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Long range particle dynamics and the linear Boltzmann equation
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Egginton, Matthew (2018) Long range particle dynamics and the linear Boltzmann equation. PhD thesis, University of Warwick.
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WRAP_Theses_Egginton_2018.pdf - Submitted Version - Requires a PDF viewer. Download (2193Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b3157873
Abstract
This thesis aims to give full and complete details of the first proof that the particle density for a tagged particle interacting with a background of particles via a long range potential ϕ converges weak-* to a weak solution of the linear Boltzmann equation for ϕ . This convergence is shown to hold for potentials where there is a τ > 0, and such that for sufficiently large [x] we have
∇∅(x)≤Ce-|x|3/2÷τ
The main difficulty in this context are the many grazing collisions in the particle dynamics which prevents a Markovian structure of the dynamics. We remove grazing collisions via the use of a regularisation parameter. This enables us to consider an associated short range evolution, which we describe on a space of marked trees to encode the collisional history of the tagged particle. This description then enables a specification of Markovian dynamics by removing a set of trees that exhibit recollisions. We then relate this evolution with the Markovian evolution of the linear Boltzmann equation on this space.
The difference between dynamics with and without grazing collisions are estimated by comparing the contribution from near collisions with a bound on the time of collision, and the contribution from grazing collisions by using an L∞ estimate on the potential.
The remaining error for the contribution of the grazing collisions on solutions of the linear Boltzmann equation are estimated by estimating the difference between deviation angles with and without grazing collisions.
| Item Type: | Thesis (PhD) | ||||
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| Subjects: | Q Science > QC Physics | ||||
| Library of Congress Subject Headings (LCSH): | Transport theory, Markov spectrum | ||||
| Official Date: | 9 February 2018 | ||||
| Dates: |
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| Institution: | University of Warwick | ||||
| Theses Department: | Mathematics Institute | ||||
| Thesis Type: | PhD | ||||
| Publication Status: | Unpublished | ||||
| Supervisor(s)/Advisor: | Theil, Florian ; Melbourne, Ian. | ||||
| Sponsors: | Engineering and Physical Sciences Research Council | ||||
| Extent: | v, 130 leaves. | ||||
| Language: | eng |
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