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Stochastic processes and thermal fluctuations in superconductors Leung, ManChiu
Abstract
The generalized Langevin equation, which is derived by H. Mori,as a general, microscopic formulation of the stochastic theory of Brownian motions, is applied to evaluate the longitudinal attenuation rate of sound in weakcoupling, pure superconductors. In order to qualify the ultrasonic absorption as stochastic processes, the relaxation time of the random interactions on the phonon must be much shorter than the relaxation time of the phonon. We show that this is indeed the case by showing that the former is of the order of the relaxation time of the distribution function of the excitations from the BGS condensate. As compared, with the earlier result derived by first order perturbation theory, the inclusion of the lifetime effect has improved the agreement with experimental data. When Ƭ≿Δ,ω, it is shown that Cooper pairs participate in the absorption processes even when ω <2 Δ. A sum rule for the longitudinal ultrasonic absorption rate is also derived. We also employ the Lamrevin approach to evaluate the spectra, variances and covariances of the magnetic flux, the currents and the magnetisation in a lohec, thin, hollow superconducting cylinder. When the wall thickness of a cylinder is arbitrary, the langevin approach becomes inconvenient. Modifying slightly Kubo's formulation of perturbation theory to take care of the fact that generalized susceptibility may not be zero at infinite frequency, we establish in the classical limit, a general relation between generalized susceptibility αBA(ω) and the crossspectrum of two quantities A and B, where ω is the frequency. Putting ω to zero, a particularly simple relation between covariance and generalized susceptibility is obtained. This relation and fluctuation dissipation theorem are then applied to evaluate the spectra, variances and covariances of magnetic flux, currents and magnetisation for the case of a long, hollow, superconducting cylinder of arbitrary thickness. Setting the inner radius of the cylinder to be zero, we obtain the particular case of a solid cylinder and our theory is compared with an experiment by VantHull et. al., which measures the fluctuations of the magnetic flux in solid, metal cylinders. According to our theory, the thermal magnetic noise does not disappear when the metal becomes superconducting. With the dimensions of the tin rod used in the experiment, we expect theoretically, however, the standard deviation of the magnetic flux to drop more than an order of magnitude from its normal state value when becomes greater than 10⁻⁵. Experiment agrees with our theoretical expectation. More accurate data, however, are needed to decide whether our theory is quantitatively adequate. In the above two approaches, we have assumed the fluctuation of the electron density ns in the BCScondensate to be negligible. To take care of the fluctuation of pair density, we make use of the fact that the probability density of ns and ϕ is proportional to exp {βGs(ns, ϕ)} where ϕ is the magnetic flux and Gs is the Gibbs free energy. To avoid, mathematical complications, we keep to the case of a long, thin, hollow cylinder in the LandauGinzburg temperature domain. We also discuss the responses of magnetic flux and currents to an external magnetic field which changes stepwise at time t=0.
Item Metadata
Title 
Stochastic processes and thermal fluctuations in superconductors

Creator  
Publisher 
University of British Columbia

Date Issued 
1970

Description 
The generalized Langevin equation, which is derived by H. Mori,as a general, microscopic formulation of the stochastic theory of Brownian motions, is applied to evaluate the longitudinal attenuation rate of sound in weakcoupling, pure superconductors. In order to qualify the ultrasonic absorption as stochastic processes, the relaxation time of the random interactions on the phonon must be much shorter than the relaxation time of the phonon. We show that this is indeed the case by showing that the former is of the order of the relaxation time of the distribution
function of the excitations from the BGS condensate.
As compared, with the earlier result derived by first order perturbation theory, the inclusion of the lifetime effect has improved the agreement with experimental data. When Ƭ≿Δ,ω, it is shown that Cooper pairs participate in the absorption processes even when ω <2 Δ. A sum rule for the longitudinal ultrasonic absorption rate is also derived.
We also employ the Lamrevin approach to evaluate the spectra, variances and covariances of the magnetic flux, the currents and the magnetisation in a lohec, thin, hollow superconducting cylinder. When the wall thickness of a cylinder is arbitrary, the langevin approach becomes inconvenient. Modifying slightly Kubo's formulation of perturbation theory to take care of the fact that generalized
susceptibility may not be zero at infinite frequency, we establish in the classical limit, a general relation between generalized susceptibility αBA(ω) and the crossspectrum of two quantities A and B, where ω is the frequency. Putting ω to zero, a particularly simple relation between covariance and generalized susceptibility is obtained. This
relation and fluctuation dissipation theorem are then applied
to evaluate the spectra, variances and covariances of magnetic
flux, currents and magnetisation for the case of a long,
hollow, superconducting cylinder of arbitrary thickness.
Setting the inner radius of the cylinder to be zero, we
obtain the particular case of a solid cylinder and our theory
is compared with an experiment by VantHull et. al., which
measures the fluctuations of the magnetic flux in solid,
metal cylinders. According to our theory, the thermal
magnetic noise does not disappear when the metal becomes
superconducting. With the dimensions of the tin rod used
in the experiment, we expect theoretically, however, the
standard deviation of the magnetic flux to drop more than
an order of magnitude from its normal state value when
becomes greater than 10⁻⁵. Experiment agrees with our theoretical expectation. More accurate data, however, are needed to decide whether our theory is quantitatively adequate.
In the above two approaches, we have assumed the fluctuation of the electron density ns in the BCScondensate to be negligible. To take care of the fluctuation of pair density, we make use of the fact that the probability density of ns and ϕ is proportional to exp {βGs(ns, ϕ)} where ϕ is the magnetic flux and Gs is the Gibbs free energy. To avoid, mathematical complications, we keep to the case of a long, thin, hollow cylinder in the LandauGinzburg temperature domain.
We also discuss the responses of magnetic flux and currents to an external magnetic field which changes stepwise at time t=0.

Genre  
Type  
Language 
eng

Date Available 
20110520

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0084800

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.