Equilibrium fluctuations for a multi-species particle system with long jumps

Abstract: In the present paper, we study the equilibrium fluctuations of a particle system in infinite volume with two conserved quantities and long-range dependence. More specifically, the model of interest is the so-called ABC model, in which three types of particles (A, B and C) exchange their locations between $x\in\mathbb{Z}$ and $x+z\in\mathbb{Z}$ at a rate that depends on the type of particles involved and is proportional to $|z|<sup>{-γ-1}$ for $γ&gt;0$. After rigorously identifying the normal modes associated to the conserved quantities (the density of particles of types $A$ and $B$, say), we prove that their fluctuations converge to independent fractional stochastic partial differential equations (SPDEs), which are either Gaussian or the Stochastic Burgers equation, and whose nature is determined by the microscopic range of dependence and the strength of the asymmetry.

• The paper establishes fluctuation limits in the multi-species ABC model with heavy-tailed jump kernels, showing convergence to fractional OU and SBE equations under distinct scaling regimes.
• Key methodology involves spectral diagonalization, martingale decomposition, and wavelet analysis to rigorously handle nonlocal interactions and multiple conserved fields.
• Implications include identifying the KPZ-class transition at γ = 3/2 and providing a blueprint for analyzing nonlocal, multi-species systems in fluctuating hydrodynamics.

Equilibrium Fluctuations in the Multi-Species ABC Model With Long-Range Interactions

Introduction and Context

This paper rigorously analyzes equilibrium fluctuations in a three-species exclusion process (the ABC model) on the lattice $\mathbb{Z}$, where particles can undergo long-range exchanges. The focus is on the hydrodynamic and fluctuation limits in the presence of two independent conserved quantities (the densities of A- and B-type particles) and a heavy-tailed jump kernel, $p(z) \propto |z|^{-\gamma-1}$, $\gamma > 0$. Such nonlocality translates microscopically into competition between superdiffusive transport and asymmetric drift, leading to anomalous hydrodynamics in the scaling limit.

This work generalizes prior results for the nearest-neighbor ABC model and long-range single-species exclusion processes to the multi-conserved case with heavy-tailed, non-nearest-neighbor interactions in infinite volume. The analysis leverages the framework of nonlinear fluctuating hydrodynamics and the dynamical theory of equilibrium SPDE limits, particularly fractional Ornstein–Uhlenbeck (OU) and fractional stochastic Burgers equations (SBE), dictated by the underlying microscopic regime.

Model Definition

The considered ABC model is a three-species exclusion process on $\mathbb{Z}$: each lattice site is occupied by exactly one particle, of type $A$, $B$, or $C$. Particles at sites $x$ and $x+z$ can exchange their types at rates depending on their types and the jump size $z$, with exchange rates $p(z) \propto |z|^{-\gamma-1}$0 for types $p(z) \propto |z|^{-\gamma-1}$1 and a kinetic factor $p(z) \propto |z|^{-\gamma-1}$2. Two parameters govern the behavior:

• The degree of asymmetry: controlled by $p(z) \propto |z|^{-\gamma-1}$3, with a corresponding macroscopic coupling constant $p(z) \propto |z|^{-\gamma-1}$4.
• The heavy-tailed nature of $p(z) \propto |z|^{-\gamma-1}$5: the tail parameter $p(z) \propto |z|^{-\gamma-1}$6 interpolates between diffusive and superdiffusive regimes.

A crucial feature is the existence of two (linearly independent) conserved fields: the densities of A and B particles. Owing to the complete occupation constraint, the C density is determined by their complement.

Main Results and Scaling Regimes

Two scaling regimes are distinguished, depending on the relative strength of asymmetry and the tail parameter $p(z) \propto |z|^{-\gamma-1}$7:

1. Gaussian Regime ("Weak Asymmetry" or Symmetric Scaling)

If the asymmetry is absent or sufficiently weak (precisely, $p(z) \propto |z|^{-\gamma-1}$8 decays quickly enough with $p(z) \propto |z|^{-\gamma-1}$9 or $\gamma > 0$0), the nonlinearity vanishes at the fluctuation scale. The main result (Theorem 1) is that the properly rescaled fluctuation fields (special linear combinations of A and B densities, called normal modes) converge to uncorrelated stationary solutions of independent fractional OU processes: $\gamma > 0$1 where:

• $\gamma > 0$2 represents the (possibly fractional) Laplacian, encapsulating the effect of the heavy-tailed kernel at the macroscopic scale: for $\gamma > 0$3, a true fractional Laplacian appears.
• $\gamma > 0$4 is an antisymmetric (drift) component, generically vanishing for $\gamma > 0$5.
• $\gamma > 0$6 scales the space-time noise ($\gamma > 0$7), capturing the fluctuation-dissipation characteristics.
• The limiting fields are independent in infinite volume—a structural difference from the periodic/finite case, reflecting the removal of spurious mass-coupling due to infinite propagation speed.

