Student's t-Distribution versus Zeldovich-Kompaneets Solution of Diffusion Problem

R. Wojnar Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawińskiego 5B, 02-106 Warszawa

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Student's t-distribution is compared to a solution of superdiffusion equation. This t-distribution is a continuous probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. Formally it can written in the form: similar to the Gaussian distribution, in which, however, instead of usual exponential function, the so called K-exponential - a form of binomial distribution - appears. Similar binomial form has the Zeldovich-Kompaneets solution of nonlinear diffusion-like problems. A superdiffusion process, similar to a Zeldovich-Kompaneets heat conduction process, is defined by a nonlinear diffusion equation in which the diffusion coefficient takes the form D=a(t)(1/f)n, where a=a(t) is an external time modulation, n is a positive constant, and f=f(x,t) is a solution to the nonlinear diffusion equation. It is also shown that a Zeldovich-Kompaneets solution still satisfies the superdiffusion equation if a=a(t) is replaced by the mean value of a. A solution to the superdiffusion equation is given. This may be useful in description of social, financial, and biological processes. In particular, the solution possesses a fat tail character that is similar to probability distributions observed at stock markets. The limitation of the analogy with the Student distribution is also indicated.

DOI: 10.12693/APhysPolA.121.B-133 PACS numbers: 02.70.Rr, 05.40.Jc, 05.40.Fb, 89.65.Gh