Differential‐Operator Approximations to the Linear Boltzmann Equation
dc.contributor.author | Siegel, Armand | en_US |
dc.date.accessioned | 2010-05-06T21:00:04Z | |
dc.date.available | 2010-05-06T21:00:04Z | |
dc.date.issued | 1960-09 | en_US |
dc.identifier.citation | Siegel, Armand (1960). "Differential‐Operator Approximations to the Linear Boltzmann Equation." Journal of Mathematical Physics 1(5): 378-390. <http://hdl.handle.net/2027.42/69697> | en_US |
dc.identifier.uri | https://hdl.handle.net/2027.42/69697 | |
dc.description.abstract | A measure of deviation from equilibrium of an ensemble of particles is proposed, which is physically appropriate and of especially simple form when expressed in terms of the expansion coefficients of the ensemble distribution function with respect to the system of orthogonal polynomials obtained by using the equilibrium distribution function as weight function. The linear Boltzmann operator can then be expanded in a series of terms which, under certain circumstances, may be regarded as of successively diminishing magnitude in their effect on the rate of approach to equilibrium. This expansion of the operator is different from the expansion due to Kramers (later discussed by Moyal) in derivate moments, commonly used in approximate stochastic treatments of irreversible processes. With the aid of a theorem on definite operators, it is possible to break off the series at any point and thereby obtain a correspondingly accurate approximation to the linear Boltzmann operator, whose temporal solutions tend to the correct equilibrium distribution function. The first approximation is the Fokker‐Planck operator, exactly. The next approximation would be the appropriate operator to use when the stochastic variable begins to deviate appreciably from a linear dissipation law, etc. The method is applied to the ``Rayleigh process'' (ensemble of particles in a rarefied gas medium, the medium itself being in internal equilibrium), and the second approximation to the linear Boltzmann operator for this case is explicitly derived. A possible form for the second approximation in more general processes, suggested by this, is also given. | en_US |
dc.format.extent | 3102 bytes | |
dc.format.extent | 1010929 bytes | |
dc.format.mimetype | text/plain | |
dc.format.mimetype | application/pdf | |
dc.publisher | The American Institute of Physics | en_US |
dc.rights | © The American Institute of Physics | en_US |
dc.title | Differential‐Operator Approximations to the Linear Boltzmann Equation | en_US |
dc.type | Article | en_US |
dc.subject.hlbsecondlevel | Physics | en_US |
dc.subject.hlbtoplevel | Science | en_US |
dc.description.peerreviewed | Peer Reviewed | en_US |
dc.contributor.affiliationum | University of Michigan, Ann Arbor, Michigan | en_US |
dc.contributor.affiliationother | Boston University, Boston, Massachusetts | en_US |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/69697/2/JMAPAQ-1-5-378-1.pdf | |
dc.identifier.doi | 10.1063/1.1703668 | en_US |
dc.identifier.source | Journal of Mathematical Physics | en_US |
dc.identifier.citedreference | The Fokker‐Planck equation for the temporal evolution of the probability density function P(ξ,t)P(ξ,t) of a scalar variable ξ reads ∂P(ξ,t)∂t=∂∂ξaξP(ξ,t)+b2∂2∂ξ2P(ξ,t). Here aξaξ is (if the equation is applicable) minus the ensemble average rate of change of ξ due to “friction” or dissipative effects in general; i.e., 〈〉 = −aξ.〈ξ̇〉=−aξ. For a particle undergoing Brownian motion, 〈〉〈ξ̇〉 is literally due to friction, being attributable to viscosity; more generally, ξ may be any thermodynamic observable in its range of linear dissipation, according to the theories referred to in footnotes 2 and 3, The constant b is (again, if the equation is applicable) a measure of the amplitude of thermal fluctuations, or “noise.” In the mathematically equivalent Langevin formalism, ξ(t)ξ(t) is a random function of time satisfying the Langevin equation ξ̇+aξ=(b)12ϵ(t), where ϵ(t)ϵ(t) is the “ideal random function” normalized so that 〈[∫0tϵ(t)dt]2〉=t. ξ(t)ξ(t) will then be found to have a probability density satisfying the Fokker‐Planck equation as just given. Introductory treatments of these matters will be found in the well‐known review articles by S. Chandrasekhar [Revs. Modern Phys. 15, 1 (1943)] and by M. C. Wang and G. E. Uhlenbeck [Revs. Modern Phys. 17, 323 (1945)]. | en_US |
dc.identifier.citedreference | N. Hashitsume, Progr. Theoret. Phys. (Kyoto) 8, 461 (1952). | en_US |
dc.identifier.citedreference | L. Onsager and S. Machlup, Phys. Rev. 91, 1505 (1953). | en_US |
dc.identifier.citedreference | D. K. C. MacDonald, Phys. Rev. 108, 541 (1957). | en_US |
dc.identifier.citedreference | N. G. van Kampen, Phys. Rev. 110, 319 (1958). | en_US |
dc.identifier.citedreference | R. O. Davies, Physica 24, 1055 (1958). | en_US |
dc.identifier.citedreference | C. T. J. Alkemade, Physica 24, 1029 (1958). | en_US |
dc.identifier.citedreference | M. Lax, Revs. Modern Phys. 32, 25 (1960). | en_US |
dc.identifier.citedreference | N. G. van Kampen (unpublished report, 1959). | en_US |
dc.identifier.citedreference | See S. Chandrasekhar or M. C. Wang and G. E. Uhlenbeck, cited in footnote 1. | en_US |
dc.identifier.citedreference | Lord Rayleigb, Scientific Papers (Cambridge University Press, New York, 1902), Vol. 3, p. 273; discussed by C. S. Wang Chang and G. E. Uhlenbeck, Kinetic Theory of a Gas in Alternating Outside Force Fields, Engineering Research Institute Report 2457‐3‐T (University of Michigan, Ann Arbor, Michigan, 1956). | en_US |
dc.identifier.citedreference | F. D. Murnaghan, The Theory of Group Representations (The Johns Hopkins Press, Baltimore, 1938), p. 20. | en_US |
dc.identifier.citedreference | See, e.g., H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1945), Chap. 30. | en_US |
dc.identifier.citedreference | H. C. Brinkman, Physica 23, 82 (1957). | en_US |
dc.identifier.citedreference | H. A. Kramers, Physica 7, 284 (11940). | en_US |
dc.identifier.citedreference | S. Chandrasekhar, Revs. Modern Phys. 15, 1 (1943). | en_US |
dc.identifier.citedreference | M. C. Wang and G. E. Uhlenbeck, Revs. Modern Phys. 17, 323 (1945). | en_US |
dc.identifier.citedreference | J E. Moyal, J. Roy. Stat. Soc. (London) B11, 150 (1949). | en_US |
dc.identifier.citedreference | J. Keilson and J. E. Storer, Quart. Appl. Math. 10, 243 (1952). | en_US |
dc.identifier.citedreference | As pointed out by H. A. Kramers (footnote 15), the denvate moments αnαn are even or odd functions as n is an even or odd number. | en_US |
dc.identifier.citedreference | A. Einstein, Ann. Physik 17, 549 (1905). | en_US |
dc.owningcollname | Physics, Department of |
Files in this item
Remediation of Harmful Language
The University of Michigan Library aims to describe library materials in a way that respects the people and communities who create, use, and are represented in our collections. Report harmful or offensive language in catalog records, finding aids, or elsewhere in our collections anonymously through our metadata feedback form. More information at Remediation of Harmful Language.
Accessibility
If you are unable to use this file in its current format, please select the Contact Us link and we can modify it to make it more accessible to you.