[email protected] Room 2032 China Canada Winnipeg Manitoba.
-
Upload
solomon-singleton -
Category
Documents
-
view
220 -
download
1
Transcript of [email protected] Room 2032 China Canada Winnipeg Manitoba.
[email protected] Room 2032
Computer Simulation Methods in Physics
Lecture 1
B.W. SouthernDepartment of Physics and Astronomy
University of ManitobaWinnipeg Manitoba
Canada
Computer Simulation• High speed computers have become more readily
available in recent years
• computer simulation is an important tool used in all areas of science
Experiment
Analytical Theory Computer Simulation
Computational Physics in theUndergraduate Curriculum
• New courses in computational physics emphasizing computer simulations, numerical methods, and symbolic manipulation have been developed
• new texts are available
• ideally we should incorporate computational methods into every course
• a beginning course in computer science is not a good substitute
• numerical methods are more meaningful when part of a simulation
Computer Simulation Courses• computer simulations provide an opportunity for
involving students in open-ended problems
• they can ‘do’ physics in the same way that research is done
• a good predictor for how students might perform in graduate work
• the process of converting an abstract model into a working program makes the model more meaningful
• a broader vision of physics can be introduced using models of interest to geologists, biologists and material scientists
Textbook
• I will follow closely the ideas presented in the textbook by H. Gould and J. Tobochnik “An Introduction to Computer Simulation Methods”, second edition, Addison-Wesley (1996)
• the website http://www.clarku.edu/~sip has extensive resources available for download such as a list of compuational physics books as well as software programs in basic, fortran and C.
Computer Simulation• Why should we use computers?
• analytical tools are best suited to the analysis of linear problems
• the study of nonlinear problems involves mathematical approximations
• comparison of realistic models with experimental data leaves many open questions
• computer simulations can deal with a model without approximations apart from statistical and controllable systematic errors
Computer simulations are computer experiments
• construct an idealized model of a physical system• develop a procedure or algorithm to study the model on a
computer• model the behaviour of a macroscopic system of 1024
particles by a small system of 102 to 105 particles• compare the results of the computer experiment with
laboratory experiments• the design of various programs relies on experience as in
the case of real experimental setups• does not rely on assumptions, approximations, or the
discarding of “small” terms
Computer Simulation
• There are two common types of simulation techniques
• Molecular Dynamics
• Monte Carlo methods
Molecular dynamics• system of classical particles interacting with each
other with two-body forces • integrate Newton’s equations of motion numerically• perform a time average of state variables over the
trajectory in phase space (r,p) or (q,p)
• r=(xi,yi,zi,…) p=(pxi,pyi,pzi,…) i=1,N
• 6N dimensional space• energy is often conserved (microcanonical)• ergodic hypothesis is crucial• all regions of phase space d3Nrd3Np equally probable
Molecular Dynamics• Consider a system of many identical particles
which are described by classical physics
• total energy E=K+U is kinetic plus potential where U= u(r12)+u(r13)+…+u(r23)+…
• sum of pairwise interactions
• a common potential is the Lennard-Jones potential characterized by two parameters and
12 6
( ) 4u rr r
dr
dt
p
mdp
dtF r
i i
iij
j i
N
( )
t t n t
x x t a t
a a t
n
n n n
n n
0
12
1
1
21
2
v
v v
n
n+1 n
( )
( )
Phase Space
r p
i Ni i( ..., ...)
,
1
VerletAlgorithm
Molecular Dynamics
• Basic steps:
• start with a random assignment of positions and velocities of particles
• ensure that the center of mass velocity is zero!=> total momentum conserved
• iterate the equations for some time to allow system to equilibrate
• then collect data at each time step and average
Molecular Dynamics• Follow trajectories in phase space for a given
set of initial conditions
• compute averages over the trajectories
• for example the kinetic energy
K m t
dNkT t
i
N1
2
2
2
1
vi ( )
( )
Can estimate the temperature <T> as a time average of K