scholarly journals Derivation of regularized Grad's moment system from kinetic equations: modes, ghosts and non-Markov fluxes

2018 ◽  
Vol 376 (2118) ◽  
pp. 20170230 ◽  
Author(s):  
Ilya Karlin
Keyword(s):  
Distribution Function ◽  
Kinetic Equations ◽  
Mode Structure ◽  
Theme Issue ◽  
Dynamic Correction ◽  
Intrinsic Structure ◽  
Nonlinear Modes ◽  
Moment System ◽  
Non Local ◽  
Convective Fluxes

Derivation of the dynamic correction to Grad’s moment system from kinetic equations (regularized Grad’s 13 moment system, or R13) is revisited. The R13 distribution function is found as a superposition of eight modes. Three primary modes, known from the previous derivation (Karlin et al. 1998 Phys. Rev. E 57 , 1668–1672. ( doi:10.1103/PhysRevE.57.1668 )), are extended into the nonlinear parameter domain. Three essentially nonlinear modes are identified, and two ghost modes which do not contribute to the R13 fluxes are revealed. The eight-mode structure of the R13 distribution function implies partition of R13 fluxes into two types of contributions: dissipative fluxes (both linear and nonlinear) and nonlinear streamline convective fluxes. Physical interpretation of the latter non-dissipative and non-local in time effect is discussed. A non-perturbative R13-type solution is demonstrated for a simple Lorentz scattering kinetic model. The results of this study clarify the intrinsic structure of the R13 system. This article is part of the theme issue ‘Hilbert’s sixth problem’.

scholarly journals Complex partial synchronization patterns in networks of delay-coupled neurons

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180128 ◽  
Author(s):  
D. Nikitin ◽  
I. Omelchenko ◽  
A. Zakharova ◽  
M. Avetyan ◽  
A. L. Fradkov ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Temporal Dynamics ◽  
Delay Systems ◽  
Theme Issue ◽  
Partial Synchronization ◽  
Multiplex Network ◽  
Control Scheme ◽  
Non Local ◽  
Spatio Temporal ◽  
Fast Oscillations

We study the spatio-temporal dynamics of a multiplex network of delay-coupled FitzHugh–Nagumo oscillators with non-local and fractal connectivities. Apart from chimera states, a new regime of coexistence of slow and fast oscillations is found. An analytical explanation for the emergence of such coexisting partial synchronization patterns is given. Furthermore, we propose a control scheme for the number of fast and slow neurons in each layer. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

scholarly journals Boltzmann Equation Theory of Charged Particle Transport in Neutral Gases: Perturbation Treatment

1999 ◽  
Vol 52 (6) ◽  
pp. 999 ◽  
Author(s):  
Slobodan B. Vrhovac ◽  
Zoran Lj. Petrovic
Keyword(s):  
Distribution Function ◽  
Perturbation Theory ◽  
Charged Particle ◽  
Boltzmann Equation ◽  
Particle Transport ◽  
Kinetic Equations ◽  
Initial Distribution ◽  
Formal Structure ◽  
Time Dependent ◽  
Charged Particle Transport

This paper examines the formal structure of the Boltzmann equation (BE) theory of charged particle transport in neutral gases. The initial value problem of the BE is studied by using perturbation theory generalised to non-Hermitian operators. The method developed by R�sibois was generalised in order to be applied for the derivation of the transport coecients of swarms of charged particles in gases. We reveal which intrinsic properties of the operators occurring in the kinetic equation are sucient for the generalised diffusion equation (GDE) and the density gradient expansion to be valid. Explicit expressions for transport coecients from the (asymmetric) eigenvalue problem are also deduced. We demonstrate the equivalence between these microscopic expressions and the hierarchy of kinetic equations. The establishment of the hydrodynamic regime is further analysed by using the time-dependent perturbation theory. We prove that for times t ? τ0 (τ0 is the relaxation time), the one-particle distribution function of swarm particles can be transformed into hydrodynamic form. Introducing time-dependent transport coecients ? *(p) (?q,t), which can be related to various Fourier components of the initial distribution function, we also show that for the long-time limit all ? *(p) (?q,t) become time and ?q independent in the same characteristic time and achieve their hydrodynamic values.

