Molecular Dynamics (2)
Molecular dynamics for continuous potentials
Short history:
The first MD simulation for a system interacting with a continuous
potential (Lennard-Jones potential) was carried out by A,Rahman
in 1964.
A,Rahman,Phys,Rev,136,A405,(1964).
Main differences between MD with continuous poentials and
MD of HS:
MD (continuous potentials)
continuous change of forces
exerted on all the particles;
approximate solution of motion
of equations;
wide applications.
MD (HS)
discontinuous changes of forces
exerted on all the particles;
exact solution of motion of
equations;
restricted applications.
Trajectory generation
Equation of motion:
mi?2ri/?t2 = miai = fi
mi,mass of particle i;
ri,position of particle i;
ai,acceleration of particle i;
fi,force on particle i,fi = -?iV
V,potential energy
Numerical solution:
Method of finite difference.
Desirable qualities for a good algorithm
It should be fast and requires little memory.
It should permit the use of a long time step,dt.
It should satisfy the known conservation laws for the energy and
momentum and be time-reversible.
It should be simple in form and easy to program.
Verlet’s algorithm
Position:
r(t+dt) = 2r(t) - r(t-dt) + (dt)2a(t)
The error on position is of order of (dt)4.
Taylor expansion:
r(t+dt) = r(t) + dtv(t) + (dt)2a(t)/2 + …
r(t-dt) = r(t) - dtv(t) + (dt)2a(t)/2 + …
Velocity:
v(t) = [r(t+dt) - r(t-dt)]/(2dt)
The error on velocity is of order of (dt)3.
How to initialize Verlet ’s algorithm?
Problem:
At t=0,r(-dt) is unknown!
Solution to the problem:
r(-dt) = r(t) - dt,v(t)
Advantages and drawbacks of Verlet’s algorithm
Advantages:
Good stability,i.e.,relatively large time step dt;
Good energy conservation;
Good time-reversibility;
Simplicity.
Drawbacks:
Not self-starting;
Position and velocity are not treated with the same precision.
How to choose time step?
Simple case:
dt must be chosen in such a way that the total energy is well conserved
and the trajectory is time reversible.
Complicated case (multi-time scales):
When there are several time scales (e.g.,mixture of particles with
different masses,polymers in solvent,both hard and soft modes exist
in molecular systems,etc.),dt must be chosen according to the
dynamics of the component or the mode which evolves most quickly,
Reduced units
Temperature,T* = kT/e
Energy,E* = E/e
Pressure,P* = Ps3/e
Time,t* = (e/ms2)1/2t
Force,f* = fs/e
Constant-temperature Molecular Dynamics
The basic MD algorithm generates a microcanonical ensemble.
Different velocity adjusting methods:
1) Andersen’s Method:
Reference,H.C,Andersen,J,Chem,Phys,72,2384,1980.
Basic idea:
mimicing the collisions between the molecules of the considered
system with those of the thermal bath.
Practical implementation:
At a preset time interval,Dt,the velocity of a randomly chosen
molecule is reset according to the Maxwell-Boltzmann
distribution with T.
Choice of Dt:
The ideal choice of Dt is such that it allows to dissipate a thermal
perturbation at the rate determined by the thermal conductivity of
the system.
It should be avoided to use a too small Dt which leads to incorrect
velocity correlation function!
A too large Dt leads to a slow sampling of the canonical distribution
of energies.
Advantage of the method:
Easy to implement.
Drawback:
Not easy to determine the suitable Dt.
2) Velocity re-scaling method:
At each time step,the velocities of all the molecules are rescaled,
i.e.,multiplied by a factor?.
2/1
11?
D T c u r rTttd?
where T is the target temperature and Tcurr is the current kinetic
temperature,
Choice of Dt:
The same criterion as in Andersen’s method.
Dt > dt
Drawback:
It cannot be established that the configurations generated by this
method belong to the canonical ensemble.
Ni iiic ur r vvmT Nk 13 1
Non-equilibrium Molecular Dynamics
Example,Diffusion
How to set up a concentration gradient?
Method of colored particles:
12
Method for maintaining the concentration gradient:
When a uncolored particle crosses the boundary 1 by periodic
boundary condition,it is colored.
When a colored particle crosses the boundary 2 by periodic
boundary condition,it is uncolored.
Calculation of diffusion coefficient:
Fick’s law:
J = - Ddrcol(z)/dz
Calculate the steady state flux,J,and the concentration gradient
from simulation,Diffusion coefficient is then obtained from,
D = -J/[drcol(z)/dz]
Illustration:
z
rcol(z),run(z)
L0
Molecular dynamics for continuous potentials
Short history:
The first MD simulation for a system interacting with a continuous
potential (Lennard-Jones potential) was carried out by A,Rahman
in 1964.
