KINETIC EQUATIONS WITH INTERMOLECULAR POTENTIALS: AVOIDING THE BOLTZMANN ASSUMPTION
|
|
- Howard Freeman
- 6 years ago
- Views:
Transcription
1 KINETIC EQUATIONS WITH INTERMOLECULAR POTENTIALS: AVOIDING THE BOLTZMANN ASSUMPTION William Greenberg Dept. of Mathematics, Virginia Tech Blacksburg, Virginia USA VT.EDU ABSTRACT The Boltzmann equation describes the time eolution of a dilute gas, and is the best known transport equation in kinetic theory. Its drawback is that, although it allows for a ariety of long and short range intermolecular potentials, it does not predict non-ideal transport coefficients. To model gas kinetics in denser regimes, the most successful Boltzmann-like equation has been the Enskog equation, which takes into account molecular diameters. Howeer, it does hae the drawback that it does not include intermolecular potentials. We present seeral extentions of the Enskog theory which model intermolecular forces. The first of these includes a piecewise-constant short range potential. The second models longrange forces by coupling the Enskog equation to the electromagnetic field ia a Vlaso collision term. Finally, we introduce discrete elocity models. All of these open new fields for numerical analysis. I. Introduction The fundamental problem of kinetic theory is to describe the properties of a gas in terms of the one-particle distribution function f(x,,t), generally gien by an appropriate eolution equation. This distribution function is defined so that f(x,,t) dx d is the number of molecules which at time t lie in a olume element dx around a point x with elocity in a olume element d around. Its moments then proide fluid-dynamical ariables characterizing the gas. One of the first eolution equations for the distribution function was the Boltzmann equation, first proposed by Ludwig Boltzmann in 872.[] It has the particularly useful feature that its bilinear collision term can model a great ariety of intermolecular potentials. This equation gies the most precise modeling of dilute gases; for example, it is the equation of choice for the precise modeling of atmospheric gases. The Boltzmann equation may be written: (, x t,) + x f (, xt,) = 3 2 = dd ' ε{ f ( x, ', t) f ( x, ', t) f ( xt,, ) f ( x,, t)} K( ε,, t) Here,, and ', ' may be thought of as the pre- and post-collision elocities of two colliding particles with unit ector ε in the direction of '. The kernel K characterizes the interaction potential between the colliding particles. The post-collision elocities are related to the pre-collision elocities by momentum and energy conseration: ' = ε ( ) ε, ' = + ε ( ) ε The Boltzmann equation is obtained by taking a limit as the diameter of molecules anishes while the density is held constant. This is a fundamental assumption of 47 MSAS'2004
2 Boltzmann, and is why this equation describes a gas of point particles; its equation of state is that of an ideal gas, and yields only triial transport coefficients. Consequently, it can not be expected to proide an adequate description of a dense gas. In 92 Enskog proposed a scheme to surmount this Boltzmann assumption, thereby obtaining a modified Boltzmann-like equation which takes into account the non-zero molecular diameter.[2] In particular, the collision transfer does not take place at a gien point, but rather momentum is transferred upon collision from the center of one molecule to the center of the other, which is at a distance equal to the molecular diameter. The Enskog equation for molecules of diameter σ may be written: ( x, t, ) + f( xt,, ) = x 3 2 = d ' d εθ( V ε) { Y( f, x+ σε) f ( x, ', t) f ( x+ σε, ', t) Y( f, x σε) f( x,, t) f( x σε,, t)} where the kernel Y is a function of the local density at the contact point of the molecules. Here, θ is the characteristic function of the positie half-line, and V =. The collision frequency factor Y, which should represent the pair correlation function corresponding to the system in uniform equilibrium, may be chosen so that the Enskog equation, like the Boltzmann equation, satisfies an H-theorem. The Enskog equation allows for the computation of transport coefficients which, compared with those measured experimentally, are more accurate than the ones proided by the Boltzmann equation. Howeer, it does hae one eident limitation. It describes only the collision of hard spheres; ie., its kernel does not allow for any intermolecular interactions. It is this limitation we wish to address. Three excellent references for both Boltzmann and Enskog theory are [3,4,5]. II. Enskog Square Well Equation A realistic an der Waals molecular potential, as a function of the intermolecular distance, consists of an exponential repulsie inner core and an adjoining attractie potential region, with the potential falling to zero as the intermolecular distance approaches infinity. The Enskog equation may be considered to represent an intermolecular potential which is infinite for x σ and zero for x > σ. As a first approximation to the qualitatie behaior of the an der Walls potential, we hae considered the Enskog equation with an attractie square well potential added to the hard repulsie core. The potential, x < σ φ( x) = Ε, σ < x < R 0 x > R must be introduced at the Liouille leel. The result is the following Enskog Square Well equation: ( x, t, ) + f( xt,, ) = x 48 MSAS'2004
