scholarly journals Aspects of the kinetic equations for a special one-dimensional system

1979 ◽  
Vol 21 (2) ◽  
pp. 145-175
Author(s):  
J. W. Evans
Keyword(s):  
Initial Conditions ◽  
Asymptotic Properties ◽  
Kinetic Equations ◽  
Continuous Distribution ◽  
Hard Core ◽  
Delta Function ◽  
Distribution Functions ◽  
Periodic Case ◽  
Dimensional System ◽  
One Dimensional

AbstractSome initial value problems are considered which arise in the treatment of a one-dimensional gas of point particles interacting with a “hard-core” potential.Two basic types of initial conditions are considered. For the first, one particle is specified to be at the origin with a given velocity. The positions in phase space of the remaining background of particles are represented by continuous distribution functions. The second problem is a periodic analogue of the first.Exact equations for the delta-function part of the single particle distribution functions are derived for the non-periodic case and approximate equations for the periodic case. These take the form of differential operator equations. The spectral and asymptotic properties of the operators associated with the two cases are examined and compared. The behaviour of the solutions is also considered.

On the stability of lattice boltzmann equations for one-dimensional diffusion equation

2017 ◽  
Vol 08 (01) ◽  
pp. 1750013 ◽  
Author(s):  
Gerasim Vladimirovich Krivovichev
Keyword(s):  
Lattice Boltzmann ◽  
Initial Conditions ◽  
Kinetic Equations ◽  
Differential Approximation ◽  
Boltzmann Equations ◽  
Von Neumann ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
The Cauchy Problem ◽  
Lattice Boltzmann Equations

Stability analysis of lattice Boltzmann equations (LBEs) on initial conditions for one-dimensional diffusion is performed. Stability of the solution of the Cauchy problem for the system of linear Bhatnaghar–Gross–Krook kinetic equations is demonstrated for the cases of D1Q2 and D1Q3 lattices. Stability of the scheme for D1Q2 lattice is analytically analyzed by the method of differential approximation. Stability of parametrical scheme is numerically investigated by von Neumann method in parameter space. As a result of numerical analysis, the correction of the hypothesis on transfer of stability conditions of the scheme for macroequation to the system of LBEs is demonstrated.

On the dynamics of the Tent function - Phase diagrams

2016 ◽  
Vol 7 (4) ◽  
pp. 261
Author(s):  
Prince Amponsah Kwabi ◽  
William Obeng Denteh ◽  
Richard Kena Boadi
Keyword(s):  
Dynamical Systems ◽  
Phase Diagrams ◽  
Initial Conditions ◽  
Asymptotic Properties ◽  
Topological Dynamical System ◽  
Topological Transitivity ◽  
Tent Map ◽  
One Dimensional ◽  
Definition Of ◽  
Topological Dynamical Systems

This paper focuses on the study of a one-dimensional topological dynamical system, the tent function. We give a good background to the theory of dynamical systems while establishing the important asymptotic properties of topological dynamical systems that distinguishes it from other fields by way of an example - the tent function. A precise definition of the tent function is given and iterates are clearly shown using the phase diagrams. By this same method, chaos in the tent map is shown as iterates become higher. We also show that the tent map has infinitely many chaotic orbits and also express some important features of chaos such as topological transitivity, boundedness and sensitivity to change in initial conditions from a topological viewpoint.

Thermodynamic properties of a finite one-dimensional system of bosons with repulsive delta-function interaction

1995 ◽  
Vol 73 (3-4) ◽  
pp. 245-247
Author(s):  
K. L. Poon ◽  
K. Young ◽  
D. Kiang
Keyword(s):  
Thermodynamic Properties ◽  
Thermodynamic Limit ◽  
General Model ◽  
Finite Size ◽  
Delta Function ◽  
One Dimension ◽  
Dimensional System ◽  
One Dimensional ◽  
Finite Size Corrections

The thermodynamics of N bosons in a length L in one dimension, with repulsive delta-function interaction, is studied numerically for finite N, L. The results show the nature of finite-size corrections and how the thermodynamic limit is approached, and hopefully will be of some guidance in seeking the solution of a more general model.

Stability of one-dimensional systems of colliding particles

1976 ◽  
Vol 13 (1) ◽  
pp. 155-158
Author(s):  
Alan F. Karr
Keyword(s):  
Initial Velocity ◽  
Initial Conditions ◽  
Random Measure ◽  
Initial Position ◽  
Dimensional System ◽  
Poisson Random Measure ◽  
Initial Particle ◽  
One Dimensional ◽  
Stability Theorems ◽  
Particle Paths

Envision a one-dimensional system of infinitely many identical particles, in which initial particle positions constitute a Poisson random measure and the initial velocity of a particle depends only on its initial position. Given its initial conditions the system evolves deterministically, by means of perfectly elastic collisions. In this note we derive conditions for continuity of the probability laws of the system and of the particle paths, as functions of the parameters of the initial conditions. These results have the physical interpretation of stability theorems.

Stability of one-dimensional systems of colliding particles

1976 ◽  
Vol 13 (01) ◽  
pp. 155-158
Author(s):  
Alan F. Karr
Keyword(s):  
Initial Velocity ◽  
Initial Conditions ◽  
Random Measure ◽  
Initial Position ◽  
Dimensional System ◽  
Poisson Random Measure ◽  
Initial Particle ◽  
One Dimensional ◽  
Stability Theorems ◽  
Particle Paths

Envision a one-dimensional system of infinitely many identical particles, in which initial particle positions constitute a Poisson random measure and the initial velocity of a particle depends only on its initial position. Given its initial conditions the system evolves deterministically, by means of perfectly elastic collisions. In this note we derive conditions for continuity of the probability laws of the system and of the particle paths, as functions of the parameters of the initial conditions. These results have the physical interpretation of stability theorems.

scholarly journals One-particle spectral densities and phase diagrams of one-dimensional proton conductors

2021 ◽  
Vol 24 (2) ◽  
pp. 23704
Author(s):  
R. Ya. Stetsiv
Keyword(s):  
Hydrogen Bonds ◽  
Phase Diagrams ◽  
Short Range ◽  
Hard Core ◽  
Proton Conductors ◽  
Dimensional System ◽  
One Dimensional ◽  
Anharmonic Potential ◽  
Diagonalization Method ◽  
Exact Diagonalization Method

The equilibrium states of one-dimensional proton conductors in the systems with hydrogen bonds are investigated. Our extended hard-core boson lattice model includes short-range interactions between hydrogen ions, their transfer along the hydrogen bonds with two-minima local anharmonic potential, as well as their inter-bond hopping, and the modulating field is taken into account. The exact diagonalization method for finite one-dimensional system with periodic boundary conditions is used. The existence of various phases of the system at T = 0, depending on the values of short-range interactions between particles and the modulating field strength, is established by analyzing the character of the obtained frequency dependence of one-particle spectral density; the phase diagrams are built.

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