Power-Law Tails from Multiplicative Noise

We show that the well-known linear Langevin equation, modeling the Brownian motion and leading to a Gaussian stationary distribution of the corresponding Fokker-Planck equation, is changed by the smallest multiplicative noise. This leads to a power-law tail of the distribution for sufficiently large momenta. At finite ratio of the correlation strength for the multiplicative and the additive noises the stationary energy distribution becomes exactly the Tsallis distribution.

Figure 1

Comparison of the generic stationary distributions of the Langevin-type equation with correlated additive and multiplicative noises. Interesting cases, as the Gauss ($B=0,C={10}^{-5},D=2,F=0,G=1$), the gamma ($B=0,C=2,D=0.01,F=1,G=1$), and the power-law ($B=0,C=1,D=2,F=2,G=1$) distributions are labeled correspondingly. The label “Zero” refers to the case with $\vartheta =0$ ($B=1,C=0.1,D=10,F=1,G=1$).

Figure 2

Graphical representations of contributions to the multiplicative and additive noise correlations and damping expectation value in non-Abelian plasma based on Eqs. (25).