Non parabolic band transport in semiconductors: closure of the production terms in the moment equations

Received June 11, 1999     

A modified Gaussian moment closure method for nonlinear stochastic differential equations

Parametric excitation of a nonlinear physical pendulum by modulation of its moment of inertia is analyzed in terms of physics as an example of the suggested approach. The modulation is provided by a redistribution of auxiliary masses. The system is investigated both analytically and with the help of computer simulations. The threshold and other characteristics of parametric resonance are found and discussed in detail. The role of nonlinear properties of the physical system in restricting the resonant swinging is emphasized. Phase locking between the drive and oscillations of the pendulum and the phenomenon of parametric autoresonance are investigated. The boundaries of parametric instability are determined as functions of the modulation depth and the quality factor. The feedback providing active optimal control of amplification and attenuation of oscillations is analyzed. An effective method of suppressing undesirable rotary oscillations of suspended constructions is suggested.     

Crucial properties of the moment closure model FENE-QE

We investigate four crucial properties for testing and evaluating a moment closure approximation of the FENE dumbbell model for dilute polymer solutions: non-negative configuration distribution function, energy dissipation, accuracy of approximation and computational expense. Through mathematical analysis, numerical experiments and comparisons with closure model FENE-P and FENE-YDL, we prove that the FENE-QE approximation has non-negative configuration distribution function, approximates the energy dissipation behavior of original kinetic theory and provides good accuracy. To improve the efficiency of this closure approximation, we introduce a piecewise linear approximation technique that greatly reduces the computational cost. This extension of FENE-QE, FENE-QE-PLA, is the closure model we recommend for simulating dilute polymer solutions.     

Stability in fully nonlinear parabolic equations

We study the stability of the null solution of a class of nonlinear evolution equations in Banach space. After stating a local existence result and the principle of linearized stability, we study the critical case, giving sufficient conditions for stability. The results are applied to second-order fully nonlinear parabolic equations in [0, + [ × R ⁿ .     

Determination of source parameter in parabolic equations

The authors consider the problem of finding u=u(x, t) and p=p(t) which satisfy u = Lu + p(t) + F(x, t, u, _x, p(t)) in Q _T=Ω×(0, T], u(x, 0)=ø(x), x∈Ω, u(x, t)=g(x, t) on ?Ω×(0, T] and either ∫_G(t) Φ(x,t)u(x,t)dx = E(t), 0 ? t ? T or u(x₀, t)=E(t), 0≤t≤T, where Ω?R ⁿ is a bounded domain with smooth boundary ?Ω, x ₀∈Ω, L is a linear elliptic operator, G(t)?Ω, and F, ø, g, and E are known functions. For each of the two problems stated above, we demonstrate the existence, unicity and continuous dependence upon the data. Some considerations on the numerical solution for these two inverse problems are presented with examples.     

Generalized heat-transport equations: parabolic and hyperbolic models

We derive two different generalized heat-transport equations: the most general one, of the first order in time and second order in space, encompasses some well-known heat equations and describes the hyperbolic regime in the absence of nonlocal effects. Another, less general, of the second order in time and fourth order in space, is able to describe hyperbolic heat conduction also in the presence of nonlocal effects. We investigate the thermodynamic compatibility of both models by applying some generalizations of the classical Liu and Coleman–Noll procedures. In both cases, constitutive equations for the entropy and for the entropy flux are obtained. For the second model, we consider a heat-transport equation which includes nonlocal terms and study the resulting set of balance laws, proving that the corresponding thermal perturbations propagate with finite speed.     

Bulk equations and Knudsen layers for the regularized 13 moment equations

The order of magnitude method offers an alternative to the Chapman-Enskog and Grad methods to derive macroscopic transport equations for rarefied gas flows. This method yields the regularized 13 moment equations (R13) and a generalization of Grad’s 13 moment equations for non-Maxwellian molecules. Both sets of equations are presented and discussed. Solutions of these systems of equations are considered for steady state Couette flow. The order of magnitude method is used to further reduce the generalized Grad equations to the non-linear bulk equations, which are of second order in the Knudsen number. Knudsen layers result from the linearized R13 equations, which are of the third order. Superpositions of bulk solutions and Knudsen layers show good agreement with DSMC calculations for Knudsen numbers up to 0.5.      

Symmetrization properties of parabolic equations in symmetric domains

Symmetry properties of positive solutions of a Dirichlet problem for a strongly nonlinear parabolic partial differential equation in a symmetric domainD R ⁿ are considered. It is assumed that the domainD and the equation are invariant with respect to a group {Q} of transformations ofD. In examples {Q} consists of reflections or rotations. The main result of the paper is the theorem which states that any compact inC(D) negatively invariant set which consists of positive functions consists ofQ-symmetric functions. Examples of negatively invariant sets are (in autonomous case) equilibrium points, omega-limit sets, alpha-limit sets, unstable sets of invariant sets, and global attractors. Application of the main theorem to equilibrium points gives the Gidas-Ni-Nirenberg theorem. Applying the theorem to omega-limit sets, we obtain the asymptotical symmetrization property. That means that a bounded solutionu(t) asr approaches subspace of symmetric functions. One more result concerns properties of eigenfunctions of linearizations of the equations at positive equilibrium points. It is proved that all unstable eigenfunctions are symmetric.     

