Symmetric Zero-Range Process Overview

Updated 31 August 2025

The symmetric zero-range process is defined on a lattice where particles hop based solely on local occupation numbers, resulting in explicit product-form stationary measures.
It exhibits hydrodynamic limits that lead to nonlinear diffusion PDEs and captures complex phenomena such as phase transitions, condensation, and metastability.
The model provides key insights into transport, fluctuations, and disorder effects, influencing research in nonequilibrium statistical mechanics and large deviation theory.

The symmetric zero-range process is a class of interacting particle systems on a lattice where the dynamics are governed by local particle-dependent hop rates, symmetric spatial transition kernels, conservation of the total particle number, and explicit product-form stationary measures. This process has been a cornerstone model for the paper of nonequilibrium statistical mechanics, phase transitions, hydrodynamic limits, current fluctuations, metastability, and processes with disorder or spatial constraints.

1. Formal Definition and Core Properties

In the symmetric zero-range process (ZRP), the configuration space consists of occupation numbers $\eta_x \in \mathbb{N}_0$ for each site $x$ of a lattice, typically the discrete torus $\mathbb{T}_N$ . Particles hop from a site $x$ with $k$ particles to a neighboring site $y$ at a rate determined solely by the occupation at $x$ ; the hop rate function $g: \mathbb{N}_0 \to \mathbb{R}_+$ is non-decreasing (i.e., the process is attractive). The transition kernel $p(y-x)$ is symmetric and has mean zero, ensuring spatial reversibility: $\mathcal{L}_N f(\eta) = \sum_{x \in \mathbb{T}_N} \sum_z p(z)\, g(\eta(x)) \left[f(\eta^{(x,x+z)}) - f(\eta)\right]$ where $\eta^{(x,x+z)}$ is the configuration with one particle moved from $x$ to $x+z$ . The total particle number is conserved, and the system admits explicit product-form invariant (steady-state) measures, parameterized by a fugacity or chemical potential.

The key object controlling the stationary state is the single-site weight: $\mu_\rho(\eta(x) = k) = \frac{1}{Z(\phi(\rho))} \frac{[\phi(\rho)]^k}{g(1)\cdots g(k)}$ where $Z(\phi)$ is the partition function and $\phi(\rho)$ is tuned so that the mean occupation matches the global density.

2. Hydrodynamic Limits and Macroscopic PDEs

Under diffusive scaling (space by $N$ , time by $N^2$ ), the microscopic empirical density profile converges to the solution of a nonlinear diffusion PDE of the form: $\partial_t \rho(t, u) = \Delta\Phi(\rho(t, u))$ where $u$ is the macroscopically rescaled spatial variable and $\Phi=\mathcal{R}^{-1}$ is the inverse of the expected density in the product invariant measure. This nonlinear equation generally describes a porous medium or nonlinear heat flow, with nonlinearity inherited from the jump rate function $g$ .

Boundary conditions reflect the details of the boundary reservoirs. If particle injection/removal rates at the boundaries are scaled as $N^{-\theta}$ with $\theta \geq 1$ , slow boundary dynamics are induced. For $\theta=1$ (slow boundaries), Robin-type boundary conditions arise; for $\theta>1$ , Neumann boundary conditions hold, indicating zero macroscopic flux across boundaries (Frómeta et al., 2020, Araújo et al., 26 Aug 2025).

Recent advances provide uniform-in-time, quantitative convergence rates in Monge-Kantorovich distance and relative entropy; for example, convergence errors scale as $N^{-1/6+\varepsilon}$ in $d=1$ and $[\log N]^{-1/8+\varepsilon}$ in $d=2$ (Marahrens et al., 21 Dec 2024). These approaches exploit microscopic contraction properties analogues to Kruzhkov's $L^1$ stability for conservation laws and do not rely on traditional block averaging.

3. Phase Transitions: Condensation and Metastability

Condensation occurs when the total density exceeds a critical value $p_c$ determined by the divergence of single-site partition sums. Above $p_c$ , nearly all excess particles localize at a single site (the condensate), while other sites maintain occupations near $p_c$ . The canonical measure, conditioned on the total number of particles, exhibits the property that, after removing the site with maximal occupation, the distribution on the remaining sites converges to the grand canonical measure at critical fugacity. Explicit for jump rates $g(k)=(k/(k-1))^\alpha$ for $k\geq2$ , condensation occurs for all $\alpha \geq 1$ (Xu, 2020).

Metastability refers to the long time localization of the condensate at a single site, with rare transitions ("tunneling") to other sites. On the accelerated time scale $L^{1+b}$ (where $b$ is related to the tail of $g$ ), the rescaled location of the condensate converges to a Markov process on the macroscopic torus with generator determined by the scaling limits of capacities of a random walk (Armendáriz et al., 2015). This effective process has stationary, independent increments, and potential-theoretic capacity estimates, as well as translation invariance, are essential ingredients for its rigorous characterization.

