# Anderson localization: Metallic behavior in two dimensions?

Save this PDF as:

Size: px
Start display at page:

Download "Anderson localization: Metallic behavior in two dimensions?"

## Transcription

1 Anderson localization: Metallic behavior in two dimensions? Vladimir Kuzovkov Institute of Solid State Physics University of Latvia 8 Kengaraga Str., LV-1063, RIGA, Latvia September 13, 2005 Page 1 of 48 Collaborators: Wolfgang von Niessen, Slava Kashcheyevs, Oliver Hein

2 From Fermat s last theorem to Anderson theorem Fermat s last theorem The equation x D + y D = z D has no non-zero integer solutions for x, y and z when D > 2. Solutions for D = 1 (trivial). Solutions for D = 2 (Pythagoras Theorem). No solutions for D > 2 (Fermat, Euler, Cauchy, Lamé, Kummer,...,Wiles-1993) Page 2 of 48

3 Phase transition in magnetics: Ising model The Ising model undergoes a phase transition between an ordered and a disordered phase in D 2. Exact solution for D = 1 (Ising) - no phase transition. Exact solution for D = 2 (Onsager) - second order phase transition. Page 3 of 48

4 Phase transition in magnetics: Heisenberg model No exact solutions for 2D! Condensed-Matter Physics (1986) We believe that the 2-D Heisenberg model has no phase transition at all - it remains paramagnetic (disordered) at all temperatures other than zero. H. E. Stanley and T. A. Kaplan, PRL, 17, 913 (1966) Abstract: We point out that the existence of a phase transition - as indicated by extrapolation from high-temperature expansions - is as well-founded for two-dimensional lattices with nearestneighbor ferromagnetic Heisenberg interactions as for threedimensional lattices, and that the well-known result that there exists no phase transition in two dimensions is not a valid conclusion from the standard spin-wave argument. Page 4 of 48

5 1.2. Ideal crystal The periodicity of the crystal Classification of electronic wave functions as Bloch waves Calculation of band structures Page 5 of 48

6 1.3. Extended state In real life the crystalline state is the exception rather than the rule. Disorder: Weak-disorder limit (few impurities or interstitials in an otherwise perfect crystalline) Strongly disordered limit (alloys or glassy structures). The weak-disorder limit is traditionally described by the scattering of Bloch waves by impurities. Page 6 of 48

7 The traditional view had been that scattering by the random potential causes the Bloch waves to lose phase coherence on the length scale of the mean free path. The wave function remains extended throughout the sample. Page 7 of 48

8 P.W. Anderson, Phys.Rev. 109, 1492 (1958) If the disorder is very strong, the wave function may become localized: the envelope of the wave function decays exponentially, Ψ(r) exp( r/ξ), and ξ is the localization length. Extended state = metallic state Localized state = insulating state Transition between a metal and an insulator = metal-insulator transition (MIT) Page 8 of 48

9 Philip Warren Anderson - Autobiography The years since the Nobel Prize (1977) have been productive ones for me. For instance, in 1978, shortly after receiving the prize in part for localization theory, I was one of the Gang of Four (with Elihu Abrahams, T.V. Ramakrishnan, and Don Licciardello) who revitalized that theory by developing a scaling theory which made it into a quantitative experimental science with precise predictions as a function of magnetic field, interactions, dimensionality, etc.; a major branch of science continues to flow from the consequences of this work. Page 9 of 48

10 1.4. Theoretical consensuns: no metallic behavior in two dimensions! All states are localized in one (1-D) dimensions. (2-D) dimen- All states are localized in two sions. MIT is a continuous one (second-order) in three (3-D) dimensions. Scaling theory of localization: Abrahams, Anderson, Licciardello and Ramakrishnan, Phys.Rev.Lett. 42, 673 (1979) Page 10 of 48

11 1.5. L. Onsager and R. Feynman Then Professor Onsager got up and said in a dour voice, Well, Professor Feynman is new in our field, and I think he needs to be educated. There s something he ought to know, and we should tell him. I thought, Geesus! What did I do wrong? Onsager said, We should tell Feynman that nobody has ever figured out the order of any transition correctly from first principles... Richard P. Feynman Surely You re Joking, Mr Feynman! Page 11 of 48

