SciPost Submission Page
Emergent Hydrodynamics in an Exclusion Process with Long-Range Interactions
by Ali Zahra, Jerome Dubail, Gunter M. Schütz
Submission summary
| Authors (as registered SciPost users): | Jérôme Dubail · Aly Zahra |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2508.09879v1 (pdf) |
| Date submitted: | Aug. 18, 2025, 11:32 a.m. |
| Submitted by: | Aly Zahra |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the symmetric Dyson exclusion process (SDEP) - a lattice gas with exclusion and long-range, Coulomb-type interactions that emerge both as the maximal-activity limit of the symmetric exclusion process and as a discrete version of Dyson's Brownian motion on the unitary group. Exploiting an exact ground-state (Doob) transform, we map the stochastic generator of the SDEP onto the spin-$1/2$ XX quantum chain, which in turn admits a free-fermion representation. At macroscopic scales we conjecture that the SDEP displays ballistic (Eulerian) scaling and non-local hydrodynamics governed by the equation $\partial_t \rho+\partial_x j[\rho]=0$ with $j[\rho]=(1/\pi)\sin(\pi\rho(x,t))\sinh(\pi\mathcal{H}\rho(x,t))$, where $\mathcal{H}$ is the Hilbert transform, making the current a genuinely non-local functional of the density. This non-local one-field description is equivalent to a local two-field "complex Hopf" system for finite particle density. Closed evolution formulas allow us to solve the melting of single and double block initial states, producing limit shapes and arctic curves that agree with large-scale Monte Carlo simulations. The model thus offers a tractable example of emergent non-local hydrodynamics driven by long-range interactions.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
(2) Doob transform in the context of stochastic particle systems.
(3) Derivation of novel hydrodynamic equations.
(4) Validation through Monte Carlo simulations.
Weaknesses
Report
detailed analytic results are based on a Doob transform yielding an integrable quantum system. The novel results provide important insights on the special feature of long-range forces. While Doob
transform as such is a known method, the particular application to stochastic many-particle systems is novel. With great care, the predictions on macroscopic behavior are validated by numerical simulations.
The article is very well written and readable for a larger community.
In view of the novelty and density of results I recommend publication in SciPost Physics.
Requested changes
I have only two comments:
(1) Eq. (32) is a coupled system, so why independent.
(2) Since the interaction is long-ranged, the local jump rates might be defined only for a restricted class of initial conditions. Maybe I overlooked, but there does not seem to be a discussion.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)