One-dimensional dislocations IV. Dynamics

1950 ◽  
Vol 201 (1065) ◽  
pp. 261-268 ◽  
Keyword(s):  
Wave Velocity ◽  
Equations Of Motion ◽  
Dislocation Model ◽  
Characteristic Wave ◽  
One Dimensional ◽  
Regular Sequences ◽  
All Solutions ◽  
Stable Equilibria ◽  
The One

Tho equations of motion of the one-dimensional dislocation model are studied, and all solutions representing disturbances propagated without change of form are obtained. These comprise: (i) dislocations, and regular sequences of the same, either of like or of alternating sign, travelling with velocity less than c , the characteristic wave velocity of the system; (ii) an iddislocations, and sequences of the spme, travelling with velocity greater than c; and (iii) waves of infinitesimal amplitude, belonging to two branches travelling respectively with velocities less than and greater than c . Only dislocations or sequences of dislocations of like sign, and waves of velocity less than c , correspond to stable equilibria. The dislocations exhibit ‘relativistic’ behaviour. The relevance of anti-dislocations to very fast slip in solids is considered, and rejected.

scholarly journals One-dimensional dislocations. I. Static theory

1949 ◽  
Vol 198 (1053) ◽  
pp. 205-216 ◽  
Keyword(s):  
Critical Conditions ◽  
Dislocation Model ◽  
Physical Parameters ◽  
Spontaneous Generation ◽  
Static Theory ◽  
One Dimensional ◽  
Lennard Jones ◽  
Regular Sequences ◽  
Crystalline Substrate ◽  
Made In

The theory of a one-dimensional dislocation model is developed. Besides acting as a pointer to developments of general dislocation theory, it has a variety of direct physical applications, particularly to monolayers on a crystalline substrate and to conditions in the edge row of a terrace of molecules in a growing crystal. Allowance is made in the theory for a difference in natural lattice-spacing between the surface layer or row and the substrate. The form and energy of single dislocations and of regular sequences of dislocations are calculated. Critical conditions for spontaneous generation (or escape) of dislocations are determined, and likewise the activation energies for such processes below the critical limits. Various physical applications of the model are discussed, and the physical parameters are evaluated with the aid of the Lennard-Jones force law for the above-mentioned principal applications.

Integration of one-dimensional equations of motion

2020 ◽  
pp. 3-5
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Potential Energy ◽  
Equations Of Motion ◽  
Field Change ◽  
One Dimensional ◽  
System A ◽  
Small Correction ◽  
The Law ◽  
The One ◽  
The Given

If the potential energy is independent of time, the energy of the system remains constant during the motion of a closed system. A system with one degree of freedom allows for the determination of the law of motion in quadrature. In this chapter, the authors consider motion of the particles in the one-dimensional fields. They discuss also how the law and the period of a particle moving in the potential field change due to adding to the given field a small correction.

CRITICAL DYNAMICS OF THE XY-MODEL ON THE ONE-DIMENSIONAL SUPERLATTICE BY POSITION SPACE RENORMALIZATION GROUP

1996 ◽  
Vol 10 (22) ◽  
pp. 1077-1083 ◽  
Author(s):  
J.P. DE LIMA ◽  
L.L. GONÇALVES
Keyword(s):  
Renormalization Group ◽  
Equations Of Motion ◽  
Xy Model ◽  
Dynamic Scaling ◽  
Critical Dynamics ◽  
Renormalization Group Theory ◽  
Position Space ◽  
One Dimensional ◽  
Homogeneous Lattice ◽  
The One

The critical dynamics of the isotropic XY-model on the one-dimensional superlattice is considered in the framework of the position space renormalization group theory. The decimation transformation is introduced by considering the equations of motion of the operators associated to the excitations of the system, and it corresponds to an extension of the procedure introduced by Stinchcombe and dos Santos (J. Phys. A18, L597 (1985)) for the homogeneous lattice. The dispersion relation is obtained exactly and the static and dynamic scaling forms are explicitly determined. The dynamic critical exponent is also obtained and it is shown that it is identical to the one of the XY-model on the homogeneous chain.

The solvability of an elliptic system under a singular boundary condition

2006 ◽  
Vol 136 (3) ◽  
pp. 509-546 ◽  
Author(s):  
J. García-Melián ◽  
J. Sabina de Lis ◽  
R. Letelier-Albornoz
Keyword(s):  
Boundary Condition ◽  
Elliptic System ◽  
Positive Solutions ◽  
Detailed Account ◽  
Singular Boundary ◽  
Dimensional Problem ◽  
One Dimensional ◽  
All Solutions ◽  
The One ◽  
Singular Boundary Condition

In this work we are considering both the one-dimensional and the radially symmetric versions of the elliptic system Δu = vp, Δv = uq in Ω, where p, q > 0, under the boundary condition u|∂Ω = +∞, v|∂Ω = +∞. It is shown that no positive solutions exist when pq ≤ 1, while we provide a detailed account of the set of (infinitely many) positive solutions if pq > 1. The behaviour near the boundary of all solutions is also elucidated, and symmetric solutions (u, v) are completely characterized in terms of their minima (u(0), v(0)). Non-symmetric solutions are also deeply studied in the one-dimensional problem.

scholarly journals The One-dimensional Coulomb-force Problem in Classical Radiation Reaction Theories

1977 ◽  
Vol 32 (7) ◽  
pp. 685-691
Author(s):  
W. Heudorfer ◽  
M. Sorg
Keyword(s):  
Coulomb Potential ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Radiation Reaction ◽  
Coulomb Force ◽  
Stationary States ◽  
One Dimensional ◽  
The One ◽  
Force Problem ◽  
Classical Radiation

Abstract Numerical solutions of the recently proposed equations of motion for the classically radiating electron are obtained for the case where the particle moves in a one-dimensional Coulomb potential (both attractive and repulsive). The solutions are discussed and found to be meaningful also in that case, where the well-known Lorentz-Dirac equation fails (attractive Coulomb force). Discrete, stationary states are found in a non-singular version of the Coulomb potential. During the transition between those stationary states the particle looses energy by emission of radiation, which results in a smaller amplitude of the stationary oscillations.

