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E-raamat: Computational Physics: An Introduction To Monte Carlo Simulations Of Matrix Field Theory

(Bm Annaba Univ, Algeria)
  • Formaat: 312 pages
  • Ilmumisaeg: 07-Feb-2017
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789813200234
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Computational Physics: An Introduction To Monte Carlo Simulations Of Matrix Field TheorySuurem pilt
  • Formaat: 312 pages
  • Ilmumisaeg: 07-Feb-2017
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789813200234
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This book is divided into two parts. In the first part we give an elementary introduction to computational physics consisting of 21 simulations which originated from a formal course of lectures and laboratory simulations delivered since 2010 to physics students at Annaba University. The second part is much more advanced and deals with the problem of how to set up working Monte Carlo simulations of matrix field theories which involve finite dimensional matrix regularizations of noncommutative and fuzzy field theories, fuzzy spaces and matrix geometry. The study of matrix field theory in its own right has also become very important to the proper understanding of all noncommutative, fuzzy and matrix phenomena. The second part, which consists of 9 simulations, was delivered informally to doctoral students who were working on various problems in matrix field theory. Sample codes as well as sample key solutions are also provided for convenience and completeness.
Preface vii
Introductory Remarks xv
Introduction to Computational Physics 1(2)
1 Euler Algorithm 3(14)
1.1 Euler Algorithm
3(1)
1.2 First Example and Sample Code
4(3)
1.2.1 Radioactive Decay
4(2)
1.2.2 A Sample Fortran Code
6(1)
1.3 More Examples
7(3)
1.3.1 Air Resistance
7(2)
1.3.2 Projectile Motion
9(1)
1.4 Periodic Motions and Euler-Cromer and Verlet Algorithms
10(3)
1.4.1 Harmonic Oscillator
10(1)
1.4.2 Euler Algorithm
11(1)
1.4.3 Euler-Cromer Algorithm
12(1)
1.4.4 Verlet Algorithm
13(1)
1.5 Exercises
13(1)
1.6 Simulation 1: Euler Algorithm-Air Resistance
14(1)
1.7 Simulation 2: Euler Algorithm-Projectile Motion
15(1)
1.8 Simulation 3: Euler, Euler-Cromer and Verlet Algorithms
16(1)
2 Classical Numerical Integration 17(6)
2.1 Rectangular Approximation
17(1)
2.2 Trapezoidal Approximation
18(1)
2.3 Parabolic Approximation or Simpson's Rule
18(2)
2.4 Errors
20(1)
2.5 Simulation 4: Numerical Integrals
21(2)
3 Newton-Raphson Algorithms and Interpolation 23(8)
3.1 Bisection Algorithm
23(1)
3.2 Newton-Raphson Algorithm
23(1)
3.3 Hybrid Method
24(1)
3.4 Lagrange Interpolation
25(1)
3.5 Cubic Spline Interpolation
26(2)
3.6 The Method of Least Squares
28(1)
3.7 Simulation 5: Newton-Raphson Algorithm
29(2)
4 The Solar System: The Runge-Kutta Methods 31(12)
4.1 The Solar System
31(4)
4.1.1 Newton's Second Law
31(1)
4.1.2 Astronomical Units and Initial Conditions
32(1)
4.1.3 Kepler's Laws
32(2)
4.1.4 The Inverse-Square Law and Stability of Orbits
34(1)
4.2 Euler-Cromer Algorithm
35(1)
4.3 The Runge-Kutta Algorithm
36(3)
4.3.1 The Method
36(1)
4.3.2 Example 1: The Harmonic Oscillator
37(1)
4.3.3 Example 2: The Solar System
37(2)
