Mapping equilibrium and non-equilibrium entropy landscapes : the - - PowerPoint PPT Presentation

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Mapping equilibrium and non-equilibrium entropy landscapes : the path-sampling approach

Manuel Athènes Service de Recherches de Métallurgie Physique - CEA Saclay Gilles Adjanor, EDF, Les Renardières Florent Calvo, Laboratoire de Physique Quantique, Toulouse

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Introduction

Statistical mechanics in ensembles of paths rather than of states Transition-path sampling implicitely computes entropies

« migration » entropies, Sinai-Kolmogorov entropy

Extentions to compute entropies in various contexts Adequate to compute non-equilibrium entropies Efficicient in rugged energy landscapes Built-in diagnosing tools in methods based on non- equilibrium work theorems

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Non-equilibrium and equilibrium entropy

( )

= - ( )ln ( )d ( ) d

eq eq eq eq

S H F H F

ρ χ ρ χ χ ρ χ β χ β χ β β = − = −    

∫ ∫

• N particle system

state with hamiltonian

• définition of ensemble : phase space + associated probabilities
• entropy
• equilibrium :Boltzmann

( )

( )=exp - +

eq

H F

ρ χ β χ    

( )

=-ln exp - d ln = −    

∫

F H Z

β β χ χ

=( ) χ

r,p H( )

r,p

= - ( , )ln ( , )d

∫

neq neq neq t

S t t

ρ χ ρ χ χ

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Nonequilibrium Entropy

• rder parameter q

F(q)

t

χ

nonequilibrium ensemble equilibrium ensemble

χ

( ) ( )

( )

+ cond z,

, = P ( )D

∈ ∫

neq eq t

t z z

χ

ρ χ ρ χ

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Nonequilibrium Entropy

q’- q’/2 q’+ q’/2

• rder parameter q

F(q)

j

χ

nonequilibrium ensemble equilibrium ensemble

i

χ

i

χ

j

χ

• cond

P (z)

eq

( ) ρ χ

+ cond

P (z) ×

neq( , )

t

ρ χ

×

+

• (

) ( )

( ) ( )

( ) ( )

+ cond z,t

• cond

z,t + cond

• cond

, ,

P (z)Dz , = P (z)Dz P (z) P (z) exp

∈ ∈

= = − + −    

∫ ∫

eq neq eq R R

t H F Q

χ χ χ χ

ρ χ ρ χ ρ χ β χ β β

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Nonequilibrium Entropy

q’- q’/2 q’+ q’/2

• rder parameter q

F(q)

j

χ

nonequilibrium ensemble 0 : ensemble à l’équilibre

i

χ ( ) ( )

j

,

, exp = − + −    

neq R

H F Q

χ

ρ χ τ β χ β β

( ) ( ) ( )

,

, ln , ln exp = − = − − + −    

∫

neq neq neq R neq

S d H F Q

χ

ρ χ τ ρ χ τ χ β χ β β ( ) ( ) ( )

1 ,0

, exp = − +    

neq

t z z

χ

ρ χ σ σ

( )

cond

ln

−

  = − + −  

H F Q P

θ β

σ θβ χ

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Space-time thermodynamic perturbation

r

t

t λ = τ

( ) ( )

s r V r F

λ β

= β −β

( ) ( ) ( )

1 ,0

ln ex , p − − +     =

s z t z

χ

σ χ σ

1

( )

,

s t

χ Σ

( )

,

m t

χ

( ) ( )

( )

1 ,

− − Σ

z z

χ θ

δ σ σ

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Space-time thermodynamic integrations

[ ] [ ] [ ] [ ] ( ) ( )

1 1 1 1 1 1 1

d exp D ln exp D ln exp D var ln exp var

= = = = = = = = = = =

− = = − − = − ⋅∂ − =   = ∂ − − − − ∂ + + ∂ ∂  +  =

∫ ∫ ∫ ∫ ∫ ∫ ∫

t neq t t t t t t t t

S Q d z S z d z d

θ τ θ τ θ θ τ θ θ χ θ θ χ τ τ θ θ χ τ θ θ τ τ

θ σ σ σ θ σ β θ θ σ θ θ σ σ σ θ σ

Method 1 Method 2 Method 3

1

(1 ) = + −

θ

σ θσ θ σ

Integration by part Implies second law

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Analogy with equilibrium thermodynamics

( )

1

var − = − − = ∂

∫

t

total total neq t

S Q S S S d

θ τ θ χ

β θ θ σ

( )

( )

1 1

var ∂ − = ∫

eq eq

d S H Sβ

β β β β

β β β

( ) ( )

