RANDOM WALKS ON CURVED SPACETIME SURFACE AND FRACTIONAL DERIVATIVES

A method of describing a scale free behavior of complex systems is proposed. The nonlinearity is formulated as the curvature of the spacetime surface on which random walks are described. In the random walks the memory effect is included by using the fractional derivatives. To clarify a physical meaning of the curvature and the nonlinearity, a virtual force is introduced. The curved spacetime surface is constructed as a net of curvilinear coordinates. On the basis of the formulation, a nonlinear and fractional Fokker-Planck (FP) equation is obtained from the corresponding linear FP equation on the curved spacetime surface. Using a method similar to the formulation, we compute the sliding motion of a block on the irregular surface filmed by lubrication.