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Nonlinear Higher-Order Thermo-Hydrodynamics II: Illustrative Examples

The construction of a generalized (higher-order) nonlinear thermo-hydrodynamics, based on a nonequilibrium ensemble formalism has been presented in the preceding article. The working of such theory is illustrated in the present one. We consider here the case of two ideal classical fluids in interaction between them, and the nonlinear equations of evolution for the density (a hyperbolic one) and for the velocity field describing motion under flow are derived. Also, now at the quantum level, it is described the nonlinear transport in the fluid of electrons in doped polar semiconductors, and comparison with Monte Carlo calculations and with experimental data is done, obtaining very good agreement.


I. INTRODUCTION
In the preceding article (hereafter called as I, and when equations present in it are indicated in what follows, we write the number preceded by I) [1], it was described the construction of a generalized (higher-order) and nonlinear thermo-hydrodynamics, based on mechano-statistical foundations. In that way it was established a kind of unification of the microscopic dynamics and a mesoscopic hydrodynamics.
In this follow-up article we attempt to illustrate the functioning of the theory by applying it to a couple of somewhat simplified situations. Both cases are such that we can describe their underlying mechanics in terms of an individuals single-particle approximation. They are: 1. The case of two ideal classical fluids in interaction between them, one being the main system of interest driven away from equilibrium, and the other acting as a thermal reservoir; the associated nonlinear generalized hydrodynamics, however restricted to a study of a first-order one, is derived in section II. 2. A similar case, but now composed of two quantum gases, namely, conduction band electrons in a doped polar semiconductor under the action of an eletric field, and the phonon system acting as a thermal bath, is dealt with in section IV. Moreover, the study of section II is complemented in section III with the analysis of diffusion-advection motion governed by a nonlinear hydrodynamic equation.

II. TWO CLASSICAL FLUIDS IN INTERACTION
Let us consider two ideal classical fluids with interaction between them, with the Hamiltonian given by m being the mass of the particles in the system of interest, M that in the second system taking as a thermal bath at temperature T 0 , and the interaction being of the type of central forces. V ext stands for the interaction energy with an external source, and it is to be understood that the bath is constantly in equilibrium with an external ideal reservoir. We analize the hydrodynamics of this system in a first order approximation, meaning to take as basic variables of the system of interest the kinetic energy, the density, and the first flux (current) of the latter, with the associated dynamical quantities beinĝ with the understanding that all three are defined in Γ-phase space. The thermal bath is assumed to remain constantly in equilibrium with a reservoir at temperature T 0 , and the system subjected to external forces driving it out of equilibrium.
According to I (the previous article) the auxiliary ("instantaneously frozen") nonequilibrium statistical operator is given by [cf. Eq.(I. 6)] where ρ R is the canonical distribution at temperature T 0 of the thermal bath, and where, we recall, φ(t) ensures the normalization and we have introduced the set of nonequilibrium thermodynamic variables (Lagrange multipliers in the variational derivation of the formalism), namely and we also recall that the nonequilibrium statistical operator is given, in terms of the auxiliary one of Eq.
n(r, t) = dΓn(r)ρ(t, 0) , where the integration is over the phase space of the system [cf. Eq.(I.31)]. These equations, Eqs. (8) to (10), are the nonequilibrium thermodynamic equations of state, meaning that they relate the basic macrovariables on the left, with the nonequilibrium thermodynamic variables of Eq.(7) present on the right-hand side in the distributionρ of Eq.(6).
Equations (8) to (10) take the form where is a Boltzmann-style one-particle distribution function, and Performing the calculations we find that Moreover, we do find that h(r, t) = u(r, t) + n(r, t) for the energy, where u is the internal energy introducing, via the definition F −1 h = k B T * , the field of quasitemperature [2,3,4] or nonequilibrium kinetic temperature [5] T * (r, t), with the last term on the right of Eq.(18) being the drift-kinetic energy, and It can be noticed that if n is the concentration of particles, it needs be satisfied the constraint that with n(r, t) given by Eq. (12).
where I h is the flux of energy and J h is the collision integral; which is the conservation equation for the density; where I [2] n (r, t) is the flux of flux (second-order flux) of particles, namely with [p j p j ] standing for tensorial product of vectors, rendering a second-rank tensor, two of them, one introducing a term proportional to the gradient of concentration (and responsible for a perturbational modification of the diffusion effect; cf. Ref. [6]) and another introducing nonlocal effects (correlations in space), are neglected and we conserve the main contribution given by introducing the inverse of a second-rank tensor, Θ [2] n having dimension of time and playing the role of a tensorial-in-character Maxwell-relaxation time [4,7,8], its explicit expression given in Eq.(A3) of Appendix A.
Finally, to close the system of Eqs. (22) to (24) coupled to the set of Eqs. (16), (18) and (20) -the equations of state -we need to express the second-order flux of Eq. (25) in terms of the basic variables, and we obtain that m I [2] n (r, t) = n(r, t) k B T * (r, t) This second-order flux is related to the field of pressure tensor by after using Eq. (19), recovering the local and instantaneous usual form of the hydrostatic pressure.
This first-order hydrodynamics of this simple modeled system sets clearly in evidence the nonlinearity referred to in I, namely, one coming out of the expression for the second flux, cf.
Eq.(27), and other the higher nonlinear expression for the collision integrals. We also stress that in Eq.(26) Θ [2] n (r, t) is also highly nonlinear in the basic thermodynamic variables. In that way it has been explicitly shown that nonlinearities are always present in the generalized hydrodynamic equations, and we further illustrate the matter first, on the basis of the results above, deriving nonlinear equations of evolution for the density and the velocity in the fluid under flow, and after that a study of electric current in a system of mobile electrons in a doped polar semiconductor, dealt with at the quantum mechanical level.

