1,899 research outputs found
Statistical Mechanics of Surjective Cellular Automata
Reversible cellular automata are seen as microscopic physical models, and
their states of macroscopic equilibrium are described using invariant
probability measures. We establish a connection between the invariance of Gibbs
measures and the conservation of additive quantities in surjective cellular
automata. Namely, we show that the simplex of shift-invariant Gibbs measures
associated to a Hamiltonian is invariant under a surjective cellular automaton
if and only if the cellular automaton conserves the Hamiltonian. A special case
is the (well-known) invariance of the uniform Bernoulli measure under
surjective cellular automata, which corresponds to the conservation of the
trivial Hamiltonian. As an application, we obtain results indicating the lack
of (non-trivial) Gibbs or Markov invariant measures for "sufficiently chaotic"
cellular automata. We discuss the relevance of the randomization property of
algebraic cellular automata to the problem of approach to macroscopic
equilibrium, and pose several open questions.
As an aside, a shift-invariant pre-image of a Gibbs measure under a
pre-injective factor map between shifts of finite type turns out to be always a
Gibbs measure. We provide a sufficient condition under which the image of a
Gibbs measure under a pre-injective factor map is not a Gibbs measure. We point
out a potential application of pre-injective factor maps as a tool in the study
of phase transitions in statistical mechanical models.Comment: 50 pages, 7 figure
Cellular automata and Lyapunov exponents
In this article we give a new definition of some analog of Lyapunov exponents
for cellular automata . Then for a shift ergodic and cellular automaton
invariant probability measure we establish an inequality between the entropy of
the automaton, the entropy of the shift and the Lyapunov exponent
Right-Permutative Cellular Automata on Topological Markov Chains
In this paper we consider cellular automata with
algebraic local rules and such that is a topological Markov
chain which has a structure compatible to this local rule. We characterize such
cellular automata and study the convergence of the Ces\`aro mean distribution
of the iterates of any probability measure with complete connections and
summable decay.Comment: 16 pages, 2 figure. A new version with improved redaction of Theorem
6.3(i)) to clearify its consequence
Some properties of cellular automata with equicontinuity points
We investigate topological and ergodic properties of cellular automata having
equicontinuity points. In this class surjectivity on a transitive SFT implies
existence of a dense set of periodic points. Our main result is that under the
action of such an automaton any shift ergodic measure converges in Cesaro Mean
Probabilistic cellular automata, invariant measures, and perfect sampling
A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The
cells are updated synchronously and independently, according to a distribution
depending on a finite neighborhood. We investigate the ergodicity of this
Markov chain. A classical cellular automaton is a particular case of PCA. For a
1-dimensional cellular automaton, we prove that ergodicity is equivalent to
nilpotency, and is therefore undecidable. We then propose an efficient perfect
sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm
does not assume any monotonicity property of the local rule. It is based on a
bounding process which is shown to be also a PCA. Last, we focus on the PCA
Majority, whose asymptotic behavior is unknown, and perform numerical
experiments using the perfect sampling procedure
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