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Register a new userLong-time self-similar asymptotic of the macroscopic quantum models
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The unipolar and bipolar macroscopic quantum models derived recently for instance in the area of charge transport are considered in spatial one-dimensional whole space in the present paper. These models consist of nonlinear fourth-order parabolic equation for unipolar case or coupled nonlinear fourth-order parabolic system for bipolar case. We show for the first time the self-similarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a self-similar wave (which is not the exact solution of the models) in large time at an algebraic time-decay rate.
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We introduce the notion of perturbations of quantum stochastic models using the series product, and establish the asymptotic convergence of sequences of quantum stochastic models under the assumption that they are related via a right series product perturbation. While the perturbing models converge to the trivial model, we allow that the individual sequences may be divergent corresponding to large model parameter regimes that frequently occur in physical applications. This allows us to introduce the concept of asymptotically equivalent models, and we provide several examples where we replace one sequence of models with an equivalent one tailored to capture specific features. These examples include: a series product formulation of the principle of virtual work; essential commutativity of the noise in strong squeezing models; the decoupling of polarization channels in scattering by Faraday rotation driven by a strong laser field; and an application to quantum local asymptotic normality.
The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendental functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attention is paid to the possibility of extrapolating the behavior of functions, with arguments tending to infinity, from the related asymptotic series at small arguments. Applications of the method are thoroughly illustrated by the examples of several functions, nonlinear differential equations, and anharmonic models.
We consider semigroups ${alpha_t: ; tgeq 0}$ of normal, unital, completely positive maps $alpha_t$ on a von Neumann algebra ${mathcal M}$. The (predual) semigroup $ u_t (rho):= rho circ alpha_t$ on normal states $rho$ of $mathcal M$ leaves invariant the face ${mathcal F}_p:= {rho : ; rho (p)=1}$ supported by the projection $pin {mathcal M}$, if and only if $alpha_t(p)geq p$ (i.e., $p$ is sub-harmonic). We complete the arguments showing that the sub-harmonic projections form a complete lattice. We then consider $r_o$, the smallest projection which is larger than each support of a minimal invariant face; then $r_o$ is subharmonic. In finite dimensional cases $sup alpha_t(r_o)={bf 1}$ and $r_o$ is also the smallest projection $p$ for which $alpha_t(p)to {bf 1}$. If ${ u_t: ; tgeq 0}$ admits a faithful family of normal stationary states then $r_o={bf 1}$ is useless; if not, it helps to reduce the problem of the asymptotic behaviour of the semigroup for large times.
Starting from a particle model describing self-propelled particles interacting through nematic alignment, we derive a macroscopic model for the particle density and mean direction of motion. We first propose a mean-field kinetic model of the particle dynamics. After diffusive rescaling of the kinetic equation, we formally show that the distribution function converges to an equilibrium distribution in particle direction, whose local density and mean direction satisfies a cross-diffusion system. We show that the system is consistent with symmetries typical of a nematic material. The derivation is carried over by means of a Hilbert expansion. It requires the inversion of the linearized collision operator for which we show that the generalized collision invariants, a concept introduced to overcome the lack of momentum conservation of the system, plays a central role. This cross diffusion system poses many new challenging questions.
Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schrodinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system which admits several conserved quantities, which allow to construct explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs.
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