scholarly journals Generalized Fock space and contextuality

2019 ◽  
Vol 377 (2157) ◽  
pp. 20190096 ◽  
Author(s):  
Sergey Rashkovskiy ◽  
Andrei Khrennikov
Keyword(s):  
Quantum Mechanics ◽  
Linear Space ◽  
Wide Class ◽  
Fock Space ◽  
Kinetic Equations ◽  
Interference Term ◽  
Context Dependence ◽  
Total Probability ◽  
Theme Issue ◽  
Formula Of Total Probability

This paper is devoted to linear space representations of contextual probabilities—in generalized Fock space. This gives the possibility to use the calculus of creation and annihilation operators to express probabilistic dynamics in the Fock space (in particular, the wide class of classical kinetic equations). In this way, we reproduce the Doi–Peliti formalism. The context-dependence of probabilities can be quantified with the aid of the generalized formula of total probability—by the magnitude of the interference term. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond'.

The effects of magnetic island on toroidal ion temperature gradient mode instability

2021 ◽  
Author(s):  
Guodong Zhang ◽  
Weixin Guo ◽  
Lu Wang
Keyword(s):  
Growth Rate ◽  
Temperature Gradient ◽  
Kinetic Equations ◽  
Plasma Pressure ◽  
Linear Growth Rate ◽  
Ion Temperature ◽  
Magnetic Island ◽  
Mode Structure ◽  
Ion Temperature Gradient ◽  
Eigen Value

Abstract In this work, we have investigated the influences of magnetic island (MI) on electrostatic toroidal ion temperature gradient (ITG) mode, where the ions are described by gyro-kinetic equations including MI, and adiabatic approximation is used for electrons. The eigen-equation for short-wavelength toroidal ITG mode in Fourier-ballooning representation is derived, and the corresponding eigen-value as well as mode structure are solved. Both the flattening effects of MI on plasma pressure and MI-scale shear flow are considered. It is found that when only considering the flattening effects of MI, ITG mode can be stabilized as compared to the case without MI. While, the effective drive of toroidal ITG mode could be enhanced by including MI-scale flow, which indicates the dominant destabilizing by MI-scale flow over the stabilizing by flattening profile and results in higher growth rate than the case without MI. It is also found that the total flow shearing may prevent the ITG turbulence spreading from X-point of MI but not strong enough to prevent spreading from the seperatrix across O-point of larger MI via comparison between the flow shearing rate and the linear growth rate. Furthermore, the corresponding width of lowest-order mode structure in ballooning angle is slightly widened (narrowed) for the case without (with) MI-scale flow, as compared to the case without MI. Besides, the shifted even symmetry in ballooning angle is not qualitatively influenced by the presence of MI. The mode structure is radially asymmetric, but is symmetric with respect to the phase of MI at the O-point.

QUANTUM TRANSPORT IN SEMICONDUCTOR SUPERLATTICES BEYOND THE KADANOFF–BAYM ANSATZ

2001 ◽  
Vol 15 (31) ◽  
pp. 4123-4143 ◽  
Author(s):  
P. KLEINERT ◽  
V. V. BRYKSIN
Keyword(s):  
Distribution Function ◽  
Quantum Transport ◽  
Transport Model ◽  
Kinetic Equations ◽  
Nonlinear Transport ◽  
Carrier Distribution ◽  
Semiconductor Superlattices ◽  
Symmetry Properties ◽  
Scattering Strength ◽  
Green Function Approach

Quantum transport in semiconductor superlattices subject to quantizing parallel electric and magnetic fields is studied based on the Kadanoff–Baym–Keldysh double-time Green function approach. Exploiting the symmetry properties of the underlying Hamiltonian, coupled kinetic equations are derived and analytically solved for the density-of-states and the carrier distribution function. Scattering giving rise to collisional broadening plays an important role in our transport model, whose unperturbed eigenstates are completely discrete due to Wannier–Stark and Landau quantization. It is shown that a correct description of the stationary quantum transport in superlattices with field-induced localized eigenstates requires the determination of a time-dependent distribution function from a kinetic equation, which emerges beyond the Kadanoff–Baym Ansatz. Depending on the scattering strength, gaps are predicted to occur in the electric and magnetic field dependence of the current density. The rigorous quantum-mechanical approach reveals the hopping nature of the nonlinear transport in narrow miniband superlattices. This is compared with results obtained recently within the density-matrix approach.

