Tonks–Girardeau gas

In physics, a Tonks–Girardeau gas is a Bose gas in which the repulsive interactions between bosonic particles confined to one dimension dominate the physics of the system. It is named after physicists Marvin D. Girardeau and Lewi Tonks. Strictly speaking this is not a Bose–Einstein condensate as it does not demonstrate any of the characteristics, such as off diagonal long range order or a unitary two body correlation function, even in a thermodynamic limit and as such cannot be described by a macroscopically occupied orbital (order parameter) in the Gross Pitaevskii formulation.

Consider a row of bosons all confined to a one-dimensional line. They cannot pass each other and therefore cannot exchange places. The resulting motion has been compared to a traffic jam: the motion of each boson would be strongly correlated with that of its two neighbours. This can be thought of the large c limit of the Delta Bose Gas.

Because the particles cannot exchange places, one might expect their behaviour to be fermionic, but it turns out that their behaviour differs from that of fermions in several important ways: the particles can all occupy the same momentum state which corresponds to neither Bose–Einstein nor Fermi–Dirac statistics. This is the phenomenon of bosonization which happens in 1+1 dimensions.

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 coincide with quantum Nonlinear Schrödinger equation for infinite repulsion, which can be efficiently analyzed by Quantum inverse scattering method. This relation help to study Correlation function (statistical mechanics). The correlation functions can be described by Integrable system. In a simple case it is Painlevé transcendents. A textbook explains in detail the description of quantum correlation functions of Tonks–Girardeau gas by means of classical completely integrable differential equations. Thermodynamics of Tonks–Girardeau gas was described by Chen Ning Yang.

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