Hydrodynamic equations for electrons in graphene obtained from the maximum entropy principle

Abstract

The maximum entropy principle is applied to the formal derivation of isothermal, Euler-like equations for semiclassical fermions (electrons and holes) in graphene. After proving general mathematical properties of the equations so obtained, their asymptotic form corresponding to significant physical regimes is investigated. In particular, the diffusive regime, the Maxwell-Boltzmann regime (high temperature), the collimation regime and the degenerate gas limit (vanishing temperature) are considered. C 2014 AIP Publishing LLC. [http://dx.

Particle Dynamics in Graphene: Collimated Beam Limit

“…According to the maxi mum entropy principle, in Barletti (2014) it is proven that in the hydrodynamic limit r -> 0 we have f± -*■ /±q> and w± -> 0 (9) and then we obtain {dt+cv-V*±F-Vp) /±q = 0.…”

“…Then, the problem of giving the hydrodjmamic model (11), (15), (16), and (17) an explicit form is reduced to the problem of computing P, Pi, Q, and Qx as functions of n and \u\, through system (20). Even though this program cannot be carried out in general, nevertheless special regimes (of physical relevance) exist where the explicit expressions can be obtained (Barletti, 2014). In next section one of these cases will be examined, namely the regime of high temperatures and, more specifically, the case of a collimated beam.…”

“…which implies that, from a semiclassical viewpoint, electrons move at constant speed c. In this article we deal with a hydrodynamic model for electron transport in graphene derived in Barletti (2014), focusing in particular on the equations for the "collimated beam" limit.…”

“…In Section 2 we summarize the derivation, obtained in Barletti (2014), of a hydrodynamic system describing a population of electrons or holes in graphene. We derive the mathematical formulation of the problem in the framework of the kinetic description of the quantum particle dynamics, rep resented by the Wigner equations endowed with a BGK (Bhatnagar-GrossKrook;Bhatnagar, Gross, and Krook, 1954) relaxation term.…”

“…We summarize here the derivation of the hydrodynamic model for graphene (more details can be found in Barletti, 2014). Let us consider the transport equations for a statistical population of electrons in graphene.…”