Phase Transitions in Cellular Automata for Cargo Transport and Kinetically Constrained Traffic

Deterministic phase transitions and self-organization in logistic cellular automata

Physical Review E ◽

10.1103/physreve.100.042216 ◽

2019 ◽

Vol 100 (4) ◽

Author(s):

M. Ibrahimi ◽

O. Gulseren ◽

S. Jahangirov

Keyword(s):

Phase Transitions ◽

Cellular Automata ◽

Self Organization

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Damage spreading in cellular automata models of nonequilibrium phase transitions

Physics Letters A ◽

10.1016/0375-9601(90)90497-c ◽

1990 ◽

Vol 148 (8-9) ◽

pp. 447-451 ◽

Cited By ~ 2

Author(s):

Michel Droz ◽

Laurent Frachebourg

Keyword(s):

Phase Transitions ◽

Cellular Automata ◽

Nonequilibrium Phase Transitions ◽

Damage Spreading ◽

Nonequilibrium Phase

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Monotone Cellular Automata in a Random Environment

Combinatorics Probability Computing ◽

10.1017/s0963548315000012 ◽

2015 ◽

Vol 24 (4) ◽

pp. 687-722 ◽

Cited By ~ 13

Author(s):

BÉLA BOLLOBÁS ◽

PAUL SMITH ◽

ANDREW UZZELL

Keyword(s):

Phase Transitions ◽

Cellular Automata ◽

Two Dimensions ◽

Bootstrap Percolation ◽

Unified Theory ◽

Critical Probability ◽

The Family ◽

Update Rules ◽

Critical Class

In this paper we study in complete generality the family of two-state, deterministic, monotone, local, homogeneous cellular automata in $\mathbb{Z}$d with random initial configurations. Formally, we are given a set $\mathcal{U}$ = {X1,. . . , Xm} of finite subsets of $\mathbb{Z}$d \ {0}, and an initial set A0 ⊂ $\mathbb{Z}$d of ‘infected’ sites, which we take to be random according to the product measure with density p. At time t ∈ $\mathbb{N}$, the set of infected sites At is the union of At-1 and the set of all x ∈ $\mathbb{Z}$d such that x + X ∈ At-1 for some X ∈ $\mathcal{U}$. Our model may alternatively be thought of as bootstrap percolation on $\mathbb{Z}$d with arbitrary update rules, and for this reason we call it $\mathcal{U}$-bootstrap percolation.In two dimensions, we give a classification of $\mathcal{U}$-bootstrap percolation models into three classes – supercritical, critical and subcritical – and we prove results about the phase transitions of all models belonging to the first two of these classes. More precisely, we show that the critical probability for percolation on ($\mathbb{Z}$/n$\mathbb{Z}$)2 is (log n)−Θ(1) for all models in the critical class, and that it is n−Θ(1) for all models in the supercritical class.The results in this paper are the first of any kind on bootstrap percolation considered in this level of generality, and in particular they are the first that make no assumptions of symmetry. It is the hope of the authors that this work will initiate a new, unified theory of bootstrap percolation on $\mathbb{Z}$d.

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