12 1.6. Experiment Colloquium: Metallic behavior and related phenomena in two dimensions E. Abrahams, S.V.Kravchenko and M.P.Sarachik, Reviews of Modern Physics, 73, 251 (2001) Abstract: For about 20 years, it has been the prevailing view that there can be no metallic state or metal-insulator transition in two-dimensions in zero megnetic field. In the last several years, however, unusual behavior suggestive of such a transition has been reported in a variety of dilute two dimensional electron and hole systems. The physics behind these observations is at present not understood. The authors review and discuss the main experimental findings und suggested theoretical models. Page 12 of 48

13 IOP Page 13 of 48

14 Referee A: This is a most unusual paper, for in it the authors claim to provide an analytical solution to one of the major problems in condensed matter theory: two-dimensional Anderson localization in a disordered tight binding model. The conventional scaling-based view of D=2 is that all states are localized for arbitrarily weak disorder. In contrast the present authors conclude that both extended and localized states may arise, and indeed may coexist in such a way that the localized states are not rendered extended (phase coexistence). This is a radical departure from the conventional view. If the present work is correct, it is highly important. Is it correct? I can only be honest and say I do not know. But if the work is correct, it will become a classic. Referee B: If the conclusions of the paper are correct then much of our understanding of this problem will have to be revised. Page 14 of 48

15 2.2. PRL Page 15 of 48

16 Referee A: The authors claim to have an exact calculation of the localization length in the two-dimensional Anderson model. If that were true, it would be an exciting result. Referee B: This paper proposes an exact analytic solution of the 2-d Anderson localization problem. If this was true, it would be a major breakthrough. Referee C: This paper deals with a very important topic, and the results of the paper are sensational. The authors claim to solve exactly the longstanding problem of the two-dimensional localization with highly unexpected results. Namely, they claim that Anderson transition in 2-d is of the first order, and that the localized and conducting states can co-exist. If the results were correct, they would constitute a breakthrough compared only probably with Anderson s original paper. Page 16 of 48

17 Referee A: In short, neither the model addressed, nor the solution presented can apply to the problem of Anderson localization in two dimensions. I cannot recommend publication. Referee C: Concluding, it is my option, that the manuscript has nothing to do with the problem of 2-d localization and, therefore, should be rejected. Referee D: I agree with the ctiticism expressed by all three referees A,B,C. The best one can say about this paper is that it possibly considers some other model. This paper should not be published in PRL. Page 17 of 48

18 Referee D: It is not the job of the referees to find out what this model might be and how it relates to the canonical localization problem. Rather, it is the responsibility of the authors to explain why they get a different result from what is accepted and why their result should be the correct one. Page 18 of 48

19 Although there is no general analytical solution availibel, there is consensuns that in twodimensions all states are localized. The fact that certain experimental observations in Si-MOSFET s and other systems appear to suggest a metallic phase in a disordered, interacting 2-D electron system should not be used to argue that we do not understand disordered systems of noninteracting electrons. This is because if there is indeed a phase transition in these systems (which remains to be proven beyond doubt) it is most likely related to the Coulomb interaction or possibly the spinorbit interaction. For weak disorder the localization length... can be calculated in the perturbation theory... (Results) are confirmed by numerical studies... Page 19 of 48

20 2.3. Paper Therefore the metal-insulator transition should be looked at from the basis of first-order phase transition theory. This opinion differs from the traditional point of view, which considers this transition as continuous (second-order). We have shown that in principle there is the possibility for the phase of delocalized states in a 2-D disordered non-interacting electron system. This phase has its proper existence region, but it belongs to the marginal type of stability. Page 20 of 48

21 Tight-binding equation Standard form for 1-D: ψ n+1 + ψ n 1 = (E ε n )ψ n. The lattice constant and the hopping matrix element are equal to unity. The on-site potentials ε n are independently and identically distributed with existing first two moments, ε n = 0 and ε n 2 = σ 2. σ - parameter of disorder. Page 21 of 48

22 3.2. Recursion relation Recursion relation and time evolution of 1-D system Semi-infinite system, n 0 Recursion relation (n = 1, 2,...): ψ n+1 = (E ε n )ψ n ψ n 1. m d2 x dt 2 = F (x, t). It turns out to be convenient to consider the index n not as a spatial coordinate, but as discrete time. Then equation describes the time evolution of a D 1 = 0 dimensional system. Page 22 of 48

