Integrability and exact large deviations of the weakly-asymmetric exclusion process
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| Format: | Preprint |
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2025
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| _version_ | 1866908369144512512 |
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| author | Krajenbrink, Alexandre Doussal, Pierre Le |
| author_facet | Krajenbrink, Alexandre Doussal, Pierre Le |
| contents | The weakly asymmetric exclusion process (WASEP) in one dimension is a paradigmatic system of interacting particles described by the macroscopic fluctuation theory (MFT) in the presence of driving. We consider an initial condition with densities $ρ_1,ρ_2$ on either side of the origin, so that for $ρ_1=ρ_2$ the gas is stationary. Starting from the microscopic description, we obtain exact formulae for the cumulant generating functions, and large deviation rate functions of the time-integrated current and the position of a tracer. As the asymmetry/driving is increased, these describe the crossover between the symmetric exclusion process (SSEP) and the weak noise regime of the Kardar-Parisi-Zhang (KPZ) equation: we recover the two limits and describe the crossover from the WASEP cubic tail to the $5/2$ and $3/2$ KPZ tail exponents. Finally, we show that the MFT of the WASEP is classically integrable, by exhibiting the explicit Lax pairs, which are obtained through a novel mapping between the MFT of the WASEP and a complex extension of the classical anisotropic Landau-Lifshitz spin chain. This shows integrability of all MFTs of asymmetric models with quadratic mobility as well as their dual versions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12034 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrability and exact large deviations of the weakly-asymmetric exclusion process Krajenbrink, Alexandre Doussal, Pierre Le Statistical Mechanics Disordered Systems and Neural Networks Mathematical Physics Probability Exactly Solvable and Integrable Systems The weakly asymmetric exclusion process (WASEP) in one dimension is a paradigmatic system of interacting particles described by the macroscopic fluctuation theory (MFT) in the presence of driving. We consider an initial condition with densities $ρ_1,ρ_2$ on either side of the origin, so that for $ρ_1=ρ_2$ the gas is stationary. Starting from the microscopic description, we obtain exact formulae for the cumulant generating functions, and large deviation rate functions of the time-integrated current and the position of a tracer. As the asymmetry/driving is increased, these describe the crossover between the symmetric exclusion process (SSEP) and the weak noise regime of the Kardar-Parisi-Zhang (KPZ) equation: we recover the two limits and describe the crossover from the WASEP cubic tail to the $5/2$ and $3/2$ KPZ tail exponents. Finally, we show that the MFT of the WASEP is classically integrable, by exhibiting the explicit Lax pairs, which are obtained through a novel mapping between the MFT of the WASEP and a complex extension of the classical anisotropic Landau-Lifshitz spin chain. This shows integrability of all MFTs of asymmetric models with quadratic mobility as well as their dual versions. |
| title | Integrability and exact large deviations of the weakly-asymmetric exclusion process |
| topic | Statistical Mechanics Disordered Systems and Neural Networks Mathematical Physics Probability Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2505.12034 |