Derivation of quantum hydrodynamics
For a concrete deterministic nonintegrable quantum lattice model with a specified conserved density, derive a diffusive scaling limit with finite positive diffusion coefficient and controlled errors for smooth initial profiles. Treat ballistic Euler limits and
Problem statement
For a concrete deterministic nonintegrable quantum lattice model with a specified conserved density, derive a diffusive scaling limit with finite positive diffusion coefficient and controlled errors for smooth initial profiles. Treat ballistic Euler limits and nonlinear fluctuations as separate subtargets.
Lean 4
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Background
Hydrodynamics — the effective large-scale description of conserved densities via diffusion or ballistic transport — is expected to emerge from microscopic quantum dynamics, but deriving it rigorously is a longstanding challenge, the quantum analogue of Hilbert's sixth problem for classical mechanics.
What is known: Generalized hydrodynamics provides a powerful Euler-scale theory for integrable systems, with rigorous results in controlled settings and extensive physical support. Its derivation and validity assumptions are model dependent; it is not a universal error-controlled diffusion theorem for nonintegrable lattices. KPZ-type behavior and ballistic transport require different scaling questions. Rigorous classical interacting-particle hydrodynamic limits supply methodology rather than an automatic quantum proof.
Deriving normal diffusion — a finite diffusion constant — from a genuinely interacting, non-integrable quantum lattice model, with quantitative error bounds in a diffusive scaling limit, remains open and is a central programme in mathematical statistical mechanics. Related: B5, B12.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
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