TRUNCATION OF THE MANY BODY HIERARCHY AND RELAXATION TIMES IN THE McKEAN MODEL

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TRUNCATION OF THE MANY BODY HIERARCHY AND RELAXATION TIMES IN THE McKEAN

MODEL

K.-J. Schmitt

To cite this version:

K.-J. Schmitt. TRUNCATION OF THE MANY BODY HIERARCHY AND RELAXATION TIMES IN THE McKEAN MODEL. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-241-C2-244.

�10.1051/jphyscol:1987234�. �jpa-00226501�

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TRUNCATION OF THE MANY BODY HIERARCHY AND RELAXATION TIMES IN THE McKEAN MODEL

K.-J. SCHMITT

Institut fur Theoretische Physik der Universitat Erlangen- Nurnberg, 0-8520 Erlangen, F.R.G.

ABSTRACT

En l e m o d s l e d e McKean l a h i k r a r c h i e d e BBGKY est k q u i v a l e n t 2 u n e h i k r a r c h i e s i m p l e d ' b q u a t i o n s c o u p l h s p o u r l e s f o n c t i o n s d e c o r r e l a t i o n d e p - p a r t i c u l e s . L e s k f f e t s d e t r o n c a t i o n e t l a c o n v e r g e n c e d e l a f o n c t i o n d e d i s t r i b u t i o n d e 4 - p a r t i c u l e e n v e r s s o n s o l u t i o n e x a c t e o n t k t k k t u d i h . Dans l a l i m i t e d e temps i n -

f i n i on p e u t t r o u v e r l a s o l u t i o n a n a l y t i q u e . On p e u t m o n t r e r q u e l a d i s s o l u t i o n d e l a f o n c t i o n d e c o r r e l a t i o n d e p - p a r t i c u l e s non e n k q u i l i b r e e s t p - f o i s p l u s v i t e q u e c e l l e d e l a f o n c t i o n d e d i s t r i b u t i o n d e A - p a r t i c u l e .

I n t h e McKean model t h e BBGKY-hierarchy i s e q u i v a l e n t t o a s i m p l e h i e r a r c h y o f c o u p l e d e q u a t i o n s f o r t h e p - p a r t i c l e c o r r e l a t i o n f u n c t i o n s . T r u n c a t i o n e f f e c t s and t h e c o n v e r g e n c e o f t h e o n e - p a r t i c l e d i s t r i b u t i o n t o w a r d s i t s e x a c t s h a p e h a v e b e e n s t u d i e d . I n t h e l o n g t i m e l i m i t t h e e q u a t i o n s c a n b e s o l v e d i n a c l o s e d form. I t t u r n s o u t t h a t t h e p - p a r t i c l e c o r r e l a t i o n d e c a y s p - t i m e s f a s t e r t h a n t h e n o n - e q u i - l i b r i u m o n e - p a r t i c l e d i s t r i b u t i o n .

1. INTRODUCTION

The e q u a t i o n s o f t h e BBGKY-hierarchy which g i v e a c o m p l e t e d i s c r i p t i o n o f a many- p a r t i c l e s y s t e m a r e h i g h l y n o n l i n e a r i n t e g r o - d i f f e r e n t i a l - e q u a t i o n s which c a n b e s o l v e d o n l y w i t h t h e h e l p o f s t r o n g a s s u m p t i o n s . I n most c a s e s , t h e s e a s s u m p t i o n s c a n b e t e s t e d o n l y e m p i r i c a l l y . I t i s t h e r e f o r e i m p o r t a n t t o s t u d y t h e s e a s s u m p t i o n s i n s i m p l e m o d e l s .

A method o f t e n u s e d t o s o l v e t h e BBGKY-hierarchy a p p r o x i m a t e l y is t o t r u n c a t e t h e h i e r a r c h y a t t h e l o w e s t l e v e l , i . e . t o n e g l e c t t h e t w o - p a r t i c l e c o r r e l a t i o n s i n t h e f i r s t h i e r a r c h y e q u a t i o n . T h i s may be a good a p p r o x i m a t i o n i n d i l u t e d s y s t e m s o r i n s y s t e m s n e a r t h e e q u i l i b r i u m b e c a u s e we m i g h t a s s u m e t h a t c o r r e l a t i o n s d e c a y f a s t e r t h a n t h e n o n e q u i l i b r i u m o n e - p a r t i c l e d i s t r i b u t i o n .

I f we s t u d y s y s t e m s f a r from t h e e q u i l i b r i u m we s h o u l d a t l e a s t e x p e c t t h a t t h e t i m e e v o l u t i o n o f t h e o n e - p a r t i c l e d i s t r i b u t i o n c o n v e r g e s t o w a r d s i t s e x a c t s o l u t i o n i f h i g h e r h i e r a r c h y e q u a t i o n s a r e t a k e n i n a c c o u n t . T h e s e q u e s t i o n s a r e s t u d i e d i n t h e f o l l o w i n g u s i n g t h e McKean model l ) a s a s i m p l e n o n t r i v i a l model o f a many p a r t i c l e s y s t e m .

