Condensing Zero-Range Processes

Updated 14 January 2026

Condensing zero-range processes are interacting particle systems where the hop rate depends only on the departure site's occupancy, leading to macroscopic condensation above a critical density.
The methodology rigorously combines hydrodynamic scaling, martingale formulations, and potential theory to derive nonlinear diffusion limits and metastable Markov chain behavior.
Applications in granular clustering, traffic flow, and phase separation illustrate the practical implications of controlling condensation and predicting drastic metastable transitions.

A condensing zero-range process (ZRP) is an interacting particle system on a fixed finite or infinite graph in which the particle hopping rate depends only on the occupation number of the departure site. Under certain conditions on the rate function and system parameters, the stationary state exhibits a condensation transition: above a critical density, a macroscopic fraction of the particles accumulates on a single site or a small set of favored sites. This phenomenon is central to the study of phase separation outside equilibrium, metastability, and dynamical effects in many-body stochastic systems. Rigorous frameworks now exist for the stationary, hydrodynamic, scaling, and large deviation characteristics of condensation, including the precise asymptotics of metastable transitions and multi-scale rate functions (Choi, 2024).

1. Model Definitions and Condensation Mechanism

A generic zero-range process is defined on a set of sites $S$ (finite or infinite) with particles hopping from site $x$ to $y$ at rate $g(\eta_x) r(x,y)$ , where $\eta_x$ is the current occupation at $x$ and $r(x,y)$ is an irreducible underlying random walk kernel. Prototypical condensing ZRPs take rate functions $g(n)$ with slow decay or sticky behavior, such as $g(n)=1 + b/n$ or more generally $g(n)=n^\alpha/(n-1)^\alpha$ for $\alpha>1$ (Choi, 2024, Beltrán et al., 6 Jan 2026, Seo, 2018).

The canonical state space is

$H_N = \{\,\eta \in \mathbb{N}^S\,:\,\sum_{x\in S} \eta_x = N\,\}$

and the dynamics conserve total particle number $N$ . The stationary measure is often of product form

$\rho_N(\eta) = \frac{a(\eta)}{Z_{S,N}}, \quad a(\eta) = \prod_{x \in S} a(\eta_x)$

with $a(n)$ the factorial weight $a(n)=n^\alpha$ or a variant (Seo, 2018).

Condensation arises whenever the critical density

$\rho_c = \lim_{\phi \to \phi_c^-} \frac{\phi\,Z'(\phi)}{Z(\phi)}$

is finite (with $Z(\phi)=\sum_k a(k)\phi^k$ and $\phi_c$ the radius of convergence). In the supercritical regime $\rho>\rho_c$ , excess mass $N - |S|\rho_c$ concentrates on few sites ("the condensate").

Finite capacity models and compartmental restrictions can interpolate between Fermi, Bose, and Gentile statistics, regularize condensation, and produce glassy kinetics (Ryabov, 2013). Non-reversible or disordered models can exhibit extended or localized condensates, tunable via interaction and disorder exponents (Godreche et al., 2012).

2. Hydrodynamic Limit and Macroscopic Evolution

For mean-zero, bounded step kernels and subcritical initial density profiles, the macroscopic evolution of the empirical density is governed by nonlinear diffusion equations of gradient type. On the $d$ -dimensional discrete torus $T_N^d$ , in the diffusive scaling limit (time rescaled by $N^2$ ), the empirical density converges to the solution of

$\partial_t \rho(t,u) = \nabla \cdot (\Sigma\,\nabla \Phi(\rho(t,u)) )$

where $\Phi(\rho)$ is the mean jump rate (flux) function given by inverting the microscopic density-fugacity relation (Stamatakis, 2014, Dirr et al., 2016).

When the initial profile stays everywhere below $\rho_c$ , macroscopic condensation is absent in the scaling limit, and local equilibrium holds. For multi-species systems, an analogous hydrodynamic limit applies as long as the combined density vector remains in the sub-critical region. Species-blind rates yield scalar PDEs with an invariant region determined by total density; global existence of classical solutions follows if initial conditions remain below $\sigma_c$ (Dirr et al., 2016).

For systems with supercritical scaling or absorption (i.e., ZRPs with rates $g(n)\sim 1 + b/n$ and $b>1$ ), the continuum process becomes a singular diffusion on the simplex whose dimension decays by absorption at the boundary $\{x_i=0\}$ :

$Lf(x) = \sum_{i\in S} b\,\mathbb{1}_{x_i>0}\frac{m_i}{x_i}\sum_j r(i,j)(\partial_{x_j}-\partial_{x_i})f(x) + \tfrac{1}{2}\sum_{i,j} m_i\,r(i,j)(\partial_{x_i}-\partial_{x_j})^2f(x)$

Once any coordinate $x_i$ reaches zero, the process is confined to the lower-dimensional face, and the generator is recursively adjusted (Beltrán et al., 6 Jan 2026, Beltrán et al., 2015). The absorption time is finite with high probability, and the process ultimately terminates in a singleton vertex (full condensation).

3. Metastability, Coarsening, and Condensate Motion

For super-critical density, condensing ZRPs exhibit metastability: the condensate is localized at a single site for long periods, then moves abruptly via large rare jumps. The equilibration within each "metastable well" occurs quickly relative to the timescale for inter-well transitions.

