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2012 | 121 | 2B | B-133-B-136
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Student's t-Distribution versus Zeldovich-Kompaneets Solution of Diffusion Problem

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