N ov 2 00 7 Econophysics , Statistical Mechanics Approach to

Probability density P (x) is defined so that the probability to find a random variable x in the interval from x to x+ dx is equal to P (x) dx. Cumulative probability C(x) is defined as the integral C(x) = ∫∞ x P (x) dx. It gives the probability that the random variable exceeds a given value x. The Boltzmann-Gibbs distribution gives the probability of finding a physical system in a state with the energy ε. Its probability density is given by the exponential function (1). The Gamma distribution has the probability density given by a product of an exponential function and a power-law function, as in Eq. (9). The Pareto distribution has the probability density P (x) ∝ 1/x and the cumulative probability C(x) ∝ 1/x given by a power law. These expressions apply only for high enough values of x and not for x → 0. The Lorenz curve was introduced by American economist Max Lorenz to describe income and wealth inequality. It is defined in terms of two coordinates x(r) and y(r) given by Eq. (19). The horizontal coordinate x(r) is the fraction of population with income below r, and the vertical coordinate y(r) is the fraction of income this population accounts for. As r changes from 0 to ∞, x and y change from 0 to 1, parametrically defining a curve in the (x, y)-plane. The Gini coefficient G was introduced by Italian statistician Corrado Gini as a measure of inequality in a society. It is defined as the area between the Lorenz curve and the straight diagonal line, divided by the area of the triangle beneath the diagonal line. For perfect equality (everybody has the same income or wealth) G = 0, and for total inequality (one person has all income or wealth, and the rest have nothing) G = 1. The Fokker-Planck equation is the partial differential equation (22) that describes evolution in time t of the probability density P (r, t) of a random variable r experiencing small random changes ∆r during short time intervals ∆t. It is also known in mathematical literature as the Kolmogorov forward equation. Diffusion equation is an example of the Fokker-Planck equation.