1 Introduction

[549.1.1]A large number of authors have recently and in the past proposed to use fractional time derivatives on heuristic or aesthetic grounds as phenomenological models for various natural [page 550, §0]   processes [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. [550.0.1]Can such use of fractional time derivatives in physics be justified from first principles ? [550.0.2]The traditional answer to this question is a firm “No”, because fractional time derivatives, contrary to integer ones, are nonlocal operators, and their use would contradict the deeply rooted principle of locality in physics [17]. [550.0.3]The first indications that fractional time derivatives have a deeper and more fundamental significance for physics than merely that of a convenient phenomenological modeling tool appeared in recent work of the present author on the classification of phase transitions [18, 19, 20, 21].

[550.1.1]My objective in this paper is to further investigate the origin of fractional time derivatives in physics, and to show that they appear generically in coarse grained descriptions of dynamical behaviour in the ultra-long-time limit [21]. [550.1.2]I shall call into question the applicability of the traditional concepts of stationarity and equilibrium in this limit. [550.1.3]The ultralong time limit is a limit in which a discretized time evolution is iterated infinitely often and the discretzation time step becomes simultaneously infinite.

[550.2.1]Dynamical descriptions of macroscopic (coarse grained) nonequilibrium phenomena typically involve a reduction in the number of underlying microscopic dynamical degrees of freedom.[550.2.2]This reduction or coarse graining amounts to a restriction of the microscopic dynamics to a subspace (i.e. a subset of measure zero) of the microscopic phase space.[550.2.3]Simultaneously the characteristic time scale of the reduced or coarse grained description is often so much longer than that of the underlying microscopic dynamics, that it may be idealized as infinite.

[550.3.1]Given these general ideas the present paper employs concepts from abstract ergodic theory to show that fractional time derivatives appear as the infinitesimal generators of reduced or coarse grained dynamical desriptions in the ultralong time limit.[550.3.2]The results of the present paper are direct consequences of a recent classification of phase transitions in statistical mechanics, and the ultralong time limit is a version of the ensemble limit [18, 19].