The Active Universe. 1 Active Motion
|
|
- Hortense Wilcox
- 5 years ago
- Views:
Transcription
1 The Active Universe Alexnder Glück, Helmuth Hüffel, Sš Ilijić, Gerld Kelnhofer Fculty of Physics, University of Vienn Deprtment of Physics, FER, University of Zgreb Abstrct Active motion is concept in complex systems theory nd ws successfully pplied to vrious problems in nonliner dynmics. Explicit studies for grvittionl potentils were missing so fr. We interpret the Friedmnn equtions with cosmologicl constnt s dynmicl system, which cn be mde ctive in strightforwrd wy. These ctive Friedmnn equtions led to cyclic universe, which is shown numericlly. 1 Active Motion To ccount for self-driven motion s observed in biologicl systems, n dditionl degree of freedom, clled the "internl energy" e, ws introduced [Schweitzer et l. (1998), Schweitzer (003)] in the context of complex systems theory. First interprettions of this formlism were given with respect to niml movement nd complex motion of prticles in generl. It describes prticles who cn convert their internl energy e into mechnicl energy nd thus exhibit complex movement. Vrious pplictions of this model were studied extensively, for instnce in swrm theory [Schweitzer et l. (001), Glück et l. (010)]. Also for Quntum Field Theory it could be shown tht ctive dynmics is importnt for the behvior of systems fr from equilibrium, nmely such which exhibit spontneous symmetry breking [Glück et l. (008)]. Obviously this introduction of n internl vrible e enbles us to describe open dissiptive systems in generl wy. The vrible e functions s dynmicl quntity which models the energy flux between n open system nd its surrounding. We now wnt to discuss the implictions of ctive dynmics lso for grvittionl systems. The ctive formlism is pplied to the cosmologicl equtions of motion strightforwrdly in order to study the behvior of universe fr from equilibrium. Dynmics of prticle with position q nd momentum p undergoing ctive motion s discussed in [Glück et l. (009)] is determined by q i = p i, p i = U (1 e), (1) q i 1
2 where the evolution of the internl energy e is given by the eqution ė = c 1 c e c 3 eu(q). () The c i remin constnts to be fixed. The first integrl of the ctive dynmicl system reds q i + U(q i) ˆ t 0 dt e(t ) U(q i (t )) = C = const, (3) which cn be viewed s generliztion of the clssicl energy eqution. In the context of ctive motion the prticle is not simply driven to the minimum of the potentil U(q), its internl energy llows it to exhibit self-driven movement. Depending on the choice of the constnts c i nd on the structure of the potentil, the systems show vrious complex motion ptterns. These cn be studied nlyticlly by bifurction theory nd numericlly by simulting the evolution of q, p nd e in time. So fr, explicit investigtions were done only for hrmonic potentils (see [Glück et l. (009)] for detiled study of the nonliner dynmics). One cn nturlly sk for ctive motion of prticles driven by grvittionl interctions. We will now give formultion of this problem within the frmework of Friedmnn equtions. Active Friedmnn Equtions The Friedmnn equtions for the dimensionless normlized cosmologicl scle prmeter (t), describing the sptil evolution of flt, homogeneous nd isotropic universe with cosmologicl constnt Λ red ( ä = H0 Ωr,0 3 + Ω ) Ω Λ, (4) ȧ ( = H 0 Ωr,0 + Ω ) + Ω Λ, (5) where Ω nd Ω r,0 re the density prmeters of mtter nd rdition t present time t = 0 nd Ω Λ = is the density prmeter of the cosmologicl Λ 3H 0 constnt. (0) = 1 per definition nd H 0 = ȧ(0) denotes the current Hubble prmeter. If we introduce the following potentil U() = H 0 + Ω ) + Ω Λ, (6) then the nlogy with clssicl dynmicl system of single prticle with coordinte (t) nd energy C = 0 is evident. According to generl reltivity the constnt C is relted to the sptil curvture of the universe which, however, is tken to be zero due to recent observtions. The density prmeter