III. NONLINEAR EQUATIONS OF EVOLUTION
Let us consider the equations of evolution for the basic variables of Eqs. (8) to (10), given by Eqs. (22), (23) and (24). Deriving in time Eq.(23) and using Eq.(24) it follows that where for simplicity we have taken the tensorial Maxwell characteristic time of Eq.(26) as a scalar (see Appendix A).
Equation (29) can be rewritten as after multiplying by Θ and the definition introducing v a which we call velocity responsible for creating a driven flow in the system.
Second, in order to close Eq.(30) we need to express the second-order flux in terms of the basic variables, n and I n , which after some calculus takes the form Using this Eq.(32) in Eq.(30) and, for further simplifying the presentation, let us take Θ n (r, t) as smoothly dependent on r and t, i.e. its gradient and time derivative can be neglected, we finally obtain that where G n contains gradients and time derivatives of Θ n (r, t) which, for simplicity we ignore in what follows, we have used that I n (r, t) = n(r, t) v(r, t), and the second contribution on the right-hand side -arising out of the contribution of the so-called convective pressure -can be written as + n(r, t)Λ [2] (r, t) ⊗Λ [2] (r, t) whereΛ [2] is the velocity-gradient tensor,Λ [2] = [∇v(r, t)].
Hence, the equation of evolution for the density, Eq. (3), is dependent on the velocity field and then it needs be coupled to the equation of evolution for the velocity, which is where the pressure tensor field is given by P [2] (r, t) = n(r, t) k B T * (r, t) 1 [2] .
Equation (33) is a nonlinear Maxwell-Cattaneo-like equation with sources. Moreover, in the strong condition of neglecting the right-hand side and the second derivative in time implying in a movement in conditions such that ω Θ n ≪ 1, we recover the standard Fick's diffusion equation Next let us introduce the effects of advection in the diffusive regime, i.e. the density follows the diffusion-advection equation of evolution, as given by Eq.(33), in the condition ω Θ n ≪ 1 (thus neglecting the second derivative in time), and for simplicity we illustrate the matter in the case of motion restricted to proceed in x-direction; hence we do have that Moreover, the diffusion coefficient has the standard expression of kinetic theory (in one where v th is the local and instantaneous thermal velocity Equation (38) can be reorganized in the following way: where that is, a diffusion coefficient whose standard kinetic theory expression is modified adding to the contributions of the thermal velocity the one of the drift velocity. This contribution together with the last three on the right-hand side of Eq.(39) have their origin in the second divergence of the convection pressure, and it can be noticed that all are nonlinear (quadratic) in v(x, t). The first term on the right of Eq.(39), we recall, is related to the divergence of the driving force producing the advective motion [cf. Eq.(31)].
Moreover, as already noticed, Eq.(39) needs be solved in conjunction with the equation of evolution for the velocity, in this case of motion under flow given by where we used that p(x, t) = n(x, t) k B T * (x, t).

