Run-and-tumble particles — lattice random walkers whose drift direction is an internal state that flips on the macroscopic time scale — have a stationary fluctuation theory that is complete only at the Gaussian level: for independent particles the layer-resolved density fluctuations converge to an infinite-dimensional Ornstein–Uhlenbeck process, and with exclusion interactions the total density is a non-Markovian Gaussian process with closed-form covariance (Redig–van Wiechen, 2024). The large-deviation layer remains unbuilt: a rate function with memory terms was computed only in a single worked example (a Schilder-type small-noise limit with no passive diffusion and two internal states), rigorous dynamical large-deviation principles exist only for mean-field switching rates and for the multispecies stirring process, and the physics literature derives heuristic non-Gaussian fluctuating-hydrodynamics actions for the active lattice gas. The open problem is to prove a dynamical large-deviation principle for the empirical densities of run-and-tumble particle systems with local exclusion interactions in the diffusive weak-asymmetry slow-flip scaling, to identify the rate function including its non-Gaussian tumble contribution, and to characterize the associated stationary rate function (quasi-potential) of the total density with its memory terms made explicit.

Background

Run-and-tumble particles (RTPs) are a minimal lattice model of active matter: each particle carries an internal state σ\sigma from a finite set S⊂ZS\subset\mathbb{Z} (for bacteria such as E. coli, S={−1,1}S=\{-1,1\} encodes the swimming direction), performs nearest-neighbour random walks, and switches its internal state ('tumbles') at random times. On the one-dimensional integer lattice Z\mathbb{Z} one studies the diffusive scaling: space is rescaled by NN, time by N2N^2, passive jumps occur at rate κN2\kappa N^2, active jumps (x,σ)→(x+σ,σ)(x,\sigma)\to(x+\sigma,\sigma) at rate λN\lambda N ('weak asymmetry'), and flips (x,σ)→(x,σ′)(x,\sigma)\to(x,\sigma') at rate c(σ,σ′)c(\sigma,\sigma') — so flips are slow, of order one on the macroscopic time scale. The hydrodynamic limit (law of large numbers for the empirical densities as N→∞N\to\infty) is then a system of reaction–diffusion equations for the layer densities ρt(x,σ)\rho_t(x,\sigma) — linear for independent particles, nonlinear when an exclusion rule (at most α\alpha particles per site) is imposed. Because the internal state relaxes only on the macroscopic time scale, the total density ϱt(x)=∑σρt(x,σ)\varrho_t(x)=\sum_\sigma \rho_t(x,\sigma) is governed by a non-closed equation: eliminating the polarization field Δt(x)=ρt(x,1)−ρt(x,−1)\Delta_t(x)=\rho_t(x,1)-\rho_t(x,-1) yields an equation with second-order time derivatives, i.e. the total-density process is non-Markovian and carries memory.

At the level of typical fluctuations this structure is now understood. For independent RTPs, Redig and van Wiechen proved that the joint fluctuation fields of the layer densities converge to an infinite-dimensional Ornstein–Uhlenbeck process (a Gaussian distribution-valued Markov process solving a linear stochastic partial differential equation, SPDE), whose noise has a conservative part from particle transport and a non-conservative part from the flips; duality (an exact algebraic relation between expectations of two Markov processes that here reduces many-particle correlations to single-particle ones) determines the stationary covariance in closed form, also for the multi-layer symmetric exclusion process with flips, where the total-density fluctuations form a non-Markovian Gaussian process with explicit covariance, and, in the independent-particle case with two internal states, a closed second-order-in-time SPDE.

