Delayed Arrow-of-Time Detection in Signed Laplacian Dynamics

TL;DR

This paper investigates how the arrow of time becomes detectable in signed Laplacian dynamics, revealing conditions under which time direction can be identified or remains forever hidden, with implications for mesoscopic thermodynamics.

Contribution

It introduces a test for detecting the arrow of time in signed Laplacian dynamics and proves its effectiveness over certain intervals, extending understanding of time detectability in complex systems.

Findings

The test correctly identifies time direction if conducted over an interval longer than τ.

The test cannot give incorrect results over shorter intervals.

In a superquantum example, the arrow of time remains permanently undetectable.

Abstract

We study how rapidly the direction of time becomes operationally detectable from mesoscopic data when state-weights may be positive or negative. In contrast with classical Markov processes -- where forward evolution is instantly distinguishable from its reverse -- signed dynamics can render the arrow of time undetectable during an initial interval. Assume the generator of the signed dynamics is a symmetric signed Laplacian with a single linear invariant and a phase-space Second Law holds in the form of non-decreasing R\'enyi-$2$ entropy. Drawing on recent results on eventual exponential positivity of signed Laplacians (Chen et al., 2021), we define a test that correctly identifies the direction of thermodynamic time if conducted over a time interval of length at least $\tau$. We go on to prove that the test cannot deliver an incorrect conclusion if conducted over a shorter interval. Dropping symmetry, we exhibit a superquantum example where $\tau = +\infty$, so that the arrow of time remains permanently undetectable under our test.