Affine geometric description of thermodynamics

Abstract

Thermodynamics provides a unified perspective of the thermodynamic properties of various substances. To formulate thermodynamics in the language of sophisticated mathematics, thermodynamics is described by a variety of differential geometries, including contact and symplectic geometries. Meanwhile, affine geometry is a branch of differential geometry and is compatible with information geometry, where information geometry is known to be compatible with thermodynamics. By combining above, it is expected that thermodynamics is compatible with affine geometry and is expected that several affine geometric tools can be introduced in the analysis of thermodynamic systems. In this paper, affine geometric descriptions of equilibrium and nonequilibrium thermodynamics are proposed. For equilibrium systems, it is shown that several thermodynamic quantities can be identified with geometric objects in affine geometry and that several geometric objects can be introduced in thermodynamics. Examples of these include the following: specific heat is identified with the affine fundamental form and a flat connection is introduced in thermodynamic phase space. For nonequilibrium systems, two classes of relaxation processes are shown to be described in the language of an extension of affine geometry. Finally, this affine geometric description of thermodynamics for equilibrium and nonequilibrium systems is compared with a contact geometric description.