Beyond Gaussian fluctuations one wants a large-deviation principle (LDP): a rate function II such that the probability of an atypical density trajectory ρ^\hat\rho behaves like exp⁡(−N I(ρ^))\exp(-N\,I(\hat\rho)) (speed NN). For gradient reversible systems this is the macroscopic fluctuation theory (MFT) of Bertini, Landim and Olla, with a quadratic rate function built from a Gaussian noise. Reaction terms with slow rates break this: the number of flip events per macroscopic space-time cell is large but Poissonian, contributing a non-quadratic term of the form Klog⁡(K/ρ)−K+ρK\log(K/\rho)-K+\rho to the action. The static counterpart is the quasi-potential — the stationary LDP rate function of the density profile, obtained by minimizing the dynamical rate function over trajectories ending at the given profile — which plays the role of a nonequilibrium free energy. For interacting RTPs the stationary measure is not known in closed form, and for sufficiently strong activity the system undergoes motility-induced phase separation (MIPS): demixing into dense and dilute phases caused by the density-dependence of the self-propulsion speed created by exclusion, with no attractive forces.

The rigorous LDP layer exists only at the edges of the problem. Casini, Redig and van Wiechen proved a dynamical LDP for the multispecies stirring process (neighbours exchange states; no exclusion, no active drift), extending the superexponential-estimate method of Varadhan, Kipnis and Olla to several species. Pulvirenti, Redig and van Wiechen proved the hydrodynamic limit and a dynamical LDP for RTPs whose flip rate depends on the global magnetization (mean-field interaction), where — as they emphasize — the mean-field structure removes the need for superexponential replacement lemmas. In parallel, the physics literature built an exact but heuristic theory for the active lattice gas (run-and-tumble particles with exclusion, stirring, weak active drift and slow tumbles): Kourbane-Houssene, Erignoux, Bodineau and Tailleur derived its exact hydrodynamics and MIPS phase diagram; Erignoux proved the hydrodynamic limit rigorously; Agranov, Ro, Kafri and Lecomte derived the exact fluctuating hydrodynamics and a non-Gaussian MFT whose action contains both a Gaussian conservative part and a Poissonian tumble contribution, which they used to compute current large-deviation functions, a dynamical phase transition into traveling-wave phase-separated states, and (with Cates and Jack) entropy-production large deviations; Neville computed the near-critical stationary large-deviation functions. These derivations rest on a fluctuating-hydrodynamics ansatz verified against simulations, not on a proof from the microscopic Markov chain, and they address specific observables rather than the general rate function and quasi-potential.

What is missing — the gap to which the source paper points when it calls hydrodynamic limits, fluctuations and large deviations of non-Markovian particle systems 'largely terra incognita', and when it notes that rate functions obtained via the contraction principle from the Markovian multi-layer system are 'very implicit' infima that do not make the memory terms explicit — is the rigorous large-deviation theory for RTP systems with local interactions: a dynamical LDP for the multi-layer exclusion process with flips (with or without active jumps) in the diffusive weak-asymmetry slow-flip scaling, the identification of its rate function (including the non-Gaussian tumble contribution and, as a comparison suggested by the physics literature rather than by the source, its relation to the fluctuating-hydrodynamics action), and the quasi-potential of the total density with its memory terms explicit beyond the single worked example of Redig and van Wiechen (which treated independent particles, set the passive diffusivity κ\kappa to zero, kept two internal states, and analyzed the small-noise limit of the Gaussian fluctuation process via Schilder's theorem — the classical LDP for Gaussian measures).

Problem Statement

Prove the large-deviation layer of the stationary fluctuation theory of run-and-tumble particle systems with memory, within the model class of the source work: particles on the one-dimensional integer lattice Z\mathbb{Z} with a finite set of internal states S⊂ZS\subset\mathbb{Z}, at most α\alpha particles per site, passive jumps at rate κN2\kappa N^2, active jumps (x,σ)→(x+σ,σ)(x,\sigma)\to(x+\sigma,\sigma) at rate λN\lambda N, and internal-state flips at rates c(σ,σ′)c(\sigma,\sigma'), so that on the diffusive time scale the hydrodynamic limit is a (for exclusion, nonlinear) reaction–diffusion system for the layer densities ρt(x,σ)\rho_t(x,\sigma) and the total density is non-Markovian. Two jointly linked deliverables serve this single objective.

