1,074 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
Entropy rate of higher-dimensional cellular automata
We introduce the entropy rate of multidimensional cellular automata. This
number is invariant under shift-commuting isomorphisms; as opposed to the
entropy of such CA, it is always finite. The invariance property and the
finiteness of the entropy rate result from basic results about the entropy of
partitions of multidimensional cellular automata. We prove several results that
show that entropy rate of 2-dimensional automata preserve similar properties of
the entropy of one dimensional cellular automata.
In particular we establish an inequality which involves the entropy rate, the
radius of the cellular automaton and the entropy of the d-dimensional shift. We
also compute the entropy rate of permutative bi-dimensional cellular automata
and show that the finite value of the entropy rate (like the standard entropy
of for one-dimensional CA) depends on the number of permutative sites.
Finally we define the topological entropy rate and prove that it is an
invariant for topological shift-commuting conjugacy and establish some
relations between topological and measure-theoretic entropy rates
From quantum cellular automata to quantum lattice gases
A natural architecture for nanoscale quantum computation is that of a quantum
cellular automaton. Motivated by this observation, in this paper we begin an
investigation of exactly unitary cellular automata. After proving that there
can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in
one dimension, we weaken the homogeneity condition and show that there are
nontrivial, exactly unitary, partitioning cellular automata. We find a one
parameter family of evolution rules which are best interpreted as those for a
one particle quantum automaton. This model is naturally reformulated as a two
component cellular automaton which we demonstrate to limit to the Dirac
equation. We describe two generalizations of this automaton, the second of
which, to multiple interacting particles, is the correct definition of a
quantum lattice gas.Comment: 22 pages, plain TeX, 9 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages); minor typographical
corrections and journal reference adde
Index theory of one dimensional quantum walks and cellular automata
If a one-dimensional quantum lattice system is subject to one step of a
reversible discrete-time dynamics, it is intuitive that as much "quantum
information" as moves into any given block of cells from the left, has to exit
that block to the right. For two types of such systems - namely quantum walks
and cellular automata - we make this intuition precise by defining an index, a
quantity that measures the "net flow of quantum information" through the
system. The index supplies a complete characterization of two properties of the
discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the
sense that there is a system S which locally acts like S_1 in one region and
like S_2 in some other region, if and only if S_1 and S_2 have the same index.
Second, the index labels connected components of such systems: equality of the
index is necessary and sufficient for the existence of a continuous deformation
of S_1 into S_2. In the case of quantum walks, the index is integer-valued,
whereas for cellular automata, it takes values in the group of positive
rationals. In both cases, the map S -> ind S is a group homomorphism if
composition of the discrete dynamics is taken as the group law of the quantum
systems. Systems with trivial index are precisely those which can be realized
by partitioned unitaries, and the prototypes of systems with non-trivial index
are shifts.Comment: 38 pages. v2: added examples, terminology clarifie
- …