IV. NONLINEAR QUANTUM TRANSPORT IN SEMICONDUCTORS
We consider the case of polar semiconductors described by a two-inverted-parabolic bands where, we recall, φ(t) ensures the normalization condition, and we have introduced the nonequilibrium thermodynamic variables interpreted as nonequilibrium quasitemperature, T * , quasi-chemical potential, µ, and drift velocity, v, of the electrons, and quasitemperature of the polar phonons, T * LO . Finally, ρ R is the statistical operator of the reservoir at temperature T 0 .
We call the basic macrovariables where N is constant in time, and we recall, the density is indicated by n, and the nonequilibrium thermodynamic equations of state are, after direct calculation, given by On the other hand the equations of evolution (see Appendix B) are for the energy of the electrons, for the electron linear momentum in the direction of the applied field, for the energy of the polar phonons; volume V has been taken as 1.
It is worth noticing that in quantities A 3 and A 4 [cf. Eqs.(58) and (59)] it can be put in evidence the quantity x 1/2 which is proportional to v, and therefore we can formally write Eq.(49) as where with after taking into account the initial condition v(0) = 0. Moreover, once the current density is given by using Eq.(61) we can write introducing a time-dependent (on the evolution of the nonequilibrium thermodynamic state of the system) Drude-type conductivity Next we illustrate numerically the matter considering the polar semiconductor GaAs and the strong polar ones and large gap GaN (of present interest for its use in blue diodes and lasers [14]). Figure 1  perature: It can be noticed a good agreement with the experimental data taken from three sources, namely, Refs. [16], [17] and [18].
Going over the particular case of the large-gap strongly polar GaN (in the cubic, i.e. zincblende, phase with n ≃ 10 18 cm −3 , and T 0 = 300K), we can see in Figure 3 the increase of the electron-drift velocity with the electric field intensity, and a good agreement can be noticed in a comparison with the NMD-Monte Carlo simulation, the dots, taken from Ref. [19].
Finally, in Figure 4 we The collision integral present on the right-hand side of Eq. (17), resulting from the collisions of the two types of particles via the potential V(|r j − R µ |) [cf. Eq.(1)] is given in the Markovian approximation [11,24,25,26] by the expression where A(r, q) = n(r, t) β(r, t) 2πm In Eqs.(A2) and (A3), V(q) is the Fourier transform of the interaction potential between the two types of particles, m r is the reduced mass, m −1 r = m −1 + M −1 , and N R is the number of particles in the thermal bath.
Introducing a shift in coordinate p, namely with β(r, t)1 [2] and performing the calculations we arrive at the result that J n (r, t) ≃ Θ [2] n (r, t) −1 I n (r, t) , where Θ is a rank-two tensor with dimensions of time, playing the role of a tensorial Maxwellrelaxation time [4,7,8], with its inverse given by with n R being the density of particles in the thermal bath.
We stress that this tensorial Maxwell-relaxation time depends on position and time through its dependence on the basic nonequilibrium thermodynamic variables β(r, t) and v(r, t): recalling that I n (r, t) = n(r, t) v(r, t), this time-relaxation-type contribution is not a linear one, but a highly nonlinear in v(r, t). In fact, performing the integration in Eq.(A8) we find that K ij = 0 for i = j, and there survive the diagonal terms with q 0 being a cut-off limit of the order of the inverse of the mean distance between particles of the system and the bath, γ is the incomplete Gamma function [12], and we can also alternatively write It may be noticed that we can write and the last term is then traceless, and accounting for anisotropic effects. Furthermore, using Eq.(A10) there follows that Conserving only this contribution in Eq.(A14), linearizing in v(r, t) all expressions, say, the case of a low density flux, assuming a good thermal contact between system and reservoir, such that β(r, t) ≃ β R , and small amplitude movement, namely n(r, t) ≃ n, we are left with a typical Maxwell-type contribution J n (r, t) ≃ −I n (r, t)/Θ n , where 1 Θ n = π 2 n R β The collision integrals are in this case given by whereÂ 1 ≡Ĥ e for energy andÂ 2 ≡ P for momentum, andĤ ′ is Fröhlich electron-phonon interaction; subindex nought indicates time dependence in interaction representation. These operators areĤ data (up triangle from Ref. [14], × from Ref. [17], empty circles from Ref. [18]).