  1. Dynamical LDP with local interactions. Prove a dynamical large-deviation principle at speed NN for the trajectory of the layer-resolved empirical densities — for the multi-layer symmetric exclusion process with flips treated in the source, and for its weakly asymmetric version with active jumps (the active exclusion process). The rate function must be identified explicitly enough to exhibit the non-quadratic, Poisson-type contribution of the flip dynamics alongside the Gaussian contribution of the conservative transport.

  2. Quasi-potential and memory terms. Characterize the stationary large-deviation rate function of the interacting system's density profile (the quasi-potential, obtained by minimizing the dynamical rate function over trajectories ending at the profile), and, after contraction to the total density, make its memory terms — the non-Markovian structure originating from the slow relaxation of the internal states — explicit, generalizing beyond the single worked example of the source (independent particles, κ=0\kappa=0, two internal states, small-noise limit of the Gaussian fluctuation process).

Deliverable 1 restricted to independent particles, to mean-field switching, or to pure stirring dynamics does not by itself answer the problem, since such restrictions bypass the local-interaction structure that the question targets.

References

  1. F. Redig and H. van Wiechen, 'Stationary Fluctuations of Run-and-Tumble Particles', Markov Processes and Related Fields 30(2), 297-331 (2024). DOI: 10.61102/1024-2953-mprf.2024.30.2.003, https://doi.org/10.61102/1024-2953-mprf.2024.30.2.003 (arXiv version: arXiv:2307.02967, https://arxiv.org/abs/2307.02967)
  2. E. Pulvirenti, F. Redig and H. van Wiechen, 'Hydrodynamic Limit and Large Deviations for Run-And-Tumble Particles with Mean-Field Switching Rates', Journal of Statistical Physics 193, 51 (2026). DOI: 10.1007/s10955-026-03609-y, https://doi.org/10.1007/s10955-026-03609-y (arXiv:2509.22254, https://arxiv.org/abs/2509.22254)
  3. F. Casini, F. Redig and H. van Wiechen, 'A large deviation principle for the multispecies stirring process', Electronic Journal of Probability 30 (2025). DOI: 10.1214/25-ejp1417, https://doi.org/10.1214/25-ejp1417 (arXiv:2410.20857, https://arxiv.org/abs/2410.20857)
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  7. T. Agranov, S. Ro, Y. Kafri and V. Lecomte, 'Exact fluctuating hydrodynamics of active lattice gases—typical fluctuations', Journal of Statistical Mechanics: Theory and Experiment 2021, 083208 (2021). DOI: 10.1088/1742-5468/ac1406, https://doi.org/10.1088/1742-5468/ac1406 (arXiv:2104.14650, https://arxiv.org/abs/2104.14650)
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  9. T. Agranov, M. E. Cates and R. L. Jack, 'Entropy production and its large deviations in an active lattice gas', Journal of Statistical Mechanics: Theory and Experiment 2022, 123201 (2022). DOI: 10.1088/1742-5468/aca0eb, https://doi.org/10.1088/1742-5468/aca0eb (arXiv:2209.03000, https://arxiv.org/abs/2209.03000)
  10. S. Jose, R. Dandekar and K. Ramola, 'Current fluctuations in an interacting active lattice gas', Journal of Statistical Mechanics: Theory and Experiment 2023, 083208 (2023). DOI: 10.1088/1742-5468/aceb53, https://doi.org/10.1088/1742-5468/aceb53 (arXiv:2301.10680, https://arxiv.org/abs/2301.10680)
  11. L. Neville, 'Hydrodynamic equations and near critical large deviations of active lattice gases', Journal of Statistical Mechanics: Theory and Experiment 2025, 033204 (2025). DOI: 10.1088/1742-5468/adbefe, https://doi.org/10.1088/1742-5468/adbefe (arXiv:2410.04105, https://arxiv.org/abs/2410.04105)