Geometric Flow appearing in Conservation Law in Classical and Quantum Mechanics

TL;DR

This paper reveals how geometric flows emerge as anomalies in conservation laws for particle number and probability on thin curved surfaces, leading to modified conservation laws in classical and quantum systems.

Contribution

It introduces a new geometric flow term in conservation laws for particles confined to curved surfaces, applicable to both classical diffusion and quantum mechanics.

Findings

The geometric flow depends on Gaussian and mean curvature.

Anomalous terms arise from surface thickness expansion.

A modified conservation law includes the geometric flow.

Abstract

The appearance of a geometric flow in the conservation law of particle number in classical particle diffusion and in the conservation law of probability in quantum mechanics is discussed in the geometrical environment of a two-dimensional curved surface with thickness $\epsilon$ embedded in $R_3$. In such a system with a small thickness $\epsilon$, the usual two-dimensional conservation law does not hold and we find an anomaly. The anomalous term is obtained by the expansion of $\epsilon$. We find that this term has a Gaussian and mean curvature dependence and can be written as the total divergence of some geometric flow. We then have a new conservation law by adding the geometric flow to the original one. This fact holds in both classical diffusion and quantum mechanics when we confine particles to a curved surface with a small thickness.