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Bernhoff, N. (2019). Discrete quantum Boltzmann equation. In: AIP Conference Proceedings: . Paper presented at 31st International Symposium on Rarefied Gas Dynamics, RGD 2018, 23 July 2018 through 27 July 2018 (pp. 1-9). American Institute of Physics (AIP), 2132, Article ID 130011.
Open this publication in new window or tab >>Discrete quantum Boltzmann equation
2019 (English)In: AIP Conference Proceedings, American Institute of Physics (AIP), 2019, Vol. 2132, p. 1-9, article id 130011Conference paper, Published paper (Refereed)
Abstract [en]

In this work, we consider a Boltzmann equation for anyons. In particular, we study a general discrete velocity model of the equation, where the velocity variable is assumed to only take values from a given finite-such that the (finite) number of velocities is arbitrary-set of velocities. Included, as two limiting cases, is the discrete quantum Boltzmann equation (Nordheim-Boltzmann/Uehling-Uhlenbeck equation) for bosons and fermions. Mass, momentum, and energy are assumed to be conserved during collisions, and considering suitable discrete velocity models, they will also be the only collision invariants. The equilibrium distributions will be given by a transcendental equation, and only in some few cases-including the two limiting cases where they are Planckians-they will be explicitly expressed. However, there is an H-theorem, and therefore one can prove that for the spatially homogeneous equation, as time tends to infinity, as well as, for the steady equation in a half-space with slab-symmetry, as the space variable tends to infinity, the distribution function converges to an equilibrium distribution. Linearizing around an equilibrium distribution in a suitable way, we find that the obtained linearized operator has similar properties as the corresponding linearized operator for the discrete Boltzmann equation: E.g. it is symmetric and positive semi-definite. Hence, previously obtained results for the spatially homogeneous Cauchy problem and the steady half-space problem in a slab symmetry for the discrete Boltzmann equation, can be applied also in the considered quantum case.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2019
Series
AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-75689 (URN)10.1063/1.5119631 (DOI)2-s2.0-85070692522 (Scopus ID)9780735418745 (ISBN)
Conference
31st International Symposium on Rarefied Gas Dynamics, RGD 2018, 23 July 2018 through 27 July 2018
Available from: 2019-11-12 Created: 2019-11-12 Last updated: 2019-11-21Bibliographically approved
Bernhoff, N. (2018). Discrete Velocity Models for Polyatomic Molecules Without Nonphysical Collision Invariants. Journal of statistical physics, 172(3), 742-761
Open this publication in new window or tab >>Discrete Velocity Models for Polyatomic Molecules Without Nonphysical Collision Invariants
2018 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 172, no 3, p. 742-761Article in journal (Refereed) Published
Abstract [en]

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. Unlike for the Boltzmann equation, for DVMs there can appear extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and hence, without spurious ones, is called normal. The construction of such normal DVMs has been studied a lot in the literature for single species, but also for binary mixtures and recently extensively for multicomponent mixtures. In this paper, we address ways of constructing normal DVMs for polyatomic molecules (here represented by that each molecule has an internal energy, to account for non-translational energies, which can change during collisions), under the assumption that the set of allowed internal energies are finite. We present general algorithms for constructing such models, but we also give concrete examples of such constructions. This approach can also be combined with similar constructions of multicomponent mixtures to obtain multicomponent mixtures with polyatomic molecules, which is also briefly outlined. Then also, chemical reactions can be added.

Place, publisher, year, edition, pages
New York: Springer, 2018
Keywords
Boltzmann equation, Discrete velocity models, Collision invariants, Polyatomic molecules
National Category
Computational Mathematics Probability Theory and Statistics Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-68641 (URN)10.1007/s10955-018-2063-4 (DOI)000438686100004 ()
Available from: 2018-08-01 Created: 2018-08-01 Last updated: 2018-12-17
Bernhoff, N. (2017). Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinetic and Related Models, 10(4), 925-955
Open this publication in new window or tab >>Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations
2017 (English)In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 10, no 4, p. 925-955Article in journal (Refereed) Published
Abstract [en]

