scholarly journals FRACTAL DIMENSION OF SPIN-GLASSES INTERFACES IN DIMENSION d = 2 AND d = 3 VIA STRONG DISORDER RENORMALIZATION AT ZERO TEMPERATURE

Fractals ◽  
2015 ◽  
Vol 23 (04) ◽  
pp. 1550042 ◽  
Author(s):  
CÉCILE MONTHUS
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Spin Glasses ◽  
Periodic Boundary ◽  
Domain Walls ◽  
Numerical Study ◽  
Periodic Boundary Conditions ◽  
Zero Temperature ◽  
Low Dimensions ◽  
Strong Disorder

For Gaussian Spin-Glasses in low dimensions, we introduce a simple Strong Disorder renormalization at zero temperature in order to construct ground states for Periodic and Anti-Periodic boundary conditions. The numerical study in dimensions [Formula: see text] (up to sizes [Formula: see text]) and [Formula: see text] (up to sizes [Formula: see text]) yields that Domain Walls are fractal of dimensions [Formula: see text] and [Formula: see text], respectively.

scholarly journals Correlation functions by separation of variables: the XXX spin chain

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Giuliano Niccoli ◽  
Hao Pei ◽  
Véronique Terras
Keyword(s):  
Integral Representation ◽  
Boundary Conditions ◽  
Thermodynamic Limit ◽  
Correlation Functions ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Periodic Boundary Conditions ◽  
Multiple Integral ◽  
Periodic Case ◽  
Zero Temperature

We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.

scholarly journals Numerical study of one-dimensional Stefan problem with periodic boundary conditions

2013 ◽  
Vol 17 (5) ◽  
pp. 1453-1458
Author(s):  
Liang-Hui Qu ◽  
Feng Ling ◽  
Lin Xing
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Boundary Conditions ◽  
Stefan Problem ◽  
Periodic Boundary ◽  
Moving Boundary ◽  
Numerical Study ◽  
Periodic Boundary Conditions ◽  
One Dimensional ◽  
Stefan Number

A finite difference approach to a one-dimensional Stefan problem with periodic boundary conditions is studied. The evolution of the moving boundary and the temperature field are simulated numerically, and the effects of the Stefan number and the periodical boundary condition on the temperature distribution and the evolution of the moving boundary are analyzed.

A shear instability mechanism for the pulsations of Rayleigh–Taylor unstable model flames

2014 ◽  
Vol 748 ◽  
pp. 618-640 ◽  
Author(s):  
E. P. Hicks
Keyword(s):  
Boundary Conditions ◽  
Flame Front ◽  
Vortex Shedding ◽  
Periodic Boundary ◽  
Domain Walls ◽  
Landau Equation ◽  
Periodic Boundary Conditions ◽  
Shear Instability ◽  
Temporal Behaviour ◽  
Spatio Temporal

AbstractPrevious studies have shown that the behaviour of Rayleigh–Taylor (RT) unstable flames depends on the boundary conditions. If the boundary conditions at the domain walls are impermeable/adiabatic or reflecting then the flame assumes a stable parabolic shape. On the other hand, periodic boundary conditions can produce unstable pulsating solutions. In this paper, we explore why periodic boundary conditions allow unstable solutions by showing the results of two-dimensional direct numerical simulations of model flames. We show that RT unstable premixed model flames pulsate at low gravity because of a shear instability of the vorticity layers behind the flame front. The resulting vortex shedding is controlled by a region of absolute-like instability which moves closer to the flame front as gravity is increased, ultimately disturbing the flame and leading to pulsations. We demonstrate that this region is ‘absolutely unstable’ by showing that the wake is dominated by pure frequency oscillations. In addition, the shear instability can be described by the Landau equation and can be represented dynamically by a Hopf bifurcation. The applicability of the Landau equation allows the apparently complex spatio-temporal behaviour of the vortex shedding to be described by a simple temporal model with only a secondary spatial dependence. We show that the flame behaviour is analogous to the initial instability downstream of a circular cylinder, which leads to the von Kármán vortex street for large enough values of the Reynolds number.

Molecular Dynamics Using Non-Variational Polarizable Force Fields: Theory, Periodic Boundary Conditions Implementation and Application to the Bond Capacity Model

2019 ◽  
Author(s):  
Pier Paolo Poier ◽  
Louis Lagardere ◽  
Jean-Philip Piquemal ◽  
Frank Jensen
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Force Fields ◽  
Periodic Boundary Conditions ◽  
Computational Effort ◽  
Polarizable Force Fields ◽  
Energy Models ◽  
Capacity Model ◽  
Energy Gradient ◽  
Polarization Models

<div> <div> <div> <p>We extend the framework for polarizable force fields to include the case where the electrostatic multipoles are not determined by a variational minimization of the electrostatic energy. Such models formally require that the polarization response is calculated for all possible geometrical perturbations in order to obtain the energy gradient required for performing molecular dynamics simulations. </p><div> <div> <div> <p>By making use of a Lagrange formalism, however, this computational demanding task can be re- placed by solving a single equation similar to that for determining the electrostatic variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p><div><div><div> </div> </div> </div> <p> </p><div> <div> <div> <p>variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p> </div> </div> </div> </div> </div> </div> </div> </div> </div>

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

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