The zero-range process with heavy-tailed long jumps on {1,…,N−1}\{1,\dots,N-1\}, in contact with two infinitely extended reservoirs of densities α,β\alpha,\beta, is one of the rare interacting particle systems with a semi-explicit non-equilibrium stationary state in a genuinely superdiffusive regime. Bernardin, Gonçalves, Oviedo-Jiménez and Scotta (J. Stat. Phys. 189, Art. 32, 2022) proved its hydrostatic limit in mean, a fractional Fick's law with the case θ=0,γ=1\theta=0,\gamma=1 excluded, and a static large-deviation principle for the empirical density under the hypothesis that the partition function is entire. The authors themselves state that the hydrodynamic limits are not derived, and ask (Remark 2.17) for the exact behaviour of the hydrostatic profile at the boundary points, where it is non-differentiable for a range of parameters. All of these gaps were still open as of 2026: the hydrodynamic limit and dynamical large deviations of this model are explicitly recorded as open in recent work, and among boundary-driven superdiffusive lattice gases with non-compact state space only static large-deviation principles are known. The problem posed here is to complete the rigorous macroscopic theory of this model, namely: (A) the hydrodynamic limit and the dynamical large-deviation principle (macroscopic fluctuation theory) for all reservoir strengths θ∈R\theta\in\mathbb{R} and jump tails 0<γ<20<\gamma<2; (B) the exact boundary asymptotics of the hydrostatic profile; (C) the doubly-critical case θ=0,γ=1\theta=0,\gamma=1; (D) the condensation regime α\alpha or β≥m∗\beta\ge m^*; and (E) the static large-deviation principle beyond an entire partition function.

Background

A zero-range process is a continuous-time Markov chain on occupation-number configurations ξ(x)∈N0\xi(x)\in\mathbb{N}_0 with x∈ΛN={1,…,N−1}x\in\Lambda_N=\{1,\dots,N-1\}, in which the rate at which a particle leaves a site depends only on the number of particles at that site: a jump x→yx\to y occurs at rate p(y−x) g(ξ(x))p(y-x)\,g(\xi(x)) with g(0)=0g(0)=0. On a fixed finite box the process has a family of invariant product measures ν~φ\tilde\nu_\varphi whose one-site marginal is ν~φ(ξ(x)=k)=φk/(g(k)! Z(φ))\tilde\nu_\varphi(\xi(x)=k)=\varphi^k/(g(k)!\,Z(\varphi)), where φ\varphi is a fugacity, g(k)!=g(1)⋯g(k)g(k)!=g(1)\cdots g(k), and Z(φ)=∑k≥0φk/g(k)!Z(\varphi)=\sum_{k\ge0}\varphi^k/g(k)! is the partition function; the mean density is R(φ)=φZ′(φ)/Z(φ)R(\varphi)=\varphi Z'(\varphi)/Z(\varphi). If ZZ has finite radius of convergence φ∗\varphi^*, then RR maps [0,φ∗)[0,\varphi^*) onto [0,m∗)[0,m^*) with m∗=lim⁡φ↑φ∗R(φ)m^*=\lim_{\varphi\uparrow\varphi^*}R(\varphi) the critical density, above which the process condenses (a macroscopic fraction of the mass accumulates on o(N)o(N) sites).

The model studied here, introduced by Bernardin, Gonçalves, Oviedo-Jiménez and Scotta (2022; BGJOS22 below), is the boundary-driven version with long jumps: p(z)=cγ∣z∣−1−γp(z)=c_\gamma|z|^{-1-\gamma} with 0<γ<20<\gamma<2 and cγc_\gamma a normalizing constant, a heavy-tailed Lévy-stable kernel with infinite variance, so the dynamics is superdiffusive (the bulk evolves on the time scale Θ(N)=Nγ\Theta(N)=N^\gamma for θ≥0\theta\ge0, and Nγ+θN^{\gamma+\theta} for θ<0\theta<0, instead of the diffusive N2N^2). The box is coupled to two infinitely extended reservoirs: the left reservoir occupies all sites ≤0\le 0 and the right one all sites ≥N\ge N, and each site xx of ΛN\Lambda_N exchanges particles with reservoir sites yy at rate κN−θp(y−x)\kappa N^{-\theta}p(y-x), κ>0\kappa>0; the reservoir densities α,β\alpha,\beta enter through their fugacities Φ(α),Φ(β)\Phi(\alpha),\Phi(\beta), where Φ=R−1\Phi=R^{-1}. The parameter θ∈R\theta\in\mathbb{R} tunes the strength of the reservoirs (θ<0\theta<0 fast reservoirs, θ=0\theta=0 critical, θ>0\theta>0 slow reservoirs). The non-equilibrium stationary state (NESS, i.e. the stationary state that carries a stationary current between the reservoirs) νNss\nu_N^{ss} exists for α,β∈(0,m∗)\alpha,\beta\in(0,m^*) and is a product measure with fugacity profile φN\varphi_N solving an explicit finite linear traffic equation.

