Long-Range Zero-Range Process

• LRZRP is a stochastic model where particles jump over long distances with heavy-tailed probabilities, leading to fractional diffusion and anomalous transport.
• It uses local rate functions and product-form invariant measures to link fugacity with density, forming a basis for studying phase transitions and hydrodynamics.
• The process exhibits unique condensation phenomena and scaling limits, including fractional Brownian motion and fractional Burgers equations, with extensions to multi-species systems.

The long-range zero-range process (LRZRP) is a stochastic interacting particle system on a lattice, characterized by particle jumps with heavy-tailed transition kernels and local (zero-range) rate functions. It provides a canonical framework for the study of fractional macroscopic behavior, condensation phenomena, and non-equilibrium statistical mechanics. LRZRPs generalize the classic zero-range process by allowing particles to jump over arbitrarily large distances with probability decaying as a power law. This produces novel transport properties, anomalous diffusion, and phase transitions absent from short-range models.

1. Mathematical Definition and Core Dynamics

The LRZRP is defined on the configuration space $\eta \in \mathbb{N}^{\mathbb{Z}^d}$, with %%%%1%%%% the number of particles at site $x \in \mathbb{Z}^d$. Each particle at site $x$ jumps to another site $y \ne x$ according to a kernel $p(y-x) = c_\alpha |y-x|^{-(d+\alpha)}$ for $\alpha \in (0,2)$, with normalization $c_\alpha$ such that $\sum_{z \ne 0} |z|^{-(d+\alpha)} < \infty$. The local interaction rate is given by a site-dependent function $g: \mathbb{N} \to [0,\infty)$, satisfying $g(0)=0$ and bounded linear increments. The generator is

$$(\mathcal{L}^{d,\alpha} f)(\eta) = \sum_{x \in \mathbb{Z}^d} \sum_{y \ne x} g(\eta(x))\,|y-x|^{-(d+\alpha)} (f(\eta^{x,y}) - f(\eta))$$

where $\eta^{x,y}$ is the configuration after moving a particle from $x$ to $y$.

The jump kernel produces superdiffusive dynamics for $\alpha<2$, and the zero-range nature ensures dynamics depend only locally on occupation numbers. Extensions include multi-species LRZRP (Zhao, 2023), generalized two-site interacting LRZRP (Khaleque et al., 2016), and boundary-driven LRZRPs with contacts to reservoirs (Bernardin et al., 2022).

2. Invariant and Non-Equilibrium Measures

For bulk LRZRP on $\mathbb{Z}^d$, invariant (stationary) measures are product-form grand-canonical distributions:

$$\nu_\beta(\eta(x) = k) = Z(\beta)^{-1} \frac{\beta^k}{g(1)\cdots g(k)}, \quad Z(\beta) = \sum_{k=0}^\infty \frac{\beta^k}{g(1)\cdots g(k)}$$

The mean density $\gamma = E_{\nu_\beta}[\eta(0)]$ increases strictly with $\beta$, providing a bijection between fugacity and density.

Under these measures, reversibility and detailed balance hold:

$$\nu_\gamma(\eta) c(\eta(x)) |x-y|^{-(d+\alpha)} = \nu_\gamma(\eta^{x,y}) c(\eta(y)+1) |y-x|^{-(d+\alpha)}$$

Stationarity and ergodicity are established; all polynomial moments are finite (in particular $\text{Var}_{\nu_\gamma}[\eta(0)] < \infty$).

In open systems with long-range jumps and boundary reservoirs, the stationary state remains a product measure with site-dependent fugacities $\phi_N(x)$ solving discrete traffic equations. The LRZRP hydrodynamic profile and non-equilibrium current laws are inherited from related exclusion processes via a mapping $$E_{ZRP}[g(\eta(x))] = (\Phi(\alpha)+\Phi(\beta))E_{Ex}[\eta^\text{ex}(x)]$$ (Bernardin et al., 2022).

3. Limit Theorems and Fractional Brownian Macroscale

Central limit theorems for additive functionals of LRZRP reveal dramatic departures from normal diffusive behavior (Xiaofeng, 25 Jan 2026). For local, centered functions $V$ of the configuration, the additive observable

$A^N_t(V) = \int_0^{tN} V(\eta_s)\,ds$

after suitable normalization, converges in law (under $\nu_\gamma$) to a Gaussian process $X^H$ with Hurst parameter $H$ determined by $\alpha$ and the spatial dimension $d$:

• For $d=1$, $\alpha < 1$: normal scaling, ordinary Brownian motion ($H=1/2$).
• For $d=1$, $1 < \alpha < 2$: anomalous scaling ($N^{1-1/(2\alpha)}$), limit is fractional Brownian motion (fBM) with $H = 1 - \frac{1}{2\alpha} \in (1/2,1)$.
• For $d=1$, $\alpha \ge 2$: nearest-neighbor exponent $H=3/4$.
• For $d=2$, similar phase transitions (Theorem 2.2), while for $d\ge3$ all regimes converge to ordinary Brownian motion.

