Microscopic Theory of Transport (Fall 2003) Lecture 6 (9/19/03) Static and Short Time Properties of Time Correlation Functions
|
|
- Lewis Rice
- 5 years ago
- Views:
Transcription
1 .03 Microscopic Theory of Trasport (Fall 003) Lecture 6 (9/9/03) Static ad Short Time Properties of Time Correlatio Fuctios Refereces -- Boo ad Yip, Chap There are a umber of properties of time correlatio fuctios which follow from their defiitios, sometimes i a straightforward maer ad other times ot so simply. It is importat to make ote of them because ot oly do they have useful physical meaigs, but also they are ofte used to formulate model descriptios of time correlatio fuctios. Actually we have already ecoutered several examples i our discussio of diffusio, for example, the relatio betwee the log-time behavior of the mea-squared displacemet fuctio ad the diffusio coefficiet. Static Correlatio Fuctios Sice a time correlatio fuctio is a fuctio of time, oe ca always ask about its value at time t = 0, also kow as the iitial value. This value is the correlatio of two variables (which may be differet or the same) at the same time but at differet spatial positios. Correlatio fuctios which have spatial variatios but do ot deped o time are called static. We have see that the iitial value of the Va Hove self correlatio fuctio is just the delta fuctio, (4.5). It is a iterestig exercise to work out the correspodig iitial value for the desity correlatio fuctio Grt (,). Settig t = 0 i (5.5) we fid Gr (,0) = δ () r + gr [ () ] (6.) where gr δ r Ri δ R j i, j () = < ( ) ( ) > = (6.) ( ) 3 3 U(0, r, R3,, R )/ kbt dr3 dre Q The fuctio g(r) is well-kow i liquid-state theory as the equilibrium pair distributio or the radial distributio fuctio (RDF). This is the quatity that oe obtais from a x- ray or eutro diffractio measuremet. Recall from the previous lecture that G is the sum of G s ad G d. Eq.(6.) shows that the first term is the iitial value of G s as we already kow from (4.5). The secod term is therefore the iitial value of G d.
2 Because g(r) is the fudametal quatity i the equilibrium theory of liquids [see McQuarrie], we digress somewhat from the preset discussio of time-depedet desity correlatio fuctios to examie its relatio to the caoical distributio fuctio. Suppose we defie βu ( ) 3 3 e 3 3 P ( r... r ) d r... d r = d r... d r Z (6.3) as the probability that atom is i d 3 r about r,, atom is i d 3 r about r. Itegratig this over the positios of atoms + through gives ( ) 3 3 βu P ( r... r)... d r... d r e + Z = (6.4) gives the probability that atom is i d 3 r about r,, atom is i d 3 r about r irrespective of the positios of atoms + through. ow the probability that ay atom is i d 3 r, ay other atom is i d 3 r, ay atom other tha the first two is i d 3 r 3,, ad ay atom other tha the previous - atoms is i d 3 r is ( )! ( ) ( r... r) = P ( r... r ) (6.5) ( )! where!/( )! is the umber of ways oe ca pick objects out of possible choices. For example, the umber of ways of pickig two atoms out of a -particle system is!/(-)! = (-). Give (6.5), we have V 3 () d r ( r) = ρ (6.6) V which is the umber desity of the system (usually deoted as ad take to be a ( ) costat). We ext defie the -particle correlatio fuctio g ( r... r ) by writig ( ) ( ) r r = g r r (... ) (... ) (6.7) ( ) otice that if the atoms are ot correlated, the ( ) =, ad g. This is what we mea operatioally whe we say there is o spatial correlatio amog the -particles i the system. Combiig (6.4), (6.5), ad (6.77) we obtai a defiitio of the -particle correlatio fuctio i terms of the caoical distributio, V! g r r d r d r e ( ) 3 3 βu (... ) = ( )! Z (6.8) which is the desired relatio.
