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A SIMPLE MODEL OF A GLASS AT LOW TEMPERATURES
N. Rivier
To cite this version:
N. Rivier. A SIMPLE MODEL OF A GLASS AT LOW TEMPERATURES. Journal de Physique
Colloques, 1978, 39 (C6), pp.C6-984-C6-985. �10.1051/jphyscol:19786436�. �jpa-00217913�
JOURNAL DE PHYSIQUE
Colloque C6, supplément au n° 8, Tome. 39, août 1978, page C6-984
A SIMPLE MODEL OF A GLASS AT LOW TEMPERATURES
N. Rivier,
Blaokett Laboratory, Physios Department, Imperial College, London SW 7. England.
Résumé.- La mécanique statistique des systèmes désordonnés bloqués peut être construite selon la formulation subjective de Jaynes. On peut exprimer le champ moyen dû aux degrés de liberté bloqués comme une marche aléatoire. Cela donne une énergie à plusieurs vallées dans un espace de configu- ration à une dimension de telle sorte que la distance dans cet espace entre différents états métas- tables (vallées) et l'entropie de point zéro peuvent être évaluées simplement.
Abstract.- The statistical mechanics of quenched disordered systems can be constructed in a logical fashion using Jaynes' subjective formulation. We obtain the mean field due to the quenched degrees of freedom as a random walk. This yields a multivalleys energy on a configuration space reduced to a one-dimensional axis, so that the distance in configuration space between various metastable states (the valleys) and the zero point entropy can be simply evaluated.
A glass can be described as a system in which several degrees of freedom have been quenched at random below some freezing temperature T . These de- grees of freedom are fixed a priori according to a given probability distribution, and are not in ther- mal equilibrium. Nevertheless, a statistical mecha- nics can be constructed /l/ following the subjecti- ve method of Jaynes, based solely on our knowledge of the system.
In a rubber, the quenched degrees of freedom correspond to the topology of its constituting po- lymer chains, i.e. to their cross-linkage 111. The cross links are fixed in number and position, even if the rubber is subsequently stretched or heated, and the rubber remains a solid with a finite shear modulus. In a spin glass, the random position of the magnetic impurities is given a priori. A sub- sequent application of a magnetic field does not al- ter this constraint and the response - magnetic sus- ceptibility - of the system remains well below that of a magnetic liquid. In this case, the freezing temperature is apparently well defined and the sus- ceptibility has a sharp cusp below T .
Quenching implies a ground state for the sys- tem which has a random structure - a random walk in the simplest case of a spin glass in the local random mean field approximation. Several authors 131 have suggested for the potential energy of a glass a random function with many valleys in confi- guration space. Despite its successes (the tunne- ling modes as elementary excitations), this picture cannot provide - in the absence of a microscopic model, however simplified - a scale for the confi-
guration space and thus the distance between valleys the metastability of the various possible glass states and the zero point entropy. Our aim is to produce a simple model which yields a multi-valleys potential for which a distance appears in a direct fashion.
The system is described by the hamiltonian H (h,S), where the degrees of freedom are divided in two classes, the dynamical variables {S}, free to reach thermal equilibrium, and the random variables {h}which are quenched in the glass state. We have in mind as an example a spin glass in the. mean field approximation for which S is given spin and h the random, local mean field due to the others. The thermodynamic state of the system is determined by an occupation frequency ir*(h,S) which is obtained by maximizing the entropy subject to constraints cor- responding to our knowledge of the system, its ave- rage energy and, in the glass, the normalized pro- bability distribution of the quenched parameters P(h). For an annealed system (liquid), it is given by
wA(h,S) = exp[-eH(h,S)] Ifdh fdS exp[-BH(h,S}] (1) (6 = 1/kT, dS = T rg. . . ) , and for a quenched sys- tem (glass) by III
TT (h,S) = P(h)T7(h|s) = P(h)expG-eH(h,S)J//dS exp
L-6H(h,si] (2) ir(h|s) is the occupation frequency of S at given h.
Eq. (2) is almost obvious : quenched systems must indeed be characterized by a structure of condi- tional probability I hi.
The system is annealed above T and quenched
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786436
below. A t To, t h e r e f o r e , nA=rrQ, and one o b t a i n s a r e l a t i o n between h a m i l t o n i a n H o r ( f r e e ) energy
*
Fo(h) = -kTolnZo(h) and p r o b a b i l i t y d i s t r i b u t i o n P(h),
Fo(h) = -kTolnP(h)
-
lnZo (3)Here, Zo(h) = I d s exp E : ' B ~ H ( ~ , s ) ] and Zo = dh Zo(h).
