Comparing linear and nonlinear hydrodynamical models for charge transport in graphene based on the Maximum Entropy Principle

Equilibrium Wigner Function for Fermions and Bosons in the Case of a General Energy Dispersion Relation

“…The equilibrium density matrix can be obtained by employing a generalisation of the Maximum Entropy Principle (hereafter MEP) in a quantum context [ 10 , 17 , 18 ] (for the semiclassical case see [ 6 , 9 , 12 , 19 , 20 , 21 , 22 ]). According to the quantum version of MEP the equilibrium density matrix is obtained by maximising the quantum entropy under suitable constraints on the expectation values.…”

“…Instead, for several material like semiconductors or semimetal, e.g., graphene, other dispersion relations must be considered [ 4 , 5 , 6 , 7 ]. From the Wigner transport equation quantum hydrodynamical models have been obtained in [ 8 ] for charge transport in silicon in the case of parabolic bands, while in [ 9 ] the same has been devised for electrons moving in graphene.…”

“…A crucial issue is the expression of the entropy in the quantum case. In [ 8 , 9 ] the standard prescription proposed by von Neumann has been adopted but it leads to a semiclassical limit represented by the Maxwell–Boltzmann distribution. For dilute gases this can constitute an acceptable approximation but for strongly degenerate gases it is necessary to include the appropriate statistics as for example done in [ 13 ] where expansions of the equilibrium Wigner functions were determined by using the Moyal formalism.…”

Quantum Energy-Transport and Drift-Diffusion Models for Electron Transport in Graphene An Approach by The Wigner Function

“…It can be obtained by using the Jaynes approach [14] of maximizing the entropy under suitable constraints on the expectation values, a crucial issue being the expression of the entropy in the quantum case. In [11,15] the standard prescription proposed by von Neumann has been adopted, which leads to a semiclassical limit represented by the Maxwell-Boltzmann distribution. That is, with this assumption, the zero-order term of the equilibrium Wigner function does not coincide with the Fermi-Dirac distribution but gives the Boltzmann-Maxwell low density limit.…”

“…In [1] a quantum hydrodynamic model for charge transport in graphene is derived from a moment expansion of the Wigner-Boltzmann equation and the needed closure relations are obtained by adding quantum corrections based on the equilibrium Wigner function to the semiclassical model formulated in [2][3][4][5][6] by exploiting the Maximum Entropy Principle. The expression of the equilibrium Wigner function which takes into account the form of the energy band of graphene has been obtained by solving the corresponding Bloch equation.…”