## 1. Introduction

[2] Boltzmann-Gibbs statistical mechanics cannot describe nonequilibrium physical systems with large variability and multifractal structure such as the solar wind. Tsallis introduced a generalized statistical mechanics with an entropy function *S*_{q} describing the statistics and a constraint expressing the physics [*Tsallis*, 1988, 2004; *Tsallis and Brigatti*, 2004]. Choosing the nonextensive pseudoadditive entropy *S*_{q} = ∑ (*p*_{i}^{q} − 1)/(1 − *q*), (*p*_{i} is the probability of the *i*th microstate, and *q* is a constant that measures the degree of nonextensivity) and extremizing *S*_{q} subject to two constraints, *Tsallis* [1988, 2004] derived the probability distribution (pdf)

where *x* is physical quantity such as energy and *A*, *C*, and *q* are constants at a given scale. In the limit *q* → 1, the statistical mechanics of Tsallis reduces to that of Boltzmann and Gibbs, where the pdf is proportional to an exponential (Gaussian) distribution and *x* is energy. In the limit of large *x*, the Tsallis pdf (1) approaches the power law *D* × *x*^{−(2/(q − 1))}, where *D* ≡ *A*[(*q* − 1)*C*]^{−1/(q − 1)}.

[3] *Tsallis* [2004] notes that there are at least 20 entropy functions in the literature, and he compares several in detail. For example, *Rényi* [1970] introduced an extensive entropy, and *Daroczy* [1970] introduced “entropies of type B” related to Rényi entropies. These entropies are different than the Tsallis entropy. For example, the Rényi entropy is extensive, nonstable, and nonconcave; the Tsallis entropy is nonextensive, stable, and concave. Generalized statistics and power law distributions were discussed by many people, including *Hasegawa et al.* [1985] and *Kaniadakis* [2001]. It is possible to interpret the Tsallis distribution as a consequence of fluctuations, described by “superstatistics” [*Beck*, 2001, 2002a, 2002b, 2004; *Beck and Cohen*, 2003]. A statistical mechanical foundation for superstatistics was given by *Tsallis and Souza* [2003].

[4] The Tsallis distribution describes the probability density functions (pdfs) of fluctuations in increments of the magnetic field strength *B* in the solar wind at 1 AU on scales from 1 hour to 128 days near both solar maximum and minimum [*Burlaga and Viñas*, 2004a, 2005a]. The Tsallis distribution also describes fluctuations in the solar wind speed at 1 AU from scales of 64 s to 128 days [*Burlaga and Viñas*, 2004b]. *Leubner and Vörös* [2005a, 2005b] and *Vörös et al.* [2006] discuss distribution functions of small-scale intermittency and turbulence in the solar wind. Distribution functions associated with small-scale turbulence were discussed by *Arimitsu and Arimitsu* [2000, 2001a, 2001b, 2002a, 2002b, 2002c, 2002d, 2002e]. *Treumann et al.* [2004] discuss stationary states far from equilibrium for systems such as the solar wind. Tsallis distributions of large-scale fluctuations in the heliosheath were observed by *Burlaga et al.* [2006]. *Burlaga and Viñas* [2005b] found Tsallis distributions of daily observations of the fluctuation in *B* throughout each of the years 1980, 1991, 2001, and 2002, when Voyager 1 (V1) was at 6.9–9.7, 43.6–47.2, 79.8–83.4, and 83.4–86.9 AU, respectively.

[5] The purpose of this paper is to show that Tsallis distributions of fluctuations in the increments of *B* between 5 and 90 AU on scales from 1 to 128 days and the parameters of these distributions are predicted by a deterministic one-dimensional time-dependent magnetohydrodynamic (MHD) model with Advanced Composition Explorer (ACE) observations at 1 AU as input. We also show that the predictions of the model agree with the observations from V1 at ∼80 AU.