Langevin equation equation for for a a system system Langevin - PowerPoint PPT Presentation

Langevin equation equation for for a a system system Langevin nonlinearly coupled coupled to a to a heat heat bath bath nonlinearly Mykhaylo Evstigneev Evstigneev Mykhaylo BEMOD 12 BEMOD 12 MPIPKS, Dresden MPIPKS, Dresden March 27, 2012 March 27, 2012

Motivation Motivation • Molecular dynamics: • Stochastic dynamics: How ? 10 N atoms of the heat bath, atoms of the heat bath N < 10 not simulated explicitly n atoms of the system, Langevin equations of motion: ~ n ~ 1…10      p f x x x x t   ( ) ( ) ( , )      t Equations of motion: Forces:   p f x F x X  ( ) ( , ) system:    • renormalized    P g X G x X bath: ( ) ( , )    • dissipative • random

Langevin simulation simulation Langevin • Langevin equation ~      p f x x x t x t   ( ) ( ) ( ) ( , )      t t t • Noise   x t ( , ) 0 – Unbiased:  t       x t x s k T x t s ( , ) ( , ) 2 ( ) ( )    – Gaussian and white: t s B t    n n v v [ 1 / 2 ] [ 1 / 2 ] ~         n n n n m f x x v r [ ] [ 1 ] [ 1 / 2 ] [ ] ( ) ( ) • Implementation:       t      n n n x [ 1 ] x [ ] v [ 1 / 2 ] t    [ n r ] with Gaussian random numbers having the statistical properties:        n n k n r r r k T x t [ ] [ ] [ ] [ ] 0 ; 2 ( ) /    B nk

Plan of the derivation Plan of the derivation • Step 1. From the heat bath equations of motion    P g X G x X ( ) ( , )    approximately evaluate    X t X x t t u t ( ) ([ ( ' )]) ( )    (systematic part + noise) • Step 2. Plug the result back into the system’s equations of motion:        p  f x F x X f x F x X x u t ( ) ( , ) ( ) , ([ ]) ( )         F x X x t u t , ([ ], ) ( ) • Step 3. Linearize to single out force renormalization,  dissipation, and noise effects • Step 4. Take the limit of zero noise correlation time

Standard recipe Standard recipe • Initial microscopic equations:      p  f x F x X P g X G x X ( ) ( , ) ; ( ) ( , )       ~      p f x x x t x t   ( ) ( ) ( ) ( , ) • Langevin equation:      t t Bogoliubov, 1945; Magalinskii, 1959; Zwanzig, 1973:      F x X ~ ,     f x f x F x X u x 0 – Renormalized force: ( ) ( ) , ( )      0 X  u u    u x G x X 0 where ( ) ( , )   0 k T B – Thermal noise:     F x X ,       x t u t u u t t ds u u s 0 ( , ) ( ) , ( 0 ) ( ) 2 ( ) ( 0 ) ( )        X 0 0  0 – Dissipation matrix: u u s    F x X ( 0 ) ( ) F x X ( , ) ( , )     s    t x 0 ds 0 0 ( )    X X k T   B 0

New recipe New recipe • Initial microscopic equations:      p  f x F x X P g X G x X ( ) ( , ) ; ( ) ( , )       ~      p f x x x t x t   ( ) ( ) ( ) ( , ) • Langevin equation:      t t M.E. and P.Reimann, Phys. Rev. B 82 , 224303 (2010):   ~    f x f x F x X u x ( ) ( ) , ( ) – Renormalized force:    0 u u    u x G x X 0 ( ) ( , ) where   0 k T B – Thermal noise:      F x X u x t , ( , )       x t u t u u t t ds u u s 0 ( , ) ( ) , ( 0 ) ( ) 2 ( ) ( 0 ) ( )        X 0 0  0 – Dissipation matrix: u u s      F x X u x ( 0 ) ( ) F x X u x ( , ( )) ( , ( ))     s    t x 0 ds 0 0 ( )    X X k T   B 0

Numerical test #1 Numerical test #1 • Model:          m x W x X W   (a) 10 ( ) ( )        W (b) 20 ( )      W x x 12 x 6 ( ) ( / ) 2 ( / )              2.0 (a) M X X W x X X t 2.0 ( ) ( ) 1.8 1.6 1.8 1.4 1.02 1.2 1.0  1.6  k T k T 50 100 150 1.00     x /  X u B dt u u t B 2 0 ; ; ( 0 ) ( ) 1.4   0 2 0 0.98 0 980 990 1000 0 1.2 ~      1.0 m x f x x x x t • Langevin equation:    ( ) ( ) ( , ) 0.8 0 200 400 600 800 1000   t [ps]    2   W x u x ( ) ~          x   f x W x u x ( ) ( ) ( )    2.4 2.4 (b) 2.2    u x W x 2.0 ( ) ( ) / 2.2 1.8 1.6 1.4 2.0 1.2 1.10 1.0 1.08 1.8 50 100 150 200 1.06     pN nm nm 1.04 x /  4 ; 0 . 5 • Parameters: 1.02 1.6 1.00 0.98 0.96   m yg M yg 1.4 0.94 200 ; 100 0.92 980 990 1000 1.2    X X ( 0 ) 0 ; ( 0 ) 0 1.0 0.8     x x m s  0 200 400 600 800 1000 ( 0 ) 4 ; ( 0 ) 10 / t [ps]  T 0

Numerical test #2 Numerical test #2 • Equations of motion:      m r   W r R ( ) i i              M R R R W r R R Ξ R t 0 ( ) ( ) ( , ) i i i i i i i ~ • Langevin equation:      m r f r r r r t    ( ) ( ) ( , )          m M      200 yg ; 100 yg W r r 12 r 6 ( ) ( / ) 2 ( / ) • Parameters:     k T γ κ M 5 pN nm ; 2 pN nm / 10 B   σ a .     2 0 4 nm 72 / 0 1.6 1.4 3.5 2 /s] t desorption [  s] 1.2 -3 x D [mm 1.0 3.0 0.8 0.6 10 0.4 2.5 0.2 0.0 0 100 200 300 400 500 0 100 200 300 400 500  /  0  /  0

Conclusions Conclusions • Langevin equation can save you a great deal of computational effort • Langevin equation is an approximation valid for – weak system-bath coupling – large time-scale separation between the (slow) system and (fast) bath degrees of freedom • The new recipe for deriving Langevin equation improves its accuracy and increases its validity range by about an order of magnitude with respect to the system-bath coupling strength • More details in M.E. and P.Reimann, Phys. Rev. B 82 , 224303 (2010) • Acknowledgements Deutsche Forschungsgemeinschaft (SFB 613) and ESF programs NATRIBO and FANAS (project Nanoparma)