664 research outputs found
Some integral representations and limits for (products of) the parabolic cylinder function
Veestraeten [1] recently derived inverse Laplace transforms for Laplace
transforms that contain products of two parabolic cylinder functions by
exploiting the link between the parabolic cylinder function and the transition
density and distribution functions of the Ornstein-Uhlenbeck process. This
paper first uses these results to derive new integral representations for
(products of two) parabolic cylinder functions. Second, as the Brownian motion
process with drift is a limiting case of the Ornstein-Uhlenbeck process also
limits can be calculated for the product of gamma functions and (products of)
parabolic cylinder functions. The central results in both cases contain, in
stylised form, D_{v}(x)D_{v}(y) and D_{v}(x)D_{v-1}(y) such that the recurrence
relation of the parabolic cylinder function straightforwardly allows to obtain
integral representations and limits also for countless other combinations in
the orders such as D_{v}(x)D_{v-3}(y) and D_{v+1}(x)D_{v}(y)
Cross-over between diffusion-limited and reaction-limited regimes in the coagulation-diffusion process
The change from the diffusion-limited to the reaction-limited cooperative
behaviour in reaction-diffusion systems is analysed by comparing the universal
long-time behaviour of the coagulation-diffusion process on a chain and on the
Bethe lattice. On a chain, this model is exactly solvable through the
empty-interval method. This method can be extended to the Bethe lattice, in the
ben-Avraham-Glasser approximation. On the Bethe lattice, the analysis of the
Laplace-transformed time-dependent particle-density is analogous to the study
of the stationary state, if a stochastic reset to a configuration of
uncorrelated particles is added. In this stationary state logarithmic
corrections to scaling are found, as expected for systems at the upper critical
dimension. Analogous results hold true for the time-integrated
particle-density. The crossover scaling functions and the associated effective
exponents between the chain and the Bethe lattice are derived.Comment: 21 pages, 5 figures; v3: Scaling arguments at beginning of Section 4
were correcte
Application of the lattice Green's function for calculating the resistance of an infinite networks of resistors
We calculate the resistance between two arbitrary grid points of several
infinite lattice structures of resistors by using lattice Green's functions.
The resistance for dimensional hypercubic, rectangular, triangular and
honeycomb lattices of resistors is discussed in detail. We give recurrence
formulas for the resistance between arbitrary lattice points of the square
lattice. For large separation between nodes we calculate the asymptotic form of
the resistance for a square lattice and the finite limiting value of the
resistance for a simple cubic lattice. We point out the relation between the
resistance of the lattice and the van Hove singularity of the tight-binding
Hamiltonian. Our Green's function method can be applied in a straightforward
manner to other types of lattice structures and can be useful didactically for
introducing many concepts used in condensed matter physics
Does a Single Zealot Affect an Infinite Group of Voters ?
A method for studying exact properties of a class of {\it inhomogeneous}
stochastic many-body systems is developed and presented in the framework of a
voter model perturbed by the presence of a ``zealot'', an individual allowed to
favour an opinion. We compute exactly the magnetization of this model and find
that in one (1d) and two dimensions (2d) it evolves, algebraically () in 1d and much slower () in 2d, towards the unanimity
state chosen by the zealot. In higher dimensions the stationary magnetization
is no longer uniform: the zealot cannot influence all the individuals.
Implications to other physical problems are also pointed out.Comment: 4 pages, 2-column revtex4 forma
Green's function of a finite chain and the discrete Fourier transform
A new expression for the Green's function of a finite one-dimensional lattice
with nearest neighbor interaction is derived via discrete Fourier transform.
Solution of the Heisenberg spin chain with periodic and open boundary
conditions is considered as an example. Comparison to Bethe ansatz clarifies
the relation between the two approaches.Comment: preprint of the paper published in Int. J. Modern Physics B Vol. 20,
No. 5 (2006) 593-60
The Cauchy-Schlomilch transformation
The Cauchy-Schl\"omilch transformation states that for a function and , the integral of and over the
interval are the same. This elementary result is used to evaluate
many non-elementary definite integrals, most of which cannot be obtained by
symbolic packages. Applications to probability distributions is also given
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