218 research outputs found
Zeros of the Potts Model Partition Function on Sierpinski Graphs
We calculate zeros of the -state Potts model partition function on
'th-iterate Sierpinski graphs, , in the variable and in a
temperature-like variable, . We infer some asymptotic properties of the loci
of zeros in the limit and relate these to thermodynamic
properties of the -state Potts ferromagnet and antiferromagnet on the
Sierpinski gasket fractal, .Comment: 6 pages, 8 figure
Exact calculations of first-passage quantities on recursive networks
We present general methods to exactly calculate mean-first passage quantities
on self-similar networks defined recursively. In particular, we calculate the
mean first-passage time and the splitting probabilities associated to a source
and one or several targets; averaged quantities over a given set of sources
(e.g., same-connectivity nodes) are also derived. The exact estimate of such
quantities highlights the dependency of first-passage processes with respect to
the source-target distance, which has recently revealed to be a key parameter
to characterize transport in complex media. We explicitly perform calculations
for different classes of recursive networks (finitely ramified fractals,
scale-free (trans)fractals, non-fractals, mixtures between fractals and
non-fractals, non-decimable hierarchical graphs) of arbitrary size. Our
approach unifies and significantly extends the available results in the field.Comment: 16 pages, 10 figure
Partition function of the Potts model on self-similar lattices as a dynamical system and multiple transitions
We present an analytic study of the Potts model partition function on two
different types of self-similar lattices of triangular shape with non integer
Hausdorff dimension. Both types of lattices analyzed here are interesting
examples of non-trivial thermodynamics in less than two dimensions. First, the
Sierpinski gasket is considered. It is shown that, by introducing suitable
geometric coefficients, it is possible to reduce the computation of the
partition function to a dynamical system, whose variables are directly
connected to (the arising of) frustration on macroscopic scales, and to
determine the possible phases of the system. The same method is then used to
analyse the Hanoi graph. Again, dynamical system theory provides a very elegant
way to determine the phase diagram of the system. Then, exploiting the analysis
of the basins of attractions of the corresponding dynamical systems, we
construct various examples of self-similar lattices with more than one critical
temperature. These multiple critical temperatures correspond to crossing phases
with different degrees of frustration.Comment: 16 pages, 12 figures, 1 table; title changed, references and
discussion on multiple transitions adde
Quantum walk approach to search on fractal structures
We study continuous-time quantum walks mimicking the quantum search based on
Grover's procedure. This allows us to consider structures, that is, databases,
with arbitrary topological arrangements of their entries. We show that the
topological structure of the database plays a crucial role by analyzing, both
analytically and numerically, the transition from the ground to the first
excited state of the Hamiltonian associated with different (fractal)
structures. Additionally, we use the probability of successfully finding a
specific target as another indicator of the importance of the topological
structure.Comment: 15 pages, 14 figure
Efficiency of encounter-controlled reaction between diffusing reactants in a finite lattice: topology and boundary effects
The role of dimensionality (Euclidean versus fractal), spatial extent,
boundary effects and system topology on the efficiency of diffusion-reaction
processes involving two simultaneously-diffusing reactants is analyzed. We
present numerically-exact values for the mean time to reaction, as gauged by
the mean walklength before reactive encounter, obtained via application of the
theory of finite Markov processes, and via Monte Carlo simulation. As a general
rule, we conclude that for sufficiently large systems, the efficiency of
diffusion-reaction processes involving two synchronously diffusing reactants
(two-walker case) relative to processes in which one reactant of a pair is
anchored at some point in the reaction space (one walker plus trap case) is
higher, and is enhanced the lower the dimensionality of the system. This
differential efficiency becomes larger with increasing system size and, for
periodic systems, its asymptotic value may depend on the parity of the lattice.
Imposing confining boundaries on the system enhances the differential
efficiency relative to the periodic case, while decreasing the absolute
efficiencies of both two-walker and one walker plus trap processes. Analytic
arguments are presented to provide a rationale for the results obtained. The
insights afforded by the analysis to the design of heterogeneous catalyst
systems are also discussed.Comment: 15 pages, 8 figures, uses revtex4, accepted for publication in
Physica
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