Suppose t h a t P(h) i s a g a u s s i a n d i s t r i b u t i o n . Then, Fo(h) = c h 2 , which i s indeed t h e s e l f - e n e r g y of t h e mean magnetic f i e l d . But t h e randomness of t h e quenched v a r i a b l e ( s ) h a s been p r o j e c t e d o u t : and must be recovered by r e t u r n i n g t o t h e random p r o c e s s a t t h e o r i g i n of P ( h ) . The s p i n g l a s s o f f e r s a p a r t i c u l a r l y ' s i m p l e example : t h e l o c a l magnetic f i e l d h i s a s u p e r p o s i t i o n of a l a r g e number of s t a t i s t i c a l l y independant c o n t r i b u t i o n s . It i s the-- r e f o r e t h e r e s u l t of a random walk 151, each s t e p being t h e c o n t r i b u t i o n of every s p i n c o n s t i t u t i n g t h e system. The random p r o c e s s h ( t )
-
t h e d i s c r e t e v a r i a b l e t l a b e l l i n g t h e s t e p s i n t h e walk, t h a t i s t h e i m p u r i t i e s of t h e system-
i s s t a t i o n a r y and F ( h ) , p r o p o r t i o n a l t o h 2 , i s a random f u n c t i o n of one v a r i a b l e t w i t h many v a l l e y s . The v a l l e y s a r e-
t h e z e r o s e t of t h e random walk, which has, i f h i s a s c a l a r f i e l d ( I s i n g s p i n g l a s s ) , f r a c t i o n a l dimen- s i o n
$
161. The z e r o p o i n t entropy i ss
= - I d h P ( h ) l n P ( h ) % : l n N (4) where N i s t h e number of i m p u r i t i e s .The f r e e energy of t h e system i s r e a d i l y ob- t a i n e d from e q . ( 2 ) ,
F = kT Idh P ( h ) l n P(h) + Idn P(h)F(h) = Fdis + F1 The f i r s t term, due t o t h e d i s o r d e r f r o z e n upon quenching, i s p u r e l y e n t r o p i c . The second term can be r e w r i t t e n 1 2 1 a s FI=
-
kT(a/an)lnZ(n) u s i n g t h e r e p l i c a method (In2 = (a/an)Zn ),
where0 $0
z(,) =
Idhids
(0'.
,.
("&-60H e-P( . . . e -BE(") = ZoZn~.dh P (h) ( h ) I n
(5 The b a r i n d i c a t e s t h e a c t u a l experimental c o n d i t i o n s i t s absence t h e c o n d i t i o n s upon f r e e z i n g . !Chis d i f - f e r e n c e a p a r t , t h e n + l r e l i c a s i n (5) a r e i d e n t i c a l : t h e y a r e d e s c r i b e d by t h e same h & i l t o n i a n , o r , using ( 3 ) , by s i m i l a r random p r o c e s s e s . Thus, F = kTfdhP(h)lnP(h)
-
kTfdh ~ ( h ) l n i ; ( h )-
k ~ l n z ( 6 ) and can be d e s c r i b e d e n t i r e l y i n terms of t h e ran-dom p r o c e s s h ( t ) .
I f t h e system has been quenched i n t h e pre-
*
A f r e e energy f o r t h e dynamical v a r i a b l e s S, an energy f o r t h e random v a r i a b l e s h.sence of a magnetic f i e l d , t h e random walk h a s a f i - n i t e propagating component. It i s no longer s t a t i o - nary. Accordingly, F (h) f l u c t u a t e s about a parabo- l a i n s t e a d of t h e a b s c i s s a a x i s . The z e r o s e t i s d r a s t i c a l l y reduced ( b u t n o t e n t i r e l y i n s p i n g l a s - s e s where t h e a p p l i e d f i e l d i s much s m a l l e r than t h e i n t e r n a l f i e l d , t h e f l u c t u a t i n g component). I f t h e f i e l d i s removed a f t e r quenching, t h e f r o z e n s t a t e i s m e t a s t a b l e and t h e g l a s s decays i n t o one of t h e v a l l e y s belonging t o t h e z e r o s e t .
The o p p o s i t e experiment i n which an e x t e r n a l f i e l d i s a p p l i e d on a system quenched i n z e r o f i e l d can be d e s c r i b e d i n s i m i l a r terms u s i n g ( 6 ) . The assymetry between t h e s i n g l e quenched r e p l i c a and t h e n thermodynamic r e p l i c a s m a n i f e s t s i t s e l f by a d i f f e r e n t r a t e of decay. Furthermore, t h e system decays h e r e from a s t a t e of h i g h d e g e n e r a c y , i n t o a s t a t e of low degeneracy, whereas
the
Converse was t h e c a s e of t h e f i r s t experiment. D e t a i l e d c a l c u l a - t i o n s remain t o b e done and w i l l b e r e p o r t e d e l s e - where.1 am g r a t e f u l t o D r . M. Warner f o r s t i m u l a t i n gl e c t u r e s and d i s c u s s i o n s , p a r t i c u l a r l y on r e f e r e n c e 121.
References
111
R i v i e r , N., Physica 86-88B (1977) 856./2/ Deam and Edwards, S.F., P h i l o s . Trans.
280
(1976) 317.
131 Anderson, P.W., H a l p e r i n , B. and Varma, C.M., P h i l . Mag,
2
(1972) 1 ./4/ Brout, R., Phys. Rev.
106
(1957) 620151 See f o r example R i v i e r , N. and Adkins, K . J . , i n Amorphous Magnetism, Hooper and de Graaf, e d s . , Plenum 1973.
161 Mandelbrodt, B. F r a c t a l s , Freeman 1977.