1

, var = ∂

∫

t

m t d

χ θ θ

χ θ θ σ

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Brownian tube proposal

• G. Stoltz, J. Comp. Phys. 2007

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First and second moment integration

r

t

t λ = τ

( ) ( )

s r V r F

λ β

= β −β

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Three path-sampling methods

[ ] ( )

1 1

var ln exp

= = = =

=

− −∆ + = ∆ + ∆ = = ∫

∫

t t t t

neq t

d S Q d

S

α θ τ τ τ

τ

β θ θ ϕ ϕ α ϕ

( ) ( ) ( )

1

1

• var

= ∫

eq eq

S S d H

β β β

β β β β

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Non-equilibrium entropy

Equilibrium entropy Non-equilibrium entropy

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Perspectives

N-particle system Entropy at glass transition Formalism for non-conservative dissipative systems

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( ) ( )

+ cond 3

1 D P ! =

∫

%

i N

Z zN z h N

χ

1 1

= %

Z Z

( ) ( )

+ cond 3

1 d D P !

Ω

=

∫ ∫

%

i

i i N

Z N z z h N

χ χ

( )

3

1 d ! = =

∫

%

i i N

Z N Z h N

χ χ

[ ]

1

exp − ∆ =

% %

Z F Z

β

Free energy calculations in path ensembles

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( ) ( )

1

D exp 2 D exp 2   − +     =   − −    

∫ ∫

% % % % % %

z U W Z Z z U W

β β

( )

1 1

ln exp ln exp   − = − −     =  

% %

F F W W

β β β

Thermodynamic perturbation

( )

1

exp 1 1 ln exp   −   − = −    

% %

W F F W

θ θ

θ β β θβ

( ) ( )

1

D exp exp 2 D exp 2     − − −       =   − −    

∫ ∫

% % % % % % %

z W U W Z Z z U W

β β β

Work distribution

( )

1

D exp 1 D exp   −   =    

∫ ∫

% % % % % %

z W K Z Z Z Z z W K

θ θ θ θ

θ β θβ

Jarzynski’s approach

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incomplete icosahedron (fivefold symmetry) E=-173,252 ε€ [r.u.] truncated

• ctahedron

(fcc symmetry ) E=-173.928 ε [r.u.]

Q4=0.19 Q4=4·10-2

• rientational order

parameter Q4

The 38-atom cluster « LJ38 »

liquid structures (desordered)

T Tmelt=0.17

(reduced units)

Tss=0.12

→ (Q4) ?

4·10-2 ≤Q4 ≤9·10-2

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Q4-Energy contour plots at decreasing temperatures

Λ(Q4-E)

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Free energy landscape of the LJ38 cluster

Q4=0.12

[u.r.]

[u.r.]

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Comparison with state-sampling methods (F. Calvo)

• Wang-Landau method:

auxiliary potential

• parallel tempering :

Monte-Carlo exchanges between N replica

• f the system at various temperatures

ln (E) ∝

paths parallel tempering Wang-Landau method

[u.r.]

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Comparison with harmonic superposition approximation

• harmonic superposition

approximation in class D →validation of the path-sampling approach

Adjanor, Athènes and Calvo, EPJB (2006) harmonic approximation paths N=2·103 paths N=2·104 parallel tempering Wang-Landau method [u.r.]

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( )

1 1

ln exp ln exp   − = − −     =  

% %

F F W W

β β β

Thermodynamic perturbation

( )

1

exp 1 1 ln exp   −   − = −    

% %

W F F W

θ θ

θ β β θβ

Work distribution

( )

1

D exp 1 D exp   −   =    

∫ ∫

% % % % % %

z W K Q Q Q Q z W K

θ θ θ θ

θ β θβ

Free energy computations: what is the

• ptimal bias?

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1 2 1 1 2

1 e x p 2 1ln 1 ex p 2   −     − =−      

% %

W F F W

β β β

Work distribution

Athènes EPJB (2004), Oberhofer, Geissler and Dellago JPCB(2005), Adjanor and Athènes, JCP (2005) Ytreberg, Zuckerman and Swendsen JCP(2006) Lechner and Dellago J. Stat. Phys. (2007)

Optimal bias distribution

Oberhofer and Dellago (2007)

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Information retrieving from the webs

Ceperley, Kalos, PRB (1977), Frenkel Boulougouris, JTCT (2005) Jourdain, Delmas (2007) Athènes, PRE (2002), EPJB (2007), Wu and Kofke (2005) Statistical error RW AIR M PIR