2. KPZ/Burgers Regime ("Critical Asymmetry")

At critical asymmetry (precisely, $\gamma > 0$8 scales as $\gamma > 0$9 so $\mathbb{Z}$0 for an explicit $\mathbb{Z}$1 depending on $\mathbb{Z}$2), the nonlinear term persists at the fluctuation level. The main result (Theorem 2) is that the rescaled normal modes converge to independent stationary energy solutions of fractional SBEs: $\mathbb{Z}$3 with explicit expressions for the nonlinearity coefficient $\mathbb{Z}$4 (given by microscopic model parameters). The limit is Gaussian (OU) unless the configuration of driving fields is such that both normal modes are in a strongly nonlinear ("KPZ") regime—the alignment criterion is directly inherited from the structure of the exchange rates.

Distinction by Tail Index $\mathbb{Z}$5:

• The strong-coupling (KPZ) regime only occurs for $\mathbb{Z}$6 (i.e., when the superdiffusive effect balances or is overpowered by the Burgers nonlinearity).
• The limiting equation is scaling-critical at $\mathbb{Z}$7: here, advanced SPDE techniques (e.g., regularity structures, paracontrolled calculus) fail to produce a pathwise solution theory, emphasizing the relevance of energy solutions.

Techniques and Analytical Innovations

The proof is built around advanced probabilistic and analytic machinery for interacting particle systems with multiple conservation laws and anomalous transport:

• Spectral Diagonalization: Extension of Spohn's theory to diagonalize fluctuations in systems with multiple conserved quantities, producing "normal modes" that evolve essentially independently.
• Martingale Methods & Tightness: Rigorous pathwise analysis of fluctuation fields, employing the martingale decomposition and the Kolmogorov–Centsov criterion for tightness in infinite-dimensional, distributional settings.
• Fractional Operators & Discrete-to-Continuum Limits: Careful discrete analysis to prove convergence of the generator and current terms to their fractional PDE counterparts, handling the heavy tails and nonlocal drift rigorously.
• Cross-Term Control via Wavelet Theory: In infinite volume, the cross-interactions between normal modes must be shown to vanish, requiring the replacement of Fourier analysis (used in finite/periodic settings) with wavelet bases. The functional-analytic properties (regularity, decay) of compactly supported wavelets are utilized to establish the required orthogonality and decay estimates.
• Nonlinearity Identification: The delicate balance between the asymmetry and the heavy-tailed jump kernel in the regime $\mathbb{Z}$8 is handled via a precise one-block/two-block replacement argument, sidestepping the extensive multiscale decompositions necessary in prior works.
• Martingale Problems and Uniqueness: The limiting SPDEs (fractional OU and SBE) are identified in the framework of stationary energy solutions, and uniqueness of the martingale problem is established (including novel adaptations for the critical fractional case), ensuring convergence in distribution to the claimed limits.

Implications and Theoretical Significance

• Physical Relevance: The results confirm and sharpen previous predictions from physics regarding the independence of normal modes in infinite systems and the persistence of KPZ universality in long-range settings under appropriate conditions. The explicit computation of the critical nonlinear coupling and its vanishing under symmetry further quantifies the parameter regimes yielding KPZ-type versus Gaussian fluctuations.
• Mathematical Advances: The work constitutes a complete infinite-volume extension of the classic ABC fluctuation results, fully incorporating long-range jumps and addressing the technical obstacles absent in periodic geometries. The robust method for controlling cross-field interactions via wavelets opens avenues for fluctuation analysis in other multi-conserved, infinite-volume systems.
• Scalability and Sharp Thresholds: The identification of the KPZ-class critical threshold at $\mathbb{Z}$9, as dictated by the interplay between jump kernel decay and nonlinear drift, quantifies the boundary between superdiffusive (Lévy) transport and turbulent KPZ behavior. The constructed solution theory for the critical fractional SBE at $A$0 (where standard pathwise methods fail) is particularly noteworthy.
• Foundational for Nonlocal Fluctuating Hydrodynamics: The methodology—particularly the treatment of coupled fields and the transition from microscopic interaction structure to macroscopic nonlinear SPDEs—provides a blueprint for analyzing a range of anomalous transport systems with multiple conservation laws and nonlocality, including broader classes of driven diffusive systems.

Conclusion

This paper rigorously establishes the scaling limits of fluctuations in a conservative, multi-species, long-range exclusion process, delineating when hydrodynamic fluctuations are governed by Gaussian (fractional OU) or nonlinear (KPZ/Burgers-type) stochastic PDEs. The results elucidate the statistical physics of systems with multiple conserved quantities and anomalous (long-range) transport, and develop new tools for handling the analytic challenges posed by nonlocality and multi-conserved fields at the macroscopic level. These advances set the stage for further exploration of nonlocal, nonlinear fluctuating hydrodynamics in both equilibrium and nonequilibrium contexts.