Ergodicity of minimal sets in scalar parabolic equations

Skew product semiflowΠ _t :X ×Y → X × Y generated by $$\left\{ \begin{gathered} u_t = u_{xx} + f(y \cdot t,x,u,u_x ), t > 0 x \in (0,1), y \in Y, \hfill \\ D or N boundary conditions \hfill \\ \end{gathered} \right.$$ is considered, whereX is an appropriate subspace ofH ²(0, 1), (Y, ?) is a compact minimal flow. By analyzing the zero crossing number for certain invariant manifolds and the linearized spectrum, it is shown that a minimal setE?X × Y ofΠ, is uniquely ergodic if and only if (Y, ?) is uniquely ergodic andμ(Y ₀)=1, whereμ is the unique ergodic measure of (Y, ?),Y ₀={ity∈Y} Card(E∩P ^?1(y))=1},P:X × Y → Y is the natural projection (it was proved in an authors' earlier paper thatY ₀ is a residual subset ofY). Moreover, if (E, ?) is uniquely ergodic, then it is topologically conjugated to a subflow ofR ¹ ×Y. A consequence of the last result is the following: in the case that (Y, ?) is almost periodic,Π, is expected to have many purely almost automorphic motions which are not ergodic.     

Finite dimensional aspects of semilinear parabolic equations

We prove that the semiflow generated by _t ,u– _xx ² u=f(x, u), withu(0, t)= u(1,t)=0, onL ²([0, 1]) isC -conjugate to a product of finite dimensional flows.     

Boltzmann equation and moment equations in curvilinear coordinates

Various forms of writing the Boltzmann equation in an arbitrary orthogonal curvilinear coordinate system are discussed. The derivation is presented of a general transport equation and moment equations containing moments of the distribution function no higher than the fourth. For a gas of Maxwellian molecules it is shown that the system of moment equations for flows which differ little from equilibrium flows transforms into the system of hydrodynamic equations. The resulting equations may be useful in solving problems on motions of a rarefied gas by the moment methods. The results are valid for both the Boltzmann equation and model kinetic equations.The author wishes to thank A. A. Nikol'skii for discussions and helpful comments.     

Conditional moment closure and large-scale fluctuations of scalar dissipation

The accuracy of the method of calculating turbulent reacting flows [1–3] which is based on the use of the conditional mean concentrations of the reacting components is analyzed. The effect of large-scale fluctuations of the dissipation of passive admixture concentration [4–6] on the accuracy of the equations for the conditional means is considered and the corresponding corrections are computed. By means of numerical calculations the models for the conditional moment and traditional calculation methods are compared, as are the various approaches to the computation of the coefficients of the equations for the conditional means.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 50–59, September–October, 1993.The author wishes to express his wann thanks to A. B. Vatazhin for discussing the work and making useful comments.     

Maximum-entropy principle for nonlinear hydrodynamic transport in semiconductors

We present, in a strong nonlinear context, a full-band hydrodynamic approach by using the first 13 moments of the distribution function in the framework of extended thermodynamics. Following this approach we show that: (1) the full-band effects of the band structure are described accurately up to high electric fields both in homogeneous and nonhomogeneous conditions; (2) the effectiveness of the dissipation processes can be properly investigated, in homogeneous conditions, only in a strong nonlinear context; and (3) the hyperbolicity region of the system is very large, also in the nonlinear conditions. In this way, by using a strong nonlinear closure, it is possible to describe accurately the transport phenomena in submicron devices, when very high electric fields and field gradients occur (E ≈ 220 kV/cm, E/(dE/dx) ≈ 100 Å).     

Blow-up at the boundary for degenerate semilinear parabolic equations

This paper treats a superlinear parabolic equation, degenerate in the time derivative. It is shown that the solution may blow up in finite time. Moreover, it is proved that for a large class of initial data, blow-up occurs at the boundary of the domain when the nonlinearity is no worse than quadratic. Various estimates are obtained which determine the asymptotic behaviour near the blow-up. The mathematical analysis is then extended to equations with other degeneracies.     

Nonlinear singularly perturbed problems of ultra parabolic equations

A class of nonlinear singularly perturbed problem of ultra parabolic equations are considered. Using the comparison theorem, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.     

On uniqueness in inverse problems for semilinear parabolic equations

Communicated by R. V. Kohn