3 of Λ is fixed by the Friedmnn eqution (5) which t present time t = 0 gives the reltion Ω r,0 + Ω + Ω Λ = 1. (7) We now wnt to propose new phenomenologicl model for the development of the universe which is bsed on the ctive generliztion of the conventionl Friedmnn equtions. The ctive Friedmnn equtions follow from equtions (1) - (3) using (6) nd re given by ä = H Ω ė = c 1 c e + c 3 e H 0 ȧ = H 0 H 0 ˆ t 0 + Ω dt e(t )ȧ(t ) ) Ω Λ (1 e), (8) + Ω ) + Ω Λ, (9) + Ω Λ ) + 3 (t ) + Ω ) (t ) Ω Λ(t ) + H 0 C, (10) where c 1, c, c 3 re rbitrry but fixed rel constnts. For nottionl convenience we hve chosen H0 C/ s the corresponding constnt for the first order integrl in (10). Differing from the conventionl cse we llow for nonvnishing C in the ctive context, which hs to be determined yet. We first hve to ssign specific vlues to the density prmeters of mtter/rdition nd the cosmologicl constnt. Notice tht we cn t set Ω Λ = 0.7, since this results from eqution (7), which looks different in the ctive formultion, nmely Ω Λ = 1 Ω r,0 Ω C, (11) which is derived from eqution (10) by setting t = 0. Hence, the constnt C determines the density prmeter Ω Λ of the cosmologicl constnt. It cn be fixed by demnding tht the current ccelertion ä(0) of the universe in the ctive scheme should gree with the ccelertion vlue given by the conventionl Friedmnn equtions (thus ccounting for the experimentl result ä(0) > 0). For ny fixed e 0 := e(0) one finds C = e ( e 0 Ω + Ω r,0 1). (1) Accordingly, Ω Λ = 1 ( 1 Ω (1 + e ) 0 1 e 0 ) Ω r,0(1 + e 0 ), (13) so tht Ω Λ is now completely determined by the density prmeters of mtter/rdition nd the initil condition for e, which is freely chosen. Depending on this initil condition Ω Λ my lso be brought to vnish. 3
4 Figure 1: Numericl result for the evolution of (t), with c 1 = 5, c = 1, c 3 = 1, e(0) = 100, H 0 = 1, Ω = 0.9 nd Ω r,0 = (These re rough representtive vlues for the density prmeters, consult [Komtsu et l. (009)] for recent overwiev of cosmologicl prmeters.) This choice of prmeters leds to Ω Λ = Discussion Figure 1 shows simultion exmple of the ctive dynmics of the normlized cosmologicl scle prmeter (t). We observe oscilltory behvior with significntly smller mplitudes nd shorter periods in the pst. The scle prmeter (t) is oscillting round certin equilibrium point, which cn be clculted nlyticlly. The ctive Friedmnn equtions cn be rewritten s follows: ȧ = p ṗ = du() (1 e) (14) d ė = c 1 c e c 3 eu(). Equilibrium points re found by serching vlues (ã, p, ẽ), for which the right hnd side of (14) vnishes. Equilibrium points with ẽ = 1 re unstble, the cse ẽ 1 leds to the conditions p = 0, 0 = Ω Λ ã 4 Ω ã Ω r,0, (15) c 1 ẽ = c + c 3 U(ã). For the specific choice of prmeters nd initil conditions mde for the simultion shown in Figure 1, the numericl vlues of (ã, p, ẽ) cn be clculted 4
5 directly. The qurtic eqution for ã hs only one positive, rel-vlued solution, nmely ã = 1.01, resulting in (ã, p, ẽ) = (1.01, 0, 4.08). Figure 1 shows tht this vlue of the scle prmeter mrks the center point of the oscilltions. The stbility nlysis of the bove equilibrium point (ã, p, ẽ) cn be mde by linerizing the system (14) nd clculting the eigenvlues of the Jcobin mtrix. Our clcultion shows the presence of purely imginry pir of eigenvlues λ 1, = ±iω nd we re ner Hopf bifurction point (for bifurction theory, see [Kuznetsov (1995)]). In our cse ω = 1.19, so tht the period of oscilltions is given by T = π ω = 5.8 in units of the Hubble time H0 1, which grees well for lrge times with our numericl simultion. Summrizing we cn sy tht interpreting the Friedmnn equtions s nonliner dynmicl system nd hndling it within the frmework of ctive motion, cn give rise to oscilltory solutions for the scle prmeter. Hence we succeeded in constructing model for cyclic universe giving rise to sequence of expnsions nd contrctions without ny singulrity, yet