Diffusion with finite-helicity field tensor: a new mechanism of generating heterogeneity

TL;DR

This paper introduces a new class of field tensors with finite helicity governed by Beltrami vectors, extending Hamiltonian mechanics to explain heterogeneity formation in constrained dynamical systems.

Contribution

It extends the framework of Hamiltonian systems to include finite-helicity field tensors and proves an H theorem for this class, revealing mechanisms for heterogeneity creation.

Findings

Introduction of finite-helicity field tensors characterized by Beltrami vectors.

Proof of an H theorem for the Beltrami class of systems.

Identification of a 'field charge' that prevents entropy maximization and leads to heterogeneity.

Abstract

Topological constraints on a dynamical system often manifest themselves as breaking of the Hamiltonian structure; well-known examples are non-holonomic constraints on Lagrangian mechanics. The statistical mechanics under such topological constraints is the subject of the present study. Conventional arguments based on phase spaces, Jacobi identity, invariant measure, or the H theorem are no longer applicable, since all these notions stem from the symplectic geometry underlying canonical Hamiltonian systems. Remembering that Hamiltonian systems are endowed with field tensors (canonical 2-forms) that have zero helicity, our mission is to extend the scope toward the class of systems governed by finite-helicity field tensors. Here we introduce a new class of field tensors that are characterized by Beltrami vectors. We prove an H theorem for this Beltrami class. The most general class of energy-conserving systems are non-Beltrami, for which we identify the "field charge" that prevents the entropy to maximize, resulting in creation of heterogeneous distributions. The essence of the theory can be delineated by classifying three-dimensional dynamics. We then generalize to arbitrary (finite) dimensions.