4.4 Precession of the Perihelion of Mercury
39(1)
4.5 Exercises
39(1)
4.6 Simulation 6: Runge-Kutta Algorithm: Solar-System
40(1)
4.7 Simulation 7: Precession of the perihelion of Mercury
41(2)
5 Chaotic Pendulum 43(12)
5.1 Equation of Motion
43(2)
5.2 Numerical Algorithms
45(2)
5.2.1 Euler-Cromer Algorithm
46(1)
5.2.2 Runge-Kutta Algorithm
46(1)
5.3 Elements of Chaos
47(3)
5.3.1 Butterfly Effect: Sensitivity to Initial Conditions
47(1)
5.3.2 Poincare Section and Attractors
48(1)
5.3.3 Period-Doubling Bifurcations
48(1)
5.3.4 Feigenbaum Ratio
49(1)
5.3.5 Spontaneous Symmetry Breaking
49(1)
5.4 Simulation 8: The Butterfly Effect
50(1)
5.5 Simulation 9: Poincare Sections
50(2)
5.6 Simulation 10: Period Doubling
52(1)
5.7 Simulation 11: Bifurcation Diagrams
53(2)
6 Molecular Dynamics 55(8)
6.1 Introduction
55(1)
6.2 The Lennard-Jones Potential
56(1)
6.3 Units, Boundary Conditions and Verlet Algorithm
57(2)
6.4 Some Physical Applications
59(1)
6.4.1 Dilute Gas and Maxwell Distribution
59(1)
6.4.2 The Melting Transition
60(1)
6.5 Simulation 12: Maxwell Distribution
60(1)
6.6 Simulation 13: Melting Transition
61(2)
7 Pseudo Random Numbers and Random Walks 63(12)
7.1 Random Numbers
63(3)
7.1.1 Linear Congruent or Power Residue Method
63(1)
7.1.2 Statistical Tests of Randomness
64(2)
7.2 Random Systems
66(3)
7.2.1 Random Walks
66(1)
7.2.2 Diffusion Equation
67(2)
7.3 The Random Number Generators RAN 0,1,2
69(3)
7.4 Simulation 14: Random Numbers
72(1)
7.5 Simulation 15: Random Walks
73(2)
8 Monte Carlo Integration 75(14)
8.1 Numerical Integration
75(3)
8.1.1 Rectangular Approximation Revisited
75(1)
8.1.2 Midpoint Approximation of Multidimensional Integrals
76(2)
8.1.3 Spheres and Balls in d Dimensions
78(1)
8.2 Monte Carlo Integration: Simple Sampling
78(2)
8.2.1 Sampling (Hit or Miss) Method
79(1)
8.2.2 Sample Mean Method
79(1)
8.2.3 Sample Mean Method in Higher Dimensions
80(1)
8.3 The Central Limit Theorem
80(2)
8.4 Monte Carlo Errors and Standard Deviation
82(2)
8.5 Nonuniform Probability Distributions
84(2)
8.5.1 The Inverse Transform Method
84(2)
8.5.2 The Acceptance-Rejection Method
86(1)
8.6 Simulation 16: Midpoint and Monte Carlo Approximations
86(1)
8.7 Simulation 17: Nonuniform Probability Distributions
87(2)
9 The Metropolis Algorithm and the Ising Model 89(16)
9.1 The Canonical Ensemble
89(1)
9.2 Importance Sampling
90(1)
9.3 The Ising Model
91(1)
9.4 The Metropolis Algorithm
92(2)
9.5 The Heat-Bath Algorithm
94(1)
9.6 The Mean Field Approximation
94(3)
9.6.1 Phase Diagram and Critical Temperature
94(2)
9.6.2 Critical Exponents
96(1)
9.7 Simulation of the Ising Model and Numerical Results
97(4)
9.7.1 The Fortran Code
97(2)
9.7.2 Some Numerical Results
99(2)
9.8 Simulation 18: The Metropolis Algorithm and the Ising Model
101(1)
9.9 Simulation 19: The Ferromagnetic Second Order Phase Transition
102(1)
9.10 Simulation 20: The 2-Point Correlator
103(1)
9.11 Simulation 21: Hysteresis and the First Order Phase Transition
104(1)
Monte Carlo Simulations of Matrix Field Theory 105(186)
10 Metropolis Algorithm for Yang-Mills Matrix Models
107(12)
10.1 Dimensional Reduction
107(5)
10.1.1 Yang-Mills Action
107(1)
10.1.2 Chern-Simons Action: Myers Term
108(4)
10.2 Metropolis Accept/Reject Step