An Application of Mixture Theory to Particulate Sedimentation

1980 ◽  
Vol 47 (2) ◽  
pp. 261-265 ◽  
Author(s):  
C. D. Hill ◽  
A. Bedford ◽  
D. S. Drumheller
Keyword(s):  
Particle Concentration ◽  
Equations Of Motion ◽  
Mixture Theory ◽  
Solid Particles ◽  
Two Phase ◽  
Small Perturbations ◽  
One Dimensional ◽  
Blood Sedimentation ◽  
The One ◽  
Extended Variational Principle

Equations for two-phase flow are used to analyze the one-dimensional sedimentation of solid particles in a stationary container of liquid. A derivation of the equations of motion is presented which is based upon Hamilton’s extended variational principle. The resulting equations contain diffusivity terms, which are linear in the gradient of the particle concentration. It is shown that the solution of the equations for steady sedimentation is stable under small perturbations. Finally, finite-difference solutions of the equations are compared to the data of Whelan, Huang, and Copley for blood sedimentation.

Green's function approach to the Ising model

1969 ◽  
Vol 47 (7) ◽  
pp. 769-777 ◽  
Author(s):  
K. C. Lee ◽  
Robert Barrie
Keyword(s):  
Ising Model ◽  
Green’S Function ◽  
Green's Function ◽  
Body Problem ◽  
Equations Of Motion ◽  
Function Technique ◽  
Many Body ◽  
One Dimensional ◽  
Spinless Fermion ◽  
The One

It is shown that the spin [Formula: see text] Ising model can be formulated as a spinless fermion many-body problem and that the Green's function technique can be applied to it. The hierarchy of Green's function equations of motion terminates at the (q + 1)-particle Green's function, where q is the coordination number. This finite number of equations yields Fisher's transformation of correlations. The technique discussed in this paper can be used to obtain exact results for the one-dimensional Ising model.

One-dimensional dislocations - III. Influence of the second harmonic term in the potential representation, on the properties of the model

1949 ◽  
Vol 200 (1060) ◽  
pp. 125-134 ◽  
Keyword(s):  
Potential Energy ◽  
Second Harmonic ◽  
Dislocation Model ◽  
Stability Conditions ◽  
One Dimensional ◽  
Harmonic Term ◽  
A Chain ◽  
The Stability ◽  
The One ◽  
Work Done

In the previous one-dimensional dislocation model, a single sinusoidal term was taken to represent the potential energy of the deposit as a function of its position on the substrate. In this model a more general representation of the potential, containing a second harmonic term as well, is used, and it is shown that the solution in this case is also expressible in terms of elliptic integrals. The displacements corresponding to a sequence of dislocations (or a single one) are calculated. The work done in generating a single dislocation by a force on a free end is derived and the stability conditions for such a chain determined. It turns out that the properties of single dislocations, especially as concerns their application to misfitting monolayers and oriented overgrowth, remain almost uninfluenced, unless the amplitude of the second harmonic term is so large as to produce a new minimum and provided the overall amplitude of the potential energy is taken to be constant. When the amplitude of the second harmonic term is large, so that the potential curve has a second minimum, a complete dislocation splits up into two halves which are the one-dimensional analogues of Shockley’s ‘half-dislocations’ in close-packed lattices. The equilibrium separation of the two halves, as well as the stability conditions for the existence of a single half, are determined.

A separated-flow model for collapsible-tube oscillations

1985 ◽  
Vol 157 ◽  
pp. 375-404 ◽  
Author(s):  
Claudio Cancelli ◽  
T. J. Pedley
Keyword(s):  
Equations Of Motion ◽  
Separated Flow ◽  
Propagation Speed ◽  
Adverse Pressure Gradient ◽  
Collapsible Tube ◽  
State Collapse ◽  
One Dimensional ◽  
Longitudinal Wall ◽  
Wave Propagation Speed ◽  
The One

A new model is presented to describe flow in segments of collapsible tube mounted between two rigid tubes and surrounded by a pressurized container. The new features of the model are the inclusion of (a) longitudinal wall tension and (b) energy loss in the separated flow downstream of the time-dependent constriction in a collapsing tube, in a manner which is consistent with the one-dimensional equations of motion. As well as accurately simulating steady-state collapse, the model predicts self-excited oscillations whose amplitude is large enough to be observable only if the flow in the collapsible tube becomes supercritical somewhere (fluid speed exceeding long-wave propagation speed). The dynamics of the oscillations is dominated by longitudinal movement of the point of flow separation, in response to the adverse pressure gradient associated with waves propagating backwards and forwards between the (moving) narrowest point of the constriction and the tube outlet.

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