3 + + R R = 3 2 d ' d εθ( ε V){ Y( x+ σε) f (, x ',) t f ( x+ σε, ',) t Y( x σε) f(,,) x t f( x σε,,)} t ( + ) + ), / /2 R d ' d εθ ε V){ Y( x σε f ( x, '', t) f ( x σε, '', t) Y( x σε f( x,, t) f( x σε, t)} dd ' εθ( ε V 2 E ){ Y( x+ σε) f( x, ''', t) f( x+ σε, ''', t) Y( x σε) f( xt,, ) f( x σε,,)} t + d ' d εθ( 2 E ε V){ Y( x+ σε) f(, x ',) t f( x+ σε, ',) t Y( x σε) f(,,) x t f( x σε,,)} t The double and triple primed elocities, representing entrance and exit into the square well, can be computed from the conseration laws. Although this might at first look intractable, it consist simply of four Enskog-type collision operators, corresponding to four `collision processes: entrance into the square well, reflection at the molecular diameter, exit out of the square well, and reflection at the well diameter if the particles, as a result of an intermediate collision, no longer hae the energy to escape the well. In principle, the collision frequency factor Y should be determined by local equilibrium considerations, such as a Mayer cluster expansion. From a numerical point of iew, it would be ery interesting to explore this equation with arious models for Y, for example a monotonic function approaching at low density and infinity at close packing density. Een the case Y, which corresponds to a regime intermediate to dilute and moderately dense, would be of great interest. The Enskog Square Well equation satisfies conseration of mass and momentum, and has a Liapuno fuctional which can be deried in a fashion analogous to that of the Enskog equation. Although energy can not be consered because of the potential well, it satisfies a growth estimate, which has been shown to be sufficient to proe the existence of (weak) global-in-time solutions to the Cauchy problem. [6] Extending the Enskog Square Well Equation to finite range piecewise constant potentials φ( x), which equal Ei for Ri < x < Ri, i=,..., N, presents no technical difficulties. Such an equation will hae 3N+ Enskog-like collision terms, and would proide a numerical model which one might expect to closely emulate a an der Waals potential. III. Vlaso-Enskog Equations Another approach to modeling intermolecular potential effects in dense gases is to add to the hard core repulsion of the Enskog equation the effects of a long range smooth attractie tail. We hae considered two different ideas to accomplish this: coupling a Coulomb potential to the hard-core repulsion, and adding a Fokker-Planck elocity diffusion. Let us write C E (f,f) for the Enskog collision operator (right hand side of the Enskog equation). By adding an extended tail potential Φ(x), which is sufficiently weak to ignore 49 MSAS'2004
4 two particle correlations as well as effects within the molecular diameter, beginning with the Liouille equation and the BBGKY hierarchy, we hae been able to derie a Vlaso- Enskog equation: (, x t,) + x f(, xt,) = CE( f, f) E(,) xt f(, xt,) which, if Φ(x) is chosen as the Coulomb potential Φ( x) = k x, is coupled to the Poisson equation 3 diext x (, ) = α df( xt,, ). This coupled pair of equations we refer to as the Vlaso Enskog equation. The deriation of the Vlaso Enskog equation neglects elocity correlations. One method of taking into account these effects is the addition of a Fokker-Planck term (or Kolmogoro forward process). Such terms hae been taken in arious forms by different authors. The simplest case is to treat it just as a Laplacian Δ f. Another possibility is to write the process as di (f). From the point of iew of physics, the Laplacian accounts for the thermal background interaction and the diergence arises from dynamical friction forces. Both of these terms, and combinations of them, are referred to as Fokker-Planck terms in the literature. We refer to the system ( x,, t) + f ( x,, t) = C ( f, f ) E( x, t) f ( x,, t) + di{ η f ( x,, t) + ω f ( x,, t)} x E along with the coupled Poisson equation, as the Diffusie Vlaso Enskog equation. We point out that from the iewpoint of either physics or mathematics, the cases η 0 and η =0 can not be inferred from each other. With either type of diffusion term, the system of equations satisfies conseration of mass and momentum, as well as an energy bound of the form 2 2 ddx f ( xt,, ) + dx Ext (, ) = K0 + Kt A Liapuno functional may be deried which includes the term t 0 ds ddx f ( x,, s) For all of these equations, (weak) existence theorems of arious forms hae been proed. [7,8] 2 IV. Discrete Velocity Models Discrete elocity models hae been studied rather extensiely for the Boltzmann equation for more than three decades. Yet the literature is nearly nonexistent for discrete model Enskog equations. We pose here a general framework which will allow the introduction of arious discrete elocity models to Enskog theory. 50 MSAS'2004
5 Assuming that all particle elocities belong to a finite set { } N i i =, let P ( ε ) be the probability that two particles of elocity i and j encounter at an angle ε and produce post-collision elocities of k and l. Then probability, conseration laws and symmetry would require: P ( ε ) 0 0 P < i j, ε > 0 < k l, ε > 0 if and only if i + j = k + l i + j = k + l P P ( ε) = P ( ε) = lk ( ε ) = P ( ε ) The discrete elocity Enskog equation is then gien by x + + = dε{ Y( x+ σε) P ( ε) f f < ε, > Y( x σε) P f f < ε, > } i i + i = j + B ji k l k l i j i j where f ± = i fi( x ± σε, t) and B + = { ε ε ε =, < ε, 0,, 0} i j > < ε k l >. This model reduces easily to the few known discrete Enskog models which hae appeared in the literature. [9,0,] It is straightforward to check that the equation satisfies conseration of mass, momentum and energy, and has a Liapuno functional t + + Γ () t = dxfilog fi ds dε dx[ fk fi fi f j ] Y( x+ σε) P ( ε) < ε, i j > 2 + i V. Numerical Problems 0 B Numerical implementation of the Boltzmann equation is an area which has attracted considerable attention for seeral decades. Because of the singular nature of the collision term, techniques hae been deeloped which are designed particularly for Boltzmann theory. Most analysts hae approached the problem by adapting Monte Carlo procedures, although some work has been done as well utilizing deterministic methods, ie, discrete ordinates, spherical harmonics, nodal methods, etc. A solid introduction to both approaches may be obtained in reference [2]. 5 MSAS'2004
