The density of a fluid on a curved surface

TL;DR

This paper investigates how the density of particles in a fluid on a curved surface behaves, establishing conditions for uniform density and interpreting curvature effects as external forces.

Contribution

It provides new conditions under which the fluid density remains constant on curved surfaces and offers a physical interpretation of curvature effects as external forces.

Findings

Density is constant in the thermodynamic limit under certain conditions.

A sufficient condition for density constancy along isometries is derived.

Curvature effects can be viewed as external forces acting on particles.

Abstract

We discuss the property of the number density of a fluid of particles living in a curved surface without boundaries to be constant in the thermodynamic limit. In particular we find a sufficient condition for the density to be constant along the Killing vector field generating a given isometry of the surface and the relevant necessary condition. We reinterpret the effect of a curvature on the fluid in a physical way as responsible of an external "force" acting on the particles.