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Level 2.5 large deviations and uncertainty relations for non-Markov self-interacting dynamics

Francesco Coghi, Amarjit Budhiraja, Juan P. Garrahan

Abstract

We address the general problem of formulating the dynamical large deviations of non-Markovian systems in a closed form. Specifically, we consider a broad class of ``self-interacting'' jump processes whose dynamics depends on the past through a functional of a state-dependent empirical observable. Exploiting a natural separation of timescales, we obtain the exact (so-called ``level 2.5'') large deviation joint statistics of the empirical measure over configurations and of the empirical flux of transitions. As an application of this general framework, we derive explicit general bounds on the fluctuations of trajectory observables, generalising to the non-Markovian case both thermodynamic and kinetic uncertainty relations. We illustrate our theory with simple examples, and discuss potential applications of these results.

Level 2.5 large deviations and uncertainty relations for non-Markov self-interacting dynamics

Abstract

We address the general problem of formulating the dynamical large deviations of non-Markovian systems in a closed form. Specifically, we consider a broad class of ``self-interacting'' jump processes whose dynamics depends on the past through a functional of a state-dependent empirical observable. Exploiting a natural separation of timescales, we obtain the exact (so-called ``level 2.5'') large deviation joint statistics of the empirical measure over configurations and of the empirical flux of transitions. As an application of this general framework, we derive explicit general bounds on the fluctuations of trajectory observables, generalising to the non-Markovian case both thermodynamic and kinetic uncertainty relations. We illustrate our theory with simple examples, and discuss potential applications of these results.
Paper Structure (9 sections, 26 equations, 1 figure)

This paper contains 9 sections, 26 equations, 1 figure.

Table of Contents

  1. Introduction.
  2. Self-interacting jump processes.
  3. Large deviations at level 2.5 for SIJPs.
  4. Kinetic uncertainty relation for SIJPs.
  5. Thermodynamic uncertainty relation for SIJPs.
  6. Simple examples.
  7. Conclusions.
  8. Acknowledgments.
  9. Appendix: notation and conventions.

Figures (1)

  • Figure 1: (a) Exact rate function for the empirical measure, $I_2(\ell)$ (full-black curve), obtained by solving Eqs. (\ref{['eq:RateFunct25']}) and (\ref{['eq:FunctionalFinal25']}) with constraints \ref{['eq:AccDistrReverse']}-\ref{['eq:RareDensity']}, for the two-level SIJP with $\alpha = 2$, Monte Carlo simulations (open-red circles), and Markovian approximation (dashed-blue curve). (b) Optimal trajectories $\rho(t)$: $(\pi_0,\pi_1)$ is an unstable fixed point, the dynamics is integrated backward in time, and by progressively truncating the trajectory at different $t$, one uncovers the most likely dynamics responsible for the fluctuation in $\ell$. Since trajectories yielding $\ell < \pi_0$ evolve slowly (see Inset), the left tail of $I_2(\ell)$ is close to that of the Markovian approximation (as the exponential discount suppresses distant-in-time contributions faster than they can accumulate). This is not the case for $\ell > \pi_0$, and the non-Markovian character becomes apparent in the higher likelihood of those fluctuations. (c) Exact rate function $I(b)$ for the flux observable $B_T=\Phi_{01}(T)$ (full-black curve) for the two-level SIJP, Monte Carlo simulations (open-red circles), and the SIJP-KUR bound (dashed-green curve). (d) Fluctuations of the particle current in the three-state model: Monte Carlo simulations (open-red circles) and the SIJP-TUR bound. All simulation results are for $10^6$ trajectories of time extent $T=1000$.