6 Although the literature on numerical implementation of the Enskog equation is more scarce, similar strategies hae been applied to obtain numerical solutions of Enskog equations. With reference to the continuous models contained in this paper, the field of numerical analysis is wide open. Indeed, the author knows of no literature which offers a numerical strategy for these equations. The reader is thus kindly inited to consider analysis of these equations. In particular, it would be of great interest to hae estimates on the effect of the potential well in the square well equation. What effect does it hae on clustering, for example? Can one estimate the effects of the Vlaso term and the elocity diffusions in the Enskog-Vlaso models? Of course, for the discrete elocity models, the numerical problem is a great deal simpler. Here, the most immediate interest would be in creating discrete models corresponding to the rich literature of discrete Boltzmann models. We are aware at present of only two such models in the literature [9,0,], but are certain the construction outlined aboe proides a framework for a number of additional models. REFERENCES [] L.Boltzmann, Sitz. Wien. Akad. Wiss. 66, (872). [2] D. Enskog, Kinetiske Theorie, Senska Akad. 63 (92). English translation in Kinetic Theory, S. Bruch, ed., ol. 3, Pergamon, New York, 972. [3] C. Cercignani, The Boltzmann Equation and its Applications (Applied Mathematical Sciences ol. 67), Springer-Verlag, Berlin, 988. [4] N. Bellomo, A. Palczewski and G. Toscani, Mathematical Topics in Nonlinear Kinetic Theory, World Scientific, Singapore, 988. [5] N. Bellomo, M. Lachowicz, J. Polewczak and G. Toscani, Mathematical Topics in Nonlinear Kinetic Theory II, World Scientific, Singapore, 99. [6] W. Greenberg and A. Yao, Jour. Transport Theor. Stat. Phys. 27, (997). [7] W. Greenberg and P. Lei, in Proceedings of International Scientific Conference, Russian Academy of Science (Institute for Mathematical Biology), Ter (997). [8] W. Greenberg, in Proceedings of II Congreso International de Matematica Aplicada y Computacional (CIMAC II), to appear. [9] G. Borgioli, V. Gerasimenko, G. Lauro and R. Monaco, Jour. Transport Theor. Stat. Phys. 25, (996). [0] G. Borgioli, V Gerasimenko, G. Lauro and R. Monaco, Reports on Mathematical Physics 40, (997). [] G. Borgioli, V. Gerasimenko and G. Lauro, Rendiconti del Seminario Matematico dell Uniersita e del Politecnico di Torino, to appear. [2] V. Aristo, Methods of Direct Soing the Boltzmann Equation and Study of NonequilibriumFlows, Kluwer, Amsterdam, MSAS'2004
Kinetic plasma description
Kinetic plasma description Distribution function Boltzmann and Vlaso equations Soling the Vlaso equation Examples of distribution functions plasma element t 1 r t 2 r Different leels of plasma description
More informationСollisionless damping of electron waves in non-maxwellian plasma 1
http:/arxi.org/physics/78.748 Сollisionless damping of electron waes in non-mawellian plasma V. N. Soshnio Plasma Physics Dept., All-Russian Institute of Scientific and Technical Information of the Russian
More informationThe Boltzmann Equation and Its Applications
Carlo Cercignani The Boltzmann Equation and Its Applications With 42 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo CONTENTS PREFACE vii I. BASIC PRINCIPLES OF THE KINETIC
More informationChem 4521 Kinetic Theory of Gases PhET Simulation
Chem 451 Kinetic Theory of Gases PhET Simulation http://phet.colorado.edu/get_phet/simlauncher.php The discussion in the first lectures centered on the ideal gas equation of state and the modifications
More informationGlobal Weak Solution to the Boltzmann-Enskog equation
Global Weak Solution to the Boltzmann-Enskog equation Seung-Yeal Ha 1 and Se Eun Noh 2 1) Department of Mathematical Science, Seoul National University, Seoul 151-742, KOREA 2) Department of Mathematical
More information4. A Physical Model for an Electron with Angular Momentum. An Electron in a Bohr Orbit. The Quantum Magnet Resulting from Orbital Motion.
4. A Physical Model for an Electron with Angular Momentum. An Electron in a Bohr Orbit. The Quantum Magnet Resulting from Orbital Motion. We now hae deeloped a ector model that allows the ready isualization
More informationThe Kinetic Theory of Gases
978-1-107-1788-3 Classical and Quantum Thermal Physics The Kinetic Theory of Gases CHAPTER 1 1.0 Kinetic Theory, Classical and Quantum Thermodynamics Two important components of the unierse are: the matter
More informationLecture 21: Physical Brownian Motion II
Lecture 21: Physical Brownian Motion II Scribe: Ken Kamrin Department of Mathematics, MIT May 3, 25 Resources An instructie applet illustrating physical Brownian motion can be found at: http://www.phy.ntnu.edu.tw/jaa/gas2d/gas2d.html
More informationWeb Resource: Ideal Gas Simulation. Kinetic Theory of Gases. Ideal Gas. Ideal Gas Assumptions
Web Resource: Ideal Gas Simulation Kinetic Theory of Gases Physics Enhancement Programme Dr. M.H. CHAN, HKBU Link: http://highered.mheducation.com/olcweb/cgi/pluginpop.cgi?it=swf::00%5::00%5::/sites/dl/free/003654666/7354/ideal_na.swf::ideal%0gas%0law%0simulation
More information0 a 3 a 2 a 3 0 a 1 a 2 a 1 0
Chapter Flow kinematics Vector and tensor formulae This introductory section presents a brief account of different definitions of ector and tensor analysis that will be used in the following chapters.
