Hydrodynamic theory of an electron gas

Abstract

The generalised hydrodynamic theory of an electron gas, which does not rely on an assumption of a local equilibrium, is derived as the long-wave limit of a kinetic equation. Apart from the common hydrodynamics variables the theory includes the tensor fields of the higher moments of the distribution function. In contrast to the Bloch hydrodynamics, the theory leads to the correct plasmon dispersion and in the low frequency limit recovers the Navies-Stocks hydrodynamics. The linear approximation to the generalised hydrodynamics is closely related to the theory of highly viscous fluids.PACS numbers: 71.10. Ca, 05.20.Dd, 47.10.+g, 62.10.+s Hydrodynamic theory of an electron gas was heuristically introduced by Bloch in 1933 [1] as an extension of Thomas-Fermi model. Only macroscopic variableselectron density n(r, t), velocity v(r, t), pressure P , and electrostatic potential ϕ(r, t), enter the theory. The set of equations (continuity equation, Euler and Poisson equations) becomes complete when the equation of state is added, and in the original paper [1] Bloch identified P with the kinetic pressure of a degenerate Fermi gas. The Bloch's hydrodynamic theory (BHT) has been applied to variety of kinetic problems [2,3,4,5,6,7,8,9] with minor improvements (inclusion of exchange, correlation and quantum gradient corrections) [5,9].From the microscopic point of view BHT cannot be a fully consistent theory since it extends the collisiondominated hydrodynamics to the electron gas where the collisionless (Vlasov) limit is most common. For example, in the plasmon dispersion law ω 2 = ω 2 p + v 2 0 q 2 BHT predicts for degenerate electron gas v 2 0 = 1 3 v 2 F instead of a correct result 3 5 v 2 F (v F is the Fermi velocity) [4,6,7,8,11,12,13,14,15] . At arbitrary degeneracy the hydrodynamics gives v 2 0 = v 2 s , where v s is velocity of sound, whereas in the kinetic theory v 2 0 equals to the mean square of the particle velocity < v 2 p > [15]. It has been realized [11,12,13] that this discrepancy originates from the assumption of a local equilibrium , which underlies the common hydrodynamic theory [16]. The assumption allows to reduce the kinetic equation for the distribution function f p (r, t)to equations for macroscopic variables n(r, t),v(r, t) and, in general case, temperature T (r, t). The requirement of the local equilibrium is fulfilled if the characteristic time of the process τ ∼ 1/ω is much longer than the inverse collision frequency 1/ν and the typical length L is greater than the mean free path l ∼ u/ν (u is the average particleIn a zero oder with respect to parameters (2) Eq. (1) reduces to I p [f p ] = 0 which means that f p must have a locally equilibrium form. The stress tensor in the comoving (Lagrange) framebecomes diagonal P ij = P δ ij with P being the local pressure. As a result, the equations for the first three moments of the distribution function i.e. density n, current j = nv and stress tensor P ij , form the closed set of hydrodynamics equations for an ideal liquid. Due to the high frequency of ...

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