Langevin Equation Based on Deformed Derivatives

Resumo

In this contribution we can consider that the dynamical evolution of the granular system follows possible anomalous dynamics, characterized by different dynamical equations and with the presence of dissipation intrinsically. By this justification, we generalize the Langevin Equations (LE), to describe granular gases dynamics as dissipative systems and, for such intend we consider different forms of deformed derivatives (DD) as derivatives which are included in the kinetic equations. As a consequence of this description, the geometry of phase-space, implicit in the choice of DD by the mapping to fractal continuous, has deep influence in the form of the solutions for the corresponding deformed LE. We claim that the dynamical evolution of the granular system follows possible anomalous dynamics, characterized by different dynamical equations and with the presence of dissipation intrinsically. By this justification, we generalize the Langevin Equations (LE), to describe granular gases dynamics as dissipative systems and, for such intend we consider different forms of DD as derivatives which are included in the kinetic equations. Here we also claim, as in Ref. [7], that new conceptions and approaches, such as DD, may allow us to understand new systems. In particular, the use of deformed derivatives (local), similarly to the (nonlocal) fractional calculus (FC), allows us to describe and emulate complex dynamics involving environmental variation, without the addition of explicit terms relating to this complexity in the dynamical equations describing the system, i.e, without explicit many-body, dissipation or geometrical terms [5]. [...]