More informationPh.D. Crina Gudelia Costea
Politecnico di Torino - Dipartimento di Scienza dei Materiali ed Ingegneria Chimica Ph.D. Crina Gudelia Costea Adisor: prof. Marco Vanni Introduction Morphology of aggregates Porosity and permeability
More informationSimulations of bulk phases. Periodic boundaries. Cubic boxes
Simulations of bulk phases ChE210D Today's lecture: considerations for setting up and running simulations of bulk, isotropic phases (e.g., liquids and gases) Periodic boundaries Cubic boxes In simulations
More informationLect-19. In this lecture...
19 1 In this lecture... Helmholtz and Gibb s functions Legendre transformations Thermodynamic potentials The Maxwell relations The ideal gas equation of state Compressibility factor Other equations of
More informationHilbert Sixth Problem
Academia Sinica, Taiwan Stanford University Newton Institute, September 28, 2010 : Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem:
More informationSLIP MODEL PERFORMANCE FOR MICRO-SCALE GAS FLOWS
3th AIAA Thermophysics Conference 3- June 3, Orlando, Florida AIAA 3-5 SLIP MODEL PERFORMANCE FOR MICRO-SCALE GAS FLOWS Matthew J. McNenly* Department of Aerospace Engineering Uniersity of Michigan, Ann
More informationGeostrophy & Thermal wind
Lecture 10 Geostrophy & Thermal wind 10.1 f and β planes These are planes that are tangent to the earth (taken to be spherical) at a point of interest. The z ais is perpendicular to the plane (anti-parallel
More informationClassical Mechanics NEWTONIAN SYSTEM OF PARTICLES MISN NEWTONIAN SYSTEM OF PARTICLES by C. P. Frahm
MISN-0-494 NEWTONIAN SYSTEM OF PARTICLES Classical Mechanics NEWTONIAN SYSTEM OF PARTICLES by C. P. Frahm 1. Introduction.............................................. 1 2. Procedures................................................
More informationKinetic Theory. Reading: Chapter 19. Ideal Gases. Ideal gas law:
Reading: Chapter 19 Ideal Gases Ideal gas law: Kinetic Theory p nrt, where p pressure olume n number of moles of gas R 831 J mol -1 K -1 is the gas constant T absolute temperature All gases behae like
More informationTHE CAUCHY PROBLEM FOR ONE-DIMENSIONAL FLOW OF A COMPRESSIBLE VISCOUS FLUID: STABILIZATION OF THE SOLUTION
GLASNIK MATEMATIČKI Vol. 4666, 5 3 THE CAUCHY POBLEM FO ONE-DIMENSIONAL FLOW OF A COMPESSIBLE VISCOUS FLUID: STABILIZATION OF THE SOLUTION Nermina Mujakoić and Ian Dražić Uniersity of ijeka, Croatia Abstract.
More informationSF Chemical Kinetics.
SF Chemical Kinetics. Lecture 5. Microscopic theory of chemical reaction kinetics. Microscopic theories of chemical reaction kinetics. basic aim is to calculate the rate constant for a chemical reaction
More informationGeneral Lorentz Boost Transformations, Acting on Some Important Physical Quantities
General Lorentz Boost Transformations, Acting on Some Important Physical Quantities We are interested in transforming measurements made in a reference frame O into measurements of the same quantities as
More informationIntroduction to Thermodynamic Cycles Part 1 1 st Law of Thermodynamics and Gas Power Cycles
Introduction to Thermodynamic Cycles Part 1 1 st Law of Thermodynamics and Gas Power Cycles by James Doane, PhD, PE Contents 1.0 Course Oeriew... 4.0 Basic Concepts of Thermodynamics... 4.1 Temperature
More informationSection 6: PRISMATIC BEAMS. Beam Theory
Beam Theory There are two types of beam theory aailable to craft beam element formulations from. They are Bernoulli-Euler beam theory Timoshenko beam theory One learns the details of Bernoulli-Euler beam
More informationMin Chen Department of Mathematics Purdue University 150 N. University Street
EXISTENCE OF TRAVELING WAVE SOLUTIONS OF A HIGH-ORDER NONLINEAR ACOUSTIC WAVE EQUATION Min Chen Department of Mathematics Purdue Uniersity 150 N. Uniersity Street 47907-067 chen@math.purdue.edu Monica
More informationGroup and Renormgroup Symmetry of Quasi-Chaplygin Media
Nonlinear Mathematical Physics 1996, V.3, N 3 4, 351 356. Group and Renormgroup Symmetry of Quasi-Chaplygin Media Vladimir F. KOVALEV Institute for Mathematical Modelling, Miusskaya 4-A, Moscow, 125047,
More informationMonte Carlo simulations of dense gas flow and heat transfer in micro- and nano-channels
Science in China Ser. E Engineering & Materials Science 2005 Vol.48 No.3 317 325 317 Monte Carlo simulations of dense gas flow and heat transfer in micro- and nano-channels WANG Moran & LI Zhixin Department
More informationTo string together six theorems of physics by Pythagoras theorem
To string together six theorems of physics by Pythagoras theorem H. Y. Cui Department of Applied Physics Beijing Uniersity of Aeronautics and Astronautics Beijing, 00083, China ( May, 8, 2002 ) Abstract
More informationv v Downloaded 01/11/16 to Redistribution subject to SEG license or copyright; see Terms of Use at
