The Harmonic Solid

2019 ◽  
pp. 331-349
Author(s):  
Robert H. Swendsen
Keyword(s):  
Boundary Conditions ◽  
Specific Heat ◽  
Periodic Boundary ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Normal Modes ◽  
Periodic Boundary Conditions ◽  
One Dimension ◽  
Debye Approximation

In Chapter 26 we return to calculating the contributions to the specific heat of a crystal from the vibrations of the atoms. The vibrations of a model of a solid, for which the interactions are quadratic in form, is investigated. Calculations are restricted to one dimension for simplicity in the derivations of the Fourier modes and the equations of motion. Both pinned and periodic boundary conditions are discussed. The representation of the Hamiltonian in terms of normal modes and the solution in terms of the equations of motion are derived. The Debye approximation is then introduced for three-dimensional systems.

INVESTIGATION OF THE CONSTRAINTS ON TWISTED AND UNTWISTED YANG–MILLS THEORIES IN A SLAB

1991 ◽  
Vol 06 (31) ◽  
pp. 2893-2900
Author(s):  
A. R. LEVI
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
One Dimension ◽  
Yang Mills ◽  
Quantum Constraints

BRST is used to investigate the consistency of the quantum constraints for Yang–Mills theories based on twisted and untwisted SU (N) in a slab with periodic boundary conditions in one dimension.

scholarly journals On the Implementation of Stream-Wise Periodic Boundary Conditions

2005 ◽  
Author(s):  
Steven B. Beale
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Equations Of Motion ◽  
Periodic Boundary Conditions ◽  
Constant Heat Flux ◽  
State Variables ◽  
Constant Heat ◽  
Mathematical Basis ◽  
Constant Wall Temperature ◽  
Primitive Variables

Fully-developed periodic boundary conditions have frequently been employed to perform calculations on the performance of typical elements of heat exchangers. Many such calculations have been achieved by transforming the equations of motion to obtain a new set of state variables which are cyclic in the stream-wise direction. In others, primitive variables, based on substitution schemes are employed. In this paper; a review of existing procedures is provided, and a new method is proposed. The method is based on the use of primitive variables with periodic boundary conditions combined with the use of slip values. Either pressure difference or mass flow rate may be prescribed, and both constant wall temperature and constant heat flux wall conditions may be considered. The example of an offset-fin plate-fin heat exchanger is used to illustrate the application of the procedure. The scope and limitations of the method are discussed in detail, and the mathematical basis by which the method may be extended to the consideration of problems involving mass transfer, with associated continuity, momentum, and species source/sinks is proposed.

Optimal feedback control problem for Bingham media motion with periodic boundary conditions

2019 ◽  
Vol 485 (2) ◽  
pp. 139-141
Author(s):  
V. G. Zvyagin ◽  
M. V. Turbin
Keyword(s):  
Feedback Control ◽  
Boundary Conditions ◽  
Control Problem ◽  
Periodic Boundary ◽  
Three Dimensional ◽  
Degree Theory ◽  
Periodic Boundary Conditions ◽  
Optimal Feedback Control ◽  
Operator Inclusion ◽  
Optimal Feedback

We study the optimal feedback control problem for the motion of Bingham media with periodic boundary conditions in two- and three-dimensional cases. First, the considered problem is interpreted as an operator inclusion with a multivalued right-hand side. Then, the approximation-topological approach to hydrodynamic problems and the degree theory for a class of multivalued maps are used to prove the existence of solutions of this inclusion. Finally, we prove that, among the solutions of the considered problem, there exists one minimizing the given cost functional.

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 Simplified CFD Model With Multi-Periodic Boundary Conditions for Cross Wavy Channels

2011 ◽  
Author(s):  
L. X. Du ◽  
M. Zeng ◽  
Q. W. Wang
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Flow Direction ◽  
Wave Length ◽  
Three Dimensional ◽  
Periodic Boundary Conditions ◽  
Navier Stokes ◽  
The Cross ◽  
Wavy Channels

The compact and efficient primary surface heat exchangers are often used as recuperators in microturbine regenerative cycle systems. In the present study, the flow and the heat transfer performance of the cross wavy (CW) ducts have been simulated by three-dimensional models. The hydrodynamic diameters of the models are 1.689mm. Navier-Stokes and energy equations are solved by COMSOL3.5. Because one single wavy cell will overlap more than one adjacent channel, multi-periodic boundary conditions are especially adopted to simplify the calculations. Multi-periodic boundary conditions have been proved to have more reasonable flow field and heat transfer coefficient compared with the literature results. A dimensionless parameter L/A (wave length L, internal height of the corrugation in flow direction A) is defined as the optimization target. The numerical results indicated that when L/A = 6, the CW channel has the best comprehensive performance in all the cases. The comprehensive performances of the CW ducts are evaluated by the j/f (heat transfer factor j and friction factor f). The flow and heat transfer characteristics are much more complex in the cross wavy channels, especially when L/A is small.

Phase Transitions in Three Dimensional Ising Systems under Free and Periodic Boundary Conditions

2004 ◽  
Vol 301 (1) ◽  
pp. 91-95
Author(s):  
M FELISA MARTÍNEZ RUIZ ◽  
FRANCISCA PAJUELO ◽  
JORGE GARCÍA ◽  
CARMEN ARAGÓ ◽  
JULIO. A. GONZALO
Keyword(s):  
Phase Transitions ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Three Dimensional ◽  
Periodic Boundary Conditions ◽  
Ising Systems

scholarly journals Uniform Asymptotics of Toeplitz Determinants with Fisher–Hartwig Singularities

2021 ◽  
Vol 383 (2) ◽  
pp. 685-730
Author(s):  
B. Fahs
Keyword(s):  
Asymptotic Formula ◽  
Boundary Conditions ◽  
Characteristic Polynomial ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Fixed Number ◽  
One Dimension ◽  
Uniform Asymptotics ◽  
Toeplitz Determinants ◽  
Unitary Ensemble

AbstractWe obtain an asymptotic formula for $$n\times n$$ n × n Toeplitz determinants as $$n\rightarrow \infty $$ n → ∞ , for non-negative symbols with any fixed number of Fisher–Hartwig singularities, which is uniform with respect to the location of the singularities. As an application, we prove a conjecture by Fyodorov and Keating (Philos Trans R Soc A 372: 20120503, 2014) regarding moments of averages of the characteristic polynomial of the Circular Unitary Ensemble. In addition, we obtain an asymptotic formula regarding the momentum of impenetrable bosons in one dimension with periodic boundary conditions.

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