1

Manuel Athènes, C.-M. Marinica

Service de Recherches de Metallurgie Physique, CEA Saclay

Gilles Adjanor, EDF-R&D les Renardières

Florent Calvo, Université de Lyon

Waste-Recycling Monte Carlo and

the calculation of free energies

Waste-recycling Monte Carlo

2

Goal : reducing the statistical variance of the estimator in

Markov Chain Monte Carlo techniques based on the

Metropolis algorithm

How? by including information within the estimator about

the states that have been sampled but rejected.

Ceperley, Chester and Kalos, Phys. Rev.1977,

Frenkel, PNAS 2004,

Delmas &Jourdain, J. Applied Probab. 2009.

Outline

3

I-Conditional expectations

Speeding-up of parallel tempering with configuration bias (col.

F. Calvo)

Control variate problem & optimal estimator (Delmas&Jourdain)

Applications : Ising systems and realistic FeCr system (col. G. Adjanor)

Free energy reconstruction from steered molecular dynamics

Vacancy in Iron, Structural transitions in LJ38 cluster (col. C. Marinica)

II-Posterior conditional expectations

Combination of waste-recycling and multistate Bennett acceptance ratio

method

4

Parallel replica simulations

Exchanges between replicas monoproposal multiproposal

1

klI

R R

kl

klki

R

R RR

min 1,

klklacc

RR R

R

min 1,

klmkl

kl

acc

RR R

R

Esselink, Loyens,

Smit, PRE 1995

5

LJ fluid

12 6

4ijE r r r

Nr ij

i j

U E r

Colluza, Frenkel

PhysChemPhys 2005

Athenes, Calvo

PhysChemPhys 2008

6

Ferromagnetic

system

ij ij ijE E E

lnF M kT p M

i ii M Mp M h

7

Maxwell construction G(c)

(c)c

2

1

c

c

2 1 2 1

(c) * dc 0

G(c ) G(c ) c c *

c

conf

Cr

c Cr Fe

A(c, *) kT ln h (conf )

Nh (conf ) c

N N

c

c

c

c

G(c)

A(c, *)

2c1c

1c 2c

A

A

8

Estimation of chemical potential differences

Gradual transmutation of a Fe atom into Cr atom

Reference system 0: nFe - (m-n)Cr

Target system 1: (n-1)Fe - (m-n+1)Cr

0λ 1λ

ex

0 11 0

1exp(- ) = exp(- G) = exp W

exp W

forward MD trajectory

backward MD trajectory

nFe,(m-n)Cr (n-1)Fe,(m-n+1)CrH = (1- ) H + H

C. Jarzynski PRL (1997), G. E. Crooks, J. Stat. Phys. (1998)

9

Exploration of alloy configurations: path sampling

ex 1/2

1/2

exp W2

exp(- )

exp W2

backward trial

transmutation

Monte Carlo test on

trial transmutations

→

0λ

1λ

0λ

forward trial

transmutation

exp W2

Acceptance rate

Measurement of chemical

potential difference

G. Adjanor, M. Athènes, J. Rodgers, J. Chem. Phys. 2011

10

Metropolis estimator

Waste-recycling estimator (Frenkel, PNAS 2004)

Optimal estimator (Delmas & Jourdain, J. Appl. Probab. 2009)

n nf exp W2

n nf exp W2

N0N n

n 1

1J (f ) f f

N

N

WR acc accN n n n n

n 1

1J (f ) f 1 p f p

N

b* 0 1N N NJ (f ) 1 b* J (f ) b*J (f ) n n 1b* 1 corr f ,f

11

Assessement of optimal estimator

in a BCC Ising-like binary system

12

Statistical variances and chemical potentials

A

G. Adjanor, M. Athènes, J. Rodgers, J. Chem. Phys. 2011

13

Empirical potential used:

two-band model (2BM) (P. Olsson et al.Phys. Rev. B 2005)

