Computational Physics - A Practical Introduction to Computational Physics and Scientific Computing

682 Pages·2014·20.97 MB·English

by  Konstantinos N. Anagnostopoulos

Computational Physics
Konstantinos N. Anagnostopoulos
COMPUTATIONAL PHYSICS
A P I  C P
 S C
A, 2014
KONSTANTINOS N. ANAGNOSTOPOULOS
National Technical University of Athens
N T U  A
PUBLISHEDBYKONSTANTINOSN.ANAGNOSTOPOULOS
PhysicsDepartment,NationalTechnicalUniversityofAthens,ZografouCampus,15780Zografou,Greece
[email protected],www.physics.ntua.gr/˜konstant/
andtheNATIONALTECHNICALUNIVERSITYOFATHENS
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www.physics.ntua.gr/˜konstant/ComputationalPhysics
©KonstantinosN.Anagnostopoulos2014
FirstPublished2014
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Contents
Foreword vii
1 The Computer 1
1.1 The Operating System . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Filesystem . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Commands . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.3 Looking for Help . . . . . . . . . . . . . . . . . . . . 14
1.2 Text Processing Tools – Filters . . . . . . . . . . . . . . . . 16
1.3 Programming with Emacs . . . . . . . . . . . . . . . . . . . 21
1.3.1 Calling Emacs . . . . . . . . . . . . . . . . . . . . . . 21
1.3.2 Interacting with Emacs . . . . . . . . . . . . . . . . 23
1.3.3 Basic Editing . . . . . . . . . . . . . . . . . . . . . . 24
1.3.4 Cut and Paste . . . . . . . . . . . . . . . . . . . . . . 27
1.3.5 Windows . . . . . . . . . . . . . . . . . . . . . . . . 29
1.3.6 Files and Buffers . . . . . . . . . . . . . . . . . . . . 30
1.3.7 Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.3.8 Emacs Help . . . . . . . . . . . . . . . . . . . . . . . 32
1.3.9 Emacs Customization . . . . . . . . . . . . . . . . . 34
1.4 The Fortran Programming Language . . . . . . . . . . . . 35
1.4.1 The Foundation . . . . . . . . . . . . . . . . . . . . 35
1.4.2 Details . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.4.3 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.5 Gnuplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.6 Shell Scripting . . . . . . . . . . . . . . . . . . . . . . . . . 62
2 Kinematics 77
2.1 Motion on the Plane . . . . . . . . . . . . . . . . . . . . . . 77
2.1.1 Plotting Data . . . . . . . . . . . . . . . . . . . . . . 85
2.1.2 More Examples . . . . . . . . . . . . . . . . . . . . . 88
2.2 Motion in Space . . . . . . . . . . . . . . . . . . . . . . . . 99
2.3 Trapped in a Box . . . . . . . . . . . . . . . . . . . . . . . . 107
iii
iv CONTENTS
2.3.1 The One Dimensional Box . . . . . . . . . . . . . . 108
2.3.2 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.3.3 The Two Dimensional Box . . . . . . . . . . . . . . 117
2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3 Logistic Map 145
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.2 Fixed Points and 2n Cycles . . . . . . . . . . . . . . . . . . 147
3.3 Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . . . 154
3.4 The Newton-Raphson Method . . . . . . . . . . . . . . . . 157
3.5 Calculation of the Bifurcation Points . . . . . . . . . . . . . 163
3.6 Liapunov Exponents . . . . . . . . . . . . . . . . . . . . . . 167
3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4 Motion of a Particle 191
4.1 Numerical Integration of Newton’s Equations . . . . . . . . 191
4.2 Prelude: Euler Methods . . . . . . . . . . . . . . . . . . . . 192
4.3 Runge–Kutta Methods . . . . . . . . . . . . . . . . . . . . . 204
4.3.1 A Program for the 4th Order Runge–Kutta . . . . . 208
4.4 Comparison of the Methods . . . . . . . . . . . . . . . . . . 212
4.5 The Forced Damped Oscillator . . . . . . . . . . . . . . . . 215
4.6 The Forced Damped Pendulum . . . . . . . . . . . . . . . . 222
4.7 Appendix: On the Euler–Verlet Method . . . . . . . . . . . 229
4.8 Appendix: 2nd order Runge–Kutta Method . . . . . . . . 233
4.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
5 Planar Motion 241
5.1 Runge–Kutta for Planar Motion . . . . . . . . . . . . . . . 241
5.2 Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . . 246
5.3 Planetary Motion . . . . . . . . . . . . . . . . . . . . . . . . 252
5.4 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
5.4.1 Rutherford Scattering . . . . . . . . . . . . . . . . . 260
5.4.2 More Scattering Potentials . . . . . . . . . . . . . . . 267