The pseudo-analytical method: application of pseudo-laplacians to acoustic and acoustic anisotropic wae propagation John T. Etgen* and Serre Brandsberg-Dahl Summary We generalize the pseudo-spectral method
More informationRelation between compressibility, thermal expansion, atom volume and atomic heat of the metals
E. Grüneisen Relation between compressibility, thermal expansion, atom olume and atomic heat of the metals Annalen der Physik 6, 394-40, 1908 Translation: Falk H. Koenemann, 007 1 Richards (1907) and his
More informationInelastic Collapse in One Dimensional Systems with Ordered Collisions
Lake Forest College Lake Forest College Publications Senior Theses Student Publications 4-4-05 Inelastic Collapse in One Dimensional Systems with Ordered Collisions Brandon R. Bauerly Lake Forest College,
More informationMATHEMATICAL MODELLING AND IDENTIFICATION OF THE FLOW DYNAMICS IN
MATHEMATICAL MOELLING AN IENTIFICATION OF THE FLOW YNAMICS IN MOLTEN GLASS FURNACES Jan Studzinski Systems Research Institute of Polish Academy of Sciences Newelska 6-447 Warsaw, Poland E-mail: studzins@ibspan.waw.pl
More informationarxiv: v1 [cond-mat.stat-mech] 5 Oct 2018
On the canonical distributions of a thermal particle in the weakly confining potential of special type Tatsuaki Wada Region of Electrical and Electronic Systems Engineering, Ibaraki Uniersity, Nakanarusawa-cho,
More informationChapter 2 A Brief Introduction to the Mathematical Kinetic Theory of Classical Particles
Chapter 2 A Brief Introduction to the Mathematical Kinetic Theory of Classical Particles 2.1 Plan of the Chapter The contents of this chapter are motivated by the second key question posed in Section 1.3,
More informationMECHANICAL FORMULATION OF THE ENTROPY OF A SYSTEM
JOHN RIUS CAMPS MECHANICAL FORMULATION OF THE ENTROPY OF A SYSTEM 6 TH FEBRUARY 009 ORDIS EDITIONS ORDIS EDITIONS GRAN VIA DE CARLOS III, 59, º, 4ª BARCELONA 0808 6 TH June 008 3 4 . INTRODUCTION. What
More informationChapter 14 Thermal Physics: A Microscopic View
Chapter 14 Thermal Physics: A Microscopic View The main focus of this chapter is the application of some of the basic principles we learned earlier to thermal physics. This will gie us some important insights
More informationTowards Universal Cover Decoding
International Symposium on Information Theory and its Applications, ISITA2008 Auckland, New Zealand, 7-10, December, 2008 Towards Uniersal Coer Decoding Nathan Axig, Deanna Dreher, Katherine Morrison,
More informationWhy does Saturn have many tiny rings?
2004 Thierry De Mees hy does Saturn hae many tiny rings? or Cassini-Huygens Mission: New eidence for the Graitational Theory with Dual Vector Field T. De Mees - thierrydemees @ pandora.be Abstract This
More informationChapter 19 Kinetic Theory of Gases
Chapter 9 Kinetic heory of Gases A gas consists of atoms or molecules which collide with the walls of the container and exert a pressure, P. he gas has temperature and occupies a olume V. Kinetic theory
More informationNumerical Simulation of the Rarefied Gas Flow through a Short Channel into a Vacuum
Numerical Simulation of the Rarefied Gas Flow through a Short Channel into a Vacuum Oleg Sazhin Ural State University, Lenin av.5, 6283 Ekaterinburg, Russia E-mail: oleg.sazhin@uralmail.com Abstract. The
More informationDiffusion. Spring Quarter 2004 Instructor: Richard Roberts. Reading Assignment: Ch 6: Tinoco; Ch 16: Levine; Ch 15: Eisenberg&Crothers
Chemistry 24b Spring Quarter 2004 Instructor: Richard Roberts Lecture 2 RWR Reading Assignment: Ch 6: Tinoco; Ch 16: Leine; Ch 15: Eisenberg&Crothers Diffusion Real processes, such as those that go on
More informationSimilarities and differences:
How does the system reach equilibrium? I./9 Chemical equilibrium I./ Equilibrium electrochemistry III./ Molecules in motion physical processes, non-reactive systems III./5-7 Reaction rate, mechanism, molecular
More informationStochastic Particle Methods for Rarefied Gases
CCES Seminar WS 2/3 Stochastic Particle Methods for Rarefied Gases Julian Köllermeier RWTH Aachen University Supervisor: Prof. Dr. Manuel Torrilhon Center for Computational Engineering Science Mathematics
More informationVelocity, Acceleration and Equations of Motion in the Elliptical Coordinate System
Aailable online at www.scholarsresearchlibrary.com Archies of Physics Research, 018, 9 (): 10-16 (http://scholarsresearchlibrary.com/archie.html) ISSN 0976-0970 CODEN (USA): APRRC7 Velocity, Acceleration
More informationLow Variance Particle Simulations of the Boltzmann Transport Equation for the Variable Hard Sphere Collision Model
Low Variance Particle Simulations of the Boltzmann Transport Equation for the Variable Hard Sphere Collision Model G. A. Radtke, N. G. Hadjiconstantinou and W. Wagner Massachusetts Institute of Technology,
More informationThe broad topic of physical metallurgy provides a basis that links the structure of materials with their properties, focusing primarily on metals.