EAM potential reproducing α and α’ phases

Fe Fe Cr Fe Fe Cr

(Olsson et al.2005, Phys. Rev. B

frustration

fig :G. Bonny et al., J. Nucl. Mat. 2008)

CDM potential : A. Caro et al

14

Calculations in FeCr

432 atoms

G. Adjanor, M. Athènes, J. Rodgers, J. Chem. Phys. 2011

15

Equilibrium phase diagram of FeCr

G. Adjanor, et al.

J. Chem. Phys. 2011

432 atoms:

Strong finite size

effects

16

Interfacial free energy

51at.% Cr

1456 atoms

T=300K

1456 & 2000 atoms

CDM potential

B. Sadigh & P. Erhart,

cond-mat.mtrl-sci 2011

Waste-recycling & steered molecular dynamics

17

nncondnselnncondnsel xPxzPzxPxPxzPzxP |||| '''

nx

'nxShooting procedure with

nonequilibrium paths

N

nnsel

W

WzxP

exp

exp|

Crooks work theorem

Summary of Monte Carlo algorithm

1. Run the following sampler

1. Proposed states in generated nonequilibrium path

2. Select new state using posterior conditional

probability

2. Evaluate average with estimator

18

M

mN

n

mn

mn

N

n

mn

W

WrA

MA

1

0

|

|

0

|

exp

exp1

Vacancy migration

in Fe Mendelev potential

Fe structure (bcc)

Single additional

steering variable

Harmonic spring on

nearest vacancy

neighbor

19

hD

hP

ln

N

n

n

N

n

nn

W

Wh

h

0

0

exp

exp

Vacancy migration in Fe Mendelev potential

Fe structure (bcc)

20

hkTF ln

Comparison with classical harmonic

approximation

21

22

incomplete

icosahedron (fivefold symmetry)

E=-173,252 [r.u.]

truncated

octahedron

(fcc symmetry )

E=-173.928 [r.u.]

Q4=0.19

Q4=4·10-2

orientational order

parameter Q4

Intermezzo: the 38-atom cluster « LJ38 »

liquid structures

(desordered)

T

Tmelt=0.17 (reduced units)

Tss=0.12

→ Λ(Q4) ?

4·10-2 ≤ Q4 ≤ 9·10-2

23

Autonomous steering with two additional coordinates

add

add

rEr

rQr

22

141

kTm

bE

mj

j

j

j

add

jjadd

jj

jadd

j

2

1

10

0

j

j

j

j

add

j dtrEzW addj

0

,1

Non-autonomous steering

Autonomous steering out of equilibrium

Equilibrium case

TAMD,( Eric Vanden-Eijden)

24

Free energy contour plot 44 ,ln, QEpkTQEFT

m

n mn

n mnmnQE

W

Wrh

MQEp

exp

exp1, 4,

4

25

Free energy landscape

26

Comparative study

PT : Parallel tempering

PS : Path sampling

WL : Wang Landau

IR : Mutiple State Estimator

Reformulation of conditional expectations

27

L

sel zPzOO1

LdPzPzO

dzdzPzPzO

dzzPzOO

1 margsel

cond

SP expmarg

zszP exp

Waste-recycling & multi-state Bennett

acceptance ratio method

28

M

mK

k kk

m

zsf

zsfOOB

11 )()(

ˆexp

ˆexp

M

mK

k kk

m

WR

Sf

SfOOB

11 )()(

ˆexp

ˆexp

zPzOO sel

L

1

Transition path sampling simulations

Bias depends on eigenvalues of

jacobian matrix

Shifting procedure

29

z

WR & MBAR

MB

AR

Fre

e e

nerg

ies

Uncert

ain

ties

30

Summary

Two points of view to see waste-recycling estimators

1. Information of the unselected states is retrieved using a

conditional expectation.

2. Information is infered from the posterior likelihood of states

(paths) in the set of generated states (paths).

First approach is more general

• can be used in combination with existing methods

• rigorous mathematical analysis (Delmas and Jourdain)

Second approach can be used in combination with post-processing

tools (MBAR). Open question: is variance reduction guaranteed?