ccounting for the observed sptil fltness nd the current ccelerted expnsion. Besides this prticulr exmple, the ctive Friedmnn equtions llow for lot of other possible scenrios which deserve further nlysis. A future tsk will be to find n interprettion of the vrible e within cosmologicl context. In prticulr, cosmologicl models which ssume the existence of extr dimensions seem to be promising cndidtes for providing n dequte nswer in this respect. Acknowledgments: We thnk Helmut Rumpf for vluble discussions. We re grteful for finncil support within the Agreement on Coopertion between the Universities of Vienn nd Zgreb. References [Schweitzer et l. (1998)] Schweitzer, F. & Ebeling W. & Tilch, B. [1998] Complex Motion of Brownin Prticles with Energy Depots, Phys. Rev. Lett. 80, [Schweitzer (003)] Schweitzer, F. [003] Brownin Agents nd Active Prticles, Springer, Berlin. [Schweitzer et l. (001)] Schweitzer, F. & Ebeling, W. & Tilch, B. [001] Sttisticl Mechnics of Cnonicl-Dissiptive Systems nd Applictions to Swrm Dynmics, Phys. Rev. E 64, [Glück et l. (008)] Glück, A. & Hüffel H. [008] Nonliner Brownin Motion nd Higgs Mechnism, Phys. Lett. B 659,
6 [Glück et l. (009)] Glück, A. & Hüffel H. & Ilijić, S. [009] Cnonicl ctive Brownin motion, Phys. Rev. E 79, [Glück et l. (010)] Glück, A. & Hüffel H. & Ilijić, S. [010] Swrms with cnonicl ctive Brownin motion, rxiv: [Komtsu et l. (009)] Komtsu, E. et l. [009] Five-Yer Wilkinson Microwve Anisotropy Probe Observtions: Cosmologicl Interprettion, Astrophys. J. Suppl. 180, 330. [Kuznetsov (1995)] Kuznetsov, Y.A. [1995] Elements of pplied Bifurction Theory, (Springer, New York). 6
4 The dynamical FRW universe
4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationA5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s
4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The
More information+ x 2 dω 2 = c 2 dt 2 +a(t) [ 2 dr 2 + S 1 κx 2 /R0
Notes for Cosmology course, fll 2005 Cosmic Dynmics Prelude [ ds 2 = c 2 dt 2 +(t) 2 dx 2 ] + x 2 dω 2 = c 2 dt 2 +(t) [ 2 dr 2 + S 1 κx 2 /R0 2 κ (r) 2 dω 2] nd x = S κ (r) = r, R 0 sin(r/r 0 ), R 0 sinh(r/r
More informationTHE INTERVAL LATTICE BOLTZMANN METHOD FOR TRANSIENT HEAT TRANSFER IN A SILICON THIN FILM
ROMAI J., v.9, no.2(2013), 173 179 THE INTERVAL LATTICE BOLTZMANN METHOD FOR TRANSIENT HEAT TRANSFER IN A SILICON THIN FILM Alicj Piseck-Belkhyt, Ann Korczk Institute of Computtionl Mechnics nd Engineering,
More informationClassical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011
Clssicl Mechnics From Moleculr to Con/nuum Physics I WS 11/12 Emilino Ippoli/ October, 2011 Wednesdy, October 12, 2011 Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory,
More informationPERFORMANCE ANALYSIS OF HARMONICALLY FORCED NONLINEAR SYSTEMS
PERFORMANCE ANALYSIS OF HARMONICALLY FORCED NONLINEAR SYSTEMS A.Yu. Pogromsky R.A. vn den Berg J.E. Rood Deprtment of Mechnicl Engineering, Eindhoven University of Technology, P.O. Box 513, 56 MB, Eindhoven,
More information221B Lecture Notes WKB Method
Clssicl Limit B Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationIntro to Nuclear and Particle Physics (5110)
Intro to Nucler nd Prticle Physics (5110) Feb, 009 The Nucler Mss Spectrum The Liquid Drop Model //009 1 E(MeV) n n(n-1)/ E/[ n(n-1)/] (MeV/pir) 1 C 16 O 0 Ne 4 Mg 7.7 14.44 19.17 8.48 4 5 6 6 10 15.4.41
More informationSTABILITY AND BIFURCATION ANALYSIS OF A PIPE CONVEYING PULSATING FLUID WITH COMBINATION PARAMETRIC AND INTERNAL RESONANCES
Mthemticl nd Computtionl Applictions Vol. 0 No. pp. 00-6 05 http://dx.doi.org/0.909/mc-05-07 STABILITY AND BIFURCATION ANALYSIS OF A PIPE CONVEYING PULSATING FLUID WITH COMBINATION PARAMETRIC AND INTERNAL
More informationNew data structures to reduce data size and search time
New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationAPPROXIMATE LIMIT CYCLES FOR THE RAYLEIGH MODEL
