201 research outputs found
Elementary Trigonometric Sums related to Quadratic Residues
Let p be a prime = 3 (mod 4). A number of elegant number-theoretical
properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) and
C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2\pi/p) are proved. For example, T(p)
equals p times the excess of the odd quadratic residues over the even ones in
the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if
p = 7 (mod 8). In this revised version the connection of these sums with the
class-number h(-p) is also discussed. For example, a very simple formula
expressing h(-p) by means of the aforementioned excess is proved. The
bibliography has been considerably enriched. This article is of an expository
nature.Comment: A number of misprints have been corrected and one or two improvements
have been done to the previous version of the paper with same title. The
paper will appear to Elem. der Mat
The Hydrodynamic Interaction in Polymer Solutions Simulated with Dissipative Particle Dynamics
We analyzed extensively the dynamics of polymer chains in solutions simulated
with dissipative particle dynamics (DPD), with a special focus on the potential
influence of a low Schmidt number of a typical DPD fluid on the simulated
polymer dynamics. It has been argued that a low Schmidt number in a DPD fluid
can lead to underdevelopment of the hydrodynamic interaction in polymer
solutions. Our analyses reveal that equilibrium polymer dynamics in dilute
solution, under a typical DPD simulation conditions, obey the Zimm model very
well. With a further reduction in the Schmidt number, a deviation from the Zimm
model to the Rouse model is observed. This implies that the hydrodynamic
interaction between monomers is reasonably developed under typical conditions
of a DPD simulation. Only when the Schmidt number is further reduced, the
hydrodynamic interaction within the chains becomes underdeveloped. The
screening of the hydrodynamic interaction and the excluded volume interaction
as the polymer volume fraction is increased are well reproduced by the DPD
simulations. The use of soft interaction between polymer beads and a low
Schmidt number do not produce noticeable problems for the simulated dynamics at
high concentrations, except that the entanglement effect which is not captured
in the simulations.Comment: 27 pages, 13 page
Application of exchange Monte Carlo method to ordering dynamics
We apply the exchange Monte Carlo method to the ordering dynamics of the
three-state Potts model with the conserved order parameter. Even for the deeply
quenched case to low temperatures, we have observed a rapid domain growth; we
have proved the efficiency of the exchange Monte Carlo method for the ordering
process. The late-stage growth law has been found to be for
the case of conserved order parameter of three-component system.Comment: 7 pages including 5 eps figures, to appear in New J. Phys.
http://www.njp.or
Lattice-Gas Simulations of Minority-Phase Domain Growth in Binary Immiscible and Ternary Amphiphilic Fluid
We investigate the growth kinetics of binary immiscible fluids and emulsions
in two dimensions using a hydrodynamic lattice-gas model. We perform
off-critical quenches in the binary fluid case and find that the domain size
within the minority phase grows algebraically with time in accordance with
theoretical predictions. In the late time regime we find a growth exponent n =
0.45 over a wide range of concentrations, in good agreement with other
simluations. In the early time regime we find no universal growth exponent but
a strong dependence on the concentration of the minority phase. In the ternary
amphiphilic fluid case the kinetics of self assembly of the droplet phase are
studied for the first time. At low surfactant concentrations, we find that,
after an early algebraic growth, a nucleation regime dominates the late-time
kinetics, which is enhanced by an increasing concentration of surfactant. With
a further increase in the concentration of surfactant, we see a crossover to
logarithmically slow growth, and finally saturation of the oil droplets, which
we fit phenomenologically to a stretched exponential function. Finally, the
transition between the droplet and the sponge phase is studied.Comment: 22 pages, 13 figures, submitted to PR
Lattice-gas simulations of Domain Growth, Saturation and Self-Assembly in Immiscible Fluids and Microemulsions
We investigate the dynamical behavior of both binary fluid and ternary
microemulsion systems in two dimensions using a recently introduced
hydrodynamic lattice-gas model of microemulsions. We find that the presence of
amphiphile in our simulations reduces the usual oil-water interfacial tension
in accord with experiment and consequently affects the non-equilibrium growth
of oil and water domains. As the density of surfactant is increased we observe
a crossover from the usual two-dimensional binary fluid scaling laws to a
growth that is {\it slow}, and we find that this slow growth can be
characterized by a logarithmic time scale. With sufficient surfactant in the
system we observe that the domains cease to grow beyond a certain point and we
find that this final characteristic domain size is inversely proportional to
the interfacial surfactant concentration in the system.Comment: 28 pages, latex, embedded .eps figures, one figure is in colour, all
in one uuencoded gzip compressed tar file, submitted to Physical Review
A scaling theory of 3D spinodal turbulence
A new scaling theory for spinodal decomposition in the inertial hydrodynamic
regime is presented. The scaling involves three relevant length scales, the
domain size, the Taylor microscale and the Kolmogorov dissipation scale. This
allows for the presence of an inertial "energy cascade", familiar from theories
of turbulence, and improves on earlier scaling treatments based on a single
length: these, it is shown, cannot be reconciled with energy conservation. The
new theory reconciles the t^{2/3} scaling of the domain size, predicted by
simple scaling, with the physical expectation of a saturating Reynolds number
at late times.Comment: 5 pages, no figures, revised version submitted to Phys Rev E Rapp
Comm. Minor changes and clarification
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