23 3.3. Causality in the tight-binding equation Initial condition: The equation ψ n+1 = (E ε n )ψ n ψ n 1 is solved with an initial condition ψ 0 = 0, ψ 1 = α. Causality: In a formal solution of this recursion relation the amplitude ψ n+1 depends only on the random variables ε n with n n. Grid amplitudes on the r.h.s. are statistically independent with respect to ε n. Page 23 of 48

24 3.4. Additive noise: Normal diffusion The equation for random walk ψ n+1 = ψ n + ε n Page 24 of 48 Bounded motion, ψ n ψ 0 = 0, for σ = 0. Divergence of the second moment for σ > 0: ψ 2 n = σ 2 n.

25 3.5. Multiplicative noise: Generalized diffusion ψ n+1 = (E ε n )ψ n ψ n 1 Page 25 of 48 Bounded motion (proper dynamics), ψ n <, for σ = 0.

26 Divergence of the second moment for σ > 0: ψ 2 n = f(n) with limn f(n) =. Page 26 of 48

27 3.6. Lyapunov exponents γ and order parameter Lyapunov exponent Asymptotic for n : f(n) exp(2γn). ψ 2 -definition: 1 γ = lim n 2n ln( ψ n 2 ). Order parameter: γ 0 - Extended states (no generalized diffusion) γ 0 - Localized states (generalized diffusion) Page 27 of 48

28 3.7. Equations for second moments We are interested in the mean behavior of ψ 2 n. : x n = ψ 2 n and y n = ψ n ψ n 1. The basic equation for the variable x n is obtained by squaring both sides of recursion relation and averaging over the ensemble. x n+1 = (E 2 + σ 2 ) x n 2Ey n + x n 1, y n = E x n y n 1. The initial conditions are: x 0 = 0, x 1 = α 2, y 0 = 0. Page 28 of 48

29 3.8. Transfer matrix The equations can be rewritten in the form w n+1 = T w n, where the vector w n and the transfer matrix T are respectively x n+1 E 2 + σ 2 1 2E w n = x n, T = y n+1 E 0 1 Initial conditions transform into w 0 = {α 2, 0, 0}. Page 29 of 48

30 3.9. Characteristic equation An explicit formula for w n is derived by diagonalizing the transfer matrix T. The characteristic equation of the eigenvalue problem for the T matrix is with function D(λ) = 0 D(z) = z 3 (E 2 + σ 2 1)z 2 + (E 2 σ 2 1)z 1. The resulting exact formula for x n is x n /α 2 = + + λ n 1(1 + λ 1 ) (λ 2 λ 1 )(λ 3 λ 1 ) + λ n 2(1 + λ 2 ) (λ 1 λ 2 )(λ 3 λ 2 ) + λ n 3(1 + λ 3 ) (λ 1 λ 3 )(λ 2 λ 3 ). Page 30 of 48

31 3.10. Z-transform An alternative solution makes use of the so-called Z-transform. X(z) = n=0 x n z, Y (z) = y n n z n. n=0 The inverse Z-transform is quite generally defined via countour integrals in the complex plane x n = 1 X(z)z ndz 2πi z. Properties: for the Z-transform x n+n0 z n 0 X(z), for the inverse Z-transform z z λ λn. Page 31 of 48

32 Determination of the poles zx = (E 2 + σ 2 )X 2EY + z 1 X + α 2 Y = Ez 1 X z 1 Y This is easily solved for X(z) X(z) = α2 z(z + 1) D(z) where the function D(z) is defined above. the solution of the characteristic equation D(λ) = 0 for the transfer matrix is equivalent to the determination of the poles of the X(z) function, D(z) = 0. The eigenvalues λ i, i.e. the poles, determine the asymptotic behavior of the solution x n for n. Page 32 of 48

33 Recursion relation Causality in the tight-binding equation for 2-D 2-D lattice with one boundary. The layers of this system are enumerated by an index n = 0, 1,... starting from the boundary, index m (, + ). Schrödinger equation in the form of a recursion relation: ψ n+1,m = (E ε n,m )ψ n,m ψ n 1,m µ=±1 ψ n,m+µ. Page 33 of 48

34 Initial condition ψ 0,m = 0, ψ 1,m = α m. Recursion relation describes the time evolution of a D 1 = 1 dimensional system. Causality: Grid amplitudes on the r.h.s. are statistically independent with respect to ε n,m. Recursion relation and time evolution of 2-D system Page 34 of 48