T h i s model is b r i e f l y d e s c r i b e d i n s e c t i o n 2 a n d t h e h i e r a r c h y o f t h e c o r r e l a t i o n f u n c t i o n s i s o b t a i n e d . I n s e c t i o n 3 t h e c o n v e r g e n c e o f t h e t i m e e v o l u t i o n o f t h e o n e p a r t i c l e d i s t r i b u t i o n t o w a r d s i t s e x a c t s h a p e i s i n v e s t i g a t e d . Near t h e e q u i - l i b r i u m t h e h i e r a r c h y e q u a t i o n s c a n b e l i n e a r i z e d a s done i n s e c t i o n 4 , s h o w i n g t h e d e c a y o f t h e c o r r e l a t i o n s . I t t u r n s o u t t h a t t h e p - p a r t i c l e c o r r e l a t i o n s d e c a y p - t i m e s f a s t e r t h a n t h e n o n - e q u i l i b r i u m o n e - p a r t i c l e d i s t r i b u t i o n . T h i s c o n f i r m s

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987234

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C2-242 JOURNAL DE PHYSIQUE

t h e a s s u m p t i o n t h a t h i g h e r c o r r e l a t i o n s become i n c r e a s i n g l y u n i m p o r t a n t when a s y s t e m a p p r o a c h e s e q u i l i b r i u m .

2. THE HIERARCHY-EOUATIONS OF THE MCKEAN MODEL

The McKean model d e s c r i b e s a s y s t e m o f p a r t i c l e s which h a v e o n l y two d e g r e e s o f f r e e d o m . They c a n move w i t h c o n s t a n t v e l o c i t i e s e = '1. The p r o b a b i l i t y o f t h e i n t e r a c t i o n b e t w e e n two p a r t i c l e s i n t h e t i m e i n t e r v a l 1 d t is

I f t h e i n t e r a c t i n g p a r t i c l e s i a n d j h a v e t h e v e l o c i t i e s e i and e . i n i t i a l l y , t h e y

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w i l l h a v e v e l o c i t i e s e* and eX a f t e r t h e i n t e r a c t i o n w i t h e q u a l p r o b a b i l i t y -

J 2

o r more e x p l i c i t l y :

The L i o u v i l l e e q u a t i o n g i v e s u s t h e t i m e e v o l u t i o n o f t h e N - p a r t i c l e d i s t r i b u t i o n which is t h e p r o b a b l l l i t y o f t h e N - p a r t i c l e s h a v i n g v e l o c i t i e s e l ,

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and o b t a i n t h e BBGKY-hierarchy o f t h e McKean model i n t h e limit N ⁺m

T h i s h i e r a r c h y o f t h e r e d u c e d d i s t r i b u t i o n f u n c t i o n s is t r a n s f o r m e d i n t o a h i e r a r c h y o f t h e p - p a r t i c l e c o r r e l a t i o n s 2, which a r e t h e i r r e d u c i b l e p a r t s o f t h e r e d u c e d p - p a r t i c l e d i s t r i b u t i o n s

Where g d e n o t e s t h e p - p a r t i c l e c o r r e l a t i o n w h e r e a l l v e l o c i t i e s e . ( l l i S p ) a r e +1 and f + P t ) s t a n d s f o r f ( e = + l ; t ) . 1 1

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A s shown by H. 5pohn3) e x a c t s o l u t i o n s o f t h e r e d u c e d d i s t r i b u t i o n f u n c t i o n s o f a n i n f i n i t e many p a r t i c l e s y s t e m a r e g i v e n a s t h e moments o f t h e s o l u t i o n o f t h e Boltzmann e q u a t i o n w i t h r e s p e c t t o a s p e c i a l m e a s u r e . The m e a s u r e d e t e r m i n e s t h e i n i t i a l v a l u e s o f t h e v a r i o u s r e d u c e d d i s t r i b u t i o n f u n c t i o n s . Now we c a n t r u n c a t e t h e h i e r a r c h y ( 2 . 4 ) a t v a r i o u s l e v e l s and compare t h e t i m e e v o l u t i o n o f t h e one- p a r t i c l e d i s t r i b u t i o n w i t h t h e e x a c t s o l u t i o n .

We o b t a i n t h e Boltzmann e q u a t i o n o f t h e McKean model by n e g l e c t i n g g 2 i n t h e f i r s t h i e r a r c h y e q u a t i o n ( 2 . 3 )

T h i s e q u a t i o n h a s two f i x e d p o i n t s , o n e o f them i s r e p u l s i v e ( f + = l ) t h e o t h e r i s a t t r a c t i v e ( f + = 1/21.