Physical Metallurgy The broad topic of physical metallurgy provides a basis that links the structure of materials with their properties, focusing primarily on metals. Crystal Binding In our discussions
More informationMathematical modelling of collective behavior
Mathematical modelling of collective behavior Young-Pil Choi Fakultät für Mathematik Technische Universität München This talk is based on joint works with José A. Carrillo, Maxime Hauray, and Samir Salem
More informationSPACE-TIME HOLOMORPHIC TIME-PERIODIC SOLUTIONS OF NAVIER-STOKES EQUATIONS. 1. Introduction We study Navier-Stokes equations in Lagrangean coordinates
Electronic Journal of Differential Equations, Vol. 2013 2013, No. 218, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SPACE-TIME HOLOMORPHIC
More informationare applied to ensure that physical principles are not iolated in the definition of the discrete transition model. The oerall goal is to use this fram
A Comprehensie Methodology for Building Hybrid Models of Physical Systems Pieter J. Mosterman Λ Institute of Robotics and System Dynamics DLR Oberpfaffenhofen P.O. Box 1116 D-8223 Wessling Germany Gautam
More information(a) During the first part of the motion, the displacement is x 1 = 40 km and the time interval is t 1 (30 km / h) (80 km) 40 km/h. t. (2.
Chapter 3. Since the trip consists of two parts, let the displacements during first and second parts of the motion be x and x, and the corresponding time interals be t and t, respectiely. Now, because
More informationA Study of the Thermal Properties of a One. Dimensional Lennard-Jones System
A Study of the Thermal Properties of a One Dimensional Lennard-Jones System Abstract In this study, the behavior of a one dimensional (1D) Lennard-Jones (LJ) system is simulated. As part of this research,
More informationE : Ground-penetrating radar (GPR)
Geophysics 3 March 009 E : Ground-penetrating radar (GPR) The EM methods in section D use low frequency signals that trael in the Earth by diffusion. These methods can image resistiity of the Earth on
More informationKinetic Molecular Theory of Ideal Gases
Lecture -3. Kinetic Molecular Theory of Ideal Gases Last Lecture. IGL is a purely epirical law - solely the consequence of experiental obserations Explains the behaior of gases oer a liited range of conditions.
More informationSELECTION, SIZING, AND OPERATION OF CONTROL VALVES FOR GASES AND LIQUIDS Class # 6110
SELECTION, SIZIN, AND OERATION OF CONTROL VALVES FOR ASES AND LIUIDS Class # 6110 Ross Turbiille Sales Engineer Fisher Controls International Inc. 301 S. First Aenue Marshalltown, Iowa USA Introduction
More informationCHAPTER 4. Basics of Fluid Dynamics
CHAPTER 4 Basics of Fluid Dynamics What is a fluid? A fluid is a substance that can flow, has no fixed shape, and offers little resistance to an external stress In a fluid the constituent particles (atoms,
More informationAlignment processes on the sphere
Alignment processes on the sphere Amic Frouvelle CEREMADE Université Paris Dauphine Joint works with : Pierre Degond (Imperial College London) and Gaël Raoul (École Polytechnique) Jian-Guo Liu (Duke University)
More informationThe existence of Burnett coefficients in the periodic Lorentz gas
The existence of Burnett coefficients in the periodic Lorentz gas N. I. Chernov and C. P. Dettmann September 14, 2006 Abstract The linear super-burnett coefficient gives corrections to the diffusion equation
More informationA. unchanged increased B. unchanged unchanged C. increased increased D. increased unchanged
IB PHYSICS Name: DEVIL PHYSICS Period: Date: BADDEST CLASS ON CAMPUS CHAPTER B TEST REVIEW. A rocket is fired ertically. At its highest point, it explodes. Which one of the following describes what happens
More informationThe purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and engineering.
Lecture 16 Applications of Conformal Mapping MATH-GA 451.001 Complex Variables The purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and
More informationElectrostatic Forces and Fields
Conductors, Insulators, and Induced Charges Electrostatic orces and ields undamental unit of charge is the Coulomb (C) electron charge is 1.60 x 10 19 C proton charge is 1.60 x 10 19 C Conductors permit
More informationReal Gas Thermodynamics. and the isentropic behavior of substances. P. Nederstigt
Real Gas Thermodynamics and the isentropic behaior of substances. Nederstigt ii Real Gas Thermodynamics and the isentropic behaior of substances by. Nederstigt in partial fulfillment of the requirements
More informationA possible mechanism to explain wave-particle duality L D HOWE No current affiliation PACS Numbers: r, w, k
A possible mechanism to explain wae-particle duality L D HOWE No current affiliation PACS Numbers: 0.50.-r, 03.65.-w, 05.60.-k Abstract The relationship between light speed energy and the kinetic energy
More informationLecture 6 Gas Kinetic Theory and Boltzmann Equation
GIAN Course on Rarefied & Microscale Gases and Viscoelastic Fluids: a Unified Framework Lecture 6 Gas Kinetic Theory and Boltzmann Equation Feb. 23 rd ~ March 2 nd, 2017 R. S. Myong Gyeongsang National
More informationTwo-sided bounds for L p -norms of combinations of products of independent random variables
Two-sided bounds for L p -norms of combinations of products of independent random ariables Ewa Damek (based on the joint work with Rafał Latała, Piotr Nayar and Tomasz Tkocz) Wrocław Uniersity, Uniersity
More informationFLUID MECHANICS EQUATIONS
FLUID MECHANIC EQUATION M. Ragheb 11/2/2017 INTRODUCTION The early part of the 18 th -century saw the burgeoning of the field of theoretical fluid mechanics pioneered by Leonhard Euler and the father and
More informationClassical dynamics on graphs
Physical Reiew E 63 (21) 66215 (22 pages) Classical dynamics on graphs F. Barra and P. Gaspard Center for Nonlinear Phenomena and Complex Systems, Uniersité Libre de Bruxelles, Campus Plaine C.P. 231,
More informationMath 425 Lecture 1: Vectors in R 3, R n
