Time-dependent Non-Equilibrium Molecular

Dynamics

Giovanni Ciccotti Dept. of Physics, Univ. of Rome “La Sapienza”

and School of Physics, UCD, Dublin

With S. Meloni (CINECA), S. Orlandini (CINECA), M. Mareschal

(ZCAM), C. Pierleoni (Sapienza), M. L. Mugnai (Austin)

•  t-dependent averages in NEMD (Onsager-Kubo) •  sampling an initial condition ensemble by MD • beyond macroscopic Hydrodynamics: Convective cells and relaxation of an interface

⇥p(⇥x) =X

i

⇥pi�(⇥x� ri)

What we mean by Non-Equilibrium

H Y D R O D Y N A M I C S

v

-v

SHEAR

x

y

g

Convective cells

T R A N S P O R T

∇T

Interface relaxation

�(⇥x, t) =< �(⇥x) >ne

⇥(⇤x) =X

i

µi�(⇤x� ri)

⇥v(⇥x, t) =< ⇥p(⇥x) >ne

�(⇥x, t)

NEMD (1) •  Assume that a given, time-dependent external local

field is coupled to our system via a suitable local property

•  The total Hamiltonian of the system is standard equilibrium Hamiltonian

H(�, t) = H0(�) +Hp(�, t)

HP (�) = �Z

dxA(x|�) (x, t) (x, t) = g�(x)�(t)

A(x|�) =X

i=1,N

Ai(�)�(ri � x)

(x, t)

= �g �(t)X

i

Ai�i

NEMD (2)

•  Equations of motion

•  Liouville equation

with

PERTURBED SYSTEM:

an observable J of the system evolves with

S†(t) = S†0(t) +

Z t

�⇥d�S†(t� �)iLp(�)S

†0(�)

�m

�t= iLm = iL0m+ iLpm � {H0,m}+ {Hp,m}

(r = �H0

�p + �Hp

�p = pµ � g �hp

�p �(t)

p = ��H0�r � �Hp

�r = F + g �hp

�r �(t)

J(t) ⌘ J(�(t)) = S(t)J(�(0))

m(�, t) = S†m(�, 0)

NEMD (2)

•  Equations of motion

•  Liouville equation

with

PERTURBED SYSTEM:

�m

�t= iLm = iL0m+ iLpm � {H0,m}+ {Hp,m}

is a given initial distribution which can be sampled by MD if it comes from a stationary state (in particular but not necessarily an equilibrium one)

(r = �H0

�p + �Hp

�p = pµ � g �hp

�p �(t)

p = ��H0�r � �Hp

�r = F + g �hp

�r �(t)

m(�, t) = S†m(�, 0)

m0(�) = m(�, 0)

The time-dependent NE average of a given observable can be obtained as follows (Onsager-Kubo equation)

Dynamical NEMD [Ciccotti, Jacucci ‘75]

Stationary MD Trajectory, m0(Γ)

NEMD t

t t

t

“Direct Computation of Dynamical Response by Molecular Dynamics: The Mobility of a Charged Lennard-Jones Particle”, G. Ciccotti and G. Jacucci, Phys. Rev. Lett. 35 (1975), 789--792

⌘DS(t)J

E

0

J(t) =DJEt

NE=

Zd�J(�)m(�, t)

⌘⇣J , S†(t)m0

⌘=

⇣S(t)J ,m0

⌘

Sampling of the initial distribution

Direct (stationary fields, boundary conditions, …)

Advanced sampling (Cond.

Prob.)

Non-Eq. stationary MD

Restraint MD/MC

by MD (or MC)

Equilibrium (transport)

e.g. e.g.

Benard cells

Interface relaxation

m0(�)

Standard practice from

the ‘70s

Equilibrium MD

Convection

(a)

(b)

(c)

(d)

FIG. 4. Setting !A", hot reservoir on the left side of the box, cold reservoiron the right side of the box, case !i": Local velocity field averaged over 1000independent initial configurations during the first oscillation of temperatureand density fields, i.e., at !a" t=4.5#! /4, !b" t=9#! /2, !c" t=13#3! /4,and !d" t=18#!.

(a)

(b)

(c)

(d)

FIG. 5. Setting !A", hot reservoir on the left side of the box, cold reservoiron the right side of the box, case !i": Local velocity field averaged over 1000independent initial configurations during the second oscillation of tempera-ture and density fields, i.e., at !a" t=22.5#5 /4!, !b" t=27#3 /2!, !c" t=31.5#7 /4!, and !d" t=36#2!.

