Thermodynamics provides a unified perspective of 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: specific heat is identified with the affine fundamental form, 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.
General disagreement between the Geometrical Description of Dynamical In-stability -using non affine parameterizations- and traditional Tangent Dynamics
Towards geometric Satake correspondence for Kac-Moody algebras -- Cherkis bow varieties and affine Lie algebras of type $A$
Research Institute for Mathematical Sciences, Kyoto University（京都大学数学科学研究所）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo（东京大学宇宙物质前沿研究中心）
Geometric foundations of thermodynamics in the quantum regime
Universidad de Granada（格拉纳达大学）；Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada（格拉纳达大学卡洛斯一世理论与计算物理研究所）；Universidad de Córdoba（科尔多瓦大学）
