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

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 1

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 2

Non-equilibrium and equilibrium entropy • N particle system χ =( r,p ) H( r,p ) state with hamiltonian • définition of ensemble : phase space + associated probabilities ∫ • entropy ρ χ ρ χ χ neq neq neq S = - ( , )ln t ( , )d t t ( ) ρ χ  β χ  eq • equilibrium :Boltzmann ( )=exp -  H + F  ( ) ∫ ∫ ρ χ ρ χ χ = ρ χ  β χ − β  χ = β − β eq eq eq eq S = - ( )ln ( )d ( ) H F d H F   ( ) ∫ β  β χ  χ = − F =-ln exp - H d ln Z   3

Nonequilibrium Entropy equilibrium χ ensemble 0 F(q) ( ) ( ) ∈ ∫ ρ χ ρ χ neq eq + , t = P ( )D z z 0 cond ( ) χ z, t χ t nonequilibrium ensemble order parameter q 4

Nonequilibrium Entropy χ i ρ χ eq ( ) 0 0 - × + equilibrium χ ensemble i F(q) + P (z) - P (z) cond cond × ρ χ neq ( , ) t χ j ( ) ∫ ρ χ eq + P (z)Dz 0 0 cond χ ( ) ( ) ∈ χ ρ χ z,t neq , t = j ∫ - P (z)Dz nonequilibrium cond ( ) ∈ χ z,t order parameter q ensemble + P (z) ( ) = ρ χ q’- q’/2 q’+ q’/2 eq cond 0 - P (z) χ cond , R ( ) =  − β χ + β − β  exp  H F Q  χ , R 5

Nonequilibrium Entropy ( ) ( ) ρ χ τ =  − β χ + β − β  neq , exp  H F Q  0 χ , R j 0 : ensemble χ ( ) ( ) ∫ = − ρ χ τ ρ χ τ χ i F(q) à l’équilibre neq neq neq S , ln , d ( ) = − − β χ + β − β   ln exp  H F Q  0 χ , R neq ( )   − σ = θβ χ − + − H F Q ln P   θ β 0 cond χ ( ) ( ) ( ) ρ χ =  − σ + σ  j neq , t exp z z   nonequilibrium 1 0 χ ,0 ensemble q’- q’/2 q’+ q’/2 order parameter q 6

Space-time thermodynamic ( ) ( ) ( ) δ σ − σ − Σ perturbation z z 1 0 χ θ , ( ) ( ) = β −β s r V r F λ β 0 1 ( ) χ s , t ( ) χ m , t t λ = t τ r Σ ( ) ( ) ( ) χ = − − σ + σ   , ln ex p s t  z z  1 0 χ ,0 7

Space-time thermodynamic integrations σ = θσ + − θ σ (1 ) Method 1 θ 1 0 [ ] ∫ − σ exp D z [ ] θ = = − − σ + σ = − 1 neq S ln exp ln [ ] = τ ∫ t 1 0 − σ = τ 0 t exp D z θ = 0 = τ t 1 1 [ ] ∫ ∫ ∫ = − θ ⋅∂ − σ = θ ∂ σ d ln exp D z d Method 2 θ θ θ θ χ = τ t = τ 0 0 t 1 1   ( ) ∫ = θ ∂ σ − θ θ − ∂ σ d var Integration by part   θ θ χ θ θ χ = τ = τ t t 0 0 1 ( ) ∫ = + β + θ θ ∂ σ S Q d var = θ θ Method 3 t 0 = τ t 0 Implies second law 8

Analogy with equilibrium thermodynamics 1 ( ) ( ) ∫ χ = θ θ ∂ σ − = − − β total total neq , var m t d S S S S Q χ θ θ τ 0 t 0 t 0 1 ( ) ∫ = θ θ ∂ σ d var θ θ χ t 0 β = ∫ ( ) 1 ( ) − β β var ∂ β eq eq S β S d H β β 1 0 β 0 9

