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.
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We are working with dimensionless Wigner functions and the constant 1/(2πℏ)2 is necessary in order to compute physical moments.4
33.
Although we have used the same notation for ν⊥ and
${\bm w}_\perp$
, the former denotes a rotated unit vector, the latter an orthogonal projection.34.
It is necessary to distinguish three cases: when both the level lines n = nm and n = nM are in the region A < −B, when both are in the region A > −B, and when n = nm is in the first region while n = nM is in the second one. Note, in fact, that the level lines of n cannot cross (asymptotically) the critical line A = −B.
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