4. Fluctuations and Transport: Current and Tagged Particle Dynamics

Away from criticality, the process exhibits standard diffusive transport, characterized by a bulk-diffusion coefficient $D(\bar{\rho})$ and a particle mobility $\chi(\bar{\rho})$ . The time-integrated current's variance grows as $\sqrt{t}$ at short times and linearly at long times. At the critical point (condensation), the bulk diffusion vanishes in the thermodynamic limit for certain parameter ranges (e.g., $2 < b < 3$ for power-law mass distributions), inducing anomalous fluctuation scaling: short-time variance growth exceeds the standard diffusive exponent and varies continuously with system parameters (Chakraborty et al., 3 Jun 2024).

Nonequilibrium tagged particle fluctuations can be rigorously described by central limit theorems and homogenization arguments. For systems with sublinear or bounded rates (e.g., $g(k)=k^\gamma$ with $0<\gamma<1$ ), the motion of a tagged particle is not Markov alone but is shown to converge to a diffusion: $dx_t = \sigma\sqrt{\psi(\rho(t, x_t))} dB_t$ where $\psi(\rho)=\phi(\rho)/\rho$ and $B_t$ is standard Brownian motion (Jara et al., 2010). These methods extend to second-class particles, which move only when all "ordinary" particles have left their site, yielding analogous limiting SDEs.

Current fluctuations, tagged particle motion, and scaling theory all reveal the interplay between microscale interactions, condensate formation, and macroscopic transport.

5. Disorder, Constraints, and Glassy Dynamics

Adding disorder (perturbed rates $g_x(n)=\exp\{\sigma\xi_x(n)+b/n^\gamma\}$ ) dramatically alters condensation criteria: condensation is restricted to $0<\gamma<1/2$ ; for $\gamma\geq1/2$ , quenched free energy diverges at criticality and condensation is suppressed (Molino et al., 2012). Finite system sizes nevertheless show rich condensation-like behavior in numerics, highlighting the disparity between thermodynamic and finite-system phase diagrams.

Spatial constraints, such as compartment size limits, modify statics and dynamics. Symmetric zero-range processes with finite compartments admit factorized steady states and valid grand canonical descriptions even in the condensed phase. Dynamics reveal "self-blocking", where boundary sites saturate, delaying condensate growth and inducing glassy relaxation (Ryabov, 2013).

When disorder produces multifractal single-particle distributions, the number of particles outside the condensate scales algebraically with system size, with exponents dependent on the disorder's strength (Miki, 2016).

6. Large Deviations, Quantitative Scaling, and Free Boundary Problems

Large deviation theory for hydrodynamic rescalings of symmetric ZRP has been completed, delivering matching upper and lower bounds on the probability of empirical density deviations from the macroscopic diffusion PDE, in any dimension, under mild hypotheses on the local jump rate $\lambda$ (Fehrman et al., 31 Jul 2025). Technical advances include superexponential concentration estimates and the extension of the parabolic-hyperbolic skeleton PDE (representing "most likely" macroscopic deviation paths) to the full space, without global convexity/concavity restrictions on the nonlinearity.

Degenerate modifications, such as the facilitated zero-range process (FZRP) with $g(k)=\mathbf{1}_{\{k\geq2\}}$ , yield hydrodynamic limits with Stefan-type free boundary conditions, reflecting regions where transport is frozen or mobile. Macroscopic mappings between exclusion and zero-range processes clarify these relationships (Erignoux et al., 2022).

7. Applications and Broader Impact

The theoretical framework of symmetric zero-range processes applies to clustering, traffic flow, heat transport, and active matter. Exact results on metastability, condensation, and fluctuation theory provide benchmarks for simulations and the analysis of experimental systems. The rich interplay between disorder, constraints, and interaction reveals new regimes, informs mathematical methods for scaling limits, and connects microscopic stochastic processes to nonlinear PDEs, free boundary problems, and large deviation principles.

Recent methodological advances, such as the use of modulated Monge-Kantorovich distance and microscopic stability properties, circumvent traditional block estimates and provide uniform-in-time quantitative convergence rates, impacting the analysis of interacting particle systems beyond ZRP.

Key Formulas Table

Concept	Formula/Expression	Reference
Generator	$\mathcal{L}_N f(\eta) = \sum_{x, z} p(z)g(\eta(x))[f(\eta^{(x,x+z)})-f(\eta)]$	(Jara et al., 2010), others
Invariant measure	$\mu_\rho(\eta(x)=k)=\frac{1}{Z(\phi(\rho))} \frac{[\phi(\rho)]^k}{g(1)\cdots g(k)}$	(Frómeta et al., 2020)
Hydrodynamic PDE	$\partial_t \rho = \Delta \Phi(\rho)$	(Araújo et al., 26 Aug 2025), etc.
Condensation criterion	Critical density $p_c = \lim_{\phi \uparrow \phi_c} p(\phi)$	(Xu, 2020)
Tagged particle SDE	$dx_t = \sigma\sqrt{\psi(\rho(t, x_t))}dB_t$ , $\psi(\rho) = \phi(\rho)/\rho$	(Jara et al., 2010)
Current fluctuation	$\langle Q^2(t)\rangle_c \sim \sqrt{t}$ (short), $\sim t$ (long), anomalous at criticality	(Chakraborty et al., 3 Jun 2024)

These formulas underpin the central results in symmetric zero-range process theory, connecting microscopic dynamics, macroscopic laws, phase transitions, and fluctuation phenomena.