The Rayleigh Model: Singular Transport Theory in One Dimension

We present a comprehensive account of the special ‘Rayleigh piston’ model for the spatial and velocity relaxation of an ensemble of labelled test-particles in a one-dimensional heat-bath of particles with identical mass. This model, originally formulated by Rayleigh in 1891 but since largely neglect...

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Bibliographic Details
Published in:Philosophical Transactions of the Royal Society of London, Series A: Mathematical and Physical Sciences Series A: Mathematical and Physical Sciences, 1982-06, Vol.305 (1490), p.383-440
Main Authors: Hoare, M. R., Raval, S., Rahman, Mizan
Format: Article
Language:English
Subjects:
Approximation
Autocorrelation
Eigenfunctions
Eigenvalues
Laplace transformation
Mathematical expressions
Mathematical functions
Particle collisions
Transport theory
Velocity
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Summary:We present a comprehensive account of the special ‘Rayleigh piston’ model for the spatial and velocity relaxation of an ensemble of labelled test-particles in a one-dimensional heat-bath of particles with identical mass. This model, originally formulated by Rayleigh in 1891 but since largely neglected, is in effect a prototype for all later models in singular particle transport theory and serves to illustrate the mathematical problems associated with the occurrence of singular eigenfunctions and continuous spectra of a scattering operator. Although other idealized scattering models are known, the Rayleigh model remains a unique example of an exactly soluble singular system which, in including conservation laws and time-reversal symmetry in scattering, retains a degree of mechanical realism.
ISSN:1364-503X
0080-4614
0962-8428
1471-2962
2054-0272
DOI:10.1098/rsta.1982.0042