Lattice gas model generalized to kramers regime

Solid State lonics 9 & 10 (1983) 1401-1408 North-Holland Publishing Company

LATT GENERALIZED

1401

II~E 13AS MODEL TO KRAMERS REBIME

Y. Boughaleb + and J.F 6ouyet Laboratoire de Physique de l a Mati~re Condens~e Ecole Polytechnique 91128 Palaiseau We propose a generalization of l a t t i c e gas models which can account f o r the dynamic of hopping p a r t i c l e s in the lcM damping Kramers regime.

I . INTRODUCTION

By l a t t i c e gas one generally means any model f o r a system of classical p a r t i c l e s whose center-of-mass coordinates are r e s t r i c t e d t o values coinciding with those of the vertices of some regular array or l a t t i c e in one, two or three dimensions. A general introduction may be found f o r instance in the Feeler and Bugganheim [1] book on s t a t i s t i c a l thermodynamics. L a t t i c e gas models have been widely used t o describe systems of " p a r t i c l e s " s i t t i n g in periodic potentials. Moreover, they are closely connected with Ising models. In p a r t i c u l a r Yang and Lee [ 2 ] pointed out that a s~del consisting of p o s i t i v e and negative spins could be interpreted in a number of d i f f e r e n t manners including ferromagnetic models as well as binary mixtures of atoms, or occupied and empty s i t e s . Such models introduce in a natural way the phase t r a n s i t i o n s of any given interacting system of " p a r t i c l e s " as well as a l l the s t a t i c thermodynamic observables. Exact results and approximate theoretical approach on l a t t i c e gas hamiltonians may be found in Domb and Breen [3]. One of the most interesting application of l a t t i c e gas model concerns superionic conductors in which systems of interacting mobile ions move in a quite r i g i d host l a t t i c e . For a general discussion on application t o such systems we r e f e r t o recent review a r t i c l e s (see f o r instance Dieterich et al [ 4 ] and references therein). When an ion jumps from one s i t e t o a nearest neighbour s i t e i t has t o overcome a b a r r i e r &V. The v a l i d i t y of a l a t t i c e gas model i s then strongly connected with the f a c t that f o r a temperature T<
+This paper is taken as a part of the 3rd cycle thesis of Y.B. (Orsay June 27, 1983).

0 167-2738/83/0000-0000/$ 03.00 ©1983 North-Holland

where Jij

H = ]Ei
¢I} between

p a r t i c l e s on s i t e s i and j and Where f i n i = N i s the t o t a l number of particles.The corresponding Ising hamiltonian i s obtained by the change ni = (1÷0i)/2

;

oi = _+ 1

(2)

On the other hand the dynamical studies suppose the knowledge of a master equati on _aP_(£_~_].t_)_ = at

I (W([¢] I [ B ] ) P ( [ B ] , t ) - W ( [ B ] I [ ~ ] ) P ( [ a ] , t ) )

(3)

B

where P ( [ l ] , t )

i s the p r o b a b i l i t y

to find this

system in configuration [¢3 at t i e s t and W([¢]I[13]) the p r o b a b i l i t y per unit time that a configuration [13] turns i n t o [ a ] . l~neral features of such master equation have been studied by a number of authors and a r e v i ~ i s given by Kawasaki [ 5 ] . At t h i s step i t i s necessary t o specify the vocabulary: we w i l l c a l l a "hop" the motion of a p a r t i c l e between two successive thersalizatiuns in the bottem of a s i t e , and a "jump" the motion over a b a r r i e r betueen t~o s i t e s without necessarily thermalizatiun. When the f r i c t i o n i s large a p a r t i c l e jumps from one s i t e j t o a nearest neighbour s i t e i and therma]izes loosing a l l emery of t h i s jump (with our d e f i n i t i o n we have a "single jump hop").This case corresponds t o the two-spin-exchange models in which matter i s conserved (N fixed) and in which a state of the system i s completely defined by the occupation or not of each s i t e . The case uith only nearest neighbours " i n t e r a c t i o n " i s the most studied case (Kawasaki, 1966, Binder, 197b, Sate and Kikuchi 1971 and others). On the contrary ÷or underdamped systems the above model i s no more v a l i d as the

1402

Y. Boughaleb, J.F. Gouyet / Lattice gas model generalized to Kramers regime

p a r t i c l e s then conserve the m~mo~y of t h e i r l a s t jump, i . e . the jumps are more or less correlated. This comes from the f a c t that v e l o c i t y i s not completely thermalized between successive jumps. Now a "hop" can contain many

"jumps'. This paper i s concerned with t h i s l a s t case.

