Level 2.5 large deviations and uncertainty relations for self-interacting jump processes: tilting constructions and the emergence of time-scale separation

Abstract

Self-interacting jump processes (SIJPs) describe systems with non-Markovian stochastic dynamics in which transition rates depend on empirical observables of the process, which gives rise to long-range memory and feedback. We derive the ``level-2.5'' large deviation (LD) principle governing the joint fluctuations of empirical occupation measure and the flux matrix for a broad class of SIJPs with general functional dependence on an empirical observable. The derivation is based on an exponential tilting construction and reveals a separation between a faster timescale of the microscopic dynamics and a slower timescale of the memory-driven evolution of transition rates, which is expressed through an exponentially discounted LD rate functional. Using this variational framework, we derive kinetic and thermodynamic uncertainty relations that extend classical Markovian bounds to non-Markovian systems, and illustrate their performance with simple examples.

Figures (5)

• Figure 1: Phase portraits of the optimal trajectories $(\rho_t,M_t)$ solving Eqs. \ref{['eq:MsDiffEx1']}–-\ref{['eq:rhosDiffEx1']} for the two-state SIJP with (a) linear feedback $h^{({\rm L})}(L_1)=L_1$ and (b) exponential feedback $h^{({\rm E})}(L_1)=\exp(\alpha L_1)$ with $\alpha=2$. Fixed points at $(0,0)$, $(\pi,\pi)$, and $(1,1)$ are shown, with $(0,0)$ and $(1,1)$ acting as saddle points (purple circles). The black solid line denotes the one-dimensional stable manifold connecting the saddle points through $(\pi,\pi)$, along which all admissible optimal trajectories lie. Arrows indicate the direction of increasing reversed time.
• Figure 2: Optimal trajectories $(\rho_t,M_t)$ along the stable manifold for the two-state SIJP with (a) linear and (b) exponential feedback. Trajectories are obtained by integrating Eqs. \ref{['eq:MsDiffEx1']}–-\ref{['eq:rhosDiffEx1']} backward in time from initial conditions close to the saddle points. Each point on the curves corresponds to a distinct initial value $M_0=\ell$, generating a fluctuation of the empirical occupation measure $L_0(T)=\ell$ in the (physical) long-time limit. Trajectories converge to $(\pi,\pi)$ as $t\to0$ ($t$ is rescaled and reversed time).
• Figure 3: Level-2 large deviation rate function $I_2(\ell)$ of the empirical occupation measure $L_0(T)$ for the two-state SIJP with (a) linear and (b) exponential feedback. Solid black lines show the exact rate function obtained by evaluating Eq. \ref{['eq:QuadraticEx1']} along the optimal trajectories $(\rho_t^*,M_t^*)$. Symbols correspond to Monte Carlo simulations. The dashed curve shows the Markov (Donsker–Varadhan) upper bound $I_{\mathrm{DV}}(\ell)$ obtained from the constant-density Ansatz $\rho_t=\ell$. Deviations from the Markov bound become pronounced for large fluctuations away from $\pi$.
• Figure 4: Large deviation rate function $I(b)$ of the unidirectional flux $B_T=\Phi_{01}(T)$ for the two-state SIJP with (a) linear and (b) exponential feedback. Solid black lines denote the exact rate functions obtained by minimising $\tilde{\mathcal{I}}[\rho,\eta]$ under the flux constraint. Symbols correspond to Monte Carlo simulations. The dashed curve shows the SIJP-KUR bound given by Eq. \ref{['eq:IbBoundSIJP']}, which correctly captures typical fluctuations and provides an upper bound on rare events.
• Figure 5: Large deviation behaviour of the total current $J_T$ in three-state SIJPs. (a) Rate function of the total current for the SIJP with state-dependent feedback $h(L_1)$ defined in Eq. \ref{['eq:TanhFeedback']}, obtained from Monte Carlo simulations (symbols) and compared with the SIJP-TUR upper bound $I^\Box(j)$ given by Eq. \ref{['eq:ScalUppBounddFinalTURSIJP']} (solid line). (b) Current fluctuations for the SIJP with jump-dependent feedback $A_t=J_t$, compared with the SIJP-UKUR bound $I^\triangle(j)$ in Eq. \ref{['eq:UKURBoundSimplified3State']} (solid line). The dashed line shows the corresponding Markov UKUR for comparison.