In the thermodynamic limit, the rescaled location process of the condensate converges to a Markov jump process on the domain of possible condensate sites. The effective jump rates are proportional to capacities of the underlying random walk:

$a(x,y) = \frac{1}{M_*\,\Gamma(\alpha)\,I_\alpha}\,\mathrm{cap}_X(x,y)$

where $M_*$ is the maximal stationary density, $\Gamma(\alpha)$ the normalizing constant, $I_\alpha$ an interaction integral, and $\mathrm{cap}_X(x,y)$ the network capacity (Seo, 2018, Armendáriz et al., 2015). For symmetric kernels, condensate jumps become a Lévy process on the torus.

Rigorous capacity bounds (Dirichlet--Thomson inequalities) and the martingale approach underpin the convergence proofs. For non-reversible systems, generalized flow decomposition allows for capacity analysis even without divergence-free test flows (Seo, 2018).

The process experiences a dynamical transition, with two distinct mechanisms for condensate motion in different phases: evaporation-recondensation through the background, or contiguous motion via macroscopic lumps with constant rate (Chleboun et al., 2014).

4. Large Deviations and Γ-Expansion Rate Function

The level-two large deviation principle for condensing ZRPs addresses the asymptotics of empirical measures in metastable systems. The rate function $\mathcal{I}_N$ for the time-averaged empirical density possesses a precise expansion at two dominant time scales (Choi, 2024):

$\mathcal{I}_N = \frac{1}{N^2}\,\mathcal{K} + \frac{1}{N^{1+\alpha}}\,\mathcal{J}$

where $\mathcal{K}$ is the rate function for the pre-metastable (absorbing) diffusion, capturing fluctuations before condensation, and $\mathcal{J}$ is the rate for the metastable Markov chain of condensate locations. The expansion is rigorous in the sense of full $\Gamma$ -convergence: $N^2\mathcal{I}_N$ converges to $\mathcal{K}$ for smooth densities, $N^{1+\alpha}\mathcal{I}_N$ to $\mathcal{J}$ for measures supported on condensate corners.

This two-scale separation clarifies the quantitative distinction between profile fluctuations prior to full condensation and rare transitions among condensate wells. Methodologically, the connection is established via the resolvent equation for metastability and suitable approximation of the Donsker–Varadhan variational formulas.

5. Typical Dynamical Scenarios and Universal Phase Behavior

The condensation transition is well characterized via explicit criteria on the rate function tail and the interaction/disorder exponents. For power-law rates $g(n)\sim 1 + b/n^\gamma$ (with $\gamma\in(0,1)$ or $b>2$ if $\gamma=1$ ), condensation occurs above $\rho_c=1/(b-2)$ (Godreche et al., 2016, Jatuviriyapornchai et al., 2015). For the class $g(n)=n^\alpha/(n-1)^\alpha$ ( $\alpha>1$ ), essentially all particles accumulate on a single site in the large $N$ limit.

Diffusion-disorder and interaction-disorder models possess rich phase diagrams: localized condensed phases (single favored site), and extended condensed phases where the condensate can reside on any site in a large sub-extensive hosting set. The hosting set size scales as $M^{1-c}$ , interpolating between spontaneous-symmetry-breaking and explicit-symmetry-breaking phases (Godreche et al., 2012).

Finite-size effects can manifest as discontinuous current overshoots and switchings between metastable fluid and condensed phases close to the critical point. These effects persist for moderate system sizes (traffic flow, granular clustering) and preclude the thermodynamic limit from faithfully capturing non-equilibrium transitions (Chleboun et al., 2010, Juntunen et al., 2010).

Memory and temporal correlations (e.g., on-off zero-range dynamics or compartmental restrictions) can promote or suppress condensation and induce glassy relaxation via self-blocking, with exponentially long relaxation times (Ryabov, 2013, Cavallaro et al., 2015).

6. Methodologies and Theoretical Frameworks

Several analytical techniques establish and characterize condensing ZRP phenomena:

Hydrodynamic scaling and entropy methods yield rigorous nonlinear diffusion equations for empirical density, with closure via block averaging and equivalence of ensembles (Stamatakis, 2014, Loulakis et al., 2019).
Martingale problem formulations and absorption properties enable the identification of limiting diffusions and recursive reduction of simplex dimension (Beltrán et al., 2015, Beltrán et al., 6 Jan 2026).
Capacity and potential theory underlie jump-rate and metastable Markov chain limits (reversible and nonreversible kernels), with Dirichlet--Thomson inequalities for large deviations (Seo, 2018).
Resolvent approaches allow for two-scale limiting theories in metastable regimes, with full $\Gamma$ -convergence for large deviation rate functions (Choi, 2024).
Analytical scaling ansatz, birth–death chains, and size-biased sampling uncover coarsening exponents, scaling forms, and the structure of bulk and condensate distributions (Jatuviriyapornchai et al., 2015, Godreche et al., 2016, Jatuviriyapornchai et al., 2024).

7. Significant Variants and Applications

Numerous condensing ZRP variants unify disparate phenomena:

Compartment models with finite capacity bridge Bose–Einstein and Gentile statistics and exhibit glassy blocking kinetics (Ryabov, 2013).
Temporally correlated ZRPs with "on-off" dynamics capture real-world cluster formation and current fluctuations (Cavallaro et al., 2015).
Randomly perturbed rates and disordered diffusion produce phase diagrams with sharp boundaries for condensation suppression or extension (Molino et al., 2012, Godreche et al., 2012).
Size-dependent rates or fast-rate modifications induce transitions from fragmented to single-site condensates, systematically controlled by rate scaling (Jatuviriyapornchai et al., 2024).

Applications range from granular clustering and traffic flow to phase separation in driven diffusive systems and stochastic transport models.

References