A)The d i f f e r e n t re~imes of t h i s system Two essential parameters define the d i f f e r e n t regimes: on one hand the r a t i o T/(VM-VA) of the temperature with respect t o the b a r r i e r height determines the degree of l o c a l i z a t i o n of the p a r t i c l e d i s t r i b u t i o n in the potential wells; on the other hand the r a t i o ¥1eA, or ¥/w!i ~ of the damping f a c t o r with

In t h i s work we f i r s t study a single p a r t i c l e motion f o r which, s t a r t i n g from knoun results on hopping transport in the underdamped case we w i l l build a generalized local state ( on one s i t e ) and i t s associated master equation from which the c h a r a c t e r i s t i c times of the system w i l l be extracted. This i s explained in section I I . Section I I I w i l l be devoted t o the straightforward consequences of such a model on the most important correlation functions. In p a r t i c u l a r we w i l l show that the model c o r r e c t l y describes the width of the structure f a c t o r function calculated s t a r t i n g from the continuous model, by Dieterich et e l . [ 6 ] . The general interacting system of part i t l e s , in the underdamped case w i l l not be treated in t h i s short paper. This case i s much more complicated than the ordinary l a t t i c e gas dynamics [14] : This i s due t o the f a c t that each c o l l i s i o n also preserves memory of the preceding state. Moreover t h i s paper w i l l be limited in practice t o the one dimensional

respect t o the characteristic o s c i l l a t i o n f r e quencies of the system separates the overdamped ¥>uA from the underdamped ¥
Case,

regime calculated by Kramers [10] and

and uM are the curvatures at the bottom of s i t e A (eA1...eAs f o r s normal modes) and at the saddle point (negative curvature along the path between t~o w e l l s ) . F i r s t l y , we w i l l always r e s t r i c t ourselves t o the case T<>uM the regime i s d i f f u s i v e and the jump p r o b a b i l i t y 4 i s defined by UAWM

=-~

VA-VM

--T--

exp

I f ¥~wM we are in the intermediate

.

l l . S i n g l e p a r t i c l e motion

=

.

m

abs

{6 )

--

~14

M

4

We consider a periodic e f f e c t i ~ potential V(~) in which each s i t e A i s separated from i t s f i r s t neighbours by a barrier AV = VM-VA, VA being the potential at the bottom

eA VA-VM where Sabs= ~ exp - - ~ - -

of a well (a s i t e A) and VM the potential at the saddle point between two neighbouring sites.We consider now a p a r t i c l e submitted t o t h i s e f f e c t i v e potential and t o random forces ~(t) due t o f l u c t u a t i o n s of the real potential seen by the p a r t i c l e around the mean V(~). The motion of the p a r t i c l e i s given by a Langevin equation d2~ d~ ~ m - - - + my - - + 9V = R ( t ) dt 2 dt

(4)

Near thermodynamic equilibrium at temperature T, the slcond f l u c t u a t i o n dissipation theorem holds and connects the

dmum~Ing l e c t o r ¥ t o the f l u c t u a t i o n of ~(t) which i s SUl~Oaed here t o be a white noise:[?]

I f ¥
--

~@~6~t}

i s t 3 ~ D i r a ~ f u n c t i o n , and < . . . > ~r~Nmnt~ s s t e t i s t i c ~ l average,

(7)

--Y- (VM-VA) WA

(e)

which supposes, at least in one dimenslon {12],

th~at t h e r e g i m e i s u n d e r d a m p e d , and t h a t

Y. Boughaleb, J.F. Gouyet / Lattice gas model generalized to Kramers regime

f l u A > T / ( V N - VA) i s r e a l i z e d , t h a t i s t o may l a r g e r than the temperature units).The Eyring regime I s t r i a n g u l a r region on f i g u r e

(9) the energy loss i s (in energy shown in a 1.