35 4.2. Concept of filters T.F.Weiss: Signals and systems. Lecture notes. Signals are variables that carry information. Systems process input signals to produce output signals. Page 35 of 48

36 A system is BIBO stable if every Bounded Input leads to a Bounded Output. Physical systems with time as independent variable are causal systems. Page 36 of 48

37 Linear and causal filter Input signal x (0) n Output signal x n Linear filter h n Causal filter (h n = 0 for n < 0) Page 37 of 48

38 Convolution property n x n = Z-transform l=0 Stable and unstable filter h n l x (0) l. X(z) = H(z)X (0) (z). Stable filter: Bounded output for bounded input Unstable filter: Unbounded output for bounded input Unstable filter: lim n h n Page 38 of 48

39 4.3. Signal theory The causal filters have always ROCs (region of convergence) outside a circle that intersects the pole with max λ i. causal filter is stable (bounded input yields a bounded output) if the unit circle z = 1 is in the ROC. Page 39 of 48

40 For 2-D Anderson localization problem: π 1 H(z) = 1 σ2 (z + 1) dk 2π(z 1) π w 2 L 2 (k), L(k) = E 2 cos(k), w 2 (z + 1)2 =. z Page 40 of 48

41 4.4. Two phases solution Page 41 of 48

42 Unstable filter H + (z), z 1 Band center (E = 0) Filter Inverse Z-transform H 1 + (z) = 1 σ2 z (z 1) 2. h + n = δ n,0 + 2 tanh(γ) sinh(2γn) (exponentially growing function, 2 sinh(γ) = σ). The localization length is ξ = γ 1. Interpretation - localized states. Page 42 of 48

43 Stable filter H (z), z 1 Band center (E = 0) Filter H 1 (z) = 1 + σ2 z (z 1) 2. Inverse Z-transform h n = δ n,0 + 2 tan(φ) sin(2φn) (bounded oscillating function, 2 sin(φ) = σ). Marginal stability. The filter exists only for small disorder (σ < 2). Interpretation - delocalized states. Page 43 of 48

44 4.5. Coexistence of phases The solution H(z) may be interpreted as representing two phases, considering that H + (z) determines the properties of the insulating state, but H (z) the metallic state. Therefore the metal-insulator transition should be looked at from the basis of first-order phase transition theory. A macroscopic system for first-order phase transition is heterogeneous and consists of macroscopically homogeneous domains of the two phases. The self-averaging quantities tend always to be unphysical because the average over an ensemble always includes an average over the phases. Page 44 of 48

45 Physically meaningful in this case are only the properties of the pure phases. The trivial example in this respect is the coexistence of water and ice. An average density of this system has no sense and depends on the relative fraction of the phases, whereas the density of pure water and pure ice are meaningful quantities. Page 45 of 48

46 Microscopic structure of the metal-insulator transition in two dimensions S.Ilani, A.Yacoby, D.Mahalu and H.Shtrikman, Science, 292, 1354 (2001) Abstract: A single electron transistor is used as a local electrostatic probe to study the underlying spatial structure of the metal-insulator transition in two-dimensions. The measurement show that as we approach the transition from the metallic side, a new phase emerges that consists of weakly coupled fragments of the two-dimensional system. These fragments consist of localized charge that coexists with the surrounding metallic phase. As the density is lowered into the insulating phase, the number of fragments increases on account of the disapearing metallic phase. Page 46 of 48

47 5. Vladimir Kuzovkov and Wolfgang von Niessen JJ II J I Page 47 of 48

48 Slava Kashcheyevs Page 48 of 48 Oliver Hein

### Accuracy of the coherent potential approximation for a onedimensional array with a Gaussian distribution of fluctuations in the on-site potential

Accuracy of the coherent potential approximation for a onedimensional array with a Gaussian distribution of fluctuations in the on-site potential I. Avgin Department of Electrical and Electronics Engineering,

### There are four common ways of finding the inverse z-transform:

Inverse z-transforms and Difference Equations Preliminaries We have seen that given any signal x[n], the two-sided z-transform is given by n x[n]z n and X(z) converges in a region of the complex plane