5.5 More Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 270
5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
6 Motion in Space 285
6.1 Adaptive Stepsize Control for Runge–Kutta Methods . . . . 286
6.2 Motion of a Particle in an EM Field . . . . . . . . . . . . . 295
6.3 Relativistic Motion . . . . . . . . . . . . . . . . . . . . . . . 296
CONTENTS v
6.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
7 Electrostatics 311
7.1 Electrostatic Field of Point Charges . . . . . . . . . . . . . . 311
7.2 The Program – Appetizer and ... Desert . . . . . . . . . . . 314
7.3 The Program – Main Dish . . . . . . . . . . . . . . . . . . . 323
7.4 The Program - Conclusion . . . . . . . . . . . . . . . . . . . 329
7.5 Electrostatic Field in the Vacuum . . . . . . . . . . . . . . . 334
7.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
7.7 Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . 342
7.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
8 Diffusion Equation 353
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
8.2 Heat Conduction in a Thin Rod . . . . . . . . . . . . . . . . 355
8.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 356
8.4 The Program . . . . . . . . . . . . . . . . . . . . . . . . . . 358
8.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
8.6 Diffusion on the Circle . . . . . . . . . . . . . . . . . . . . . 363
8.7 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
8.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
9 The Anharmonic Oscillator 373
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
9.2 Calculation of the Eigenvalues of H ((cid:21)) . . . . . . . . . . 375
nm
9.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
9.4 The Double Well Potential . . . . . . . . . . . . . . . . . . . 390
9.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
10 Time Independent Schrödinger Equation 401
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
10.2 The Infinite Potential Well . . . . . . . . . . . . . . . . . . . 404
10.3 Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . 413
10.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 422
10.5 The Anharmonic Oscillator - Again... . . . . . . . . . . . . 428
10.6 The Lennard–Jones Potential . . . . . . . . . . . . . . . . . 431
10.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
11 The Random Walker 447
11.1 (Pseudo)Random Numbers . . . . . . . . . . . . . . . . . . 448
11.2 Using Pseudorandom Number Generators . . . . . . . . . . 460
vi CONTENTS
11.3 Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . 467
11.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
12 Monte Carlo Simulations 479
12.1 Statistical Physics . . . . . . . . . . . . . . . . . . . . . . . . 480
12.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
12.3 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
12.4 Correlation Functions . . . . . . . . . . . . . . . . . . . . . 489
12.5 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
12.5.1 Simple Sampling . . . . . . . . . . . . . . . . . . . . 492
12.5.2 Importance Sampling . . . . . . . . . . . . . . . . . 493
12.6 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . 494
12.7 Detailed Balance Condition . . . . . . . . . . . . . . . . . . 495
12.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
13 Simulation of the d = 2 Ising Model 499
13.1 The Ising Model . . . . . . . . . . . . . . . . . . . . . . . . 499
13.2 Metropolis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
13.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 508
13.3.1 The Program . . . . . . . . . . . . . . . . . . . . . . 513
13.3.2 Towards a Convenient User Interface . . . . . . . . . 519
13.4 Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . 531
13.5 Autocorrelations . . . . . . . . . . . . . . . . . . . . . . . . 533
13.6 Statistical Errors . . . . . . . . . . . . . . . . . . . . . . . . 540
13.6.1 Errors of Independent Measurements . . . . . . . . 541
13.6.2 Jackknife . . . . . . . . . . . . . . . . . . . . . . . . 544
13.6.3 Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . 546
13.7 Appendix: Autocorrelation Function . . . . . . . . . . . . . 547
13.8 Appendix: Error Analysis . . . . . . . . . . . . . . . . . . . 555
13.8.1 The Jackknife Method . . . . . . . . . . . . . . . . . 555
13.8.2 The Bootstrap Method . . . . . . . . . . . . . . . . . 559