The proof proceeds via a martingale–Poisson decomposition, local central limit theorems for the underlying long-range random walk, and relaxation-to-equilibrium estimates. In $1 < \alpha < 2$ regimes, fractional Brownian motion arises from the joint limit of martingale and initial-field contributions with independent Gaussian fluctuations (Xiaofeng, 25 Jan 2026).

4. Condensation Phenomena and Phase Structure

Long-range interaction profoundly alters condensation regimes. For generalized (two-site) zero-range models with infinite-range interactions, the steady-state occupation distribution $P(n)$ displays non-universal behavior in the condensate phase (Khaleque et al., 2016). In mean-field (infinite-range) LRZRP, condensation persists for all $p < p_c \simeq 0.33$, with no finite lower bound $p_l = 0$. Above $p_c$ the system is in a fluid phase.

Condensate-phase scaling follows $$P(n) \simeq L^{-\gamma-1}F\left(\frac{n-cL}{L^\gamma}\right)$$, with $\gamma$ and the condensate mass fraction $c$ continuously dependent on $p$. In sharp contrast, short-range zero-range processes display universal scaling with $\gamma \simeq 1$ and fixed condensation thresholds $p_l \simeq 0.15$, $p_c \simeq 0.44$. In both cases, above $p_c$ the occupation distribution transitions from exponential to Gaussian forms as $p$ increases.

5. Multi-Species Interactions and Fractional Noise PDEs

Multi-species LRZRP with long jumps and vector occupation numbers $\eta^i(x)$ (for $i = 1,\dots,n$) admit coupled fractional macroscopic dynamics (Zhao, 2023). Under appropriate scaling, the species-specific density fluctuation fields converge to solutions of coupled fractional Ornstein-Uhlenbeck processes for $0<\alpha<3/2$:

$$\partial_t Y^i_t = \sum_{j=1}^n \partial_{\rho^j}\tilde g_i(\rho)\,L^* Y^j_t + 2\sqrt{\tilde g_i(\rho)(-S)}\,\dot{W}^i_t$$

where $L$ is the fractional generator, and $S = (L+L^*)/2$ is essentially the fractional Laplacian.

At the critical exponent $\alpha=3/2$ the limit points are stationary energy solutions to the coupled fractional Burgers system:

$$\partial_t Y^i_t = (\partial_{\rho^i}\tilde g_i) L^* Y^i_t + \frac{m}{2} \sum_{j,k=1}^n \partial_{\rho^j \rho^k} \tilde g_i \nabla (Y^j_t Y^k_t) + 2\sqrt{\tilde g_i(\rho)(-S)}\,\dot{W}^i_t$$

where nonlinearities introduce genuinely non-Gaussian macroscopic behavior analogous to the KPZ universality class but in the fractional context.

6. Non-Equilibrium, Hydrodynamics, and Large Deviations

Boundary-driven LRZRPs with long jumps, coupled to reservoirs at both ends, yield stationary non-equilibrium states governed by fractional macroscopic PDEs. The hydrostatic limit produces deterministic density profiles $m(u)$, determined via solutions to fractional Laplacian equations, with boundary conditions dependent on reservoir densities and interaction parameters (Bernardin et al., 2022).

The stationary current in the scaling limit is given by a nonlocal Fick's law:

$J(u) = -D(\gamma)\,\nabla^{\gamma-1} m(u)$

with the fractional gradient

$$\nabla^{\gamma-1} m(u) = \int_0^1 \frac{m(w) - m(u)}{|w-u|^{\gamma}}\,\mathrm{sgn}(w-u)\,dw$$

and diffusion coefficient $D(\gamma) = C_\gamma (\Phi(\alpha) + \Phi(\beta))$.

Empirical density measures satisfy large-deviation principles at speed $N$, with rate function and non-equilibrium free energy given locally by

$$I(\rho) = \int_0^1 \left\{ \rho(u) \log \frac{\Phi(\rho(u))}{\phi(u)} - \log \frac{Z(\Phi(\rho(u)))}{Z(\phi(u))} \right\} du$$

This quantifies the cost of density fluctuations and rigorously encodes non-equilibrium thermodynamic structure.

7. Probabilistic and Physical Significance

Long-range jump kernels with exponent $\alpha \in (0,2)$ generate fractional Laplacians on the macroscopic scale, replacing classical diffusive behavior with anomalous transport and sub- or super-diffusive fluctuations. The appearance of fractional Brownian motion and fractional Burgers type equations in scaling limits highlights connections to universality phenomena and anomalous hydrodynamics.

In multi-species LRZRP, coupling between conserved densities leads to collective noise and nonlinear macroscopic equations with rich interspecies interactions and scaling crossovers. The identification with exclusion processes provides systematic derivation of hydrostatics and transport laws. The breakdown of universality in infinite-range models elucidates the effect of interaction topology on phase transitions and scaling regimes.

A plausible implication is that the long-range zero-range process forms a theoretical backbone for understanding fractional diffusion and collective stochastic phenomena in systems with non-local transport, with rigorous connection to the underlying microscopic dynamics and deep analogies to KPZ- and Burgers-type universality classes (Xiaofeng, 25 Jan 2026, Zhao, 2023, Bernardin et al., 2022, Khaleque et al., 2016).