### The Quantum Harmonic Oscillator Stephen Webb

The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems

### EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist

### Electrical Conductivity

Advanced Materials Science - Lab Intermediate Physics University of Ulm Solid State Physics Department Electrical Conductivity Translated by Michael-Stefan Rill January 20, 2003 CONTENTS 1 Contents 1 Introduction

### (Refer Slide Time: 01:11-01:27)

Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 6 Digital systems (contd.); inverse systems, stability, FIR and IIR,

### phys4.17 Page 1 - under normal conditions (pressure, temperature) graphite is the stable phase of crystalline carbon

Covalent Crystals - covalent bonding by shared electrons in common orbitals (as in molecules) - covalent bonds lead to the strongest bound crystals, e.g. diamond in the tetrahedral structure determined

### Elementary Differential Equations

Elementary Differential Equations EIGHTH EDITION Earl D. Rainville Late Professor of Mathematics University of Michigan Phillip E. Bedient Professor Emeritus of Mathematics Franklin and Marshall College

### Chapter 7. Lyapunov Exponents. 7.1 Maps

Chapter 7 Lyapunov Exponents Lyapunov exponents tell us the rate of divergence of nearby trajectories a key component of chaotic dynamics. For one dimensional maps the exponent is simply the average

### Critical Phenomena and Percolation Theory: I

Critical Phenomena and Percolation Theory: I Kim Christensen Complexity & Networks Group Imperial College London Joint CRM-Imperial College School and Workshop Complex Systems Barcelona 8-13 April 2013

### Understanding Poles and Zeros

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

### 0.1 Phase Estimation Technique

Phase Estimation In this lecture we will describe Kitaev s phase estimation algorithm, and use it to obtain an alternate derivation of a quantum factoring algorithm We will also use this technique to design

### Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

### Perpetual motion and driven dynamics of a mobile impurity in a quantum fluid

and driven dynamics of a mobile impurity in a quantum fluid Oleg Lychkovskiy Russian Quantum Center Seminaire du LPTMS, 01.12.2015 Seminaire du LPTMS, 01.12.2015 1 / Plan of the talk 1 Perpetual motion

### Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

### Some stability results of parameter identification in a jump diffusion model

Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss

### Statistical Machine Learning

Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

### Exercises in Signals, Systems, and Transforms

Exercises in Signals, Systems, and Transforms Ivan W. Selesnick Last edit: October 7, 4 Contents Discrete-Time Signals and Systems 3. Signals...................................................... 3. System

### A wave lab inside a coaxial cable

INSTITUTE OF PHYSICS PUBLISHING Eur. J. Phys. 25 (2004) 581 591 EUROPEAN JOURNAL OF PHYSICS PII: S0143-0807(04)76273-X A wave lab inside a coaxial cable JoãoMSerra,MiguelCBrito,JMaiaAlves and A M Vallera

### The Z transform (3) 1

The Z transform (3) 1 Today Analysis of stability and causality of LTI systems in the Z domain The inverse Z Transform Section 3.3 (read class notes first) Examples 3.9, 3.11 Properties of the Z Transform

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

### ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.

### = N 2 = 3π2 n = k 3 F. The kinetic energy of the uniform system is given by: 4πk 2 dk h2 k 2 2m. (2π) 3 0

Chapter 1 Thomas-Fermi Theory The Thomas-Fermi theory provides a functional form for the kinetic energy of a non-interacting electron gas in some known external potential V (r) (usually due to impurities)

### - develop a theory that describes the wave properties of particles correctly

Quantum Mechanics Bohr's model: BUT: In 1925-26: by 1930s: - one of the first ones to use idea of matter waves to solve a problem - gives good explanation of spectrum of single electron atoms, like hydrogen

### RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A

RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:

### Till now, almost all attention has been focussed on discussing the state of a quantum system.

Chapter 13 Observables and Measurements in Quantum Mechanics Till now, almost all attention has been focussed on discussing the state of a quantum system. As we have seen, this is most succinctly done

### Inverse problems. Nikolai Piskunov 2014. Regularization: Example Lecture 4

Inverse problems Nikolai Piskunov 2014 Regularization: Example Lecture 4 Now that we know the theory, let s try an application: Problem: Optimal filtering of 1D and 2D data Solution: formulate an inverse

### Lecture 5: Finite differences 1

Lecture 5: Finite differences 1 Sourendu Gupta TIFR Graduate School Computational Physics 1 February 17, 2010 c : Sourendu Gupta (TIFR) Lecture 5: Finite differences 1 CP 1 1 / 35 Outline 1 Finite differences

### Stability. Chapter 4. Topics : 1. Basic Concepts. 2. Algebraic Criteria for Linear Systems. 3. Lyapunov Theory with Applications to Linear Systems

Chapter 4 Stability Topics : 1. Basic Concepts 2. Algebraic Criteria for Linear Systems 3. Lyapunov Theory with Applications to Linear Systems 4. Stability and Control Copyright c Claudiu C. Remsing, 2006.

