Fractional calculus as a macroscopic manifestation of randomness

Istituto di Biofisica del Consiglio Nazionale delle Ricerche, Via S. Lorenzo 26, 56127 Pisa, Italy(February 1, 2008)We generalize the method of Van Hove so as to deal with the case ofnon-ordinary statistical mechanics, that being phenomena with no time-scaleseparation. We show that in the case of ordinary statistical mechanics, even ifthe adoption of the Van Hove method imposes randomness upon Hamiltoniandynamics, the resulting statistical process is described using normal calculustechniques. On the other hand, in the case where there is no time-scaleseparation, this generalized version of Van Hove’s method not only imposesrandomness upon the microscopic dynamics, but it also transmits randomnessto the macroscopic level. As a result, the correct description of macroscopicdynamics has to be expressed in terms of the fractional calculus.PACS number(s): 05.40.+j, 05.45.+b, 05.60.+wI. INTRODUCTION