3 To reduce to the pair distributio fuctio g(r) we ote that i a liquid composed () spherical atoms, g ( r, r ) depeds oly o the separatio distace r = r r. Thus we ca write g () ( r, r) g( r), or ( r, r ) = g( r) (6.9) () which is a defiitio of g(r) i terms of the -particle correlatio fuctio (58) with =. It the follows from the above probability distributios that 3 g() r d r = probability (ot ormalized) of fidig a atom i 3 dr about r give a atom is at the origi To fid the ormalizatio we take 3 d r g() r = g()4 r π r dr = 0 (6.0) Thus, π g r dr = 4 r ( ) umber of atoms i a shell of radius r ad thickess dr cetered about a atom at the origi A sketch of g(r), which is also kow as the pair correlatio fuctio or the radial distributio fuctio, for a typical liquid shows its most sigificat features - a promiet peak at the distace of earest eighbor separatio, ad a weaker peak at the secodearest separatio, see Fig. 6.. Below a certai separatio distace g(r) vaishes, sigifyig a hard core i the iteratomic iteractio which gives rise to a excluded volume i the spatial distributio of particles. I other words, particles repeal each other at very short rage, a basic property associated with the existece of a certai desity of the system. At the opposite extreme of large separatios, g approaches uity, sigifyig the loss of spatial correlatio (particles are ucorrelated whe they are far apart). It is useful to compare the behavior of g(r) to that of a typical pair potetial like the Leard- ( ) Joes. Ideed at low desities a virial expasio of g(r) gives gr () e βv r, where V(r) is the pair potetial. This leads to the defiitio of a 'potetial of mea force' V ( r) g( r)/ β which oe ca apply eve to hard-sphere fluids. Sice the static eff structure factor S(k) ad g(r) are related by Fourier trasform, a osciallatory behavior i oe fuctio meas the other fuctio will also show oscillatios. The small wiggles i g(r) at small r are umerical artifact of the Fourier trasformatio. The value of S(k) i 3
4 the limit of zero k is the compressibility factor, which becomes large i the viciity of the critical poit. Image Removed Yarell et al., Physical Review A7, p.30 (973) top graph is fig 6 i paper; bottom graph is fig 4. Fig. 6.. Radial distribtutio fuctio g(r) of liquid argo as obtaied by Fourier trasform of the measured static structure S(Q). Data poits are eutro diffractio measuremets, the curve is the result of molecular dyamics simulatio usig the Leard-Joes potetial (Yarell et al., Phys. Rev. A7, 330 (973)). The oscillatory shape of g(r) has a simple physical meaig. Suppose we sit o ay oe of the particles ad look aroud to see how the eighborig particles are distributed. We ca do this by drawig a sphere of radius r aroud us, with some thickess dr, ad ask how may of the eighbors lie iside this spherical shell, doig this repeatedly with icreasig r. Sice for a give r ad dr, the umber of eighbors is just 4 π rg( rdr ), the fact that g(r) is a oscillatory suggests that the eighbors are ot uiformly distributed, rather they like to arrage themselves i a shell-like structure. Thus the first peak i g(r), which is by far the strogest, sigifies the positio of the earest-eighbors, the secod peak, the secod earest eighbors, ad so o. The peaks are see to damp out fairly quickly, this is a reflectio of the fact that there is oly local order i a liquid ad o log-rage order. Oe expects that for a crystal the correspodig g(r) would be a set of very sharp lies, each sittig precisely at the positio for the particular eighbors. 4
5 Returig to the desity correlatio fuctios, we ote that just as g(r) ca be measured by eutro ad x-ray diffractio, G ad G ca be directly measured by eutro ielastic s scatterig ad light scatterig. eutro ielastic scatterig experimets actually give the double Fourier trasforms of the two desity correlatio fuctios, (, ω) 3 i( k r ωt ) (, ) Sk dt dre Grt = (6.) ad the same relatio exists betwee S ( k, ω ) ad G (,) r t. Skω (, ) is geerally kow as s s the dyamic structure factor, which is reasoable sice the Fourier trasform of g(r), Sk ( ) = + dre [ gr ( ) ] (6.) is called the static structure factor. To see the coectio betwee (6.) ad (6.), we itroduce aother quatity which is related to Skω (, )(ad therefore Grt (,)) by writig (6.) as iωt Sk (, ω) = dte Fkt (, ) (6.3) so that the itermediate scatterig fuctio F is just the Fourier spatial trasform of G(r,t) (for isotropic systems such as a simple fluid, G depeds oly the magitude of r ), F kt = dre Grt (, ) (, ) with * 3 =< (0) ( t) > ( π) δ( k) k i () t k i= k (6.4) ik R ( t) = e (6.5) We ow ask what is the iitial value of F(k,t). At t = 0, (6.4) gives where F( k,0) = d re G( r,0) (6.6) δ δ, ' G(,0) r =< (' r R (0)) ( r R (0)) > (6.7) Settig r ' = 0 without ay loss of geerality, we ca write out the double sum explicitly, 5
