6,611 research outputs found
Uniform tiling with electrical resistors
The electric resistance between two arbitrary nodes on any infinite lattice
structure of resistors that is a periodic tiling of space is obtained. Our
general approach is based on the lattice Green's function of the Laplacian
matrix associated with the network. We present several non-trivial examples to
show how efficient our method is. Deriving explicit resistance formulas it is
shown that the Kagom\'e, the diced and the decorated lattice can be mapped to
the triangular and square lattice of resistors. Our work can be extended to the
random walk problem or to electron dynamics in condensed matter physics.Comment: 22 pages, 14 figure
A characterization of the locally finite networks admitting non-constant harmonic functions of finite energy
We characterize the locally finite networks admitting non-constant harmonic
functions of finite energy. Our characterization unifies the necessary
existence criteria of Thomassen and of Lyons and Peres with the sufficient
criterion of Soardi. We also extend a necessary existence criterion for
non-elusive non-constant harmonic functions of finite energy due to
Georgakopoulos
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
This article is a mini-review about electrical current flows in networks from
the perspective of statistical physics. We briefly discuss analytical methods
to solve the conductance of an arbitrary resistor network. We then turn to
basic results related to percolation: namely, the conduction properties of a
large random resistor network as the fraction of resistors is varied. We focus
on how the conductance of such a network vanishes as the percolation threshold
is approached from above. We also discuss the more microscopic current
distribution within each resistor of a large network. At the percolation
threshold, this distribution is multifractal in that all moments of this
distribution have independent scaling properties. We will discuss the meaning
of multifractal scaling and its implications for current flows in networks,
especially the largest current in the network. Finally, we discuss the relation
between resistor networks and random walks and show how the classic phenomena
of recurrence and transience of random walks are simply related to the
conductance of a corresponding electrical network.Comment: 27 pages & 10 figures; review article for the Encyclopedia of
Complexity and System Science (Springer Science
Perturbation of infinite networks of resistors
The resistance between arbitrary nodes of infinite networks of resistors is
studied when the network is perturbed by removing one bond from the perfect
lattice. A connection is made between the resistance and the lattice Green's
function of the perturbed network. Solving Dyson's equation the Green's
function and the resistance of the perturbed lattice are expressed in terms of
those of the perfect lattice. Numerical results are presented for a square
lattice. Our method of the lattice Green's function in studying resistor
networks can also be applied in the field of random walks as well as electrical
and mechanical breakdown phenomena in insulators, thin films and modern
ceramics.Comment: 10 pages, 4 figures, submitted to American Journal of Physic
Restoration of Macroscopic Isotropy on -Simplex Fractal Conductor Networks
Restoration of macroscopic isotropy has been investigated in (d+1)-simplex
fractal conductor networks via exact real space renormalization group
transformations. Using some theorems of fixed point theory, it has been shown
very rigoroursly that the macroscopic conductivity becomes isotropic for large
scales and anisotropy vanishes with a scaling exponent which is computed
exactly for arbitrary values of d and decimation numbers b=2,3,4 and
5.Comment: 27 Pages, 3 Figure
Classical magnetotransport of inhomogeneous conductors
We present a model of magnetotransport of inhomogeneous conductors based on
an array of coupled four-terminal elements. We show that this model generically
yields non-saturating magnetoresistance at large fields. We also discuss how
this approach simplifies finite-element analysis of bulk inhomogeneous
semiconductors in complex geometries. We argue that this is an explanation of
the observed non-saturating magnetoresistance in silver chalcogenides and
potentially in other disordered conductors. Our method may be used to design
the magnetoresistive response of a microfabricated array.Comment: 12 pages, 13 figures. Minor typos correcte
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