molecular dynamics (2) langevin dynamics nvt and npt ensembles
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Molecular dynamics (2)
Langevin dynamicsNVT and NPT ensembles
randi
ii
i
ii f
dt
dx
x
V
dt
xdm
2
2
wwii rr )(6
tRTttWtW
Nt
RTttWtf
ijiji
i
ti
t
randi
2
)1,0(2
limlim00
Langevin (stochastic) dynamics
Stokes’ law
Wiener process
gi – the friction coefficient of the ith atomri, rw – the radii of the ith atom and of water, respectivelyhw – the viscosity of water.
The average kinetic energy of Langevin MD simulation corresponds to absolute temperature T and the velocities obey the proper Gaussian distribution with zero mean and RT/m variance.
RTN
mEN
iiik 2
3
2
1 3
1
2
v
We can also define the momentary temperature T(t)
N
iiik tm
NRtE
NRtT
3
1
2
3
1
3
2v
Integration of the stochastic equations of motion(velocity-Verlet integrator)
i
i
i
i
i
i
i
i
i
i
m
t
i
im
t
iii
m
t
ii
m
t
iii
m
t
iii
etm
etttm
tettt
etttm
tettttt
22
4
2
2
)(
2
)()(
WrFrF
vv
WrF
vrr
Ricci and Ciccotti, Mol. Rhys., 2003, 101, 1927-1931.Ciccotti and Kalibaeva, Phil. Trans. R. Soc. Lond. A, 2004, 362, 1583-1594.
When Dt and the friction coefficient are small, the exponential terms can be expanded into the Taylor series and the integrator becomes velocity-Verlet integrator with friction and stochastic forces
i
iiiii
i
ii
iiiii
iii
m
ttttt
m
tttt
tttttm
tttttt
2)(2
2
)(
)(2
)()(
WvrFrF
vv
WvrF
vrr
Brownian dynamics
randi
i
ii f
x
E
dt
dx
Ignore the inertia term; assume that the motion results from the equilibrium between the potential and fritction+stochastic forces.
Advantage: first-order instead of second-order ODE.
Disadvantages: constraints must be imposed on bonds; energy often grows uncontrollably.
Andersen thermostat
1.Perform a regular integration step in microcanonical mode.
2.Select a number of particles, n, to undergo collision with the thermal bath.
3.Replace the velocities of these particles with those drawn from the Maxwell-Boltzmann distribution corresponding to the bath temperature T0.
Berendsen thermostat: derivation from Langevin equations
N
iii
N
iii
t
k tmttmtdt
dE
1
2
1
2
0 2
1lim vv
TTNRt
ERTN
t
dt
dtm
dt
dE
N
iii
k
N
iii
N
i
iii
k
01
01
1
3
2
32
Fv
Fv
vv
Therefore:
11
2...12
1
1
0
10
0
T
TtT
Tt
T
Tmm
T
TT
ii
iiiii
vv
vFv
Berendsen thermostat
n
iii
k
ii
tmNRNR
tEtT
tT
Tt
1
2
0
3
1
3
2
11
v
vv
t – coupling parameter
Dt – time step
Ek – kinetic energy
: velocities reset to maintain the desired temperature
: microcanonical run
1
Berendsen et al., J. Chern. Phys., 1984, 81(8) 3684-3690
Pressure control (Berendsen barostat)
n
iii
n
ijijij tmtt
Vtp
tppt
LL
1
2
1
3
1
0
2
1
3
1
1
vrF
L – the length of the system (e.g., box sizes)
b – isothermal compressibility coefficient
t – coupling parameter
Dt – time step
p0 – external pressure
Extended Lagrangian method to control temperature and pressure
