On the entropy production of time series with unidirectional linearity

TL;DR

This paper explores the connection between time-inversion asymmetry in certain non-Gaussian time series and entropy production, demonstrating that linearity asymmetries relate to entropy generation in physical systems.

Contribution

It establishes a quantitative link between the non-linearity of backward conditionals in time series and entropy production in physical models.

Findings

Backward-time conditionals are non-linear due to environment-induced dependencies.

Linearity asymmetry correlates with the minimal entropy generated during interactions.

Physical toy system analysis confirms the theoretical link between non-linearity and entropy production.

Abstract

There are non-Gaussian time series that admit a causal linear autoregressive moving average (ARMA) model when regressing the future on the past, but not when regressing the past on the future. The reason is that, in the latter case, the regression residuals are only uncorrelated but not statistically independent of the future. In previous work, we have experimentally verified that many empirical time series indeed show such a time inversion asymmetry. For various physical systems, it is known that time-inversion asymmetries are linked to the thermodynamic entropy production in non-equilibrium states. Here we show that such a link also exists for the above unidirectional linearity. We study the dynamical evolution of a physical toy system with linear coupling to an infinite environment and show that the linearity of the dynamics is inherited to the forward-time conditional probabilities, but not to the backward-time conditionals. The reason for this asymmetry between past and future is that the environment permanently provides particles that are in a product state before they interact with the system, but show statistical dependencies afterwards. From a coarse-grained perspective, the interaction thus generates entropy. We quantitatively relate the strength of the non-linearity of the backward conditionals to the minimal amount of entropy generation.