I f w e c l o s e t h e h i e r a r c h y a t t h e n e x t l e v e l , i . e . n e g l e c t i n g g3 i n t h e s e c o n d h i e r a r c h y e q u a t i o n , we o b t a i n

I n t h i s s y s t e m t h e o l d f i x e d p o i n t s a r e p r e s e r v e d i f g2=0, b u t a n a d d i t i o n a l s a d d l e p o i n t a p p e a r s f o r t h e v a l u e s f = 3 / 4 a n d g - 1 / 1 6 . The b e h a v i o u r o f t h e t r a j e c t o r i e s i n t h e f

+

- g 2 - p l a n e i s sh6wn i n ~ i ~ . ' ; .

Fig.1: The solid c u r v e s labelled by the initial values (f (t=O), g2(t=0)) a r e the trajectories o f the system. The dashed c u r v e sgparates the physical region below from the unphysical region above. Outside the area bbunded by the dotted curve the reduced two particle distribbtion function becomes negative for s o m e values o f i t s arguments. T h e dashed-dotted curve peaked at f ~ 0 . 5 bounds the physical initial values o f the exact t w o particle dist:ibution.

Now we e x a m i n e t h e c o n v e r g e n c e o f t h e t r u n c a t i o n method t o w a r d s t h e e x a c t s o l u t i o n . The u n p h y s i c a l s a d d l e p o i n t v a n i s h e s i f h i g h e r h i e r a r c h y e q u a t i o n s a r e t a k e n i n a c c o u n t . But d o e s t h e s a d d l e p o i n t h a v e a n i n f l u e n c e on t h e c o n v e r g e n c e o f t h e t r u n c a t i o n method? T h i s q u e s t i o n is i n v e s t i g a t e d i n F i g . 2 .

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C2-244 JOURNAL DE PHYSIQUE

I . . .

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^, ^,

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Fig.2: Time evolution of f with initial values lying on the right and on the left hand side of the peQk of the dashed-dotted curve of Fig.1. The drawn curves are labelled according to the number of hierarchy equations taken in account The dashed curves describe the exact solution corresponding to the initial values of the drawn curves.

The i n i t i a l v a l u e s o f f + a n d g 2 , 1 i e on t h e r i g h t a n d on t h e l e f t b r a n c h o f t h e d a s h e d - d o t t e d c u r v e o f F i g . 1 , 1 . e . n e a r a n d f a r from t h e s a d d l e p o i n t i n t h e a p p r o x i m a t i o n w h e r e g3 is n e g l e c t e d i n t h e s e c o n d h i e r a r c h y e q u a t i o n . We s e e t h a t i h e c o n v e r g e n c e i s much b e t t e r f o r i n i t i a l v a l u e s l y i n g f a r from t h e s a d d l e p o i n t t h a n f o r t h o s e l y i n g n e a r t h e s a d d l e p o i n t . Thus t h e s a d d l e p o i n t h a s a n i n f l u e n c e on t h e c o n v e r g e n c e o f t h e t r u n c a t i o n method e v e n i n a p p r o x i m a t i o n s i n which i t d o e s n o t e x i s t anymore.

4. DECAY OF THE P-PARTICLE CORRELATION

To i n v e s t i g a t e t h e d e c a y o f t h e c o r r e l a t i o n s n e a r t h e e q u i l i b r i u m v a l u e , we l i n e a r - i z e t h e h i e r a r c h y ( 2 . 4 ) a b o u t t h e a t t r a c t i v e f i x e d p o i n t a n d t r u n c a t e t h e h i e r a r c h y a t t h e p - t h l e v e l . I n t h i s way we r e d u c e t h e s y s t e m o f c o u p l e d n o n l i n e a r d i f f e r e n - t i a l e q u a t i o n s t o a l i n e a r e i g e n v a l u e p r o b l e m which c a n b e s o l v e d e x a c t l y . I t c a n be shown t h a t t h e s o l u t i o n o f t h e l i n e a r i z e d s y s t e m h a s t h e form:

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< < l ( 1 S i S p ) .

The c o n s t a n t s C . a r e d e t e r m i n e d by t h e i n i t i a l v a l u e s . The v e c t o r X . i s t h e e i g e n - v e c t o r b e l o n g i n ; t o t h e i - t h e i g e n v a l u e . E v e r y c o r r e l a t i o n d e c a y s ?Aster t o z e r o t h a n t h e o n e - p a r t i c l e distribution a p p r o a c h e s t h e e q u i l i b r i u m v a l u e . The r e l a x a t i o n t i m e s o f t h e c o r r e l a t i o n s a r e e q u a l t o t h e i n v e r s e o f t h e l o w e s t e i g e n v a l u e

o c c u r i n g i n t h e p a r t i c u l a r e q u a t i o n s o f ( 4 . 1 ) :

REFERENCES

1) McKean, J r . , H.P., J. of Comb. Theory 2 (1967) 358 2 ) S c h m i t t , K.J., J. S t a t . P h y s . , t o b e p u b l i s h e d

3 ) Spohn, H . F l u c t u a t i o n t h e o r y f o r t h e Boltzmann e q u a t i o n , i n : N o n e q u i l i b r i u m Phenomena I , e d s . J . L . L e b o w i t z a n d E.W. M o n t r o l l , N o r t h H o l l a n d , 1 9 8 3