We consider some extensions of the classical discrete Boltzmann equation to the cases of multicomponent mixtures, polyatomic molecules (with a finite number of different internal energies), and chemical reactions, but also general discrete quantum kinetic Boltzmann-like equations; discrete versions of the Nordheim-Boltzmann (or Uehling-Uhlenbeck) equation for bosons and fermions and a kinetic equation for excitations in a Bose gas interacting witha Bose-Einstein condensate. In each case we have an H-theorem and so for the planar stationary half-space problem, we have convergence to an equilibrium distribution at infinity (or at least a manifold of equilibrium distributions). In particular, we consider the nonlinear half-space problem of condensation and evaporation for these discrete Boltzmann-like equations. We assume that the flow tends to a stationary point at infinity and that the outgoing flow is known at the wall, maybe also partly linearly depending on the incoming flow. We find that the systems we obtain are of similar structures as for the classical discrete Boltzmann equation (for single species), and that previously obtained results for the discrete Boltzmann equation can be applied after being generalized. Then the number of conditions on the assigned data at the wall needed for existence of a unique solution is found. The number of parameters to be specified in the boundary conditions depends on if we have subsonic or supersonic condensation or evaporation. All our results are valid for any finite number of velocities.

Place, publisher, year, edition, pages
American Institute of Mathematical Sciences, 2017
Keywords
Nordheim-Boltzmann equation, discrete velocity models, Boltzmann equation, mixtures, polyatomic molecules, chemical reactions, boundary layers, half-space problems
National Category
Other Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-48057 (URN)10.3934/krm.2017037 (DOI)000396010800003 ()
Available from: 2017-03-02 Created: 2017-03-02 Last updated: 2018-12-17Bibliographically approved
Bernhoff, N. (2016). Boundary Layers and Shock Profiles for the Broadwell Model. International Journal of Differential Equations, 2016, 1-8, Article ID 5801728.
Open this publication in new window or tab >>Boundary Layers and Shock Profiles for the Broadwell Model
2016 (English)In: International Journal of Differential Equations, ISSN 1687-9643, E-ISSN 1687-9651, Vol. 2016, p. 1-8, article id 5801728Article in journal (Refereed) Published
Abstract [en]

We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for the Broadwell model, the solutions are (at least partly) in qualitatively good agreement with the results for the discrete Boltzmann equation, that is the general discrete velocity model, and the full Boltzmann equation.

Place, publisher, year, edition, pages
Hindawi Publishing Corporation, 2016
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-45345 (URN)10.1155/2016/5801728 (DOI)000380712800001 ()
Available from: 2016-08-16 Created: 2016-08-16 Last updated: 2018-12-17Bibliographically approved
Bernhoff, N. & Vinerean, M. (2016). Discrete Velocity Models for Mixtures Without Nonphysical Collision Invariants. Journal of statistical physics, 165(2), 434-453
Open this publication in new window or tab >>Discrete Velocity Models for Mixtures Without Nonphysical Collision Invariants
2016 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 165, no 2, p. 434-453Article in journal (Refereed) Published
Abstract [en]

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that DVMs can also have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. For binary mixtures also the concept of supernormal DVMs was introduced, meaning that in addition to the DVM being normal, the restriction of the DVM to any single species also is normal. Here we introduce generalizations of this concept to DVMs for multicomponent mixtures. We also present some general algorithms for constructing such models and give some concrete examples of such constructions. One of our main results is that for any given number of species, and any given rational mass ratios we can construct a supernormal DVM. The DVMs are constructed in such a way that for half-space problems, as the Milne and Kramers problems, but also nonlinear ones, we obtain similar structures as for the classical discrete Boltzmann equation for one species, and therefore we can apply obtained results for the classical Boltzmann equation.

Place, publisher, year, edition, pages
New York: Springer, 2016
Keywords
Boltzmann equation, discrete velocity models, collision invariants, mixtures, boundary layers
National Category
Other Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-46394 (URN)10.1007/s10955-016-1624-7 (DOI)000385182000010 ()
Available from: 2016-10-05 Created: 2016-10-05 Last updated: 2019-07-12Bibliographically approved
Bernhoff, N. (2016). Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles. In: Andrew Ketsdever, Henning Struchtrup (Ed.), 30th International Symposium on Rarefied Gas Dynamics: RGD 30: . Paper presented at 30th International Symposium on Rarefied Gas Dynamics, 10-15 July 2016, Victoria, BC, Canada (pp. 040005-1-040005-8). American Institute of Physics (AIP), Article ID 040005.
Open this publication in new window or tab >>Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles
2016 (English)In: 30th International Symposium on Rarefied Gas Dynamics: RGD 30 / [ed] Andrew Ketsdever, Henning Struchtrup, American Institute of Physics (AIP), 2016, p. 040005-1-040005-8, article id 040005Conference paper, Published paper (Refereed)
Abstract [en]

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that, in difference to in the continuous case, DVMs can have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. The construction of such normal DVMs has been studied a lot in the literature for single species as well as for binary mixtures. For binary mixtures also the concept of supernormal DVMs has been introduced by Bobylevand Vinerean. Supernormal DVMs are defined as normal DVMs such that both restrictions to the different species are normal as DVMs for single species.

