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