13.8.3 Comparing the Methods . . . . . . . . . . . . . . . . 562
13.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
14 Critical Exponents 577
14.1 Critical Slowing Down . . . . . . . . . . . . . . . . . . . . . 579
14.2 Wolff Cluster Algorithm . . . . . . . . . . . . . . . . . . . . 580
14.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 587
14.3.1 The Program . . . . . . . . . . . . . . . . . . . . . . 589
14.4 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
14.5 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 597
CONTENTS vii
14.6 Autocorrelation Times . . . . . . . . . . . . . . . . . . . . . 605
14.7 Temperature Scaling . . . . . . . . . . . . . . . . . . . . . . 608
14.8 Finite Size Scaling . . . . . . . . . . . . . . . . . . . . . . . 614
14.9 Calculation of (cid:12) . . . . . . . . . . . . . . . . . . . . . . . . 616
c
14.10Studying Scaling with Collapse . . . . . . . . . . . . . . . . 621
14.11Binder Cumulant . . . . . . . . . . . . . . . . . . . . . . . . 631
14.12Appendix: Scaling . . . . . . . . . . . . . . . . . . . . . . . 634
14.12.1Binder Cumulant . . . . . . . . . . . . . . . . . . . . 634
14.12.2Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 640
14.12.3Finite Size Scaling . . . . . . . . . . . . . . . . . . . 642
14.13Appendix: Critical Exponents . . . . . . . . . . . . . . . . . 645
14.13.1Definitions . . . . . . . . . . . . . . . . . . . . . . . . 645
14.13.2Hyperscaling Relations . . . . . . . . . . . . . . . . . 646
14.14Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Bibliography 650
Index 658
viii CONTENTS
This book has been written assuming that the reader executes
all the commands presented in the text and follows all the
instructions at the same time. If this advice is neglected, then
the book will be of little help and some parts of the text may
seem incomprehensible.
The book’s website is at
http://www.physics.ntua.gr/˜konstant/ComputationalPhysics/
From there, you can can download the accompanying software, which con-
tains, among other things, all the programs presented in the book.
Some conventions: Text using the font shown below refers to com-
mands given using a shell (the “command line”), input and output of
programs, code written in Fortran (or any other programming language),
as well as to names of files and programs:
> echo Hello world
Hello world
When a line starts with the prompt
>
then the text that follows is a command, which can be given from the
command line of a terminal. The second line, Hello World, is the output
of the command.
The contents of a file with Fortran code is listed below:
program add
z = 1.0
y = 2.0
x = z + y
print *, x
end program add
What you need in order to work on your PC:
CONTENTS ix
• An operating system of the GNU/Linux family and its basic tools.
• A Fortran compiler. The gfortran compiler is freely available
for all major operating systems under an open source license at
http://www.gfortran.org.
• An advanced text editor, suitable for editing code in several pro-
gramming languages, like Emacs².
• A good plotting program, suitable for data analysis, like gnuplot³.
• The shell tcsh⁴.
• The programs awk⁵, grep, sort, cat, head, tail, less. Make sure
that they are available in your computer environment.
If you have installed a GNU/Linux distribution on your computer,
all of the above can be installed easily. For example, in a Debian like
distribution (Ubuntu, ...) the commands
> sudo apt(cid:0)get install tcsh emacs gnuplot(cid:0)x11 gnuplot(cid:0)doc
> sudo apt(cid:0)get install gfortran gawk gawk(cid:0)doc binutils
> sudo apt(cid:0)get install manpages(cid:0)dev coreutils liblapack3
install all the necessary tools.
If you don’t wish to install GNU/Linux on your computer, you can
try the following:
• Bootyourcomputerusingausb/DVDliveGNU/Linux, likeUbuntu⁶.
This will not make any permanent changes in your hard drive but
it will start and run slower. On the other hand, you may save all
your computing environment and documents and use it on any
computer you like.
• Install Cygwin⁷ in your Microsoft Windows. It is a very good solu-
tion for Microsoft-addicted users. If you choose the full installation,
then you will find all the tools needed in this book.
²http://www.gnu.org/software/emacs/
³http://www.gnuplot.info
⁴http://www.tcsh.org
⁵http://www.gnu.org/software/gawk
⁶http://www.ubuntu.com
⁷http://www.cygwin.com