Fractional Stochastic Hydrodynamics and Fokker-Planck Dynamics of 2D Graphene Dirac Quasiparticle Models

We formulate a unified theoretical framework for the fractional dynamics of graphene Dirac quasiparticles within the Riemann-Liouville (RL) fractional calculus approach. In this case, by starting from the fractional graphene Dirac equation, we derive an exact hydrodynamic formulation through the systematic construction of the density, current, and pseudospin evolution equations without applying perturbative assumptions or phenomenological approximations. Moreover, the fractional relativistic Langevin description is subsequently established to incorporate stochastic contributions into the relativistic hydrodynamic dynamics, allowing a consistent treatment of fluctuations in fractional quantum systems. Within this formulation, the corresponding fractional Fokker-Planck equation and relativistic stochastic Hamilton-Jacobi equation are derived, providing a complete statistical framework for analyzing fractional quantum transport processes. Also, the resulting hydrodynamic variables are then used to reconstruct the fractional graphene Dirac equation exactly, confirming the equivalence between the original spinor representation and the hydrodynamic description at the mathematical level. Furthermore, we introduce a fractional geometric hydrodynamic reconstruction operator and extend the developed formalism to curved spacetime in the RL framework, leading to a covariant description of fractional graphene dynamics in nontrivial geometries. In this context, this formulation provides a systematic theoretical structure connecting fractional relativistic quantum dynamics, stochastic transport mechanisms, and geometric hydrodynamic methods, enabling the investigation of nonlocal transport behavior and anomalous diffusion effects in graphene and other Dirac materials.