ROMAI J, 4, 228, 73 8 APPROXIMATE LIMIT CYCLES FOR THE RAYLEIGH MODEL Adelin Georgescu, Petre Băzăvn, Mihel Sterpu Acdemy of Romnin Scientists, Buchrest Deprtment of Mthemtics nd Computer Science, University
More informationChapter 3 The Schrödinger Equation and a Particle in a Box
Chpter 3 The Schrödinger Eqution nd Prticle in Bo Bckground: We re finlly ble to introduce the Schrödinger eqution nd the first quntum mechnicl model prticle in bo. This eqution is the bsis of quntum mechnics
More informationMath Lecture 23
Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More informationA Vectors and Tensors in General Relativity
1 A Vectors nd Tensors in Generl Reltivity A.1 Vectors, tensors, nd the volume element The metric of spcetime cn lwys be written s ds 2 = g µν dx µ dx ν µ=0 ν=0 g µν dx µ dx ν. (1) We introduce Einstein
More informationAnalytical Based Truncation Principle of Higher- Order Solution for a x 1/3 Force Nonlinear Oscillator
World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl, Computtionl, Sttisticl, Nturl nd Physicl Engineering Vol:7, No:, Anlyticl Bsed Trunction Principle of Higher- Order Solution
More informationDYNAMIC CONTACT PROBLEM OF ROLLING ELASTIC WHEELS
DYNMIC CONTCT PROBLEM OF ROLLING ELSTIC WHEELS Dénes Tkács nd Gábor Stépán Deprtment of pplied Mechnics Budpest Uniersity of Technology nd Economics Budpest, H-151, Hungry BSTRCT: The lterl ibrtion of
More information221A Lecture Notes WKB Method
A Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using ψ x, t = e
More information13: Diffusion in 2 Energy Groups
3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups
More information4- Cosmology - II. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 1
4- Cosmology - II introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 1 4.1 - Solutions of Friedmnn Eqution As shown in Lecture 3, Friedmnn eqution is given by! H 2 = # " & % 2 = 8πG 3 ρ k 2 +
More informationHomework # 4 Solution Key
PHYSICS 631: Generl Reltivity 1. 6.30 Homework # 4 Solution Key The metric for the surfce of cylindr of rdius, R (fixed), for coordintes z, φ ( ) 1 0 g µν = 0 R 2 In these coordintes ll derivtives with
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationLinear and Non-linear Feedback Control Strategies for a 4D Hyperchaotic System
Pure nd Applied Mthemtics Journl 017; 6(1): 5-13 http://www.sciencepublishinggroup.com/j/pmj doi: 10.11648/j.pmj.0170601.1 ISSN: 36-9790 (Print); ISSN: 36-981 (Online) Liner nd Non-liner Feedbck Control
More informationAMATH 731: Applied Functional Analysis Fall Some basics of integral equations
AMATH 731: Applied Functionl Anlysis Fll 2009 1 Introduction Some bsics of integrl equtions An integrl eqution is n eqution in which the unknown function u(t) ppers under n integrl sign, e.g., K(t, s)u(s)
More informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationGAUGE THEORY ON A SPACE-TIME WITH TORSION
GAUGE THEORY ON A SPACE-TIME WITH TORSION C. D. OPRISAN, G. ZET Fculty of Physics, Al. I. Cuz University, Isi, Romni Deprtment of Physics, Gh. Aschi Technicl University, Isi 700050, Romni Received September
More information1 1D heat and wave equations on a finite interval
1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion
More informationQuantum Physics I (8.04) Spring 2016 Assignment 8
Quntum Physics I (8.04) Spring 206 Assignment 8 MIT Physics Deprtment Due Fridy, April 22, 206 April 3, 206 2:00 noon Problem Set 8 Reding: Griffiths, pges 73-76, 8-82 (on scttering sttes). Ohnin, Chpter
More informationTravelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing
Applied Mthemtics E-Notes 8(8) - c IN 67-5 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge
More informationIntroduction to Finite Element Method
Introduction to Finite Element Method Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pn.pl/ tzielins/ Tble of Contents 1 Introduction 1 1.1 Motivtion nd generl concepts.............
