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Mathematical Problems of Computer Science 29, 2007, 36{50.

N ovel M athematical Appr oach for Random 3D Spin

System Under the I n°uence of E xter nal Field.

Gener alization of Clausius-M ossotti E quation

A s h o t S . Ge vo r kya n a n d A r a x A . Ge vo r kya n

Institue for Informatics and Automation Problems of NAS of RA

e-mail: g ashot@sci.am

Abstract

A dielectric medium consisting of roughly polarized molecules has been treated
as a 3D disordered spin system. For investigation of statistical properties of this
system on scales of space-time periods of standing electromagnetic wave a microscopic
approach has been developed. Using the Birgo® ergodic hypothesis the initial 3D
spin problem is reduced to two conditionally separate 1D problems along external
electromagnetic ¯eld propagation. The ¯rst problem describes a quantum dynamics of
disordered N -particles system with relaxation, while the second one describes statistical
properties of steric disordered spin chain system. Based on developed in both problems
constructions, the coe±cient of polarizability related to collective orientational e®ects
was calculated. The Clausius-Mossotti equation for dielectric constant was generalized
on the micrometer space and nanosecond time scales.

Refer ences

[1 ] Ch . K it t e l, In t r o d u c t io n t o S o lid S t a t e P h ys ic s , J. W ile y a n d s o n s , In c ., N e w Y o r k,
L o n d o n , S yd n e y, To r o n t o , 1 9 6 2 .

[2 ] D . J. Gr i± t h , In t r o d u c t io n t o E le c t r o d yn a m ic s , P r e n t ic H a ll, N e w Je r s y, 1 9 2 ( 1 9 8 9 ) .

[3 ] R . B e c ke r , E le c t r o m a g n e t ic Fie ld s a n d In t e r a c t io n s , D o ve r , N e w Y o r k, 9 5 ( 1 9 7 2 ) .

[4 ] Y . Tu , J. Te r s o ® a n d G. Gr in s t e in , P h ys . R e v. L e t t ., 81, 4 8 9 9 ( 1 9 9 8 ) .

[5 ] I. M. L ifs h it s , S . A . Gr e d e s ku l, L . A . P a s t u r , In t r o d u c t io n t o t h e t h e o r y o f d is o r d e r e d
s ys t e m s , Mo s c o w, N a u ka , ( in R u s s ia n ) 1 9 8 2 .

[6 ] A . V . B o g d a n o v, A .S . Ge vo r kya n a n d A . G. Gr ig o r ya n , A MS / IP S t u d ie s in A d v. Ma t h -
e m a t ic s , 13, 8 1 ( 1 9 9 9 ) .

[7 ] A . S . Ge vo r kya n a n d Ch in -K u n H u , P r o c e e d in g s o f t h e IS A A C Co n f. o n A n a lys is ,
Y e r e va n , A r m e n ia , E d s b y G. A . B a r s e g ia n e t a l, 1 6 4 ( 2 0 0 4 ) .

[8 ] L . B e r t h ie r a n d A . P . Y o u n g , Tim e a n d le n g t h s c a le s in s p in g la s s , a r X iv;c o n d -
n a t / 0 3 1 0 7 2 1 v1 3 0 Oc t 2 0 0 3 .

3 6



A. S. Gevorkyan, Ar. A. Gevorkyan 3 7

[9 ] P . V . P a vlo v, A . F. K h o kh lo v, S o lid S t a t e P h ys ic s , H ig h S c h o o l B o o k Co m p a n y, Mo s c o w,
( in R u s s ia n ) 2 0 0 0 .

[1 0 ] G. A . K o r n a n d T. M. K o r n , Ma t h e m a t ic a l H a n d b o o k, fo r s c ie n t is t a n d e n g in e e r s ,
Mc Gr a w-H ill B o o k Co m p a n y, N e w Y o r k, 1 9 6 8 .

[1 1 ] W . H . Za c h a r ia s e n , J. A m . Ch e m . S o c . 54, 3 8 4 1 ( 1 9 3 2 ) .

[1 2 ] K . B in d e r a n d A .P . Y o u n g , R e v. Mo d . P h ys ic s , 58, 4 , 8 0 1 ( 1 9 8 6 ) .

[1 3 ] Ch . D a s g u r t a , S h a n g -ke n g Ma , a n d Ch in -K u n H u , P h ys . R e v. B 20, 3 8 3 7 ( 1 9 7 9 ) .

[1 4 ] J. L . va n H e m m e n , In P r o c e e d in g s o f t h e H e id e lb e r g Co lo qu iu m o n S p in Gla s s e s , e d . b y
J. L . va n H e m m e n a n d I. Mo r g e n s t e r n , L e c t u r e N o t e s in P h ys ic s 192, ( S p in g e r , B e r lin ,
1 9 8 3 ) .

[1 5 ] B . A . B e r g a n d T. N e u h a u s ,, P h ys . R e v. L e t t . 68, 9 ( 1 9 9 2 ) .

[1 6 ] M. A b r a m o wit z a n d I. S t e g u n , H a n d b o o k o f m a t h e m a t ic a l fu n c t io n s , N a t . B u r e a u o f
s t a n d a r d s A p p lie d Ma t h e m a t ic s S e r ie s -5 5 , 1 9 6 4 .

[1 7 ] S . F. E d wa r d s a n d P . W . A n d e r s o n , J. P h ys . F 9, 9 6 5 ( 1 9 7 5 ) .

[1 8 ] M. V . Fe d o r yu k, Me t h o d o f s a d d le -p o in t , Mo s c o w, N a u ka , ( in R u s s ia n ) 1 9 7 7 .

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