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Browsing Department of Mathematics by Author "Coffey, William T."
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Item The Effective Eigenvalue Method and Its Application to Stochastic Problems in Conjunction with the Nonlinear Langevin Equation(1993) Coffey, William T.; Kalmykov, Yuri P.; Massawe, Estomih S.The concept of the effective eigenvalue appears to have been originally introduced into the study of relaxation problems in statistical physics by Leontovich.’ It was later developed and applied (sometimes implicitly) to a variety of stochastic problems in laser polar fluids,’ polymers,8 nematic liquid crystals? lo etc. In the present context, namely the theory of the Brownian motion, the method constitutes a truncation procedure which allows one using simple assumptions to obtain closed-form approximations to the solution of certain infinite hierarchies of differential-difference equations in the time variables. These magnetic domains, equations govern the time behavior of the statistical averages characterizing the relaxation of nonlinear stochastic systems. Thus, their solution is needed to calculate observable quantities such as the relaxation times and dynamic susceptibilities of the systemItem Effective-Eigenvalue Approach to the Nonlinear Langevin Equation for the Brownian motion in a Tilted Periodic Potential. II. Application to the Ring-Laser Gyroscope(1993) Coffey, William T.; Kalmykov, Yuri P.; Massawe, Estomih S.The effective-eigenvalue method is used to obtain an approximate solution for the mean beat-signal spectrum for the ring-laser gyroscope in the presence of quantum noise. The accuracy of the effective-eigenvalue method is demonstrated by comparing the exact and approximate calculations. It shows clearly that the effective-eigenvalue method yields a simple and concise analytical description of the solution of the problem under consideration.Item Effective-Eigenvalue Approach to the Nonlinear Langevin Equation for the Brownian motion in a Tilted Periodic Potential: Application to the Josephson Tunneling Junction(1993) Coffey, William T.; Kalmykov, Yuri P.; Massawe, Estomih S.The concept of the effective eigenvalue appears to have been originally introduced into the study of relaxation problems in statistical physics by Leontovich.’ It was later developed and applied (sometimes implicitly) to a variety of stochastic problems in laser polar fluids,’ polymers,8 nematic liquid crystals? lo etc. In the present context, namely the theory of the Brownian motion, the method constitutes a truncation procedure which allows one using simple assumptions to obtain closed-form approximations to the solution of certain infinite hierarchies of differential-difference equations in the time variables. These magnetic domains, equations govern the time behavior of the statistical averages characterizing the relaxation of nonlinear stochastic systems. Thus, their solution is needed to calculate observable quantities such as the relaxation times and dynamic susceptibilities of the system.Item Exact Analytic Formula for the Correlation Time of a Single-Domain Ferromagnetic Particle(1994) Coffey, William T.; Crothers, D. S. F.; Kalmykov, Yuri P.; Massawe, Estomih S.; Waldron, J. T.Exact solutions for the longitudinal relaxation time T∥ and the complex susceptibility χ∥(ω) of a thermally agitated single-domain ferromagnetic particle are presented for the simple uniaxial potential of the crystalline anisotropy considered by Brown [Phys. Rev. 130, 1677 (1963)]. This is accomplished by expanding the spatial part of the distribution function of magnetic-moment orientations on the unit sphere in the Fokker-Planck equation in Legendre polynomials. This leads to the three-term recurrence relation for the Laplace transform of the decay functions. The recurrence relation may be solved exactly in terms of continued fractions. The zero-frequency limit of the solution yields an analytic formula for T∥ as a series of confluent hypergeometric (Kummer) functions which is easily tabulated for all potential-barrier heights. The asymptotic formula for T∥ of Brown is recovered in the limit of high barriers. On conversion of the exact solution for T∥ to integral form, it is shown using the method of steepest descents that an asymptotic correction to Brown’s high-barrier result is necessary. The inadequacy of the effective-eigenvalue method as applied to the calculation of T∥ is discussed.Item Exact Analytic Formula for the Correlation Times for Single Domain Ferromagnetic Particles.(Elsevier, 1993) Coffey, William T.; Crothers, D. S. F.; Kalmykov, Yuri P.; Massawe, Estomih S.; Waldron, J. T.Exact solutions for the longitudinal relaxation time T∥ and the complex susceptibility χ∥(ω) of a thermally agitated single domain ferromagnetic particle are presented for the simple uniaxial (Maier-Saupe) potential of the crystalline anisotropy considered by Brown [Phys. Rev. 130 (1963) 1677].Item Exact Solution for the Correlation Times of Dielectric Relaxation of a Single Axis Rotator with Two Equivalent Sites(1993) Coffey, William T.; Kalmykov, Yuri P.; Massawe, Estomih S.; Waldron, J. T.It is shown how exact formulas for the longitudinal and transverse dielectric correlation times and complex polar&ability tensor, of a single axis rotator with two equivalent sites may be found. This is accomplished by writing the Laplace transforms of the dipole autocorrelation functions as three term recurrence relations and solving them in terms of continued fractions. The solution of these recurrence relations, in the zero frequency limit, yields the correlation times in terms of modified Bessel functions of the first kind. The previous result of Lauritzen and Zwanzig for the longitudinal relaxation time, based on an asymptotic expansion of the SturmLiouville equation, is regained in the limit of high potential barriers.