A path integral approach to disordered systems

1969 ◽  
Vol 309 (1499) ◽  
pp. 457-472 ◽  
Keyword(s):  
Disordered Systems ◽  
Three Dimensional ◽  
Delta Function ◽  
Exact Expression ◽  
Integral Approach ◽  
Disordered System ◽  
Feynman Measure ◽  
Screened Coulomb Potential ◽  
Linear Integral ◽  
Linear Integral Equations

The exact expression for the average propagator of a completely disordered system is evaluated by using a method of expansion in terms of cumulants defined over the Feynman measure. The expansions are evaluated for a one-dimensional model of delta function potentials and for a three-dimensional screened Coulomb potential to give the asymptotic form of the density of states for | E | →∞. Using a functional Taylor expansion it is shown that the average propagator may be expanded to give approximate non-linear integral equations for either the average propagator or average Green function. Using the cumulant formulation it is shown that an effective quantum potential may be defined in terms of which the propagator may be calculated.

On water waves produced by ground motions

1983 ◽  
Vol 126 ◽  
pp. 27-58 ◽  
Author(s):  
Pierre C. Sabatier
Keyword(s):  
Ground Motion ◽  
Water Waves ◽  
Ground Motions ◽  
Three Dimensional ◽  
Surface Displacement ◽  
Physical Parameters ◽  
Response Pair ◽  
Small Times ◽  
Linear Integral ◽  
Linear Integral Equations

A linear and irrotational model is constructed to represent the formation of water waves by ground motions of a sloping bed. A survey of the constant depth case, given first, helps in understanding the mechanism of formation, and, in this oversimplified case, wave propagation away from a source, which is usually very asymmetric. The importance of asymmetry, which may produce trapped waves, is illustrated by an estimate of the propagation in a three-dimensional case. The formation of waves by a ground motion on a slope is then studied in detail. The problem is reduced to linear integral equations of the first kind. Using an inversion technique one constructs a source–response pair in which the source is ‘δ-like’ and the response is close to that which would be found if the depth was constant around the source. A general approximate solution is then derived, in both the two-dimensional and three-dimensional cases. Results for the sloping-bottom case are given for small times. They give initial values of surface displacement. They also enable one to determine the important physical parameters in the ground motion and to evaluate the efficiency of wave production.

Один подход к численному решению основного уравнения магнитостатики для конечного цилиндра в произвольном внешнем поле

2021 ◽  
pp. 22-34
Author(s):  
В.В. Дякин ◽  
О.В. Кудряшова ◽  
В.Я. Раевский
Keyword(s):  
Magnetic Field ◽  
Hypergeometric Functions ◽  
Three Dimensional ◽  
Legendre Functions ◽  
Arbitrary Configuration ◽  
Special Cases ◽  
Linear Integral ◽  
Magnetostatic Equation ◽  
Homogeneous Cylinder ◽  
Linear Integral Equations

The magnetostatics direct problem of calculating the resulting magnetic field strength from a homogeneous cylinder of finite dimensions placed in an external magnetic field of arbitrary configuration is considered. With the help of sufficiently voluminous analytical transformations using the basic properties of hypergeometric functions and Legendre functions, the solution of the basic three-dimensional magnetostatic equation for this configuration is reduced to solving of a certain number of systems of three one-dimensional linear integral equations. A simplified form of these systems for special cases of a constant external field and the resulting field on the cylinder axis is obtained.

scholarly journals Electromagnetic Waves Through Disordered Systems: Comparison of Intensity, Transmission and Conductance

2001 ◽  
Vol 694 ◽  
Author(s):  
Fredy R Zypman ◽  
Gabriel Cwilich
Keyword(s):  
Probability Distribution ◽  
Disordered Systems ◽  
Electromagnetic Waves ◽  
Delta Function ◽  
Rayleigh Distribution ◽  
Dirac Delta Function ◽  
Universal Form ◽  
Dirac Delta ◽  
Diffusive Regime ◽  
Analytical Expressions

AbstractWe obtain the statistics of the intensity, transmission and conductance for scalar electromagnetic waves propagating through a disordered collection of scatterers. Our results show that the probability distribution for these quantities x, follow a universal form, YU(x) = xne−xμ. This family of functions includes the Rayleigh distribution (when α=0, μ=1) and the Dirac delta function (α →+ ∞), which are the expressions for intensity and transmission in the diffusive regime neglecting correlations. Finally, we find simple analytical expressions for the nth moment of the distributions and for to the ratio of the moments of the intensity and transmission, which generalizes the n! result valid in the previous case.

scholarly journals On Solvability Conditions for a Certain Conjugation Problem

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 234
Author(s):  
Vladimir Vasilyev ◽  
Nikolai Eberlein
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Mellin Transform ◽  
Algebraic Equations ◽  
Conjugation Problem ◽  
Solvability Conditions ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

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