FIG. 6. Setting !A", hot reservoir on the left side of the box, cold reservoiron the right side of the box, case !i": Local velocity field averaged over 1000independent initial configurations at t=250.

(a)

(b)

FIG. 7. !a" Setting !A", case !i": Horizontal profile of the vertical componentof the velocity field vz!x" !red symbols" and vertical profile of the horizontalcomponent of the velocity field vx!z" !green symbols". The red symbols referto values in a row of cells at the midheight of the simulation box !nz=8".The green symbols refer to a column of cells at the midlength of the simu-lation box !nx=8". The values of the velocity are sampled in the final steadystate, at t#250, and averaged over 1000 independent initial configurations.!b" Setting !B", case !i": Horizontal profile of the vertical component of thevelocity field vz!x" !red symbols" and vertical profile of the horizontal com-ponent of the velocity field vx!z" !green symbols". The red symbols refer tovalues in a row of cells at nz=8. The green symbols refer to a column ofcells at nx=6. The values of the velocity are sampled in the final steadystate, at t#2500, and averaged over 480 independent initial configurations.

064106-7 Transient hydrodynamical behavior by D-NEMD J. Chem. Phys. 131, 064106 !2009"

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rT(x) 6= 0

• Sampling : standard stationary Non-Equilibrium MD with ,

• Sampling : standard segments of MD, starting at t= 0, with ,

m0(�) = m(�, t = 0)

rT(x) = const.

rT(x) = const.

J(x, t) g

g = 0

g = const.

Convection

l > rcut of the potential

Confining field

Convection Cells

l > rcut of the potential

Confining field

The particles interact via a WCA potential (a LJ truncated and shifted at the minimum). A completely repulsive potential has only solid and fluid states and

the thermodynamic conditions are such that the system is everywhere fluid.

Convection Cells

l > rcut of the potential

Confining field The particles interact via a WCA potential (a LJ

truncated and shifted at the minimum). A completely repulsive potential has only solid and fluid states and

the thermodynamic conditions are such that the system is everywhere fluid.

The system is divided in 15*15 cells, numbered from left to right 1-15. A generic cell is

Establishing Convective Cells

Circulation of the velocity field. The inset shows the path.

Establishing Convective Cells =

I Pv(x)ds

[LJunits]

Circulation

Interface Relaxation

• Sampling , where : restrained MD with for and a given initial interface

• Sampling : standard segments of MD with system Hamiltonian

J(x, t)

�⇢(x) = ⇢

A(x)� ⇢

B(x) = 0

S

m0(�) = m(�|�⇢(x) = 0, x 2 S)

�⇢(x) = 0x 2 S S

A

B

•  Field at the grid point of a discrete decomposition of the simulation box, with the atoms in the phase space point :

Where: •  for the density •  for the momentum density •  …

€

€

ˆ O ( x α ; r, p) = (1/Ωα ) d x δ x − r i( )Oi(r, p)

i=1

N

∑Ωα

∫

Hydrodynamics by NEMD

€

r, p ≡ ( r 1,…, p N )

€

x β

€

Oi(r, p) = µi

€

Oi(r, p) = pi

•  The field at time t given initial macroscopic conditions is (Onsager-Kubo)

where and

unrestrained MD started from a sample of points taken along a restrained MD (next slide)

€

Hydrodynamics by NEMD

O x, t |Δρ xα( ) = 0, xα ∈ S#$ %&=

dr0 dp0m0 r0, p0 |Δρ xα( ) = 0, xα ∈ S#$ %&∫ O x; r(t), p(t)( )

m0 r0, p0 |Δρ(xα ;r

0 ) = 0, xα ∈ S#$ %&=

exp −βH (r0, p0 )#$ %& δ Δρ(xα ;r0 )( )

xα∈ S∏

Z*PΔρ[Δρ(xα ;r0 ) = 0, xα ∈ S]

€

r(t)p(t)

"

# $

%

& ' ≡

r(t; r0, p0)p(t; r0, p0)

"

# $

%

& '

•  is smoothed by a Gaussian and the sampling of the ensemble at

on the surface is performed by restrained MD

where:

