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. 2025;191(1-2):361-420.
doi: 10.1007/s00440-024-01314-z. Epub 2024 Oct 22.

From ABC to KPZ

G Cannizzaro  1 , P Gonçalves  2 , R Misturini  3 , A Occelli  4
Affiliations

From ABC to KPZ

G Cannizzaro et al. Probab Theory Relat Fields. 2025.

Abstract

We study the equilibrium fluctuations of an interacting particle system evolving on the discrete ring with N ∈ N points, denoted by T N , and with three species of particles that we name A, B and C, but such that at each site there is only one particle. We prove that proper choices of density fluctuation fields (that match those from nonlinear fluctuating hydrodynamics theory) associated to the (two) conserved quantities converge, in the limit N → ∞ , to a system of stochastic partial differential equations, that can either be the Ornstein-Uhlenbeck equation or the Stochastic Burgers equation. To understand the cross interaction between the two conserved quantities, we derive a general version of the Riemann-Lebesgue lemma which is of independent interest.

Keywords: Crossover weakly asymmetric exclusion; KPZ equation; Multi-component; Ornstein-Uhlenbeck process; Stochastic Burgers equation; Two species.

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Conflict of interest statement

Conflict of interestThe authors declare no competing interests.

Figures

Fig. 1
Fig. 1
Dynamics of the model
Fig. 2
Fig. 2
Classification of the universal behaviour of the two modes observed in the ABC model by the structure of the mode coupling matrices. A star denotes a non-zero entry, a dot represents an arbitrary value
See this image and copyright information in PMC

References

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