scholarly journals Tensor-Product Discretization for the Spatially Inhomogeneous and Transient Boltzmann Equation in Two Dimensions

2017 ◽  
Vol 3 ◽  
pp. 219-248 ◽  
Author(s):  
Philipp Grohs ◽  
Ralf Hiptmair ◽  
Simon Pintarelli
Keyword(s):  
Boltzmann Equation ◽  
Tensor Product ◽  
Two Dimensions ◽  
Spatially Inhomogeneous

RENORMALIZATION AND BOLTZMANN EQUATIONS IN THERMAL QUANTUM FIELD THEORY

1995 ◽  
Vol 10 (11) ◽  
pp. 1693-1700 ◽  
Author(s):  
H. CHU ◽  
H. UMEZAWA
Keyword(s):  
Boltzmann Equation ◽  
Renormalization Scheme ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Boltzmann Equations ◽  
Self Energy ◽  
Spatially Inhomogeneous ◽  
Thermo Field Dynamics ◽  
Self Consistent ◽  
Consistent Manner

The renormalization scheme in nonequilibrium thermal quantum field theories is reexamined. Instead of the self-energy diagonalization scheme, we propose to diagonalize Green’s function at equal time. This eliminates the problem of on-shell definition related to time-dependent energies and spatially inhomogeneous situations, and yields a Boltzmann equation that contains memory effect. The new diagonalization scheme and the derivation of the Boltzmann equation from it can be applied to any thermal situation. It allows the treatment of a nonequilibrium problem beyond perturbational calculations in a self-consistent manner. The results are applicable to both thermo field dynamics and the closed time path formalism.

UNIFORM Lp-STABILITY ESTIMATE FOR THE SPATIALLY INHOMOGENEOUS BOLTZMANN EQUATION NEAR VACUUM

2008 ◽  
Vol 05 (04) ◽  
pp. 713-739 ◽  
Author(s):  
SEUNG-YEAL HA ◽  
MITSURU YAMAZAKI ◽  
SEOK-BAE YUN
Keyword(s):  
Stability Analysis ◽  
Boltzmann Equation ◽  
Stability Estimate ◽  
Functional Approach ◽  
Classical Solutions ◽  
Nonlinear Functional ◽  
Spatially Inhomogeneous ◽  
Nonlinear Functionals ◽  
Spatially Inhomogeneous Boltzmann Equation

We present a new uniform Lp-stability theory for the spatially inhomogeneous Boltzmann equation near vacuum via the nonlinear functional approach proposed by the first author. Our stability analysis is based on new nonlinear functionals which are equivalent to the pth power of Lp-distance. The L1-nonlinear functionals play the key role of "modulators" which make the accumulative functional be non-increasing in time t along classical solutions.

General solution of a Boltzmann equation, and the formation of Maxwellian tails

1981 ◽  
Vol 374 (1758) ◽  
pp. 371-400 ◽  
Keyword(s):  
Boltzmann Equation ◽  
General Solution ◽  
Cross Section ◽  
Relative Velocity ◽  
Collision Cross Section ◽  
Two Dimensions ◽  
Time Interval ◽  
Initial Condition ◽  
Finite Time Interval ◽  
Classical Boltzmann Equation

A classical Boltzmann equation is studied. The equation describes the evolution towards the Maxwellian equilibrium state of a homogeneous, isotropic gas where the collision cross section is inversely proportional to the relative velocity of the colliding particles. After Tjon & Wu (1979), the problem is transformed into a mathematically equivalent one, itself a model Boltzmann equation in two dimensions. Working in the context of the latter equation, a formal derivation of the general solution is presented. First a countable ensemble of particular solutions, called pure solutions , is constructed. From these, via a non-linear combination mechanism, the general solution is obtained in a form appropriate for direct numerical computation. The validity of the solution depends upon its containment in a well defined Hilbert space H~ Given that the initial condition lies within H~ it is proved that at least for a small finite time interval it remains in H~.

scholarly journals Variance Measures for Symmetric Positive (Semi-) Definite Tensors in Two Dimensions

2021 ◽  
pp. 3-22
Author(s):  
Magnus Herberthson ◽  
Evren Özarslan ◽  
Carl-Fredrik Westin
Keyword(s):  
Tensor Product ◽  
Fourth Order ◽  
Equivalence Problem ◽  
Elasticity Tensor ◽  
Two Dimensions ◽  
Second Order ◽  
Two Dimensional ◽  
Order Tensor ◽  
Symmetry Properties ◽  
Fourth Order Tensor

AbstractCalculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor $$R_{abcd}$$ R abcd . To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor $$R_{abcd}$$ R abcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors $$R_{abcd}$$ R abcd and $$\widetilde{R}_{abcd}$$ R ~ abcd . In terms of components, such an equivalence means that components $$R_{ijkl}$$ R ijkl of the first tensor will transform into the components $$\widetilde{R}_{ijkl}$$ R ~ ijkl of the second tensor for some change of the coordinate system.

THE BOLTZMANN EQUATION AND THE GROUP TRANSFORMATIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 443-476 ◽  
Author(s):  
A.V. BOBYLEV
Keyword(s):  
Boltzmann Equation ◽  
Special Role ◽  
Spatially Inhomogeneous ◽  
The Boltzmann Equation ◽  
Symmetry Properties ◽  
Full Equation ◽  
Spatially Inhomogeneous Boltzmann Equation ◽  
Spatially Homogeneous ◽  
Spatially Homogeneous Boltzmann Equation

This paper is devoted to the investigation of group properties of the nonlinear Boltzmann equation. The complete Lie group of invariant transformations for the spatially inhomogeneous Boltzmann equation is constructed. The generalization to the Lie-Backlund groups is given for the spatially homogeneous case. It is shown that there are only two non-trivial group transformations for the Boltzmann equation in the wide class of Lie and Lie-Backlund transformations. Some consequences of these symmetry properties are discussed. The special role of Galileo group and the analogy between the spatially homogeneous Boltzmann equation and the full equation are also investigated.

Sign in / Sign up

Export Citation Format

Share Document