Classical Scattering Theory. Elastic Collisions

Miles, John; Dahler, John S.

doi:10.1063/1.1673031

Cited by 30 publications

(7 citation statements)

References 2 publications

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“…In nuclear physics, Remler and collaborators initiated and developed a program in which nuclear reaction theory is formulated in the language of the Wigner distribution function (Remler, 1975(Remler, , 1981Remler andSathe, 1975, 1978). Meanwhile, phase-space methods (sometimes classical) were being developed by quantum chemists in order to elucidate chemical reaction problems (Brown and Heller, 1982;Eu, 1971Eu, , 1975Heller, 1976Heller, , 1977Lee and Scully, 1980;Miles and Dahler, 1970). We hope that the present work will encourage Copyright © 1983 The Nobel Foundation 245 communication among these disciplines and will lead to the recognition of the phase-space method as one of universal utility.…”

Section: Introductionmentioning

confidence: 98%

“…(2.26) may be regarded as a quantum Liouville equation. Information on classical scattering theory in the phase-space language can be found in Prigogine (1959) as well as in Miles and Dahler (1970) and Eu (1971Eu ( , 1973. Equation (2.29) is equivalent to the constancy of the phase-space density (df /dt =0), provided the particles move on classical orbits specified by Newton's laws: dr.!dt =p /m, dp !dt =-VV.…”

Section: A Definitionsmentioning

confidence: 99%

“…and pure quantum theory have led to analyses of semiclassical, or just classical, approaches to scattering and bound-state problems. The classical Liouville appoach to potential scattering is discussed by Prigogine ( 1959); the formal structure of this approach is analyzed in detail by Miles and Dahler (1970) in close analogy to the quantum treatment of the scattering problem. For a discussion of the three-body problem, we refer the reader to Eu (1971).…”

Section: Atomic and Molecular Collisionsmentioning

confidence: 99%

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Quantum collision theory with phase-space distributions

1983

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Quantum-mechanical phase-space distributions, introduced by Wigner in 1932, provide an intuitive alternative to the usual wave-function approach to problems in scattering and reaction theory. The aim of the present work is to collect and extend previous efforts in a unified way, emphasizing the parallels among problems in ordinary quantum theory, nuclear physics, chemical physics, and quantum field theory. The method is especially useful in providing easy reductions to classical physics and kinetic regimes under suitable conditions. Section II, dealing in detail with potential scattering of a spinless nonrelativistic particle, provides the background for more complex problems. Following a brief description of the two-body problem, the authors address the N-body problem with special attention to hierarchy closures, BoltzmannVIasov equations, and hydrodynamic aspects. The final section sketches past and possibly future applications to a wide variety of problems.

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Section: Introductionmentioning

confidence: 98%

Section: A Definitionsmentioning

confidence: 99%

Section: Atomic and Molecular Collisionsmentioning

confidence: 99%

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Quantum collision theory with phase-space distributions

1983

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“…Formal classical theory of scattering 1,3,7,9 can be formulated in the phase space in a manner parallel to the quantum mechanical scattering theory, and it holds some advantages for statistical mechanics and, especially, for kinetic theory as has been frequently demonstrated in kinetic theory investigations 2,3,11,12 in which classical collision operators are used in a formalism analogous to quantum scattering theory. Nevertheless, the meanings of such collision operators have not been studied beyond the formal definition level.…”

Section: Classical Scattering Theory In Phase Spacementioning

confidence: 99%

“…Classical collision operators are generally defined in formal analogy 7,8 to quantum mechanical collision operators in Hilbert space, but their definitions have been made without much attention paid to the space of functions in which the operators live. Clearly, study of classical collision operators would require a space of eigenfunctions for the underlying Liouville operator in the phase space.…”

Section: Introductionmentioning

confidence: 99%

Eigenfunctions for Liouville Operators, Classical Collision Operators, and Collision Bracket Integrals in Kinetic Theory Made Amenable to Computer Simulations

Eu¹

2012

Bulletin of the Korean Chemical Society

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In the kinetic theory of dense fluids the many-particle collision bracket integral is given in terms of a classical collision operator defined in the phase space. To find an algorithm to compute the collision bracket integrals, we revisit the eigenvalue problem of the Liouville operator and re-examine the method previously reported [Chem. Phys. 1977, 20, 93]. Then we apply the notion and concept of the eigenfunctions of the Liouville operator and knowledge acquired in the study of the eigenfunctions to cast collision bracket integrals into more convenient and suitable forms for numerical simulations. One of the alternative forms is given in the form of time correlation function. This form, on a further manipulation, assumes a form reminiscent of the ChapmanEnskog collision bracket integrals, but for dense gases and liquids as well as solids. In the dilute gas limit it would give rise precisely to the Chapman-Enskog collision bracket integrals for two-particle collision. The alternative forms obtained are more readily amenable to numerical simulation methods than the collision bracket integrals expressed in terms of a classical collision operator, which requires solution of classical Lippmann-Schwinger integral equations. This way, the aforementioned kinetic theory of dense fluids is made fully accessible by numerical computation/simulation methods, and the transport coefficients thereof are made computationally as accessible as those in the linear response theory.

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