The one-dimensional cyclic cellular automaton: A system with deterministic dynamics that emulates an interacting particle system with stochastic dynamics

1990 ◽  
Vol 3 (2) ◽  
pp. 311-338 ◽  
Author(s):  
Robert Fisch
Keyword(s):  
Cellular Automaton ◽  
Stochastic Dynamics ◽  
Particle System ◽  
Interacting Particle System ◽  
One Dimensional ◽  
Deterministic Dynamics ◽  
Interacting Particle ◽  
The One

The annihilating process on random trees and the square lattice

2004 ◽  
Vol 41 (3) ◽  
pp. 816-831
Author(s):  
Aidan Sudbury
Keyword(s):  
Survival Probability ◽  
Particle System ◽  
Random Trees ◽  
Interacting Particle System ◽  
Random Tree ◽  
Square Lattice ◽  
One Dimensional ◽  
Interacting Particle

An annihilating process is an interacting particle system in which the only interaction is that a particle may kill a neighbouring particle. Since there is no birth and no movement, once a particle has no neighbours its site remains occupied for ever. The survival probability is calculated for a random tree and for the square lattice. A connection is made between annihilating processes and the adsorption of molecules onto surfaces. A one-dimensional adsorption problem is solved in the case in which the two neighbours do not act independently.

The annihilating process on random trees and the square lattice

2004 ◽  
Vol 41 (03) ◽  
pp. 816-831
Author(s):  
Aidan Sudbury
Keyword(s):  
Survival Probability ◽  
Particle System ◽  
Random Trees ◽  
Interacting Particle System ◽  
Random Tree ◽  
Square Lattice ◽  
One Dimensional ◽  
Interacting Particle

An annihilating process is an interacting particle system in which the only interaction is that a particle may kill a neighbouring particle. Since there is no birth and no movement, once a particle has no neighbours its site remains occupied for ever. The survival probability is calculated for a random tree and for the square lattice. A connection is made between annihilating processes and the adsorption of molecules onto surfaces. A one-dimensional adsorption problem is solved in the case in which the two neighbours do not act independently.

scholarly journals A spatial model for selection and cooperation

2017 ◽  
Vol 54 (2) ◽  
pp. 522-539
Author(s):  
Peter Czuppon ◽  
Peter Pfaffelhuber
Keyword(s):  
Spatial Model ◽  
Nearest Neighbor ◽  
Particle System ◽  
Winning Strategy ◽  
The Other ◽  
Evolution Of Cooperation ◽  
Special Choice ◽  
One Dimensional ◽  
Interacting Particle ◽  
The One

Abstract We study the evolution of cooperation in an interacting particle system with two types. The model we investigate is an extension of a two-type biased voter model. One type (called defector) has a (positive) bias α with respect to the other type (called cooperator). However, a cooperator helps a neighbor (either defector or cooperator) to reproduce at rate γ. We prove that the one-dimensional nearest-neighbor interacting dynamical system exhibits a phase transition at α = γ. A special choice of interaction kernels yield that for α > γ cooperators always die out, but if γ > α, cooperation is the winning strategy.

scholarly journals A Local Barycentric Version of the Bak–Sneppen Model

2021 ◽  
Vol 182 (2) ◽  
Author(s):  
Philip Kennerberg ◽  
Stanislav Volkov
Keyword(s):  
Real Number ◽  
Uniform Distribution ◽  
Particle System ◽  
Interacting Particle System ◽  
Interacting Particle

AbstractWe study the behaviour of an interacting particle system, related to the Bak–Sneppen model and Jante’s law process defined in Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018). Let $$N\ge 3$$ N ≥ 3 vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called fitness (we use this term, as it is quite standard for Bak–Sneppen models). Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution $$\zeta $$ ζ . We show that in case where $$\zeta $$ ζ is a finitely supported or continuous uniform distribution, all the fitnesses except one converge to the same value.

scholarly journals Hydrodynamic limit for a nongradient interacting particle system with stochastic reservoirs

2000 ◽  
Vol 45 (4) ◽  
pp. 694-717 ◽  
Author(s):  
Claudio Landim ◽  
Claudio Landim ◽  
Claudio Landim ◽  
Claudio Landim ◽  
Mustapha Mourragui ◽  
...  
Keyword(s):  
Hydrodynamic Limit ◽  
Particle System ◽  
Interacting Particle System ◽  
Interacting Particle

Segregation in an interacting particle system

2000 ◽  
Vol 271 (1-2) ◽  
pp. 92-99 ◽  
Author(s):  
Kei-ichi Tainaka ◽  
Nariyuki Nakagiri
Keyword(s):  
Particle System ◽  
Interacting Particle System ◽  
Interacting Particle

scholarly journals A multitype contact process with frozen sites: a spatial model of allelopathy

2005 ◽  
Vol 42 (04) ◽  
pp. 1109-1119
Author(s):  
Nicolas Lanchier
Keyword(s):  
Phase Transitions ◽  
Spatial Model ◽  
Biological Process ◽  
Particle System ◽  
Contact Process ◽  
Interacting Particle System ◽  
Interacting Particle

In this paper, we introduce a generalization of the two-color multitype contact process intended to mimic a biological process called allelopathy. To be precise, we have two types of particle. Particles of each type give birth to particles of the same type, and die at rate 1. When a particle of type 1 dies, it gives way to a frozen site that blocks particles of type 2 for an exponentially distributed amount of time. Specifically, we investigate in detail the phase transitions and the duality properties of the interacting particle system.

A coupling of finite particle systems

1993 ◽  
Vol 30 (01) ◽  
pp. 258-262 ◽  
Author(s):  
T. S. Mountford
Keyword(s):  
Invariant Measure ◽  
Branching Process ◽  
Interacting Particle Systems ◽  
Particle Systems ◽  
Initial Configuration ◽  
Limit Measure ◽  
One Dimensional ◽  
Interacting Particle ◽  
The One ◽  
Finite Particle

We show that for a large class of one-dimensional interacting particle systems, with a finite initial configuration, any limit measure , for a sequence of times tending to infinity, must be invariant. This result is used to show that the one-dimensional biased annihilating branching process with parameter > 1/3 converges in distribution to the upper invariant measure provided its initial configuration is almost surely finite and non-null.

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