ is a Markovian diffusion in the free energy landscape UðqÞ [8]. In the absence of a time-scale separation, however, both STðtÞ and hUðtÞiT contain contributions from the (hidden) relaxation of the latent degrees of freedom.…”