In the case of a Tonks–Girardeau gas (TG), so many properties of this one-dimensional string of bosons would be sufficiently fermion-like that the situation is often referred to as the ' fermionization ' of bosons. Tonks–Girardeau gas matches quantum Nonlinear Schrödinger equation for infinite repulsion, which can be efficiently analyzed by quantum inverse scattering method. This relation helps to study correlation functions. The correlation functions can be described by an Integrable system. In a simple case, it is a Painlevé transcendent. The quantum correlation functions of a Tonks–Girardeau gas can be described by means of classical, completely integrable, differential equations. Thermodynamics of Tonks–Girardeau gas was described by Chen Ning Yang.