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

Langevin Langevin equation equation for for a a system system nonlinearly nonlinearly coupled coupled to a to a heat heat bath bath

Mykhaylo Mykhaylo Evstigneev Evstigneev BEMOD 12 BEMOD 12 MPIPKS, Dresden MPIPKS, Dresden March 27, 2012 March 27, 2012

Motivation Motivation

• Molecular dynamics:

) , ( ) ( ) , ( ) ( X x X X x x

     

G g P F f p       ) , ( ) ( ) ( ~ t x f p

t

x x x

    

      

• Stochastic dynamics:

10N atoms of the heat bath, N < 10 n atoms of the system, n ~ 1…10 atoms of the heat bath not simulated explicitly Equations of motion: system: bath: Langevin equations of motion: Forces:

• renormalized
• dissipative
• random

How ?

Langevin Langevin simulation simulation

• Langevin equation
• Noise

– Unbiased: – Gaussian and white:

• Implementation:

with Gaussian random numbers having the statistical properties:

) , ( ) ( ) ( ) ( ~ t t x f p

t t t

x x x

    

       ) ( ) ( 2 ) , ( ) , ( s t T k s t

t B s t

     

  

x x x ) , (  t

t

x





] [ ] 2 / 1 [ ] 1 [ ] [ ] 2 / 1 [ ] 2 / 1 [

) ( ) ( ~

n n n n n n

r v f t v v m

      

     

   

x x t v x x

n n n

  

  ] 2 / 1 [ ] [ ] 1 [   

t T k r r r

nk n B k n n

   / ) ( 2 ;

] [ ] [ ] [ ] [

 

  

x

] [n

r



Plan of the derivation Plan of the derivation

• Step 1. From the heat bath equations of motion

approximately evaluate (systematic part + noise)

• Step 2. Plug the result back into the system’s equations of motion:
• Step 3. Linearize

to single out force renormalization, dissipation, and noise effects

• Step 4. Take the limit of zero noise correlation time

) , ( ) ( X x X

  

G g P    ) ( )]) ' ( ([ ) ( t u t t X t X

  

   x

 

) ( ]) ([ , ) ( ) , ( ) ( t F f F f p u x X x x X x x     

    



 

) ( ) ], ([ , t t F u x X x 



Standard recipe Standard recipe

• Initial microscopic equations:
• Langevin equation:

Bogoliubov, 1945; Magalinskii, 1959; Zwanzig, 1973: – Renormalized force: where – Thermal noise: – Dissipation matrix:

) , ( ) ( ) ( ) ( ~ t t x f p

t t

x x x

    

      

   

) ( , , ) ( ) ( ~ x X x X x x x

     

u X F F f f     





     ) ( ) ( ) , ( ) , ( ) ( T k s u u ds X F X F

B s t       

 X x X x x

 





    ) ( ) ( ) ( 2 ) ( ) ( , ) ( , ) , ( s u u ds t t u u t u X F t

       

  X x x ) , ( ) ( ; ) , ( ) ( X x X X x x

     

G g P F f p       ) , ( ) ( X x x

   

G T k u u u

B



New recipe New recipe

• Initial microscopic equations:
• Langevin equation:

M.E. and P.Reimann, Phys. Rev. B 82, 224303 (2010): – Renormalized force: where – Thermal noise: – Dissipation matrix:

) , ( ) ( ) ( ) ( ~ t t x f p

t t

x x x

    

      

 

) ( , ) ( ) ( ~ x u X x x x   

  

F f f





       ) ( ) ( )) ( , ( )) ( , ( ) ( T k s u u ds X F X F

B s t       

 x u X x x u X x x

 





     ) ( ) ( ) ( 2 ) ( ) ( , ) ( ) , ( , ) , ( s u u ds t t u u t u X t F t

       

  x u X x x ) , ( ) ( ; ) , ( ) ( X x X X x x

     

G g P F f p       ) , ( ) ( X x x

   

G T k u u u

B



Numerical test #1 Numerical test #1

• Model:
• Langevin equation:
• Parameters:

200 400 600 800 1000 0.8 1.0 1.2 1.4 1.6 1.8 2.0

50 100 150 1.0 1.2 1.4 1.6 1.8 2.0 980 990 1000 0.98 1.00 1.02

(a)

x / t [ps]

) ( ) ( t X X x W X X M              ) ( X x W x m      

 

) ( ) ( ~ x u x W x f    

 

2

) ( ) (              x u x W x ) , ( ) ( ) ( ~ t x x x x f x m         / 10 ) ( ; 4 ) ( ) ( ; ) ( 100 ; 200 5 . ; 4           T s m x x X X yg M yg m nm nm pN     

200 400 600 800 1000 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

50 100 150 200 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 980 990 1000 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10

(b)

x / t [ps]

(a) (b)

) ( 20 ) ( 10     W W        / ) ( ) ( x W x u  

 

6 12

) / ( 2 ) / ( ) ( x x x W     

2 2

) ( ) ( ; ;    T k t u u dt T k u X

B B

  





Numerical test #2 Numerical test #2

• Equations of motion:
• Langevin equation:
• Parameters:

100 200 300 400 500 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

tdesorption [s]

 /0



   

i i

W m ) ( R r r   ) , ( ) ( ) ( t W M

i i i i i i i

R Ξ R R r R R R             

 

nm 4 nm pN 2 ; nm pN 5 ) / ( 2 ) / ( ) (

6 12

. a σ T k r r W

B

          r

2

/ 72 10 / yg 100 ; yg 200        κM γ M m ) , ( ) ( ) ( ~ t r f r m r r r

    

       

100 200 300 400 500 2.5 3.0 3.5

10

• 3 x D [mm

2/s]

 /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)