### System Identification for Acoustic Comms.:

System Identification for Acoustic Comms.: New Insights and Approaches for Tracking Sparse and Rapidly Fluctuating Channels Weichang Li and James Preisig Woods Hole Oceanographic Institution The demodulation

### Physics 551: Solid State Physics F. J. Himpsel

Physics 551: Solid State Physics F. J. Himpsel Background Most of the objects around us are in the solid state. Today s technology relies heavily on new materials, electronics is predominantly solid state.

### degrees of freedom and are able to adapt to the task they are supposed to do [Gupta].

1.3 Neural Networks 19 Neural Networks are large structured systems of equations. These systems have many degrees of freedom and are able to adapt to the task they are supposed to do [Gupta]. Two very

### Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems

Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems Wolfgang Wagner Mohrenstrasse 39 10117 Berlin Germany Tel. +49 30 20372 0 www.wias-berlin.de WIAS workshop,

### Dynamical Systems Analysis II: Evaluating Stability, Eigenvalues

Dynamical Systems Analysis II: Evaluating Stability, Eigenvalues By Peter Woolf pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative

### The Schrödinger Equation

The Schrödinger Equation When we talked about the axioms of quantum mechanics, we gave a reduced list. We did not talk about how to determine the eigenfunctions for a given situation, or the time development

### Taylor Series and Asymptotic Expansions

Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.

### Chapter 18 Statistical mechanics of classical systems

Chapter 18 Statistical mechanics of classical systems 18.1 The classical canonical partition function When quantum effects are not significant, we can approximate the behavior of a system using a classical

### Overview of Math Standards

Algebra 2 Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse

### The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

### 7. DYNAMIC LIGHT SCATTERING 7.1 First order temporal autocorrelation function.

7. DYNAMIC LIGHT SCATTERING 7. First order temporal autocorrelation function. Dynamic light scattering (DLS) studies the properties of inhomogeneous and dynamic media. A generic situation is illustrated

### Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Course objectives and preliminaries Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### Positive Feedback and Oscillators

Physics 3330 Experiment #6 Fall 1999 Positive Feedback and Oscillators Purpose In this experiment we will study how spontaneous oscillations may be caused by positive feedback. You will construct an active

### Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Chapter 1: Initial value problems in ODEs Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical

### Lecture 7: Finding Lyapunov Functions 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1

### Section 5.0 : Horn Physics. By Martin J. King, 6/29/08 Copyright 2008 by Martin J. King. All Rights Reserved.

Section 5. : Horn Physics Section 5. : Horn Physics By Martin J. King, 6/29/8 Copyright 28 by Martin J. King. All Rights Reserved. Before discussing the design of a horn loaded loudspeaker system, it is

### Lecture 3: Linear methods for classification

Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,

### 3. Diffusion of an Instantaneous Point Source

3. Diffusion of an Instantaneous Point Source The equation of conservation of mass is also known as the transport equation, because it describes the transport of scalar species in a fluid systems. In this

### CCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York

BME I5100: Biomedical Signal Processing Linear Discrimination Lucas C. Parra Biomedical Engineering Department CCNY 1 Schedule Week 1: Introduction Linear, stationary, normal - the stuff biology is not

### Sound. References: L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol. 2, Gas Dynamics, Chapter 8

References: Sound L.D. Landau & E.M. Lifshitz: Fluid Mechanics, Chapter VIII F. Shu: The Physics of Astrophysics, Vol., Gas Dynamics, Chapter 8 1 Speed of sound The phenomenon of sound waves is one that

### Nonlinear evolution of unstable fluid interface

Nonlinear evolution of unstable fluid interface S.I. Abarzhi Department of Applied Mathematics and Statistics State University of New-York at Stony Brook LIGHT FLUID ACCELERATES HEAVY FLUID misalignment

