waste-recycling monte carlo and the calculation of free ...lelievre/cecam/m_athenes.pdf ·...
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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
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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
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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
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Parallel replica simulations
Exchanges between replicas monoproposal multiproposal
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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
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LJ fluid
12 6
4ijE r r r
Nr ij
i j
U E r
Colluza, Frenkel
PhysChemPhys 2005
Athenes, Calvo
PhysChemPhys 2008
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Ferromagnetic
system
ij ij ijE E E
lnF M kT p M
i ii M Mp M h
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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
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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)
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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
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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
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Assessement of optimal estimator
in a BCC Ising-like binary system
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Statistical variances and chemical potentials
A
G. Adjanor, M. Athènes, J. Rodgers, J. Chem. Phys. 2011
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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
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Calculations in FeCr
432 atoms
G. Adjanor, M. Athènes, J. Rodgers, J. Chem. Phys. 2011
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Equilibrium phase diagram of FeCr
G. Adjanor, et al.
J. Chem. Phys. 2011
432 atoms:
Strong finite size
effects
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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
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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
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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
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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)
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hkTF ln
Comparison with classical harmonic
approximation
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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
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Autonomous steering with two additional coordinates
add
add
rEr
rQr
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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)
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Free energy contour plot 44 ,ln, QEpkTQEFT
m
n mn
n mnmnQE
W
Wrh
MQEp
exp
exp1, 4,
4
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Free energy landscape
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Comparative study
PT : Parallel tempering
PS : Path sampling
WL : Wang Landau
IR : Mutiple State Estimator
Reformulation of conditional expectations
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L
sel zPzOO1
LdPzPzO
dzdzPzPzO
dzzPzOO
1 margsel
cond
SP expmarg
zszP exp
Waste-recycling & multi-state Bennett
acceptance ratio method
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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
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z
WR & MBAR
MB
AR
Fre
e e
nerg
ies
Uncert
ain
ties
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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?