Numerical Study of Nonequilibrium Diagram Theory

1972 ◽  
Vol 50 (4) ◽  
pp. 317-335 ◽  
Author(s):  
Gary R. Dowling ◽  
H. Ted Davis
Keyword(s):  
Distribution Function ◽  
Correlation Function ◽  
Kinetic Equation ◽  
Numerical Study ◽  
Pair Correlation ◽  
Transport Coefficients ◽  
Kinetic Equations ◽  
Collision Operator ◽  
Dense Fluid ◽  
Equilibrium Pair

In this paper we numerically analyze the first few diagrams in a Boltzmann-like collision operator that occurs in Severne's exact kinetic equation for the singlet distribution function. A similar analysis was used by Allen and Cole in deriving their singlet and doublet kinetic equations. Our analysis shows that the diagrams neglected by Allen and Cole in their kinetic equations are not negligible and these should be incorporated into dense fluid theories. The Allen–Cole kinetic transport coefficients and equilibrium pair correlation function are presented and calculated for dense argon. These results are not promising.

scholarly journals The use of an adaptive mesh based on a quadtree for modeling the final state of a quantum field system under pulsed external action

2020 ◽  
Vol 11 (1) ◽  
pp. 93-105
Author(s):  
Anatolii Dmitrievich Panferov ◽  
Alexey Vladimirovich Makhankov ◽  
Alexander Vladimirovich Trunov
Keyword(s):  
Distribution Function ◽  
Momentum Space ◽  
Kinetic Equations ◽  
Adaptive Mesh ◽  
External Action ◽  
Uniform Mesh ◽  
Quantum Field ◽  
Residual Distribution ◽  
Final State ◽  
High Computational Complexity

The success of using mathematical models that determine the behavior of quantum field systems in parametric spaces critically depends on the level of optimization of the procedure of finding the solution. The paper considers the problem of calculating the density of carriers arising in graphene as a result of the action of a pulsed electric field. The basis of the model is a system of kinetic equations that provide the calculation of the residual distribution function. Its integration over momentum space gives the desired carrier density. The problem lies in the high computational complexity of covering the momentum space with a uniform mesh, which provides an accurate calculation of the density for various parameters of the field momentum. Moreover, the model does not contain criteria for determining satisfactory mesh parameters. The article proposes and implements a procedure for constructing an adaptive mesh in the form of a quadtree having a variable size of covering squares. The procedure is iterative and combined with the process of calculating the values of the distribution function.

scholarly journals A criterion for asymptotic preserving schemes of kinetic equations to be uniformly stationary preserving

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Casimir Emako ◽  
Farah Kanbar ◽  
Christian Klingenberg ◽  
Min Tang
Keyword(s):  
Distribution Function ◽  
Kinetic Models ◽  
Kinetic Equations ◽  
Asymptotic Preserving ◽  
Asymptotic Preserving Schemes

<p style='text-indent:20px;'>In this work we are interested in the stationary preserving property of asymptotic preserving (AP) schemes for kinetic models. We introduce a criterion for AP schemes for kinetic equations to be uniformly stationary preserving (SP). Our key observation is that as long as the Maxwellian of the distribution function can be updated explicitly, such AP schemes are also SP. To illustrate our observation, three different AP schemes for three different kinetic models are considered. Their SP property is proved analytically and tested numerically, which confirms our observations.</p>

Distribution function and diffusion of α-particles in DT fusion plasma

1984 ◽  
Vol 31 (3) ◽  
pp. 369-380 ◽  
Author(s):  
M. A. Liberman ◽  
A. L. Velikovich
Keyword(s):  
Distribution Function ◽  
Energy Transport ◽  
Plasma Heating ◽  
Particle Energy ◽  
Fusion Plasma ◽  
Special Cases ◽  
Critical Ignition ◽  
Non Local ◽  
And Diffusion ◽  
Α Particles

In the Fokker-Planck approximation, a general solution of the kinetic equation for the distribution function of non-thermal α-particles in spatially uniform DT plasma is obtained. The previously published solutions are then derived as special cases. Transport equations describing diffusion of non-thermal α-particles are obtained and diffusion coefficients are calculated. To compare the hydro-dynamic and kinetic descriptions of non-thermal α-particle energy transport, the critical ignition length of the overheating instability in DT fusion plasma, determined by non-local plasma heating due to α-particle diffusion, is calculated using both approaches. The two results are in good agreement.

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