A,Rahman,Phys,Rev,136,A405,(1964).
Main differences between MD with continuous poentials and
MD of HS:
MD (continuous potentials)
continuous change of forces
exerted on all the particles;
approximate solution of motion
of equations;
wide applications.
MD (HS)
discontinuous changes of forces
exerted on all the particles;
exact solution of motion of
equations;
restricted applications.
Trajectory generation
Equation of motion:
mi?2ri/?t2 = miai = fi
mi,mass of particle i;
ri,position of particle i;
ai,acceleration of particle i;
fi,force on particle i,fi = -?iV
V,potential energy
Numerical solution:
Method of finite difference.
Desirable qualities for a good algorithm
It should be fast and requires little memory.
It should permit the use of a long time step,dt.
It should satisfy the known conservation laws for the energy and
momentum and be time-reversible.
It should be simple in form and easy to program.
Verlet’s algorithm
Position:
r(t+dt) = 2r(t) - r(t-dt) + (dt)2a(t)
The error on position is of order of (dt)4.
Taylor expansion:
r(t+dt) = r(t) + dtv(t) + (dt)2a(t)/2 + …
r(t-dt) = r(t) - dtv(t) + (dt)2a(t)/2 + …
Velocity:
v(t) = [r(t+dt) - r(t-dt)]/(2dt)
The error on velocity is of order of (dt)3.
How to initialize Verlet ’s algorithm?
Problem:
At t=0,r(-dt) is unknown!
Solution to the problem:
r(-dt) = r(t) - dt,v(t)
Advantages and drawbacks of Verlet’s algorithm
Advantages:
Good stability,i.e.,relatively large time step dt;
Good energy conservation;
Good time-reversibility;
Simplicity.
Drawbacks:
Not self-starting;
Position and velocity are not treated with the same precision.
How to choose time step?
Simple case:
dt must be chosen in such a way that the total energy is well conserved
and the trajectory is time reversible.
Complicated case (multi-time scales):
When there are several time scales (e.g.,mixture of particles with
different masses,polymers in solvent,both hard and soft modes exist
in molecular systems,etc.),dt must be chosen according to the
dynamics of the component or the mode which evolves most quickly,
Reduced units
Temperature,T* = kT/e
Energy,E* = E/e
Pressure,P* = Ps3/e
Time,t* = (e/ms2)1/2t
Force,f* = fs/e
Constant-temperature Molecular Dynamics
The basic MD algorithm generates a microcanonical ensemble.
Different velocity adjusting methods:
1) Andersen’s Method:
Reference,H.C,Andersen,J,Chem,Phys,72,2384,1980.
Basic idea:
mimicing the collisions between the molecules of the considered
system with those of the thermal bath.
Practical implementation:
At a preset time interval,Dt,the velocity of a randomly chosen
molecule is reset according to the Maxwell-Boltzmann
distribution with T.
Choice of Dt:
The ideal choice of Dt is such that it allows to dissipate a thermal
perturbation at the rate determined by the thermal conductivity of
the system.
It should be avoided to use a too small Dt which leads to incorrect
velocity correlation function!
A too large Dt leads to a slow sampling of the canonical distribution
of energies.
Advantage of the method:
Easy to implement.
Drawback:
Not easy to determine the suitable Dt.
2) Velocity re-scaling method:
At each time step,the velocities of all the molecules are rescaled,
i.e.,multiplied by a factor?.
2/1
11?
D T c u r rTttd?
where T is the target temperature and Tcurr is the current kinetic
temperature,
Choice of Dt:
The same criterion as in Andersen’s method.
Dt > dt
Drawback:
It cannot be established that the configurations generated by this
method belong to the canonical ensemble.
Ni iiic ur r vvmT Nk 13 1
Non-equilibrium Molecular Dynamics
Example,Diffusion
How to set up a concentration gradient?
Method of colored particles:
12
Method for maintaining the concentration gradient:
When a uncolored particle crosses the boundary 1 by periodic
boundary condition,it is colored.
When a colored particle crosses the boundary 2 by periodic
boundary condition,it is uncolored.
Calculation of diffusion coefficient:
Fick’s law:
J = - Ddrcol(z)/dz
Calculate the steady state flux,J,and the concentration gradient
from simulation,Diffusion coefficient is then obtained from,
D = -J/[drcol(z)/dz]
Illustration:
z
rcol(z),run(z)
L0
