The free quon gas suffers Gibbs' paradox.

where f, g are test functions, i.e., elements of the oneparticle space with inner product 〈f, g〉, a(f) is a Hilbert space operator with adjoint a†(f) depending linearly on f . The deformation parameter q must be real, and we take it in the interval −1 < q < 1. At the endpoints of this interval the relations become the canonical commutation relations (q = 1), and the canonical anticommutation relations (q = −1), respectively. The Quons with q = 0 were the first example of this type considered by Greenberg [1]. The relations (1) with q = 0 also play a central rôle in the theory of freely independent random variables [4–6] initiated by Voiculescu [7]. Such systems have been used as a driving noise [5] in quantum stochastic differential equations [8]. The extension of this work to other values of q has led to an independent proposal of the relations (1) [9]. The relations also arise as the commutation relations of collective degrees of freedom in a system of many components obeying anomalous statistics [10].