6 G(,0) r = < δ( R ) δ( r R ) >+ ( ) < δ( R ) δ( r R ) > 3 3 βu ( ) 3 3 βu = dr... dre δ( R ) δ( r) dr... dre δ( R ) δ( r R) Q + Q () r g () r = V δ + (6.8) Isertig this result ito (67) we obtai F( k,0) = + d re [ g( r) ] S( k ) (6.9) Sice the iformatio eeded to calculate the desity correlatio fuctios are the timedepedet particle positios, the output of MD simulatios, it follows that MD provides a direct meas of determiig the correlatio fuctios ad their Fourier trasforms. I this way, MD is able to provide results that ca be used to iterpret experimets, as well as results that ca be used to test various dyamical models of atomic motios i gases ad liquids. Examples of both kids of applicatios will be discussed i class. Short-Time Behavior ad Sum Rules The iitial value of a time correlatio is the leadig term i a Taylor series expasio i time. Because the Fourier trasform of a time correlatio fuctio is ofte also of iterest, we ca make use of the coectio betwee the coefficiets of the Taylor expasio ad the frequecy momets of the correspodig Fourier trasform of the time correlatio fuctio. Take the example of the desity correlatio fuctio where we write F kt = dre Grt (, ) (, ) The Taylor series expasio is i t = dte ω S( k, ω) (6.0) Fkt Fk t t dfkt (, ) dfkt (, ) (, ) = (,0) dt t= 0! dt t= 0 + (6.) while from (6.0) we have 6
7 ( it) Fkt d Sk it d Sk d Sk! (, ) = ω (, ω) + ωω (, ω) + ωω (, ω)... + (6.) Matchig terms with the same power of t i (6.) ad (6.) gives relatio betwee the frequecy momets of S, kow as sum rules, to the time derivatives of F, the coefficiets i the Taylor series i (6.). The first few such relatios are: ω o ( k) = dωs( k, ω) π = S(k) (6.3) o ωs = dωss( k, ω) π = (6.4) ω ωω ω ( k) = d S( k, ) = ( kv o) π = ωs( k) = dωω Ss( k, ω) π (6.5) where v k T/ m, with v beig the thermal speed of the particle. o = B o The relatios betwee the coefficiets of the time expasio ad the frequecy momets of the Fourier trasform of the time correlatio fuctio is quite geeral; it is applicable to ay pair of fuctios which are Fourier trasforms of each other. Sice we are dealig with classical time correlatio fuctios, they are eve fuctios i time ad frequecy. So terms that are odd i t vaish i (6.) ad (6.). There are a umber of properties of classical correlatio fuctios which do ot carry over to quatum mechaical defiitios of correlatio fuctios. We do ot take the time to discuss them here except to refer the reader to Sec.,7 Liear Respose Theory for some of the details. To fiish up this lecture, we ote that there are properties such as log time behavior ad relatio to trasport coefficiets, the cocept of a memory fuctio, liear respose theory, ad kietic theory which are treated i Chap of Boo ad Yip. We will retur to them at various times i future lectures. 7
TIME-CORRELATION FUNCTIONS
p. 8 TIME-CORRELATION FUNCTIONS Time-correlatio fuctios are a effective way of represetig the dyamics of a system. They provide a statistical descriptio of the time-evolutio of a variable for a esemble
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationPhysics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.
Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;
More informationx a x a Lecture 2 Series (See Chapter 1 in Boas)
Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More informationSolutions to Final Exam Review Problems
. Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the
More informationRotationally invariant integrals of arbitrary dimensions
September 1, 14 Rotatioally ivariat itegrals of arbitrary dimesios James D. Wells Physics Departmet, Uiversity of Michiga, A Arbor Abstract: I this ote itegrals over spherical volumes with rotatioally
More informationSimilarity between quantum mechanics and thermodynamics: Entropy, temperature, and Carnot cycle
Similarity betwee quatum mechaics ad thermodyamics: Etropy, temperature, ad Carot cycle Sumiyoshi Abe 1,,3 ad Shiji Okuyama 1 1 Departmet of Physical Egieerig, Mie Uiversity, Mie 514-8507, Japa Istitut
More information4.3 Growth Rates of Solutions to Recurrences
4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.
