A new formalism in statistical mechanics, that of the nonextensive statistical mechanics (NESM), was proposed in 1998 by physicist Constantino Tsallis. Although NESM remains controversial, it has generated considerable research interest due to its success in describing systems that do not obey Boltzmann-Gibbs statistice. To date there are over 2000 publications on Tsallis statistics, including over 80 in the areas of finance and time series.

Following an overview of the emerging field of econophysics, the Tsallis approach will be introduced incrementally. First, a brief review of classical statistical mechanics that references the work of Maxwell-Boltzmann, Shannon and Jaynes is given. Next, the maximum entropy principle is stated, and its connection with the standard Brownian motion is given. Several examples of experimental and observational data are offered as evidence that a classical application of Boltzmann-Gibbs fails. Tsallis nonextensive statistics are then introduced. Since one of the main criticisms against the Tsallis statistics is that there are no physical models showing how they might arise, four etiological models are discussed:
(1) the Bening-Korolev nonextensive limit theorem,
(2) the Beck model of turbulence,
(3) the dynamic correlations model of Kodama, et. al. and
(4) the Lambiotte-Ausloos fluctuating mass model. A brief discussion of applications of NESM to stock price movement and options pricing concludes the discussion.