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Hydrodynamic limits of the asymmetric Brownian energy process and its dynamic generalization

  • Frank Redig
  • Carel Wagenaar
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Abstract

We prove that the hydrodynamic limit of the asymmetric Brownian energy process (ABEP) is the viscous Burgers equation. We also prove the hydrodynamic limit of the dynamic ABEP which leads to a nonlinear equation for the macroscopic tail-energy potential. Both processes are related to the Brownian energy process (BEP) through invertible, non-local transformations involving discrete tail energies, which can be thought of as the microscopic equivalent of a transformation of Cole-Hopf type. Our main tool is a general micro-to-macro result showing that, under suitable conditions, such microscopic tail transformations induce corresponding transformations of the limiting density profiles. Combining this result with the fact that the hydrodynamic limit of BEP is the heat equation, we obtain the limiting profiles of ABEP and dynamic ABEP as explicit nonlinear transformations of solutions of the heat equation.

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Cite

BibTeXRIS
BibTeX
@misc{redig2026hydrodynamic,
  title = {{Hydrodynamic limits of the asymmetric Brownian energy process and its dynamic generalization}},
  author = {Frank Redig and Carel Wagenaar},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06361},
  eprint = {2611.06361},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06361},
}
RIS
TY  - UNPB
TI  - Hydrodynamic limits of the asymmetric Brownian energy process and its dynamic generalization
AU  - Frank Redig
AU  - Carel Wagenaar
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06361
UR  - https://arxiv.org/abs/2611.06361
AB  - We prove that the hydrodynamic limit of the asymmetric Brownian energy process (ABEP) is the viscous Burgers equation. We also prove the hydrodynamic limit of the dynamic ABEP which leads to a nonlinear equation for the macroscopic tail-energy potential. Both processes are related to the Brownian energy process (BEP) through invertible, non-local transformations involving discrete tail energies, which can be thought of as the microscopic equivalent of a transformation of Cole-Hopf type. Our main tool is a general micro-to-macro result showing that, under suitable conditions, such microscopic tail transformations induce corresponding transformations of the limiting density profiles. Combining this result with the fact that the hydrodynamic limit of BEP is the heat equation, we obtain the limiting profiles of ABEP and dynamic ABEP as explicit nonlinear transformations of solutions of the heat equation.
ER  -