More informationHolistic Approach to the Periodic System of Elements
Holistic Approach to the Periodic System of Elemets N.N.Truov * D.I.Medeleyev Istitute for Metrology Russia, St.Peterburg. 190005 Moskovsky pr. 19 (Dated: February 20, 2009) Abstract: For studyig the objectivity
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationRandom Models. Tusheng Zhang. February 14, 2013
Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the
More informationPractice Problems: Taylor and Maclaurin Series
Practice Problems: Taylor ad Maclauri Series Aswers. a) Start by takig derivatives util a patter develops that lets you to write a geeral formula for the -th derivative. Do t simplify as you go, because
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationFirst, note that the LS residuals are orthogonal to the regressors. X Xb X y = 0 ( normal equations ; (k 1) ) So,
0 2. OLS Part II The OLS residuals are orthogoal to the regressors. If the model icludes a itercept, the orthogoality of the residuals ad regressors gives rise to three results, which have limited practical
More informationApplication to Random Graphs
A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let
More informationLecture 3. Digital Signal Processing. Chapter 3. z-transforms. Mikael Swartling Nedelko Grbic Bengt Mandersson. rev. 2016
Lecture 3 Digital Sigal Processig Chapter 3 z-trasforms Mikael Swartlig Nedelko Grbic Begt Madersso rev. 06 Departmet of Electrical ad Iformatio Techology Lud Uiversity z-trasforms We defie the z-trasform
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationPHYS-3301 Lecture 9. CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I. 5.3: Electron Scattering. Bohr s Quantization Condition
CHAPTER 5 Wave Properties of Matter ad Quatum Mecaics I PHYS-3301 Lecture 9 Sep. 5, 018 5.1 X-Ray Scatterig 5. De Broglie Waves 5.3 Electro Scatterig 5.4 Wave Motio 5.5 Waves or Particles? 5.6 Ucertaity
More informationEconomics 250 Assignment 1 Suggested Answers. 1. We have the following data set on the lengths (in minutes) of a sample of long-distance phone calls
Ecoomics 250 Assigmet 1 Suggested Aswers 1. We have the followig data set o the legths (i miutes) of a sample of log-distace phoe calls 1 20 10 20 13 23 3 7 18 7 4 5 15 7 29 10 18 10 10 23 4 12 8 6 (1)
More informationPHYS-3301 Lecture 10. Wave Packet Envelope Wave Properties of Matter and Quantum Mechanics I CHAPTER 5. Announcement. Sep.
Aoucemet Course webpage http://www.phys.ttu.edu/~slee/3301/ PHYS-3301 Lecture 10 HW3 (due 10/4) Chapter 5 4, 8, 11, 15, 22, 27, 36, 40, 42 Sep. 27, 2018 Exam 1 (10/4) Chapters 3, 4, & 5 CHAPTER 5 Wave
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More informationCastiel, Supernatural, Season 6, Episode 18
13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio
More information8. IRREVERSIBLE AND RANDOM PROCESSES Concepts and Definitions
8. IRREVERSIBLE ND RNDOM PROCESSES 8.1. Cocepts ad Defiitios I codesed phases, itermolecular iteractios ad collective motios act to modify the state of a molecule i a time-depedet fashio. Liquids, polymers,
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More information. (24) If we consider the geometry of Figure 13 the signal returned from the n th scatterer located at x, y is
.5 SAR SIGNA CHARACTERIZATION I order to formulate a SAR processor we first eed to characterize the sigal that the SAR processor will operate upo. Although our previous discussios treated SAR cross-rage
More informationSECTION 2 Electrostatics
SECTION Electrostatics This sectio, based o Chapter of Griffiths, covers effects of electric fields ad forces i static (timeidepedet) situatios. The topics are: Electric field Gauss s Law Electric potetial
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationMatsubara-Green s Functions
Matsubara-Gree s Fuctios Time Orderig : Cosider the followig operator If H = H the we ca trivially factorise this as, E(s = e s(h+ E(s = e sh e s I geeral this is ot true. However for practical applicatio
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More informationSummary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.
Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios
More information17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)
7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationPHYC - 505: Statistical Mechanics Homework Assignment 4 Solutions
PHYC - 55: Statistical Mechaics Homewor Assigmet 4 Solutios Due February 5, 14 1. Cosider a ifiite classical chai of idetical masses coupled by earest eighbor sprigs with idetical sprig costats. a Write
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More information11 Correlation and Regression
11 Correlatio Regressio 11.1 Multivariate Data Ofte we look at data where several variables are recorded for the same idividuals or samplig uits. For example, at a coastal weather statio, we might record
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationExample: Find the SD of the set {x j } = {2, 4, 5, 8, 5, 11, 7}.
