Effect of Periodic Boundary Conditions on the Harmonic Free Energy of the Kagome Lattice: An Exact Solution

1971 ◽  
Vol 55 (11) ◽  
pp. 5412-5413 ◽  
Author(s):  
Dale A. Huckaby
Keyword(s):  
Exact Solution ◽  
Free Energy ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Kagome Lattice ◽  
Kagomé Lattice

scholarly journals Thermodynamic limit for a system with dipole-dipole interactions

1975 ◽  
Vol 19 (1) ◽  
pp. 116-128 ◽  
Author(s):  
E. R. Smith ◽  
J. W. Perram
Keyword(s):  
Free Energy ◽  
Variational Principle ◽  
Boundary Conditions ◽  
Thermodynamic Limit ◽  
Periodic Boundary ◽  
Three Dimensional ◽  
Periodic Boundary Conditions ◽  
Free Energies ◽  
Dipole Interactions ◽  
Simple Boundary

AbstractIt is shown that for the three dimensional Ising model with dipole-dipole interactions, the thermodynamic limit of the free energy with simple boundary conditions is not the same as the thermodynamic limit of the free energy with periodic boundary conditions. A variational principle is developed to connect the two free energies.

A NOTE ON INTERFACES WITH PERIODIC BOUNDARIES

1992 ◽  
Vol 03 (05) ◽  
pp. 889-896 ◽  
Author(s):  
B. BUNK
Keyword(s):  
Free Energy ◽  
Gauge Theory ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Lattice Gauge Theory ◽  
Periodic Boundary Conditions ◽  
Large Area ◽  
Universal Form ◽  
Lattice Gauge ◽  
Numerical Coefficient

The partition funtion of a fluctuating interface is argued to approach a universal form at large area, including the numerical coefficient, provided that the interface has periodic boundary conditions und continuum behavior. This is demonstrated by explicit calculations, using effective actions with different regularisations. The result is applied to an analysis of the vortex free energy in SU(2) lattice gauge theory.

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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