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 BSDEs with Logarithmic Growth Driven by Brownian Motion and Poisson Random Measure and Connection to Stochastic Control Problem

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid Oufdil
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Control Problem ◽  
Stochastic Optimal Control ◽  
Random Measure ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
Stochastic Optimal Control Problem ◽  
Uniqueness Of The Solution ◽  
Logarithmic Growth

Abstract In this paper, we study one-dimensional backward stochastic differential equations under logarithmic growth in the 𝑧-variable ( | z | ⁢ | ln ⁡ | z | | ) (\lvert z\rvert\sqrt{\lvert\ln\lvert z\rvert\rvert}) . We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the control problem.

Two extreme value processes arising in hydrology

1976 ◽  
Vol 13 (1) ◽  
pp. 190-194 ◽  
Author(s):  
Alan F. Karr
Keyword(s):  
Markov Process ◽  
Random Measure ◽  
Extreme Value ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
Flood Peak ◽  
Base Level ◽  
Strong Markov Process ◽  
Hydrological System ◽  
Strong Markov

Let Tn be the time of occurrence of the nth flood peak in a hydrological system and Xn the amount by which the peak exceeds a base level. We assume that ((Tn, Xn)) is a Poisson random measure with mean measure μ(dx) K(x, dy). In this note we characterize two extreme value processes which are functionals of ((Tn, Xn)). The set-parameterized process {MA} defined by MA = sup {Xn:Tn ∈ A} is additive and we compute its one-dimensional distributions explicitly. The process (Mt), where Mt = sup{Xn: Tn ≦ t}, is a non-homogeneous strong Markov process. Our results extend but computationally simplify those of previous models.

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.

The initiation of detonation from general non-uniformly distributed initial conditions

1995 ◽  
Vol 353 (1702) ◽  
pp. 173-203 ◽  
Keyword(s):  
Initial Velocity ◽  
High Speed ◽  
Mass Concentration ◽  
Initial Conditions ◽  
Fluid Particle ◽  
Time Dependent ◽  
Hot Gas ◽  
Velocity Pressure ◽  
Reaction Wave ◽  
Particle Paths

The chemical evolution of a hot gas subject to initial non-uniformities in velocity, pressure, temperature and reactant mass concentration is studied for moderate activation energies. It is demonstrated that such initial non-uniformities generate gradients in the distribution of chemical ignition times for each fluid particle, resulting in the creation of a high-speed, shockless reaction wave. If these gradients are sufficiently large, a transition from the high speed reaction wave to a strong detonation occurs. Time-dependent generalizations of the spontaneous flame concept, where the evolution of each fluid particle is determined by integration along particle paths only, are derived for initial velocity and pressure disturbances under the assumption of slowly varying initial disturbances. The unsteady structure of the high-speed reaction wave arising due to reaction from initial non-uniformities is investigated for the moderate activation energy used.

Two extreme value processes arising in hydrology

1976 ◽  
Vol 13 (01) ◽  
pp. 190-194 ◽  
Author(s):  
Alan F. Karr
Keyword(s):  
Markov Process ◽  
Random Measure ◽  
Extreme Value ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
Flood Peak ◽  
Base Level ◽  
Strong Markov Process ◽  
Hydrological System ◽  
Strong Markov

Let Tn be the time of occurrence of the nth flood peak in a hydrological system and Xn the amount by which the peak exceeds a base level. We assume that ((Tn , Xn )) is a Poisson random measure with mean measure μ(dx) K(x, dy). In this note we characterize two extreme value processes which are functionals of ((Tn , Xn )). The set-parameterized process {MA } defined by MA = sup {Xn :Tn ∈ A} is additive and we compute its one-dimensional distributions explicitly. The process (Mt ), where Mt = sup{Xn : Tn ≦ t}, is a non-homogeneous strong Markov process. Our results extend but computationally simplify those of previous models.

STOCHASTIC EQUATIONS OF PROCESSES WITH JUMPS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350006
Author(s):  
S. BOUHADOU ◽  
Y. OUKNINE
Keyword(s):  
Differential Equations ◽  
Random Measure ◽  
Stochastic Equations ◽  
Comparison Theorems ◽  
Pathwise Uniqueness ◽  
Poisson Random Measure ◽  
New Techniques ◽  
Approximation Approach ◽  
One Dimensional ◽  
White Noises

We consider one-dimensional stochastic differential equations driven by white noises and Poisson random measure. We introduce new techniques based on local time prove new results on pathwise uniqueness and comparison theorems. Our approach is very easy to handle and do not need any approximation approach. Similar equations without jumps were studied in the same context by [8, 12] and other authors.

Predictable solution for reflected BSDEs when the obstacle is not right-continuous

2020 ◽  
Vol 28 (4) ◽  
pp. 269-279
Author(s):  
Mohamed Marzougue ◽  
Mohamed El Otmani
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Random Measure ◽  
Backward Stochastic Differential Equations ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
General Filtration ◽  
Reflected Bsdes

AbstractIn the present paper, we consider reflected backward stochastic differential equations when the reflecting obstacle is not necessarily right-continuous in a general filtration that supports a one-dimensional Brownian motion and an independent Poisson random measure. We prove the existence and uniqueness of a predictable solution for such equations under the stochastic Lipschitz coefficient by using the predictable Mertens decomposition.

Stabilization of Dominant Structures in an Ionic Reaction-Diffusion System

1998 ◽  
Vol 63 (6) ◽  
pp. 761-769 ◽  
Author(s):  
Roland Krämer ◽  
Arno F. Münster
Keyword(s):  
Reaction Diffusion ◽  
Dominant Mode ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Perturbation ◽  
Dimensional System ◽  
One Dimensional ◽  
Brusselator Model ◽  
The One ◽  
Dominant Structure

We describe a method of stabilizing the dominant structure in a chaotic reaction-diffusion system, where the underlying nonlinear dynamics needs not to be known. The dominant mode is identified by the Karhunen-Loeve decomposition, also known as orthogonal decomposition. Using a ionic version of the Brusselator model in a spatially one-dimensional system, our control strategy is based on perturbations derived from the amplitude function of the dominant spatial mode. The perturbation is used in two different ways: A global perturbation is realized by forcing an electric current through the one-dimensional system, whereas the local perturbation is performed by modulating concentrations of the autocatalyst at the boundaries. Only the global method enhances the contribution of the dominant mode to the total fluctuation energy. On the other hand, the local method leads to simple bulk oscillation of the entire system.

scholarly journals Topological phase transition between a normal insulator and a topological metal state in a quasi-one-dimensional system

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Milad Jangjan ◽  
Mir Vahid Hosseini
Keyword(s):  
Phase Transition ◽  
Dimensional System ◽  
Inversion Symmetry ◽  
Topological Phase ◽  
Edge States ◽  
Zero Energy ◽  
One Dimensional ◽  
Topological Phase Transition ◽  
Metal State ◽  
Topological Metal

AbstractWe theoretically report the finding of a new kind of topological phase transition between a normal insulator and a topological metal state where the closing-reopening of bandgap is accompanied by passing the Fermi level through an additional band. The resulting nontrivial topological metal phase is characterized by stable zero-energy localized edge states that exist within the full gapless bulk states. Such states living on a quasi-one-dimensional system with three sublattices per unit cell are protected by hidden inversion symmetry. While other required symmetries such as chiral, particle-hole, or full inversion symmetry are absent in the system.

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