1 (*) If a lot of the data is far from the mea, the may of the (x j x) 2 terms will be quite large, so the mea of these terms will be large ad the SD of the data will be large. (*) I particular, outliers
More informationFinally, we show how to determine the moments of an impulse response based on the example of the dispersion model.
5.3 Determiatio of Momets Fially, we show how to determie the momets of a impulse respose based o the example of the dispersio model. For the dispersio model we have that E θ (θ ) curve is give by eq (4).
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationOlli Simula T / Chapter 1 3. Olli Simula T / Chapter 1 5
Sigals ad Systems Sigals ad Systems Sigals are variables that carry iformatio Systemstake sigals as iputs ad produce sigals as outputs The course deals with the passage of sigals through systems T-6.4
More informationThe axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c.
5.4 Applicatio of Perturbatio Methods to the Dispersio Model for Tubular Reactors The axial dispersio model for tubular reactors at steady state ca be described by the followig equatios: d c Pe dz z =
More informationThe Method of Least Squares. To understand least squares fitting of data.
The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationSignal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform
Sigal Processig i Mechatroics Summer semester, 1 Lecture 3, Covolutio, Fourier Series ad Fourier rasform Dr. Zhu K.P. AIS, UM 1 1. Covolutio Covolutio Descriptio of LI Systems he mai premise is that the
More informationMechatronics. Time Response & Frequency Response 2 nd -Order Dynamic System 2-Pole, Low-Pass, Active Filter
Time Respose & Frequecy Respose d -Order Dyamic System -Pole, Low-Pass, Active Filter R 4 R 7 C 5 e i R 1 C R 3 - + R 6 - + e out Assigmet: Perform a Complete Dyamic System Ivestigatio of the Two-Pole,
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationHE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT. Examples:
5.6 4 Lecture #3-4 page HE ATOM & APPROXIMATION METHODS MORE GENERAL VARIATIONAL TREATMENT Do t restrict the wavefuctio to a sigle term! Could be a liear combiatio of several wavefuctios e.g. two terms:
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationThe Growth of Functions. Theoretical Supplement
The Growth of Fuctios Theoretical Supplemet The Triagle Iequality The triagle iequality is a algebraic tool that is ofte useful i maipulatig absolute values of fuctios. The triagle iequality says that
More informationMA131 - Analysis 1. Workbook 2 Sequences I
MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More informationTrue Nature of Potential Energy of a Hydrogen Atom
True Nature of Potetial Eergy of a Hydroge Atom Koshu Suto Key words: Bohr Radius, Potetial Eergy, Rest Mass Eergy, Classical Electro Radius PACS codes: 365Sq, 365-w, 33+p Abstract I cosiderig the potetial
More informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More information1. Linearization of a nonlinear system given in the form of a system of ordinary differential equations
. Liearizatio of a oliear system give i the form of a system of ordiary differetial equatios We ow show how to determie a liear model which approximates the behavior of a time-ivariat oliear system i a
More informationProblem 1. Problem Engineering Dynamics Problem Set 9--Solution. Find the equation of motion for the system shown with respect to:
2.003 Egieerig Dyamics Problem Set 9--Solutio Problem 1 Fid the equatio of motio for the system show with respect to: a) Zero sprig force positio. Draw the appropriate free body diagram. b) Static equilibrium
More information6.867 Machine learning, lecture 7 (Jaakkola) 1
6.867 Machie learig, lecture 7 (Jaakkola) 1 Lecture topics: Kerel form of liear regressio Kerels, examples, costructio, properties Liear regressio ad kerels Cosider a slightly simpler model where we omit
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationNumerical Methods in Fourier Series Applications
Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets
More informationMath 2784 (or 2794W) University of Connecticut
ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler
More informationCOURSE INTRODUCTION: WHAT HAPPENS TO A QUANTUM PARTICLE ON A PENDULUM π 2 SECONDS AFTER IT IS TOSSED IN?
COURSE INTRODUCTION: WHAT HAPPENS TO A QUANTUM PARTICLE ON A PENDULUM π SECONDS AFTER IT IS TOSSED IN? DROR BAR-NATAN Follows a lecture give by the author i the trivial otios semiar i Harvard o April 9,
More informationThe Phi Power Series
The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.