Math 425 Lecture 1: Vectors in R 3, R n Motiating Questions, Problems 1. Find the coordinates of a regular tetrahedron with center at the origin and sides of length 1. 2. What is the angle between the
More informationUNDERSTAND MOTION IN ONE AND TWO DIMENSIONS
SUBAREA I. COMPETENCY 1.0 UNDERSTAND MOTION IN ONE AND TWO DIMENSIONS MECHANICS Skill 1.1 Calculating displacement, aerage elocity, instantaneous elocity, and acceleration in a gien frame of reference
More informationKinetic Molecular Theory of Ideal Gases
Kinetic Molecular Theory of Ideal Gases Bernoulli et al. (178) Theoretical deelopment of ideal gas laws that were determined empirically Boyle Charles Gay-Lussac Aogadro (167 1691) (1746-18) (1778 1850)
More informationLittlewood-Paley Decomposition and regularity issues in Boltzmann homogeneous equations I. Non cutoff and Maxwell cases
Littlewood-Paley Decomposition and regularity issues in Boltzmann homogeneous equations I. Non cutoff and Maxwell cases Radjesarane ALEXANDRE and Mouhamad EL SAFADI MAPMO UMR 668 Uniersité d Orléans, BP
More informationSimpler form of the trace formula for GL 2 (A)
Simpler form of the trace formula for GL 2 (A Last updated: May 8, 204. Introduction Retain the notations from earlier talks on the trace formula and Jacquet Langlands (especially Iurie s talk and Zhiwei
More informationarxiv: v1 [math.gt] 2 Nov 2010
CONSTRUCTING UNIVERSAL ABELIAN COVERS OF GRAPH MANIFOLDS HELGE MØLLER PEDERSEN arxi:101105551 [mathgt] 2 No 2010 Abstract To a rational homology sphere graph manifold one can associate a weighted tree
More informationCases of integrability corresponding to the motion of a pendulum on the two-dimensional plane
Cases of integrability corresponding to the motion of a pendulum on the two-dimensional plane MAXIM V. SHAMOLIN Lomonoso Moscow State Uniersity Institute of Mechanics Michurinskii Ae.,, 99 Moscow RUSSIAN
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS AP PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS AP PHYSICS LSN 7-4: CONSERVATION OF ENERGY AND MOMENTUM IN COLLISIONS LSN 7-5: ELASTIC COLLISIONS IN ONE DIMENSION LSN 7-6: INELASTIC COLLISIONS Questions From
More information4 Fundamentals of Continuum Thermomechanics
4 Fundamentals of Continuum Thermomechanics In this Chapter, the laws of thermodynamics are reiewed and formulated for a continuum. The classical theory of thermodynamics, which is concerned with simple
More informationSOLIDS AND LIQUIDS - Here's a brief review of the atomic picture or gases, liquids, and solids GASES
30 SOLIDS AND LIQUIDS - Here's a brief review of the atomic picture or gases, liquids, and solids GASES * Gas molecules are small compared to the space between them. * Gas molecules move in straight lines
More informationResidual migration in VTI media using anisotropy continuation
Stanford Exploration Project, Report SERGEY, Noember 9, 2000, pages 671?? Residual migration in VTI media using anisotropy continuation Tariq Alkhalifah Sergey Fomel 1 ABSTRACT We introduce anisotropy
More informationAn Elastic Contact Problem with Normal Compliance and Memory Term
Machine Dynamics Research 2012, Vol. 36, No 1, 15 25 Abstract An Elastic Contact Problem with Normal Compliance and Memory Term Mikäel Barboteu, Ahmad Ramadan, Mircea Sofonea Uniersité de Perpignan, France
More informationOn some properties of the Lyapunov equation for damped systems
On some properties of the Lyapuno equation for damped systems Ninosla Truhar, Uniersity of Osijek, Department of Mathematics, 31 Osijek, Croatia 1 ntruhar@mathos.hr Krešimir Veselić Lehrgebiet Mathematische
More informationKinetic Molecular Theory of. IGL is a purely empirical law - solely the
Lecture -3. Kinetic Molecular Theory of Ideal Gases Last Lecture. IGL is a purely epirical law - solely the consequence of experiental obserations Explains the behaior of gases oer a liited range of conditions.
More informationOn Weak Solutions to the Linear Boltzmann Equation with Inelastic Coulomb Collisions
On Weak Solutions to the Linear Boltzmann Equation with Inelastic Coulomb Collisions Rolf Pettersson epartment of Mathematics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden Abstract. This
More informationA spectral Turán theorem
A spectral Turán theorem Fan Chung Abstract If all nonzero eigenalues of the (normalized) Laplacian of a graph G are close to, then G is t-turán in the sense that any subgraph of G containing no K t+ contains
More informationEntropy and irreversibility in gas dynamics. Joint work with T. Bodineau, I. Gallagher and S. Simonella
Entropy and irreversibility in gas dynamics Joint work with T. Bodineau, I. Gallagher and S. Simonella Kinetic description for a gas of hard spheres Hard sphere dynamics The system evolves under the combined
More informationYao Liu Department of Physics, Columbia University New York, New York 10027
Magnetic Flux Diffusion and Expulsion with Thin Conducting Sheets a) Yao Liu Department of Physics, Columbia Uniersity New York, New York 17 John W. Belcher Department of Physics, Massachusetts Institute
More informationThe dynamics of small particles whose size is roughly 1 µmt or. smaller, in a fluid at room temperature, is extremely erratic, and is
1 I. BROWNIAN MOTION The dynamics of small particles whose size is roughly 1 µmt or smaller, in a fluid at room temperature, is extremely erratic, and is called Brownian motion. The velocity of such particles
More informationFluid Physics 8.292J/12.330J
Fluid Phsics 8.292J/12.0J Problem Set 4 Solutions 1. Consider the problem of a two-dimensional (infinitel long) airplane wing traeling in the negatie x direction at a speed c through an Euler fluid. In
More informationRelativistic Energy Derivation
Relatiistic Energy Deriation Flamenco Chuck Keyser //4 ass Deriation (The ass Creation Equation ρ, ρ as the initial condition, C the mass creation rate, T the time, ρ a density. Let V be a second mass
More informationdifferent formulas, depending on whether or not the vector is in two dimensions or three dimensions.