### problem arises when only a non-random sample is available differs from censored regression model in that x i is also unobserved

4 Data Issues 4.1 Truncated Regression population model y i = x i β + ε i, ε i N(0, σ 2 ) given a random sample, {y i, x i } N i=1, then OLS is consistent and efficient problem arises when only a non-random

### 6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

### An Introduction to Partial Differential Equations

An Introduction to Partial Differential Equations Andrew J. Bernoff LECTURE 2 Cooling of a Hot Bar: The Diffusion Equation 2.1. Outline of Lecture An Introduction to Heat Flow Derivation of the Diffusion

### Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Quantum Monte Carlo and the negative sign problem

Quantum Monte Carlo and the negative sign problem or how to earn one million dollar Matthias Troyer, ETH Zürich Uwe-Jens Wiese, Universität Bern Complexity of many particle problems Classical 1 particle:

### Lecture 13 Linear quadratic Lyapunov theory

EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time

### F en = mω 0 2 x. We should regard this as a model of the response of an atom, rather than a classical model of the atom itself.

The Electron Oscillator/Lorentz Atom Consider a simple model of a classical atom, in which the electron is harmonically bound to the nucleus n x e F en = mω 0 2 x origin resonance frequency Note: We should

### SCHWEITZER ENGINEERING LABORATORIES, COMERCIAL LTDA.

Pocket book of Electrical Engineering Formulas Content 1. Elementary Algebra and Geometry 1. Fundamental Properties (real numbers) 1 2. Exponents 2 3. Fractional Exponents 2 4. Irrational Exponents 2 5.

### Electrostatic Fields: Coulomb s Law & the Electric Field Intensity

Electrostatic Fields: Coulomb s Law & the Electric Field Intensity EE 141 Lecture Notes Topic 1 Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University

### Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics

Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group dimitri.vandeville@epfl.ch

### WAVES AND FIELDS IN INHOMOGENEOUS MEDIA

WAVES AND FIELDS IN INHOMOGENEOUS MEDIA WENG CHO CHEW UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor IEEE Antennas and Propagation Society,

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Lecture 2: Semiconductors: Introduction

Lecture 2: Semiconductors: Introduction Contents 1 Introduction 1 2 Band formation in semiconductors 2 3 Classification of semiconductors 5 4 Electron effective mass 10 1 Introduction Metals have electrical

### Université Lille 1 Sciences et Technologies, Lille, France Lille Laboratoire de Physique des Lasers, Atomes et Molécules Équipe Chaos Quantique

Université Lille 1 Sciences et Technologies, Lille, France Lille Laboratoire de Physique des Lasers, Atomes et Molécules Équipe Chaos Quantique FRISNO 11 Aussois 1/4/011 Quantum simulators: The Anderson

### Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

### arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014

Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic

### Notes for AA214, Chapter 7. T. H. Pulliam Stanford University

Notes for AA214, Chapter 7 T. H. Pulliam Stanford University 1 Stability of Linear Systems Stability will be defined in terms of ODE s and O E s ODE: Couples System O E : Matrix form from applying Eq.

### Graduate Studies in Mathematics. The Author Author Two

Graduate Studies in Mathematics The Author Author Two (A. U. Thor) A 1, A 2 Current address, A. U. Thor: Author current address line 1, Author current address line 2 E-mail address, A. U. Thor: author@institute.edu

### There are certain things that we have to take into account before and after we take an FID (or the spectrum, the FID is not that useful after all).

Data acquisition There are certain things that we have to take into account before and after we take an FID (or the spectrum, the FID is not that useful after all). Some deal with the detection system.