BGJOS22 established three classes of stationary (hydrostatic) results. First, a hydrostatic limit in mean: the macroscopic density profile is mˉ(u)=R((Φ(α)+Φ(β)) ρˉ(u))\bar m(u)=R\big((\Phi(\alpha)+\Phi(\beta))\,\bar\rho(u)\big), where ρˉ\bar\rho is the stationary profile of the companion boundary-driven exclusion process with long jumps, characterized case by case as the weak solution of stationary equations for the regional fractional Laplacian LL (the generator of the symmetric Lévy process censored to [0,1][0,1], (Lf)(u)=cγlim⁡ε→0∫[0,1]1∣u−v∣≥ε(f(v)−f(u))∣u−v∣−1−γ dv(Lf)(u)=c_\gamma\lim_{\varepsilon\to0}\int_{[0,1]}\mathbf 1_{|u-v|\ge\varepsilon}(f(v)-f(u))|u-v|^{-1-\gamma}\,dv): a reaction-diffusion equation with non-homogeneous Dirichlet boundary data when θ=0\theta=0, a Dirichlet problem when 0<θ<γ−10<\theta<\gamma-1 with γ>1\gamma>1, fractional Robin conditions when θ=γ−1\theta=\gamma-1, and fractional Neumann conditions (constant profile) for slower reservoirs. Second, a fractional Fick's law: the mean stationary current across the macroscopic point uu, normalized by N1−γN^{1-\gamma} for θ≥0\theta\ge0 or N1−θ−γN^{1-\theta-\gamma} for θ<0\theta<0, converges to an explicit functional of mˉ\bar m and of the reservoir reaction rates r∓(u)∝u−γ,(1−u)−γr^\mp(u)\propto u^{-\gamma},(1-u)^{-\gamma} — with the doubly-critical case θ=0\theta=0, γ=1\gamma=1 excluded, and in that same case even the exclusion profile ρˉ\bar\rho is left uncharacterized. Third, a static large-deviation principle (LDP) at speed NN for the empirical density under the NESS, with explicit rate function Λ∗\Lambda^* (the non-equilibrium free energy), proved under the analyticity hypothesis φ∗=+∞\varphi^*=+\infty, i.e. for an entire partition function.

These are static results only. Macroscopic fluctuation theory (MFT) — the dynamical LDP for the empirical density from which the static rate function should emerge as a quasi-potential, and the general framework organizing non-equilibrium stationary states of diffusive systems — is not available for boundary-driven superdiffusive lattice gases; BGJOS22 explicitly presents its results as a first step toward an MFT for superdiffusive systems. The connection it exploits between the zero-range and exclusion NESS is a static, expectation-level identity that does not transfer to time-dependent evolution, and the hydrodynamic limit available for the companion exclusion process (Bernardin–Cardoso–Gonçalves–Scotta 2023) does not yield one for the zero-range process. The problem below asks to close the gaps that BGJOS22 itself acknowledges.