112(1)
10.3 Statistical Errors
113(1)
10.4 Auto-Correlation Time
114(1)
10.5 Code and Sample Calculation
115(2)
References
117(2)
11 Hybrid Monte Carlo Algorithm for Yang-Mills Matrix Models
119(12)
11.1 The Yang-Mills Matrix Action
119(1)
11.2 The Leap Frog Algorithm
120(2)
11.3 Metropolis Algorithm
122(1)
11.4 Gaussian Distribution
123(1)
11.5 Physical Tests
123(1)
11.6 Emergent Geometry: An Exotic Phase Transition
124(5)
References
129(2)
12 Hybrid Monte Carlo Algorithm for Noncommutative Phi-Four
131(10)
12.1 The Matrix Scalar Action
131(1)
12.2 The Leap Frog Algorithm
132(1)
12.3 Hybrid Monte Carlo Algorithm
132(1)
12.4 Optimization
132(2)
12.4.1 Partial Optimization
132(2)
12.4.2 Full Optimization
134(1)
12.5 The Non-Uniform Order: Another Exotic Phase
134(5)
12.5.1 Phase Structure
134(1)
12.5.2 Sample Simulations
135(4)
References
139(2)
13 Lattice HMC Simulations of Phi42: A Lattice Example
141(12)
13.1 Model and Phase Structure
141(4)
13.2 The HM Algorithm
145(2)
13.3 Renormalization and Continuum Limit
147(2)
13.4 HMC Simulation Calculation of the Critical Line
149(2)
References
151(2)
14 (Multi-Trace) Quartic Matrix Models
153(10)
14.1 The Pure Real Quartic Matrix Model
153(1)
14.2 The Multi-Trace Matrix Model
154(2)
14.3 Model and Algorithm
156(2)
14.4 The Disorder-to-Non-Uniform-Order Transition
158(2)
14.5 Other Suitable Algorithms
160(2)
14.5.1 Over-Relaxation Algorithm
160(1)
14.5.2 Heat-Bath Algorithm
161(1)
References
162(1)
15 The Remez Algorithm and the Conjugate Gradient Method
163(18)
15.1 Minimax Approximations
163(8)
15.1.1 Minimax Polynomial Approximation and Chebyshev Polynomials
163(5)
15.1.2 Minimax Rational Approximation and Remez Algorithm
168(3)
15.1.3 The Code "AlgRemez"
171(1)
15.2 Conjugate Gradient Method
171(8)
15.2.1 Construction
171(4)
15.2.2 The Conjugate Gradient Method as a Krylov Space Solver
175(2)
15.2.3 The Multi-Mass Conjugate Gradient Method
177(2)
References
179(2)
16 Monte Carlo Simulation of Fermion Determinants
181(20)
16.1 The Dirac Operator
181(4)
16.2 Pseudo-Fermions and Rational Approximations
185(2)
16.3 More on The Conjugate-Gradient
187(5)
16.3.1 Multiplication by Mu' and (Mu')+
187(3)
16.3.2 The Fermionic Force
190(2)
16.4 The Rational Hybrid Monte Carlo Algorithm
192(5)
16.4.1 Statement
192(1)
16.4.2 Preliminary Tests
193(4)
16.5 Other Related Topics
197(2)
References
199(2)
17 U(1) Gauge Theory on the Lattice: Another Lattice Example
201(16)
17.1 Continuum Considerations
201(2)
17.2 Lattice Regularization
203(6)
17.2.1 Lattice Fermions and Gauge Fields
203(2)
17.2.2 Quenched Approximation
205(1)
17.2.3 Wilson Loop, Creutz Ratio and Other Observables
206(3)
17.3 Monte Carlo Simulation of Pure U(1) Gauge Theory
209(7)
17.3.1 The Metropolis Algorithm
209(3)
17.3.2 Some Numerical Results
212(3)
17.3.3 Coulomb and Confinement Phases
215(1)
References
216(1)
18 Codes
217(74)
9.1 metropolis-ym.f
219(6)
9.2 hybrid-ym.f
225(7)
9.3 hybrid-scalar-fuzzy.f
232(10)
9.4 phi-four-on-lattice.f
242(7)
9.5 metropolis-scalar-multitrace.f
249(7)
9.6 remez.f
256(2)
9.7 conjugate-gradient.f
258(3)
9.8 hybrid-supersymmetric-ym.f
261(18)
9.9 u-one-on-the-lattice.f
279(12)
Index 291
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