Tractable Model for Tunable Non-Markovian Dynamics

TL;DR

This work introduces a minimal, tractable non-Markovian copy model where the current state tries to replicate a past state chosen according to a tunable history distribution $\rho(k)$. It derives general relations for autocorrelations, entropy, and dynamical information, and analyzes both finite- and infinite-order memory across multiple history classes (delta, binomial, Poisson, exponential, power-law, ERW). A key finding is that autocorrelations can exhibit exponential decay even when true long-range dependencies are present, whereas the dynamical information more faithfully tracks the underlying history. The results span exact solutions (where possible) and simulation-based estimates, revealing rich behaviors including phase-like transitions, crossover scalings, and nontrivial decompositions into Markovian and non-Markovian information. Overall, the model provides a versatile, analytically tractable sandbox for understanding how history dependence shapes correlations and information flow in non-Markovian dynamics, with potential parallels to attention mechanisms in learning systems and broader coarse-grained descriptions in physics and biology.

Abstract

Non-Markovian dynamics are ubiquitous across physics, biology, and engineering. Yet our understanding of non-Markovian processes significantly lags that of simpler Markovian processes, due largely to a lack of tractable models. In this article, we present a minimal model of non-Markovian dynamics in which the current state copies past states with arbitrary history dependence. We show that many properties of this process can be studied analytically, providing insight into the relationships between history dependence, autocorrelations, and information-theoretic metrics like entropy and dynamical information. Strikingly, we find that autocorrelations can fail, even qualitatively, to capture the underlying dependencies. Ultimately, this model serves as a tractable sandbox for exploring non-Markovian dynamics.

Figures (3)

• Figure 1: Schematic illustrating the non-Markovian copy process. (a) At each time $t$, the state $k$ steps in the past $x_{t-k}$ is selected with probability $\rho(k)$ and then copied to the current state $x_t$ with probability $\alpha$ (yielding error probability $1-\alpha$). (b) The strength of the $k^\text{th}$-order dependence is defined by the distribution $\rho(k)$, which can be tuned arbitrarily to produce different non-Markovian dynamics of both finite and infinite order.
• Figure 2: Copy process with finite-order history dependence. (a) Binomial history dependence distributions $\rho(k)$ [Eq. \ref{['eq:rho_binom']}] with order $N=10$ and shape parameters $q = \{0,0.1,0.5,0.9,1.0\}$. Parameters $q = 0,1$ are equivalent to delta history dependence [Eq. \ref{['eq:rho_delta']}] with $N = 1,10$, respectively. (b-c) Autocorrelations $C(k)$ with points reflecting numerical values and dashed lines in (c) depicting theoretical predictions for exponential scaling Eq. \ref{['eq:binomialCk']}. (d-e) Conditional entropy $h_k$ (d) and dynamical information $I_k$ (e) as functions of order $k$. (f) Total, Markovian, and non-Markovian dynamical information versus shape parameter $q$. Values in (d-f) are computed exactly, and $\alpha = 0.9$ for all analyses.
• Figure 3: Copy process with infinite-order history dependence. (a) History dependencies $\rho(k)$ defined by Poisson ($\lambda=5$, purple), exponential ($q=0.7$, blue), power-law ($s=4$, green; $s=1.5$, orange), and uniform ($t=1000$, red) distributions. (b) Autocorrelations computed from simulations (points), with dashed lines representing analytic calculations from Eq. \ref{['eq:poissonCk']} (Poisson), Eq. \ref{['eq:expCk']} (exponential), and Eq. \ref{['eq:zetaCk']} (power-law). For Poisson, we use exponential scaling from Eq. \ref{['eq:poissonCk']} and match the intercept to simulations at $k=100$. (c) Dynamical information $I_k$ computed from simulations using the first three terms of Eq. \ref{['eq:dynamicalinfoseries']}. For ERW, $I_k$ is computed using the recursive algorithm in Appendix C up to $k=20$ (due to computational limitations). (d-f) Total, Markovian, and non-Markovian dynamical information for exponential dependence with varying $q$ (d), power-law dependence with varying $s$ (e), and uniform dependence with varying $\alpha$ and $t = 1000$ (f). In (d-e), lines are computed from simulations using Eq. \ref{['eq:dynamicalinfoseries']}, and shaded regions reflect analytic upper and lower bounds from Eq. \ref{['eq:expIkULB']}. In (f), $I_1$ is computed exactly using Eq. \ref{['eq:ERW_I1']}, while $I_\mathrm{tot}$ and $I_{>1} = I_\mathrm{tot}-I_1$ are approximated using Eq. \ref{['eq:Itot_ERW']} (solid lines) and bounded exactly using Eqs. \ref{['eq:dynamicalinfoapprox']} and \ref{['eq:ItotUB']} (shaded regions). Note that bounds are tight enough to be nearly indistinguishable from numerical calculations. In (a-e), we set $\alpha=0.9$.