More informationTHE KALMAN FILTER RAUL ROJAS
THE KALMAN FILTER RAUL ROJAS Abstract. This paper provides a getle itroductio to the Kalma filter, a umerical method that ca be used for sesor fusio or for calculatio of trajectories. First, we cosider
More informationPosition Time Graphs 12.1
12.1 Positio Time Graphs Figure 3 Motio with fairly costat speed Chapter 12 Distace (m) A Crae Flyig Figure 1 Distace time graph showig motio with costat speed A Crae Flyig Positio (m [E] of pod) We kow
More informationC. Complex Numbers. x 6x + 2 = 0. This equation was known to have three real roots, given by simple combinations of the expressions
C. Complex Numbers. Complex arithmetic. Most people thik that complex umbers arose from attempts to solve quadratic equatios, but actually it was i coectio with cubic equatios they first appeared. Everyoe
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationHauptman and Karle Joint and Conditional Probability Distributions. Robert H. Blessing, HWI/UB Structural Biology Department, January 2003 ( )
Hauptma ad Karle Joit ad Coditioal Probability Distributios Robert H Blessig HWI/UB Structural Biology Departmet Jauary 00 ormalized crystal structure factors are defied by E h = F h F h = f a hexp ihi
More informationSimilarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall
Iteratioal Mathematical Forum, Vol. 9, 04, o. 3, 465-475 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/imf.04.48 Similarity Solutios to Usteady Pseudoplastic Flow Near a Movig Wall W. Robi Egieerig
More informationAdditional Notes on Power Series
Additioal Notes o Power Series Mauela Girotti MATH 37-0 Advaced Calculus of oe variable Cotets Quick recall 2 Abel s Theorem 2 3 Differetiatio ad Itegratio of Power series 4 Quick recall We recall here
More informationLecture 2: April 3, 2013
TTIC/CMSC 350 Mathematical Toolkit Sprig 203 Madhur Tulsiai Lecture 2: April 3, 203 Scribe: Shubhedu Trivedi Coi tosses cotiued We retur to the coi tossig example from the last lecture agai: Example. Give,
More informationChapter 5 Vibrational Motion
Fall 4 Chapter 5 Vibratioal Motio... 65 Potetial Eergy Surfaces, Rotatios ad Vibratios... 65 Harmoic Oscillator... 67 Geeral Solutio for H.O.: Operator Techique... 68 Vibratioal Selectio Rules... 7 Polyatomic
More information1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.
.3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(
More informationChapter 12 Correlation
Chapter Correlatio Correlatio is very similar to regressio with oe very importat differece. Regressio is used to explore the relatioship betwee a idepedet variable ad a depedet variable, whereas correlatio
More informationEE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course
Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL
More information1the 1it is said to be overdamped. When 1, the roots of
Homework 3 AERE573 Fall 08 Due 0/8(M) ame PROBLEM (40pts) Cosider a D order uderdamped system trasfer fuctio H( s) s ratio 0 The deomiator is the system characteristic polyomial P( s) s s (a)(5pts) Use
More informationThe time evolution of the state of a quantum system is described by the time-dependent Schrödinger equation (TDSE): ( ) ( ) 2m "2 + V ( r,t) (1.
Adrei Tokmakoff, MIT Departmet of Chemistry, 2/13/2007 1-1 574 TIME-DEPENDENT QUANTUM MECHANICS 1 INTRODUCTION 11 Time-evolutio for time-idepedet Hamiltoias The time evolutio of the state of a quatum system
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationx c the remainder is Pc ().
Algebra, Polyomial ad Ratioal Fuctios Page 1 K.Paulk Notes Chapter 3, Sectio 3.1 to 3.4 Summary Sectio Theorem Notes 3.1 Zeros of a Fuctio Set the fuctio to zero ad solve for x. The fuctio is zero at these
More informationWHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT
WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still
More informationDistribution of Random Samples & Limit theorems
STAT/MATH 395 A - PROBABILITY II UW Witer Quarter 2017 Néhémy Lim Distributio of Radom Samples & Limit theorems 1 Distributio of i.i.d. Samples Motivatig example. Assume that the goal of a study is to
More informationQuadratic Functions. Before we start looking at polynomials, we should know some common terminology.
Quadratic Fuctios I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively i mathematical
More informationII. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation
II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio
More informationQuantum Computing Lecture 7. Quantum Factoring
Quatum Computig Lecture 7 Quatum Factorig Maris Ozols Quatum factorig A polyomial time quatum algorithm for factorig umbers was published by Peter Shor i 1994. Polyomial time meas that the umber of gates
More informationIntroduction to Signals and Systems, Part V: Lecture Summary
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive
More information