In this presentation we extend the concept of supernormal DVMs to the case of multicomponent mixtures and introduce it for polyatomic molecules. By polyatomic molecules we mean here that each molecule has one of a finite number of different internal energies, which can change, or not, during a collision. We will present some general algorithms for constructing such models, but also give some concrete examples of such constructions.

The two different approaches above can be combined to obtain multicomponent mixtures with a finite number of different internal energies, and then be extended in a natural way to chemical reactions.

The DVMs are constructed in such a way that we for the shock-wave problem obtain similar structures as for the classical discrete Boltzmann equation (DBE) for one species, and therefore will be able to apply previously obtained results for the DBE. In fact the DBE becomes a system of ordinary dierential equations (dynamical system) and the shock profiles can be seen as heteroclinic orbits connecting two singular points (Maxwellians). The previous results for the DBE then give us the existence of shock profiles for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. For binary mixtures this extension has already been addressed before by the author.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2016
Series
AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616 ; 1786
Keywords
Boltzmann equation, discrete velocity models, polyatomic molecules, mixtures, chemical reactions, shock profiles
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-47170 (URN)10.1063/1.4967543 (DOI)000389513200007 ()978-0-7354-1448-8 (ISBN)
Conference
30th International Symposium on Rarefied Gas Dynamics, 10-15 July 2016, Victoria, BC, Canada
Available from: 2016-11-16 Created: 2016-11-16 Last updated: 2018-12-17Bibliographically approved
Bernhoff, N. (2015). Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation. Journal of statistical physics, 159(2), 358-379
Open this publication in new window or tab >>Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation
2015 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 159, no 2, p. 358-379Article in journal (Refereed) Published
Abstract [en]

We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for a general discrete model of a quantum kinetic equation for excitations in a Bose gas. In the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations. These systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete Boltzmann equation. Then a classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete kinetic equation can be made. The number of additional conditions that need to be imposed for well-posedness is given by some characteristic numbers. These characteristic numbers are calculated for discrete models axially symmetric with respect to the x-axis. When the characteristic numbers change is found in the discrete as well as the continuous case. As an illustration explicit solutions are found for a small-sized model.

Place, publisher, year, edition, pages
Springer, 2015
Keywords
Bose-Einstein condensate, Low temperature kinetics, Discrete kinetic equation, Milne problem, Kramer problem
National Category
Other Mathematics Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-35795 (URN)10.1007/s10955-015-1190-4 (DOI)000351690500010 ()
Available from: 2015-04-10 Created: 2015-04-10 Last updated: 2018-12-17Bibliographically approved
Bernhoff, N. (2012). Boundary Layers and Shock Profiles for the Discrete Boltzmann Equation for Mixtures. Kinetic and Related Models, 5(1), 1-19
Open this publication in new window or tab >>Boundary Layers and Shock Profiles for the Discrete Boltzmann Equation for Mixtures
2012 (English)In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 5, no 1, p. 1-19Article in journal (Refereed) Published
Abstract [en]

We consider the discrete Boltzmann equation for binary gas mixtures. Some known results for half-space problems and shock profile solutions of the discrete Boltzmann for single-component gases are extended to the case of two-component gases. These results include well-posedness results for half-space problems for the linearized discrete Boltzmann equation, existence results for half-space problems for the weakly non-linear discrete Boltzmann equation, and existence results for shock profile solutions of the discrete Boltzmann equation. A characteristic number, corresponding to the speed of sound in the continuous case, is calculated for a class of symmetric models. Some explicit calculations are also made for a simplified 6+4 -velocity model.