€

Conditional Averages by Restrained MD

S

��(⇥x↵; r0) = 0

H 0(r, p) = H(r, p) +k

2

X

⇥x↵2S

��(⇥x�; r)2

hO(⇤x⇥ ; r(t))icond = limk!1

⌦O(⇤x⇥ ; r(t))

Q⇤x↵2S exp[�� k

2�⇥(⇤x�; r0)2]↵H⌦Q

⇤x↵2S exp[�� k2�⇥(⇤x�; r0)2]

↵H

= limk!1

hO(�x� ; r)iH0

δ Δρ(xα ;r0 )( )

•  171500 particles: 88889 particles A, 82611 particles B

•  Pair potential:

•  Simulation box: ~(90 x 45 x 45) σ

•  Average density: 1.024 particles*σ3

•  Temperature: 1.5 ε/kb

•  Simulation time: •  Restrained MD: 75000 steps •  Unrestrained MD: 600000 steps

•  fields are averaged over (only) 40 unrestrained trajectories (for the moment)

€

Simulation Details

€

uAA (r) = uBB (r) = 4ε σr

$

% &

'

( ) 12

−σr

$

% &

'

( ) 6$

% &

'

( )

€

uAB (r) = 4ε σr

$

% &

'

( ) 12

In the fluid domain of pure L-J

v(�x�,max

) ⇠ 80 m/s

��(⇥x↵; t) = 0

t

t

v(�x�,max

) ⇠ 80 m/s

��(⇥x↵; t) = 0

•  The surface relaxes to the equilibrium by forming initially a two-tail profile of the velocity field that then stabilizes into a double-roll profile •  The velocity field obtained via the local time average technique violates the symmetry of the problem

Rigorous non-equilibrium ensemble averages vs local time averages

⇥v(⇥x, t) =< ⇥p(⇥x, t) >H0

�(⇥x, t)

!"#$

%&'()*(+$,$-..$

!/#$

%&'()*(+$,$01-.$

!2#$

%&'()*(+$,$1-..$

!3#$

%&'()*(+$,$-..$

!4#$

%&'()*(+$,$01-.$

!5#$

%&'()*(+$,$1-..$

~vl(~x, t) =12⌧

R t+⌧t�⌧ ds p(~x, s)

12⌧

R t+⌧t�⌧ ds ⇢(~x, s)

•  The surface relaxes to the equilibrium by forming initially a two-tail profile of the velocity field that then stabilizes into a double-roll profile •  The velocity field obtained via the local time average technique violates the symmetry of the problem

Rigorous non-equilibrium ensemble averages vs local time averages

!"#$

%&'()*(+$,$-..$

!/#$

%&'()*(+$,$01-.$

!2#$

%&'()*(+$,$1-..$

!3#$

%&'()*(+$,$-..$

!4#$

%&'()*(+$,$01-.$

!5#$

%&'()*(+$,$1-..$

⇥v(⇥x, t) =< ⇥p(⇥x, t) >H0

�(⇥x, t)

~vl(~x, t) =12⌧

R t+⌧t�⌧ ds p(~x, s)

12⌧

R t+⌧t�⌧ ds ⇢(~x, s)

•  The surface relaxes to the equilibrium by forming initially a two-tail profile of the velocity field that then stabilizes into a double-roll profile •  The velocity field obtained via the local time average technique violates the symmetry of the problem

Rigorous non-equilibrium ensemble averages vs local time averages

!"#$

%&'()*(+$,$-..$

!/#$

%&'()*(+$,$01-.$

!2#$

%&'()*(+$,$1-..$

!3#$

%&'()*(+$,$-..$

!4#$

%&'()*(+$,$01-.$

!5#$

%&'()*(+$,$1-..$

⇥v(⇥x, t) =< ⇥p(⇥x, t) >H0

�(⇥x, t)

~vl(~x, t) =12⌧

R t+⌧t�⌧ ds p(~x, s)

12⌧

R t+⌧t�⌧ ds ⇢(~x, s)

Conclusions

①  It is possible to compute, numerically but, otherwise, rigorously, time-dependent non-equilibrium responses, i.e. responses in non-stationary regimes. Local time-averages should be avoided

②  Nonequilibrium atomistic dynamics possibly combined with the ability to compute conditional averages (restrained MD) allows to simulate hydrodynamic phenomena ab initio (without using phenomenological approximations).

③  Coupling non-equilibrium non-stationary systems is a fundamental question of multi-scale approaches

Acknowledgements

Funding Computer resources