ectors The word ector comes from the Latin word ectus which means carried. It is best to think of a ector as the displacement from an initial point P to a terminal point Q. Such a ector is expressed as
More informationChapter 16. Kinetic Theory of Gases. Summary. molecular interpretation of the pressure and pv = nrt
Chapter 16. Kinetic Theory of Gases Summary molecular interpretation of the pressure and pv = nrt the importance of molecular motions elocities and speeds of gas molecules distribution functions for molecular
More informationPlasma Astrophysics Chapter 1: Basic Concepts of Plasma. Yosuke Mizuno Institute of Astronomy National Tsing-Hua University
Plasma Astrophysics Chapter 1: Basic Concepts of Plasma Yosuke Mizuno Institute of Astronomy National Tsing-Hua University What is a Plasma? A plasma is a quasi-neutral gas consisting of positive and negative
More informationThe number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.
ADVANCED GCE UNIT 76/ MATHEMATICS (MEI Mechanics MONDAY MAY 7 Additional materials: Answer booklet (8 pages Graph paper MEI Examination Formulae and Tables (MF Morning Time: hour minutes INSTRUCTIONS TO
More informationLecture 11 - Phonons II - Thermal Prop. Continued
Phonons II - hermal Properties - Continued (Kittel Ch. 5) Low High Outline Anharmonicity Crucial for hermal expansion other changes with pressure temperature Gruneisen Constant hermal Heat ransport Phonon
More informationFUNDAMENTALS OF CHEMISTRY Vol. II - Irreversible Processes: Phenomenological and Statistical Approach - Carlo Cercignani
IRREVERSIBLE PROCESSES: PHENOMENOLOGICAL AND STATISTICAL APPROACH Carlo Dipartimento di Matematica, Politecnico di Milano, Milano, Italy Keywords: Kinetic theory, thermodynamics, Boltzmann equation, Macroscopic
More informationLECTURE 2: CROSS PRODUCTS, MULTILINEARITY, AND AREAS OF PARALLELOGRAMS
LECTURE : CROSS PRODUCTS, MULTILINEARITY, AND AREAS OF PARALLELOGRAMS MA1111: LINEAR ALGEBRA I, MICHAELMAS 016 1. Finishing up dot products Last time we stated the following theorem, for which I owe you
More informationStefania Ferrari 1 and Fausto Saleri 2
ESAIM: M2AN Vol. 38, N o 2, 2004, pp. 2 234 DOI: 0.05/m2an:200400 ESAIM: Mathematical Modelling and Numerical Analysis A NEW TWO-DIMENSIONAL SHALLOW WATER MODEL INCLUDING PRESSURE EFFECTS AND SLOW VARYING
More informationA theoretical study of the energy distribution function of the negative hydrogen ion H - in typical
Non equilibrium velocity distributions of H - ions in H 2 plasmas and photodetachment measurements P.Diomede 1,*, S.Longo 1,2 and M.Capitelli 1,2 1 Dipartimento di Chimica dell'università di Bari, Via
More informationThe Kinetic-Molecular Theory of Gases
The Kinetic-Molecular Theory of Gases kinetic-molecular theory of gases Originated with Ludwig Boltzman and James Clerk Maxwell in the 19th century Explains gas behavior on the basis of the motion of individual
More informationN10/4/PHYSI/SPM/ENG/TZ0/XX PHYSICS STANDARD LEVEL PAPER 1. Monday 8 November 2010 (afternoon) 45 minutes INSTRUCTIONS TO CANDIDATES
N1/4/PHYSI/SPM/ENG/TZ/XX 881654 PHYSICS STANDARD LEVEL PAPER 1 Monday 8 Noember 21 (afternoon) 45 minutes INSTRUCTIONS TO CANDIDATES Do not open this examination paper until instructed to do so. Answer
More informationBound states of two particles confined to parallel two-dimensional layers and interacting via dipole-dipole or dipole-charge laws
PHYSICAL REVIEW B VOLUME 55, NUMBER 8 15 FEBRUARY 1997-II Bound states of two particles confined to parallel two-dimensional layers and interacting via dipole-dipole or dipole-charge laws V. I. Yudson
More informationPhysical Modeling of Multiphase flow. Boltzmann method
with lattice Boltzmann method Exa Corp., Burlington, MA, USA Feburary, 2011 Scope Scope Re-examine the non-ideal gas model in [Shan & Chen, Phys. Rev. E, (1993)] from the perspective of kinetic theory
More informationEntropic structure of the Landau equation. Coulomb interaction
with Coulomb interaction Laurent Desvillettes IMJ-PRG, Université Paris Diderot May 15, 2017 Use of the entropy principle for specific equations Spatially Homogeneous Kinetic equations: 1 Fokker-Planck:
More information