scholarly journals Exact non-equilibrium solutions of the Einstein--Boltzmann equations

1994 ◽  
Vol 11 (1) ◽  
pp. 203-225 ◽  
Author(s):  
R Maartens ◽  
F P Wolvaardt
Keyword(s):  
Equilibrium Solutions ◽  
Boltzmann Equations ◽  
Non Equilibrium

scholarly journals Stationary Non-equilibrium Solutions for Coagulation Systems

2021 ◽  
Vol 240 (2) ◽  
pp. 809-875
Author(s):  
Marina A. Ferreira ◽  
Jani Lukkarinen ◽  
Alessia Nota ◽  
Juan J. L. Velázquez
Keyword(s):  
Power Law ◽  
Source Term ◽  
Continuous Model ◽  
Stationary Solutions ◽  
Weight Functions ◽  
Equilibrium Solutions ◽  
Coagulation Rate ◽  
Diffusion Limited Aggregation ◽  
Coagulation Equations ◽  
Non Equilibrium

AbstractWe study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.

scholarly journals KINETICS QUASIINVARIANTS OF CHEMICAL REACTIONS IN CLOSED SYSTEMS

2019 ◽  
Vol 62 (6) ◽  
pp. 47-52 ◽  
Author(s):  
Vladislav Kh. Fedotov ◽  
Nikolay I. Kol'tsov
Keyword(s):  
Chemical Reactions ◽  
Multiple Equilibria ◽  
Dual Method ◽  
Boundary Values ◽  
Equilibrium Solutions ◽  
Correct Operation ◽  
Exact Time ◽  
Experiment Method ◽  
One Step ◽  
Non Equilibrium

The limitations of the dual-method and its extended version of the multi-experiment method in determining the exact time kinetic (thermodynamic) invariants and approximate invariants (quasiinvariants) of chemical reactions in closed isothermal systems are discussed. It is shown that for reactions, which allow except for internal equilibria, also boundary equilibria (multiple equilibria, multiequilibrium), for example, autocatalytic ones, there are always some "inconvenient" boundary values of reagent concentrations. These "uncomfortable" values cannot be used as the initial concentrations (conditions) for non-equilibrium multi-experiments (forward, reverse or intermediate), because for these values of non-equilibrium solutions cease to exist and, consequently, the reaction can proceed only in the equilibrium regime. As a result, the "usual " method of multi-experiments, using only the boundary values of the equilibrium concentrations of reagents, is not applicable. In this paper, a generalization of this method is proposed and a technique for conducting multi-experiments is developed, which is applicable for wider classes of reactions, including those with boundary equilibria, as well as autocatalytic reactions. This generalized method of multi-experiments (MME) allows one to bypass the limitations of the conventional multi-experiment method (dual-method) and to determine the exact time thermodynamic (kinetic) invariants of linear and some nonlinear chemical reactions, as well as approximate time invariants of any nonlinear chemical reactions in closed isothermal systems. The conditions of multi-experiments which are necessary for the correct operation of this method are determined. Examples of using the generalized method of multi-experiments for one-step and two-step nonlinear reactions with one and two independent reagents, respectively, are given. The kinetic time invariants and quasinvariants found with this method are compared with the exact solutions for the cases where they exist.

scholarly journals Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

2021 ◽  
Author(s):  
Marina A. Ferreira ◽  
Jani Lukkarinen ◽  
Alessia Nota ◽  
Juan J. L. Velázquez
Keyword(s):  
Source Term ◽  
Equilibrium Solutions ◽  
Equilibrium Conditions ◽  
Measure Concentration ◽  
Localization Property ◽  
Smoluchowski Coagulation Equation ◽  
Coagulation Equation ◽  
Asymptotic Scaling ◽  
Coagulation Kernel ◽  
Non Equilibrium

AbstractWe consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

Equilibrium and Non-Equilibrium Solution of Nonlinear Population Evolution Systems

2013 ◽  
Vol 423-426 ◽  
pp. 2244-2248
Author(s):  
Jin Mei Wang
Keyword(s):  
Equilibrium Solution ◽  
Equilibrium Solutions ◽  
Existence Of Equilibrium ◽  
Population Evolution ◽  
Evolution Systems ◽  
Fixed Population ◽  
Non Equilibrium

We provide conditions for the existence of equilibrium and non-equilibrium solutions about nonlinear population evolution systems with fixed population.

A Parametric Study of Phonon Energy Transport in Semiconductor Devices

2000 ◽  
Author(s):  
D. Greg Walker ◽  
Tim S. Fisher ◽  
Jeremy Ralston-Good ◽  
Ron D. Schrimpf
Keyword(s):  
Gallium Arsenide ◽  
Time Scales ◽  
Parametric Study ◽  
Particle Physics ◽  
Energy Transport ◽  
Semiconductor Devices ◽  
Continuum Models ◽  
Boltzmann Equations ◽  
Variation Of Parameters ◽  
Non Equilibrium

Abstract Microscale energy transport typically can not be described by traditional continuum models. This is especially true for semiconductor devices where length and time scales are continually becoming smaller. These conditions tend to place the physics of the energy transport into the non-equilibrium regime. Moments of the Boltzmann equations which describe average quantities of particle physics can be used under these conditions to obtain solutions to the transport problem. Most simulations, using this approach neglect non-equilibrium energy transport through phonon interactions. The goal of this work is to identify parameters crucial to non-equilibrium energy transport in semiconductor devices specifically where phonon interactions are concerned through a systematic variation of parameters. A Gallium Arsenide device that has been examined in previous research will be used as a benchmark for comparisons.

Ghost Fluid Based Immersed Boundary Treatment for Lattice Boltzmann Method

2009 ◽  
Author(s):  
Arpit Tiwari ◽  
Rahul Samala ◽  
S. P. Vanka
Keyword(s):  
Distribution Function ◽  
Couette Flow ◽  
Lattice Boltzmann ◽  
Image Point ◽  
Immersed Boundary ◽  
Normal Velocity ◽  
Boltzmann Equations ◽  
Boundary Treatment ◽  
Boltzmann Method ◽  
Non Equilibrium

An immersed boundary method has been developed for Lattice Boltzmann Equations via ghost fluid approach. Image points of the ghost points inside the fluid domain are obtained by extrapolation along the boundary normal. Velocity, density and non-equilibrium value of the distribution function at ghost points are extrapolated from image point values which are calculated by interpolation from the boundary and fluid domain. The distribution function at ghost points is computed from the extrapolated non-equilibrium part and the equilibrium part which is obtained from extrapolated values of the velocities. The method is found to be second order accurate. The method is applied to concentric 2D Couette flow and 3D Taylor–Couette flow.

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