The fractional Langevin equation

Abstract

We begin with an overview of fractional derivatives, which have many different definitions, not all of which are equivalent. For some of the most commonly used definitions, we present a few properties and techniques for solving fractional differential equations. Furthermore, we show some of the key differences when solving identical equations using a different definition. There are already applications of fractional derivatives, but each application requires a critical assessment for which definition is most suitable. We show a new application of fractional derivatives in the field of glasses, making use of Caputo fractional derivatives. An analytical solution of the fractional Langevin equation is obtained, where the first-order friction term is replaced by a Caputo fractional derivative of order s. Then, we show that for 0<s<0.1, the ground state of the fractional Langevin solutions exhibits emergent periodic glassy behaviour, thus characterising the recently conjectured time glass. Finally, we present a semi-classical microscopic model, which, in the low-temperature limit, is effectively described by the fractional Langevin equation, thus establishing the link between sub-ohmic open systems and fractional derivative equations.