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Article Dans Une Revue SIAM Journal on Mathematical Analysis Année : 2015

Well-Posedness of the Cauchy Problem for a Space-Dependent Anyon Boltzmann Equation

Résumé

A fully non-linear kinetic Boltzmann equation for anyons is studied in a periodic 1d setting with large initial data. Strong L 1 solutions are obtained for the Cauchy problem. The main results concern global existence, uniqueness and stabililty. We use the Bony functional, the two-dimensional velocity frame specific for anyons, and an initial layer analysis that moves the solution away from a critical value. 1 Anyons and the Boltzmann equation. Let us first recall the definition of anyon. Consider the wave function ψ(R, θ, r, ϕ) for two identical particles with center of mass coordinates (R, θ) and relative coordinates (r, ϕ). Exchanging them, ϕ → ϕ + π, gives a phase factor e 2πi for bosons and e πi for fermions. In three or more dimensions those are all possibilities. Leinaas and Myrheim proved in 1977 [10], that in one and two dimensions any phase factor is possible in the particle exchange. This became an important topic after the first experimental confirmations in the early 1980-ies, and Frank Wilczek in analogy with the terms bos(e)-ons and fermi-ons coined the name any-ons for the new quasi-particles with any phase. Anyon quasi-particles with e.g. fractional electric charge, have since been observed in various types of experiments. By moving to a definition in terms of a generalized Pauli exclusion principle, Haldane [9] extended this to a fractional exclusion statistics valid for any dimension, and coinciding with the anyon definition in the one and two dimensional cases. Haldane statistics has also been realized for neutral fermionic atoms at ultra-low temperatures in three dimensions [3]. Wu later derived [17] occupation-number distributions for ideal gases under Haldane statistics by counting states under the new fractional exclusion principle. From the number of quantum states of N identical particles occupying G states being
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Dates et versions

hal-01261187 , version 1 (24-01-2016)

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Leif Arkeryd, Anne Nouri. Well-Posedness of the Cauchy Problem for a Space-Dependent Anyon Boltzmann Equation. SIAM Journal on Mathematical Analysis, 2015, 47 (6), pp.4720-4742. ⟨10.1137/15M1012335⟩. ⟨hal-01261187⟩
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