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationarxiv:astro-ph/ v4 7 Jul 2006
Cosmologicl models with Gurzdyn-Xue drk energy rxiv:stro-ph/0601073v4 7 Jul 2006 G. V. Vereshchgin nd G. Yegorin ICRANet P.le dell Repubblic 10 I65100 Pescr Itly nd ICRA Dip. Fisic Univ. L Spienz P.le
More information-S634- Journl of the Koren Physicl Society, Vol. 35, August 999 structure with two degrees of freedom. The three types of structures re relted to the
Journl of the Koren Physicl Society, Vol. 35, August 999, pp. S633S637 Conserved Quntities in the Perturbed riedmnn World Model Ji-chn Hwng Deprtment of Astronomy nd Atmospheric Sciences, Kyungpook Ntionl
More informationDo the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?
1 Problem 1 Do the one-dimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More informationpotentials A z, F z TE z Modes We use the e j z z =0 we can simply say that the x dependence of E y (1)
3e. Introduction Lecture 3e Rectngulr wveguide So fr in rectngulr coordintes we hve delt with plne wves propgting in simple nd inhomogeneous medi. The power density of plne wve extends over ll spce. Therefore
More informationLecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationFEM ANALYSIS OF ROGOWSKI COILS COUPLED WITH BAR CONDUCTORS
XIX IMEKO orld Congress Fundmentl nd Applied Metrology September 6 11, 2009, Lisbon, Portugl FEM ANALYSIS OF ROGOSKI COILS COUPLED ITH BAR CONDUCTORS Mirko Mrrcci, Bernrdo Tellini, Crmine Zppcost University
More information1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation
1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationPhysics 202H - Introductory Quantum Physics I Homework #08 - Solutions Fall 2004 Due 5:01 PM, Monday 2004/11/15
Physics H - Introductory Quntum Physics I Homework #8 - Solutions Fll 4 Due 5:1 PM, Mondy 4/11/15 [55 points totl] Journl questions. Briefly shre your thoughts on the following questions: Of the mteril
More informationMonte Carlo method in solving numerical integration and differential equation
Monte Crlo method in solving numericl integrtion nd differentil eqution Ye Jin Chemistry Deprtment Duke University yj66@duke.edu Abstrct: Monte Crlo method is commonly used in rel physics problem. The
More informationMath 124A October 04, 2011
Mth 4A October 04, 0 Viktor Grigoryn 4 Vibrtions nd het flow In this lecture we will derive the wve nd het equtions from physicl principles. These re second order constnt coefficient liner PEs, which model
More informationCHAPTER 4a. ROOTS OF EQUATIONS
CHAPTER 4. ROOTS OF EQUATIONS A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskk Spring 00 ENCE 03 - Computtion Methods in Civil Engineering II Deprtment
More informationINTRODUCTION. The three general approaches to the solution of kinetics problems are:
INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The
More informationVibrational Relaxation of HF (v=3) + CO
Journl of the Koren Chemicl Society 26, Vol. 6, No. 6 Printed in the Republic of Kore http://dx.doi.org/.52/jkcs.26.6.6.462 Notes Vibrtionl Relxtion of HF (v3) + CO Chng Soon Lee Deprtment of Chemistry,
More informationEXAMPLES OF QUANTUM INTEGRALS
EXAMPLES OF QUANTUM INTEGRALS Stn Gudder Deprtment of Mthemtics University of Denver Denver, Colordo 88 sgudder@mth.du.edu Abstrct We first consider method of centering nd chnge of vrible formul for quntum
More informationProblem Set 3 Solutions
Chemistry 36 Dr Jen M Stndrd Problem Set 3 Solutions 1 Verify for the prticle in one-dimensionl box by explicit integrtion tht the wvefunction ψ ( x) π x is normlized To verify tht ψ ( x) is normlized,
More informationHeteroclinic cycles in coupled cell systems
Heteroclinic cycles in coupled cell systems Michel Field University of Houston, USA, & Imperil College, UK Reserch supported in prt by Leverhulme Foundtion nd NSF Grnt DMS-0071735 Some of the reserch reported
More informationarxiv:gr-qc/ v1 14 Mar 2000
The binry blck-hole dynmics t the third post-newtonin order in the orbitl motion Piotr Jrnowski Institute of Theoreticl Physics, University of Bi lystok, Lipow 1, 15-2 Bi lystok, Polnd Gerhrd Schäfer Theoretisch-Physiklisches
More informationJack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer pprox.- energy surfces 2. Men-field (Hrtree-Fock) theory- orbitls 3. Pros nd cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usully does HF-how? 6. Bsis sets nd nottions 7. MPn,
More informationPopulation Dynamics Definition Model A model is defined as a physical representation of any natural phenomena Example: 1. A miniature building model.