Brownian tube proposal G. Stoltz, J. Comp. Phys. 2007 10

First and second moment integration ( ) ( ) = β −β s r V r F λ β t λ = t τ r 11

Three path-sampling methods neq S = τ t [ ] = − −∆ ϕ ln exp = τ 0 t 1 = ∫ α ∆ ϕ d α = τ t 0 1 ( ) ∫ = + β + θ θ ∆ ϕ S Q d var = θ 0 t = τ t 0 β = ∫ 1 ( ) ( ) ( ) β β β β eq eq S - S d var H β 1 0 β 0 12

Non-equilibrium entropy Non-equilibrium entropy Equilibrium entropy 13

Perspectives N-particle system Entropy at glass transition Formalism for non-conservative dissipative systems 14

Free energy calculations in path ensembles 1 ( ) ( ) % ∫ = χ + Z D zN P z 0 0 i cond 3 N h N ! 1 ( ) ( ) % ∫ ∫ = χ χ + Z d N D P z z 0 i 0 i cond 3 N h N ! Ω i 1 ( ) % ∫ = χ χ = Z d N Z 0 i 0 i 0 3 N h N ! % % Z [ ] = exp − ∆ β = Z Z 1 F % 1 1 Z 0 15

Jarzynski’s approach Work distribution β  ( )  ∫ % % − + D exp z U W   %   Z 2 = 1 % β   ( ) Z ∫ % % − − 0 D exp z U W     2 β   ( ) % % % ∫  − β  − − D exp exp z W U W     %   Z 2 = 1 % β   ( ) Z % % ∫ − − D exp 0 z U W     2 ( ) % β − = −  − β  F F ln exp W   1 0 0 % =  β  ln exp W   ( ) ∫ %  θ − β  % % 1 D exp z 1 W K   Z Z θ = θ 1 % % Thermodynamic perturbation % ∫  θβ  Z Z D exp z W K   θ θ 0 ( ) %  θ − β  exp 1 W   1 ln − = − θ F F % β 1 0  θβ  exp W   θ 16

The 38-atom cluster « LJ 38 » orientational order parameter Q 4 liquid structures ≤ Q 4 ≤ 9·10 -2 4·10 -2 (desordered) T melt =0.17 (reduced units) incomplete icosahedron Q 4 =4·10 -2 (fivefold symmetry) E=-173,252 ε€ [r.u.] T ss =0.12 truncated octahedron Q 4 =0.19 (fcc symmetry ) E=-173.928 ε [r.u.] → (Q 4 ) ? T 17

Λ (Q 4 -E) Q 4 -Energy contour plots at decreasing temperatures 18

Free energy landscape of the LJ 38 cluster Q 4 =0.12 [u.r.] [u.r.] 19

Comparison with state-sampling methods (F. Calvo) [u.r.] ● Wang-Landau method: ∝ ln (E) auxiliary potential ● parallel tempering : Monte-Carlo exchanges between N replica of the system at various temperatures paths parallel tempering Wang-Landau method 20

Comparison with harmonic superposition approximation harmonic approximation paths N=2·10 3 paths N=2·10 4 parallel tempering Wang-Landau method ● harmonic superposition approximation in class D [u.r.] → validation of the path-sampling approach Adjanor, Athènes and Calvo, EPJB (2006) 21

Work distribution Free energy computations: what is the optimal bias? ( ) %   β − = − − β F F ln exp W   1 0 0 %   = β ln exp W   1 ( ) ∫ %  θ − β  % % D exp z 1 W K   Q Q θ = θ 1 % % Thermodynamic perturbation % ∫  θβ  Q Q D exp z W K   θ θ 0 ( ) %  θ − β  exp 1 W   1 ln − = − θ F F % β 1 0  θβ  exp W   θ 22

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)   1 % − β e x p W     2 1 1ln − =− 2 F F β   1 0 1 % β ex p 2 W     1 2 23

Information retrieving from the webs Statistical error PIR M AIR RW Ceperley, Kalos, PRB (1977), Frenkel Boulougouris, JTCT (2005) Jourdain, Delmas (2007) Athènes, PRE (2002), EPJB (2007), Wu and Kofke (2005) 24