Lagrange formulation of molecular dynamics
N
iNii qqqVqmVTL
3
1321
2 ,,,2
1
A physical trajectory minimizes L (minimum action principle). This leads to Euler equations known from functional analysis:
iqi
iiii
Fq
Vqm
q
L
q
L
dt
d
Nose Hamiltonian and Nose Lagrangian
Qss
Lp
smL
sgRTsQ
VsmL
sgRTQ
pV
smH
s
iii
N
i
NiiNose
N
i
sN
i
iNose
i
rp
rr
rp
r2
2
1
22
1
2
2
2
ln22
1
ln22
s – the coordinate that corresponds to the coupling with the thermostatQ – the „mass” of the thermostatg – the number of the degrees of freedom (=3N)
Equations of motion (Nose-Hoover scheme)
dt
sd
s
s
gRTmQ
Vm
N
i i
i
iN
i
i
ii
i
ln2
1
1
2
p
prp
pr
r
Velocity-Verlet algorithm
Q
t
gTtvm
gTttvmttt
Q
tgTtvmtttstts
t
tvtm
tf
ttvttm
tf
tvttv
ttvt
m
tfttvtrttr
N
iii
N
iii
N
iii
ii
i
ii
i
ii
ii
iii
2
2lnln
2
2
3
1
2
3
1
2
23
1
2
The NH thermostat has ergodicity problem
position
velo
city
Microcanonical Andersen thermostat Nose-Hoover thermostat
position position
Test of the NH thermostat with a one-dimensional harmonic oscillator
Nose-Hoover chains
M
j
M
jj
jjN
i i
iNNHC
MjM
M
jjjjj
j
N
i i
i
iN
i
i
ii
RTsgRTsQ
mVH
RTQQ
RTQQ
g
mQ
Vm
i
1 21
2
1
2
211
12
11
211
2
11
1
22
1
1
2
1
pr
p
prp
pr
r
Improvement of ergodicity for the NH chains thermostat
Test with a one-dimensional harmonic oscillator
Relative extended energy errors for the 108-particle LJ fluid
Kleinerman et al., J. Chem. Phys., 2008, 128, 245103
Performance of Nose-Hoover thermostat for the Lennard-Jones fluid
Kleinerman et al., J. Chem. Phys., 2008, 128, 245103
Performance various termostat on decaalanine chain
Kleinerman et al., J. Chem. Phys., 2008, 128, 245103
Extended system for pressure control (Andersen barostat)
0
13
13
1
PtPW
V
V
VV
V
m
iii
ii
ii
pFp
rp
r
W the „mass” corresponding to the barostat (can be interpreted as the mass of the „piston”)V is the volume of the system
Isothermal-isobaric ensemble
N
i i
i
N
i i
i
iiiiii
ii
RTgW
p
mp
Q
p
pQ
p
mg
dPtPdVp
W
dVpV
W
p
W
p
g
d
W
p
m
1
221
2
0
1,
,
1,
p
p
ppFprp
r
d is the dimension of the system (usually 3)g is the number of degrees of freedomW the „mass” corresponding to the barostatQ is the”mass” corresponding to the thermostat
Martyna, Tobias, and Klein, J. Chem. Phys., 1994, 101(5), 4177-4189
Martyna-Tobias-Klein NPT algorithm: tests with model systems
Model 1-dimensional system: position distribution
Model 3-dimensional system: volume distribution
Martyna, Tobias, and Klein, J. Chem. Phys., 1994, 101(5), 4177-4189
The Langevin piston method (stochastic)
tW
RTttftf
tfVptpW
V
V
VV
V
m
randrand
rand
iii
ii
ii
2
13
13
1
0
pFp
rp
r
Feller, Zhang, Pastor, and Brooks, J. Chem. Phys., 1994, 103(11), 4613-4621
Extended Hamiltonian method
Langevin piston method (W=25)
Berendsen barostat
W=5
W=25
W=225
g=20 ps-1
g=0 ps-1
g=50 s-1
tp=1 ps
tp=5 ps
Feller, Zhang, Pastor, and Brooks, J. Chem. Phys., 1994, 103(11), 4613-4621