Popultion Dynmics Definition Model A model is defined s physicl representtion of ny nturl phenomen Exmple: 1. A miniture building model. 2. A children cycle prk depicting the trffic signls 3. Disply of
More informationTerminal Velocity and Raindrop Growth
Terminl Velocity nd Rindrop Growth Terminl velocity for rindrop represents blnce in which weight mss times grvity is equl to drg force. F 3 π3 ρ L g in which is drop rdius, g is grvittionl ccelertion,
More informationOutline. I. The Why and What of Inflation II. Gauge fields and inflation, generic setup III. Models within Isotropic BG
Outline I. The Why nd Wht of Infltion II. Guge fields nd infltion, generic setup III. Models within Isotropic BG Guge-fltion model Chromo-nturl model IV. Model within Anisotropic BG Infltion with nisotropic
More informationu t = k 2 u x 2 (1) a n sin nπx sin 2 L e k(nπ/l) t f(x) = sin nπx f(x) sin nπx dx (6) 2 L f(x 0 ) sin nπx 0 2 L sin nπx 0 nπx
Chpter 9: Green s functions for time-independent problems Introductory emples One-dimensionl het eqution Consider the one-dimensionl het eqution with boundry conditions nd initil condition We lredy know
More informationApplication of Exp-Function Method to. a Huxley Equation with Variable Coefficient *
Interntionl Mthemticl Forum, 4, 9, no., 7-3 Appliction of Exp-Function Method to Huxley Eqution with Vrible Coefficient * Li Yo, Lin Wng nd Xin-Wei Zhou. Deprtment of Mthemtics, Kunming College Kunming,Yunnn,
More informationMassachusetts Institute of Technology Quantum Mechanics I (8.04) Spring 2005 Solutions to Problem Set 6
Msschusetts Institute of Technology Quntum Mechnics I (8.) Spring 5 Solutions to Problem Set 6 By Kit Mtn. Prctice with delt functions ( points) The Dirc delt function my be defined s such tht () (b) 3
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationVector potential quantization and the photon wave-particle representation
Vector potentil quntiztion nd the photon wve-prticle representtion Constntin Meis, Pierre-Richrd Dhoo To cite this version: Constntin Meis, Pierre-Richrd Dhoo. Vector potentil quntiztion nd the photon
More informationOn the Linear Stability of Compound Capillary Jets
ILASS Americs, th Annul Conference on Liquid Atomiztion nd Spry Systems, Chicgo, IL, My 7 On the Liner Stbility of Compound Cpillry Jets Mksud (Mx) Ismilov, Stephen D Heister School of Aeronutics nd Astronutics,
More informationThe Riemann-Lebesgue Lemma
Physics 215 Winter 218 The Riemnn-Lebesgue Lemm The Riemnn Lebesgue Lemm is one of the most importnt results of Fourier nlysis nd symptotic nlysis. It hs mny physics pplictions, especilly in studies of
More informationChaos in drive systems
Applied nd Computtionl Mechnics 1 (2007) 121-126 Chos in drive systems Ctird Krtochvíl, Mrtin Houfek, Josef Koláčný b, Romn Kříž b, Lubomír Houfek,*, Jiří Krejs Institute of Thermomechnics, brnch Brno,
More informationESCI 343 Atmospheric Dynamics II Lesson 14 Inertial/slantwise Instability
ESCI 343 Atmospheric Dynmics II Lesson 14 Inertil/slntwise Instbility Reference: An Introduction to Dynmic Meteorology (3 rd edition), J.R. Holton Atmosphere-Ocen Dynmics, A.E. Gill Mesoscle Meteorology
More informationBig Bang/Inflationary Picture
Big Bng/Infltionry Picture big bng infltionry epoch rdition epoch mtter epoch drk energy epoch DISJOINT Gret explntory power: horizon fltness monopoles entropy Gret predictive power: Ω totl = 1 nerly scle-invrint
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationAtwood s machine with a massive string
Atwood s mchine with mssive string rxiv:1710.01263v1 [physics.gen-ph] 1 Oct 2017 Nivldo A. Lemos Instituto de Físic - Universidde Federl Fluminense Av. Litorâne, S/N, Bo Vigem, Niterói, 24210-340, Rio
More informationChapter 3 Polynomials
Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationMethod of Localisation and Controlled Ejection of Swarms of Likely Charged Particles
Method of Loclistion nd Controlled Ejection of Swrms of Likely Chrged Prticles I. N. Tukev July 3, 17 Astrct This work considers Coulom forces cting on chrged point prticle locted etween the two coxil,
More informationPhysics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018
Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.
