scholarly journals Convergence of a three-dimensional quantum lattice Boltzmann scheme towards solutions of the Dirac equation

2011 ◽  
Vol 369 (1944) ◽  
pp. 2155-2163 ◽  
Author(s):  
Denis Lapitski ◽  
Paul J. Dellar
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Lattice Boltzmann ◽  
Schrodinger Equation ◽  
Fourier Transforms ◽  
Operator Splitting ◽  
Three Dimensional ◽  
Relativistic Limit ◽  
Quantum Lattice ◽  
Lattice Boltzmann Scheme

We investigate the convergence properties of a three-dimensional quantum lattice Boltzmann scheme for the Dirac equation. These schemes were constructed as discretizations of the Dirac equation based on operator splitting to separate the streaming along the three coordinate axes, but their output has previously only been compared against solutions of the Schrödinger equation. The Schrödinger equation arises as the non-relativistic limit of the Dirac equation, describing solutions that vary slowly compared with the Compton frequency. We demonstrate first-order convergence towards solutions of the Dirac equation obtained by an independent numerical method based on fast Fourier transforms and matrix exponentiation.

ONE-DIMENSIONAL QUANTUM LATTICE BOLTZMANN SCHEME FOR THE NONLINEAR DIRAC EQUATION

2013 ◽  
Vol 24 (12) ◽  
pp. 1340001 ◽  
Author(s):  
SILVIA PALPACELLI ◽  
PAUL ROMATSCHKE ◽  
SAURO SUCCI
Keyword(s):  
Dirac Equation ◽  
Lattice Boltzmann ◽  
Spontaneous Breaking ◽  
Mass Generation ◽  
Quantum Lattice ◽  
One Dimensional ◽  
Dynamic Mass ◽  
Nonlinear Dirac Equation ◽  
Njl Model ◽  
Lattice Boltzmann Scheme

We develop a quantum lattice Boltzmann (QLB) scheme for the Dirac equation with a nonlinear fermion interaction provided by the Nambu–Jona-Lasinio (NJL) model. Numerical simulations in 1 + 1 space-time dimensions, provide evidence of dynamic mass generation, through spontaneous breaking of chiral symmetry.

Curvilinear coordinate lattice Boltzmann simulation for necklace-ring beams in the nonlinear Schrödinger equation

2020 ◽  
Vol 31 (10) ◽  
pp. 2050136
Author(s):  
Boyu Wang ◽  
Jianying Zhang ◽  
Guangwu Yan
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Lattice Boltzmann ◽  
Schrodinger Equation ◽  
Dimensional Space ◽  
Time Derivative ◽  
Three Dimensional ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Curvilinear Coordinate

Necklace-ring solitons have gained much attention due to their potential applications in optics and other scientific areas. In this paper, the numerical investigation of the nonlinear Schrödinger equation by using the curvilinear coordinate lattice Boltzmann method is proposed to study necklace-ring solitons. Different from those used in the general curvilinear coordinate lattice Boltzmann models, the lattices used in this work are uniform in two- and three-dimensional space. Furthermore, the model contains spatial evolution rather than time evolution to avoid the complexity of dealing with higher-order time derivative terms as well as to maintain the simplicity of the algorithm. Numerical experiments reproduce the evolution of two- and three-dimensional necklace-ring solitons. The truncation error analysis indicates that our model is equivalent to the Crank–Nicolson difference scheme.

scholarly journals Non-relativistic and relativistic scattering by short-range potentials

2011 ◽  
Vol 369 (1939) ◽  
pp. 1228-1244 ◽  
Author(s):  
H. Arnbak ◽  
P. L. Christiansen ◽  
Yu. B. Gaididei
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Short Range ◽  
Perturbation Technique ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Relativistic Limit ◽  
Scattering Coefficients ◽  
Relativistic Scattering

Relativistic and non-relativistic scattering by short-range potentials is investigated for selected problems. Scattering by the δ ′ potential in the Schrödinger equation and δ potentials in the Dirac equation must be solved by regularization, efficiently carried out by a perturbation technique involving a stretched variable. Asymmetric regularizations yield non-unique scattering coefficients. Resonant penetration through the potentials is found. Approximative Schrödinger equations in the non-relativistic limit are discussed in detail.

scholarly journals Iterants, Majorana Fermions and the Majorana-Dirac Equation

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1373
Author(s):  
Louis H. Kauffman
Keyword(s):  
Dirac Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Clifford Algebras ◽  
Discrete Systems ◽  
Complex Numbers ◽  
Majorana Fermions ◽  
Matrix Algebras ◽  
Dirac Equations ◽  
Points Of View

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

Isotropy of three-dimensional quantum lattice Boltzmann schemes

2011 ◽  
Vol 83 (4) ◽  
Author(s):  
P. J. Dellar ◽  
D. Lapitski ◽  
S. Palpacelli ◽  
S. Succi
Keyword(s):  
Lattice Boltzmann ◽  
Three Dimensional ◽  
Quantum Lattice
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