### Université Lille 1 Sciences et Technologies, Lille, France Laboratoire de Physique des Lasers, Atomes et Molécules Équipe Chaos Quantique

Université Lille 1 Sciences et Technologies, Lille, France Laboratoire de Physique des Lasers, Atomes et Molécules Équipe Chaos Quantique 16 years of experiments on the atomic kicked rotor! Chaos, disorder

### Structure Factors 59-553 78

78 Structure Factors Until now, we have only typically considered reflections arising from planes in a hypothetical lattice containing one atom in the asymmetric unit. In practice we will generally deal

### 5.4 The Heat Equation and Convection-Diffusion

5.4. THE HEAT EQUATION AND CONVECTION-DIFFUSION c 6 Gilbert Strang 5.4 The Heat Equation and Convection-Diffusion The wave equation conserves energy. The heat equation u t = u xx dissipates energy. The

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### arxiv:cond-mat/0308498v1 [cond-mat.soft] 25 Aug 2003

1 arxiv:cond-mat/38498v1 [cond-mat.soft] 2 Aug 23 Matter-wave interference, Josephson oscillation and its disruption in a Bose-Einstein condensate on an optical lattice Sadhan K. Adhikari Instituto de

### Gaussian Conjugate Prior Cheat Sheet

Gaussian Conjugate Prior Cheat Sheet Tom SF Haines 1 Purpose This document contains notes on how to handle the multivariate Gaussian 1 in a Bayesian setting. It focuses on the conjugate prior, its Bayesian

### A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses. Michael R. Powers[ 1 ] Temple University and Tsinghua University

A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses Michael R. Powers[ ] Temple University and Tsinghua University Thomas Y. Powers Yale University [June 2009] Abstract We propose a

### Supplement to Call Centers with Delay Information: Models and Insights

Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290

### 1 Variational calculation of a 1D bound state

TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK FÖRDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN- ICS Kvantmekanik fördjupningskurs SI38 för F4 Thursday December, 7, 8. 13. Write on each page: Name,

### Lecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2)

Lecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2) In this lecture How does turbulence affect the ensemble-mean equations of fluid motion/transport? Force balance in a quasi-steady turbulent boundary

### Dimensional analysis is a method for reducing the number and complexity of experimental variables that affect a given physical phenomena.

Dimensional Analysis and Similarity Dimensional analysis is very useful for planning, presentation, and interpretation of experimental data. As discussed previously, most practical fluid mechanics problems

ECSE 4440 Control System Engineering Fall 2001 Project 3 Controller Design in Frequency Domain TA 1. Abstract 2. Introduction 3. Controller design in Frequency domain 4. Experiment 5. Colclusion 1. Abstract

### A Second Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A Second Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved Contents 8 Calculus of Matrix-Valued Functions of a Real Variable

### THE DYING FIBONACCI TREE. 1. Introduction. Consider a tree with two types of nodes, say A and B, and the following properties:

THE DYING FIBONACCI TREE BERNHARD GITTENBERGER 1. Introduction Consider a tree with two types of nodes, say A and B, and the following properties: 1. Let the root be of type A.. Each node of type A produces

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### Free Electron Fermi Gas (Kittel Ch. 6)

Free Electron Fermi Gas (Kittel Ch. 6) Role of Electrons in Solids Electrons are responsible for binding of crystals -- they are the glue that hold the nuclei together Types of binding (see next slide)

### TTT4120 Digital Signal Processing Suggested Solution to Exam Fall 2008

Norwegian University of Science and Technology Department of Electronics and Telecommunications TTT40 Digital Signal Processing Suggested Solution to Exam Fall 008 Problem (a) The input and the input-output

### PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004

PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall

### 1.5 / 1 -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de. 1.5 Transforms

.5 / -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de.5 Transforms Using different summation and integral transformations pmf, pdf and cdf/ccdf can be transformed in such a way, that even

### DO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS

DO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS Quantum Mechanics or wave mechanics is the best mathematical theory used today to describe and predict the behaviour of particles and waves.

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### The continuous and discrete Fourier transforms

FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1

### EE4367 Telecom. Switching & Transmission. Prof. Murat Torlak

Path Loss Radio Wave Propagation The wireless radio channel puts fundamental limitations to the performance of wireless communications systems Radio channels are extremely random, and are not easily analyzed

### Fuzzy Differential Systems and the New Concept of Stability

Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

### 221A Lecture Notes Variational Method

1 Introduction 1A Lecture Notes Variational Method Most of the problems in physics cannot be solved exactly, and hence need to be dealt with approximately. There are two common methods used in quantum

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### A LOGNORMAL MODEL FOR INSURANCE CLAIMS DATA

REVSTAT Statistical Journal Volume 4, Number 2, June 2006, 131 142 A LOGNORMAL MODEL FOR INSURANCE CLAIMS DATA Authors: Daiane Aparecida Zuanetti Departamento de Estatística, Universidade Federal de São