Download Computational simulation of Molecular dynamics

Computational
simulation of
Molecular dynamics
Talk given at
the theory group, School of Physics,
USM
31 Oct 2008
3.30 pm
Plan of the talk







Scope of investigation
General idea and simulation strategy
Simplest simulation: 2D, rigid boundary
condition, without interaction
Some details of Mathematica programming
Periodic boundary condition
Generalisation to 3D case, with interactions
Abstraction of physical quantities from the
simulation (work still in progress)
General idea: Interactions
determine physical behavior





Consider a system of particles interacting among themselves via
some inter-particle forces arises from some effective potential.
Well known example of such potential are the Lennard –Jones
potential (6-12 potential) of intermolecular force in Argon gas
 VLJ = k(a/r6 – b/r12)
These forces determine the behaviour of the particle system –
heat capacities, pressure, crystal structures, phase transition etc.
Different system is characterised by different interactions
(different type of potential, parameters)
Once the exact form of interactions is identified, the physical
consequences could be worked out either theoretically, or using
computational methods.
Empirical interactions






The LJ potential is a so-called ‘empirical potential’
Their 6-12 form is an educated ‘guess’ that fits
experimental data of the system investigated (e.g.
Argon and other inert gases)
The parameters in the LJ potential have to be
extracted from experimental data
Not predicted from first principle
Different system has different values of a, b, k
System of different interactions may not be
described by the LJ potential – other empirical form
of potential is required.
System of particles


Main purpose: To predict the behaviour and
extract essential physical observables of a
system of interacting particles (given a known
potential) at a given temperature.
The ‘particle’ could be entities with internal
structures such as complex molecules, or
simply a ‘point’ particle.
System evolved as a function
of time at a fixed temperature
Total
energy
At a fixed temperature, the (i) evolution of
the system from i to equilibrium and (ii) the
energy of the equilibrium state is
determined by the interactions among the
particles (and possible external influences
e.g. external magnetic or electric field, or
pressure)
Equilibrium
state
corresponds
to T1
Initial
configuration
t1
Temperature
fixed at T1
If temperature changes
at t1 to T2, system will
evolve to a new
equilibrium state
corresponds to T2
Equilibrium
state
corresponds to
T2
t2
time
Temperature
fixed at T2
Tracing the evolution is possible –
computational simulation


Essentially, the dynamical development as a
function of time from some initial (nonequilibrium) state to a final (equilibrium) state
can be traced in principle if the interactions
are known.
Trace the evolution using computational
methods.
First principle derivation of
interaction







The potential or forces among the particles are quantummechanical in nature.
In principle these force can be derived from first-principle (abinitio) by solving the many-body quantum-mechanical
Schroedinger equation.
Then these forces are fed into a computer code to predict the
behaviour of the system.
No experimental input is required in principle.
Parameter free.
Quantum-molecular dynamics – computationally expensive –
further work.
Only classical, empirical molecular dynamics will be discussed.
Instantaneous information
of each particles


If we can trace (i) the instantaneous position and (ii)
instantaneous velocities of each particle, as a
function of time at a fixed temperature, we then can
(i) Abstract time averages of physical quantities
(e.g. correlation length, heat capacities etc.)
(ii) Known if phase transition has occured
(iii) Display the motion of these particle on the
computer screen (merely for visualisation
purpose is possible)
(iv) Many other things…
Simplest case: 2D free particle
bouncing in a box





Imagine a 2-D box of size L x L, containing N free
particles bouncing against the wall
Forces among the particle is F = 0
How would you simulate their behavior and visually
display them on the computer screen?
Essentially, we wish to (i) track instantaneous
position, r(np)[n] and (ii) instantaneous velocities
v(np)[n]
r(np)[n], v(np)[n] positional vector and velocity of
particle indexed by np at the time step n.
Size of the box
y=L
r(np)[n]
r(np=1)[n]
r(np=2)[n]
y=0
x=0
x=L
Mathematica Simulation
of rigid boundary box containing
free particles
Algorithm







Initialisation: Number of particles, masses, positions,
velocity
Need to make sure that the total momentum of the
system is zero  a trick is required:
Let mom[np] is the momentum of particle np,
Calculate the sum of all momentum, SumMom =
Sum[mom[np],{np, 1, N}]
Divided SumMom by N  give average momentum
per particle, mom_ave
Then replace the mom[np]  mom[np] – mom_ave
When sum over mom[np] for all np from 1 to N, we
will be assured that SumMomentum
Kinematical equations of r



Evolve the particle’s position according to the
kinematical relation
r[np][n] = r[np][n-1] + v[np] deltat
v[np] is a constant in time for each particle
since no interactions act on the particle to
modify the velocity.
Loop the time step, n, from n=0 to n=nmax.
Problems with the simulation






Not physical
No interaction
Box size is too small
Number of particle is too small
Computer hangs if N1023, L~ m
Edge effect: Bouncing at edges is significant
whereas for real cases it is otherwise for real
case value of L, N In real case, L ~ m, N~1023
Computational tricks to
circumvent edge effect





Force ignored for the moment (take care of it
later)
Use periodic boundary condition:
If x > L, x (x - L)
If x < 0, x (x +L)
This trick essentially eliminates the bouncing
boundary effects at the edges – artificial
simulation of an ‘infinite’ volume
Mathematica Simulation
of a box containing free particles
with periodic boundary condition
Trick to simulate large number
of particle: Kaleidoscoping







Problem still exist to simulate large N
The trick: Kaleidoscoping 8 cell surrounding the central cell to
simulate the presence of other particles (which are mirror image
of the ones in the central cell)
Keleidoscoping creates a total of 9N evolving particles despite
only N are actually involved in the kinematical time-stepping
“fake” reality approximates true reality by using larger N
Needed if forces are taken into account later
Save computational resources: get 9N particles in the picture
despite only evolve N of them (for 2D case)
Can generalise to 3D case easily  Keleidoscoping 26 cells
surrounding the central cell  get 27N particles in the picture
despite only evolve N of them
Mathematica Simulation
of kelediasoped box containing
free particles
May the force be with you…






Forces among the particle is switched on  no more r[np][n] =
r[np][n-1] + v[np] deltat
Force between them can be then numerically evaluated via
Fr = - dV(r)/dr if the form of the potentialV(r) between two
particles with r[np1] and r[np2] are known
VLJ = k(a/r6 – b/r12)
Let DR =|r[np1]- r[np2]| ( r in VLJ)
Fr = - k( -6a / DR7 + 12 b/DR13)
The evolution of kinematical coordinates can be derived from
Newton second law  Verlet algorithm
Verlet algorithm
v' (n)  v(n)  deltat  F (t ) / 2 
r (n  1)  v(n)  deltat  v' (n) 
Calcualte F (n  1) based r (n  1) 
v(n  1)  v' (n)  deltat  F (n  1) / 2
Next n
Mathematica Simulation
of kelediasoped box containing
interacting particles
Work to be done





It’s a project still in progress.
The evolution from initial state to equilibrium state
has at fixed temperature is yet to carry out
Abstraction of physical information such as heat
capacity, transport coefficients is yet to be done
Investigation of phase transition will be possible
since the critical parameters can be numerically
obtained
Incooperating some internal degree of freedom into
the ‘particle’, simulation of structural phase transition
of liquid crystal is possible.