Problem Statement

Complete the rigorous macroscopic theory of the boundary-driven zero-range process with long jumps of BGJOS22 — the same microscopic model (occupation numbers on ΛN={1,…,N−1}\Lambda_N=\{1,\dots,N-1\}; bulk jumps x→yx\to y at rate p(y−x) g(ξ(x))p(y-x)\,g(\xi(x)) with p(z)=cγ∣z∣−1−γp(z)=c_\gamma|z|^{-1-\gamma}, 0<γ<20<\gamma<2; exchanges with the infinitely extended reservoirs at rates κN−θrN±\kappa N^{-\theta}r_N^\pm with θ∈R\theta\in\mathbb{R}, κ>0\kappa>0; time scale Θ(N)\Theta(N); reservoir densities α,β\alpha,\beta through their fugacities Φ(α),Φ(β)\Phi(\alpha),\Phi(\beta)) and the same conventions — by resolving the following five linked open questions, which together close every regime and every statement that paper leaves open:

(A) Hydrodynamic limit and dynamical large deviations. Derive, for all θ∈R\theta\in\mathbb{R} and 0<γ<20<\gamma<2, the hydrodynamic limit of the empirical density starting from general initial conditions (identify the limiting evolution — expected to be given by regional fractional operators with the boundary conditions that appear as the stationary cases of the hydrostatic classification — and prove convergence in the standard weak sense), and prove the dynamical large-deviation principle for the empirical measure, thereby establishing macroscopic fluctuation theory for this boundary-driven superdiffusive lattice gas and recovering the static rate function Λ∗\Lambda^* of BGJOS22 as the quasi-potential of the dynamical functional.

(B) Boundary behaviour of the hydrostatic profile (Remark 2.17 of BGJOS22). In the parameter ranges with γ<2\gamma<2, θ≥0\theta\ge0 in which the hydrostatic profile mˉ(⋅)\bar m(\cdot) is non-differentiable at u=0u=0 and u=1u=1 (for instance θ=0\theta=0 with γ≠1\gamma\ne1, and 0<θ≤γ−10<\theta\le\gamma-1 with γ∈(1,2)\gamma\in(1,2)), determine the exact asymptotics of mˉ(u)\bar m(u) as u↓0u\downarrow0 and u↑1u\uparrow1: the rate, including any logarithmic corrections, and the constants.

(C) The doubly-critical case θ=0\theta=0, γ=1\gamma=1. Characterize the stationary profile (ρˉ\bar\rho and hence mˉ\bar m) and the N1−γN^{1-\gamma}-scaled stationary current in the case excluded from the hydrostatic limit and the fractional Fick's law of BGJOS22, where the reservoir reaction rates carry logarithmic corrections.

(D) The condensation regime. For α≥m∗\alpha\ge m^* or β≥m∗\beta\ge m^*, prove or refute the assertion of Remark 2.4 of BGJOS22 that condensation appears and the invariant measure does not exist because mass grows at the boundaries.

(E) Static large deviations beyond an entire partition function. Extend the static LDP of Theorem 2.20 to jump rates gg for which ZZ has finite radius of convergence φ∗\varphi^* (condensing zero-range dynamics), including the treatment of the singular behaviour of the rate function Λ∗\Lambda^* as the density approaches the critical density m∗m^*.

Components (A)–(E) are the explicitly acknowledged limitations of BGJOS22 for this single model and jointly support one overarching objective, the completion of its rigorous macroscopic description from hydrostatics to a full macroscopic fluctuation theory; an answer that resolves every one of them — where resolving a component may consist in rigorously establishing that it is vacuous (for example non-existence of the invariant measure in (D)) — completes the program. Partial answers resolving individual components are meaningful steps toward it.

References

  1. Cédric Bernardin, Patrícia Gonçalves, Byron Oviedo-Jiménez, Stefano Scotta, "Non-equilibrium Stationary Properties of the Boundary Driven Zero-Range Process with Long Jumps", Journal of Statistical Physics 189(3), Article 32 (2022). DOI: 10.1007/s10955-022-02987-3; arXiv:2201.07573. https://doi.org/10.1007/s10955-022-02987-3
  2. Cédric Bernardin, Pedro Cardoso, Patrícia Gonçalves, Stefano Scotta, "Hydrodynamic limit for a boundary driven super-diffusive symmetric exclusion", Stochastic Processes and their Applications 165, 43–95 (2023). DOI: 10.1016/j.spa.2023.08.002; arXiv:2007.01621. https://doi.org/10.1016/j.spa.2023.08.002
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