Geometric Flow Appearing in Conservation Law in Classical and Quantum Mechanics

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.