Place, publisher, year, edition, pages
Springfield, MO: American Institute of Mathematical Sciences, 2012
Keywords
Boltzmann equation, discrete velocity models, mixtures, half-space problems, boundary layers, shock profiles
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-9201 (URN)10.3934/krm.2012.5.1 (DOI)000299867300001 ()
Available from: 2012-01-13 Created: 2012-01-13 Last updated: 2018-12-17Bibliographically approved
Bernhoff, N. (2012). Boundary layers for the nonlinear discrete Boltzmann equation: Condensing vapor flow in the presence of a non-condensable gas (1ed.). In: Michel Mareschal, Andrés Santos (Ed.), Michel Mareschal, Andrés Santos (Ed.), Proceedings of 28th International Symposium on Rarefied Gas Dynamics 2012: . Paper presented at 28th International Symposium on Rarefied Gas Dynamics 2012, July 9 - 13, Zaragoza (pp. 223-230). Melville, New York: American Institute of Physics (AIP)
Open this publication in new window or tab >>Boundary layers for the nonlinear discrete Boltzmann equation: Condensing vapor flow in the presence of a non-condensable gas
2012 (English)In: Proceedings of 28th International Symposium on Rarefied Gas Dynamics 2012 / [ed] Michel Mareschal, Andrés Santos, Melville, New York: American Institute of Physics (AIP), 2012, 1, p. 223-230Conference paper, Published paper (Refereed)
Abstract [en]

Half-space problems for the Boltzmann equation are of great importance in the study of the asymptotic behaviorof the solutions of boundary value problems of the Boltzmann equation for small Knudsen numbers. Half-space problems provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluid-dynamic-type equations in a neighborhood of the boundary. Here we consider a half-space problem of condensation for apure vapor in the presence of a non-condensable gas by using discrete velocity models (DVMs) of the Boltzmann equation. The Boltzmann equation can be approximated by DVMs up to any order, and these DVMs can be applied for numerical methods,but also for mathematical studies to bring deeper understanding and new ideas. For one-dimensional half-space problems,the discrete Boltzmann equation (the general DVM) reduces to a system of ODEs. We obtain that the number of parametersto be specified in the boundary conditions depends on whether the condensing vapor flow is subsonic or supersonic. Thisbehavior has earlier been found numerically. We want to stress that our results are valid for any finite number of velocities.This is an extension of known results for single-component gases (and for binary mixtures of two vapors) to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian, with a flow velocity towards thecondensed phase, at infinity, while the non-condensable gas tends to zero at infinity. Steady condensation of the vapor takes place at the condensed phase, which is held at a constant temperature. We assume that the vapor is completely absorbed, that the non-condensable gas is diffusively reflected at the condensed phase, and that vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution at the condensed phase, needed for the existence of a unique solution of the problem are investigated, assuming that the given distribution at the condensed phase is sufficiently close to the Maxwellian at infinity and that the total mass of the non-condensable gas is sufficiently small. Exact solutions and solvability conditions are found for a specific simplified discrete velocity model (with few velocities).

Place, publisher, year, edition, pages
Melville, New York: American Institute of Physics (AIP), 2012 Edition: 1
Series
AIP Conference Proceedings, ISSN 0094-243X ; 1501
Keywords
Boltzmann equation, boundary layers, discrete velocity models, half-space problem, non-condensable gas
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-16034 (URN)10.1063/1.4769509 (DOI)000312411200028 ()978-0-7354-1115-9 (ISBN)
Conference
28th International Symposium on Rarefied Gas Dynamics 2012, July 9 - 13, Zaragoza
Available from: 2012-12-02 Created: 2012-12-02 Last updated: 2018-12-17Bibliographically approved
Bernhoff, N. (2012). Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-condensable Gas. Journal of statistical physics, 147(6), 1156-1181
Open this publication in new window or tab >>Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-condensable Gas
2012 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 147, no 6, p. 1156-1181Article in journal (Refereed) Published
Abstract [en]

We consider a non-linear half-space problem related to the condensation problem for the discrete Boltzmann equation and extend some known results for a single-component gas to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian at infinity, as the non-condensable gas tends to zero at infinity. We assume that the vapor is completely absorbed and that the non-condensable gas is diffusively reflected at the condensed phase and that the vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution, needed for the existence of a unique solution of the problem are investigated. We also find exact solvability conditions and solutions for a simplified six+four-velocity model, as the given distribution is a Maxwellian at rest, and study a simplified twelve+six-velocitymodel.

Place, publisher, year, edition, pages
Berlin: Springer, 2012
Keywords
Boltzmann equation, boundary layers, discrete velocity models, half-space problem, non-condensable gas
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-14507 (URN)10.1007/s10955-012-0513-y (DOI)000305792300007 ()
Available from: 2012-08-16 Created: 2012-08-16 Last updated: 2018-12-17Bibliographically approved
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ORCID iD: ORCID iD iconorcid.org/0000-0003-1232-3272

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