2,809 research outputs found
Some properties of Ising automata
AbstractIn this work, we shall present some arithmetical and topological properties of Ising automata. More precisely, we shall study many different notions, such as faithful and strictly faithful automata, factor and product automata, irreducible and weakly irreducible automata, prime automata, homogeneous automata, minimal automata, invertible automata, etc., and discuss their related properties. We shall also define and study three different topologies over the set of all minimal automata, and discuss the topological closure property of automatic sequences. As application, we shall use the obtained results to give a somewhat detailed analysis of Ising automata
Magnetic order in the Ising model with parallel dynamics
It is discussed how the equilibrium properties of the Ising model are
described by an Hamiltonian with an antiferromagnetic low temperature behavior
if only an heat bath dynamics, with the characteristics of a Probabilistic
Cellular Automaton, is assumed to determine the temporal evolution of the
system.Comment: 9 pages, 3 figure
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
Transfer matrix analysis of one-dimensional majority cellular automata with thermal noise
Thermal noise in a cellular automaton refers to a random perturbation to its
function which eventually leads this automaton to an equilibrium state
controlled by a temperature parameter. We study the 1-dimensional majority-3
cellular automaton under this model of noise. Without noise, each cell in this
automaton decides its next state by majority voting among itself and its left
and right neighbour cells. Transfer matrix analysis shows that the automaton
always reaches a state in which every cell is in one of its two states with
probability 1/2 and thus cannot remember even one bit of information. Numerical
experiments, however, support the possibility of reliable computation for a
long but finite time.Comment: 12 pages, 4 figure
- …