Geometric contact provides a precise characterization of the approximate equality of two functions in physics, and its mathematical locality can be interpreted as physical intervality. In physics, the failure to introduce this geometric contact and provide it with a physical interpretation has led to unnecessary controversies in understandings and descriptions of the same phenomenon across various pieces of literature. By addressing two specific problems in thermodynamics and statistical physics, some long-standing unresolved conceptual issues have been resolved.

In the ideal gas of diatomic molecules, the simplest coupling of vibrational motion and rotational motion inside the molecule makes the vibratory motion deviate from the equipartition theorem of energy. At this time, the contribution of vibrational degrees of freedom to energy is not necessarily kT. For hydrogen molecules, even if the temperature is close to the characteristic temperature of dissociation, the contribution of vibration degrees of freedom to energy is approximately kT, that is, the equipartition theorem holds.

At low temperature, the vibrational degrees of freedom of diatomic molecules do not contribute to the heat capacity of the gas. When the temperature is higher than the vibrational characteristic temperature of diatomic molecules, the vibrational degrees of freedom are excited. The vibration of this diatomic molecule is actually a simple harmonic approximation of the interaction between atoms. However, the simple harmonic approximation criteria given in the literature are all mechanical results, and these results are qualitative. In this study, the statistical physics criterion is given: the validity of the harmonic approximation is that the temperature must be much lower than the molecular bond energy characteristic temperature.

Assuming that the interaction between the magnetic moment and the external magnetic field obeys quantum mechanics, the microscopic model of paramagnetic medium can be established. The simple microscopic model is a nearly independent spin particle system in the external magnetic field. Statistical physical calculation results show that the relationship between magnetic susceptibility and magnetic field is generally nonlinear.

At zero temperature, the amount of decrease of the Gibbs function accompanying a chemical reaction is the heat released from the thermodynamic process, and the heat is defined as the zero-point heat. The zero-point heat consists in two parts, one is the difference between the thermodynamic zero-point internal energy of the reactant and that of the product in the chemical reaction, and another the mechanical energy of the volume changes of the reactant the product. For some systems, the Thomsen-Berthelot principle is applicable at room temperature and even higher, which means that zero-point heat plays significant roles at relatively high temperatures.