More informationElectron Correlation Methods
Electron Correltion Methods HF method: electron-electron interction is replced by n verge interction E HF c E 0 E HF E 0 exct ground stte energy E HF HF energy for given bsis set HF Ec 0 - represents mesure
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationTHERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION
XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es
More informationExplain shortly the meaning of the following eight words in relation to shells structures.
Delft University of Technology Fculty of Civil Engineering nd Geosciences Structurl Mechnics Section Write your nme nd study number t the top right-hnd of your work. Exm CIE4143 Shell Anlysis Tuesdy 15
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationThe Periodically Forced Harmonic Oscillator
The Periodiclly Forced Hrmonic Oscilltor S. F. Ellermeyer Kennesw Stte University July 15, 003 Abstrct We study the differentil eqution dt + pdy + qy = A cos (t θ) dt which models periodiclly forced hrmonic
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationLecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations
18330 Lecture Notes: Orthogonl Polynomils, Gussin Qudrture, nd Integrl Equtions Homer Reid My 1, 2014 In the previous set of notes we rrived t the definition of Chebyshev polynomils T n (x) vi the following
More informationSolutions of Klein - Gordan equations, using Finite Fourier Sine Transform
IOSR Journl of Mthemtics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 6 Ver. IV (Nov. - Dec. 2017), PP 19-24 www.iosrjournls.org Solutions of Klein - Gordn equtions, using Finite Fourier
More informationNonlocal Gravity and Structure in the Universe
Nonlocl rvity nd Structure in the Universe Sohyun Prk Penn Stte University Co-uthor: Scott Dodelson Bsed on PRD 87 (013) 04003, 109.0836, PRD 90 (014) 000000, 1310.439 August 5, 014 Chicgo, IL Cosmo 014
More informationSufficient condition on noise correlations for scalable quantum computing
Sufficient condition on noise correltions for sclble quntum computing John Presill, 2 Februry 202 Is quntum computing sclble? The ccurcy threshold theorem for quntum computtion estblishes tht sclbility
More informationVariational problems of some second order Lagrangians given by Pfaff forms
Vritionl problems of some second order Lgrngins given by Pfff forms P. Popescu M. Popescu Abstrct. In this pper we study the dynmics of some second order Lgrngins tht come from Pfff forms i.e. differentil
More informationRemark on boundary value problems arising in Ginzburg-Landau theory
Remrk on boundry vlue problems rising in Ginzburg-Lndu theory ANITA KIRICHUKA Dugvpils University Vienibs Street 13, LV-541 Dugvpils LATVIA nit.kiricuk@du.lv FELIX SADYRBAEV University of Ltvi Institute
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationLyapunov function for cosmological dynamical system
Demonstr. Mth. 207; 50: 5 55 Demonstrtio Mthemtic Open Access Reserch Article Mrek Szydłowski* nd Adm Krwiec Lypunov function for cosmologicl dynmicl system DOI 0.55/dem-207-0005 Received My 8, 206; ccepted
More informationOn the Decomposition Method for System of Linear Fredholm Integral Equations of the Second Kind
Applied Mthemticl Sciences, Vol. 2, 28, no. 2, 57-62 On the Decomposition Method for System of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi 1 nd M. Mokhtri Deprtment of Mthemtics, Shhr-e-Rey
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
More information