If T/u A < T/(VM-V A) t h e dynamical i s c a l l e d Kramers r e g i m e and corresponds t o successive jumps over n wells: when the p a r t i c l e i s therealized in a well the probabil i t y of e x t r a c t i o n (standing jump p r o b a b i l i t y ) is :

1403

The purpose of t h i s paper w i l l be then t o generalize the dynamical l a t t i c e gas model t o a l l the hopping regime. However a l i m i t a t i o n appears necessarily at the f r o n t i e r of d i f f e r e n t regimes, t h a t i s t o say, Nhen &=T. In t h i s case the d e t a i l s of the p o t e n t i a l are e s s e n t i a l . Our model w i l l represent in t h i s region a smoothed i n t e r p o l a t i o n betNeen the extreme cases.

regies

~A ~ VA - VM = ~ x ~ exp T = T ~abs

(10)

hence l ~ r e d with r e s p e c t t o ~ abs (which i s the maximum possible j u m p p r o b a b i l i t y ) . Nevert h e l e s s the m o b i l i t y i~ i s increased in t h i s case as the number of crossed wells i s n=T/6>1 and as = ~ [(n + 1)a] 2 T

T ~ ~ P abs

The problem consists in s e t t i n g a model Nhich w i l l c o r r e c t l y represent the underdamped hopping regime : a p a r t i c l e in a well i s then in two possible s i t u a t i o n s , a) The p a r t i c l e i s t h e r s a l i z e d and has l o s t every memory of the preceeding jumps : I t has then a p r o b a b i l i t y 0 t o jump i s o t r o p i c a l l y in any of the neighbouring s i t e s . I f T>>w - - ~ - A VM-VA and i f

Diffusive Regime

then O=~abs=VAOo and ~>>1. T

(11)

where a i s the distance betwe~n two wells and ~abs the m o b i l i t y in the absolute regime.

Y/oJa

B.The oriented dynamical l a t t i c e gas model.

/j~///~/~

$

'<<'A V ~TA

then ~=~VAOo,

6

and ~ <<1.

I t i s convenient t o t r e a t in a single Nay the underdaaped hopping regimes, t o i n t r o d u c e t h e parameter

¢=exp(-6/T) such t h a t ~=(I-¢)VAOo=~ABOo 8/'r ~ .ol

~o

(12)

.

~tnterruediate Regime • ~jJ~jj/~

N e l l r e p r e s e n t c o r r e c t l y t h e i r d i f f e r e n t behav i o u r s as i t N e l l be shmm below. In t h e intermediate region 6--T and a only gives a p a r t i c u l a r r e a l i z a t i o n of the general physical problems f o r Nhich no general theory can be given. b) The p a r t i c l e has j u s t lade a Jump in the d i r e c t i o n ~ f r ~ s i t e ~-~ t o s i t e ~. Ms suppose then t h a t i t has a p r o b a b i l i t y ~ t o continue in the same d i r e c t i o n and B = 1-¢ t o be thermalized. The residence time • in a mall when no t h e r e a l i z a t i o n occurs i s around 1/2~A where VA= WA/2W corresponds t o the o s c i l l a t i o n

.oo,

.o, ., ' Cv,,:v,/ v~-vA T/ Fig. I : Figure showing the d i f f e r e n t possible regimes f o r a single p a r t i c l e in a periodic p o t e n t i a l . The hopping regime corresponds t o T<~A. The underdamp~d regime T>T and t h e l ( r a ~ r s ~me, $<
frequency in a well.

The p r o b a b i l i t y per u n i t t i m e t o c o n t i n u e i n t h e s e me d i r e c t i o n ( f l y i n g juap p r o b a b i l i t y ) i s then S~A/W 1~

(13) ~ ~

t r a p p i n g s of t h e p a r t i c l e

path

betmmn

is e a s i l y

be

~.= a /(1-~)

(14)

two

found

Y. Boughaleb, J.F. Gouyet / Lattice gas model generalized to Kramers regime

1404

which v e r i f y the Eyring (~=a) and Kraeers (~= aTIE) regimes in the two 1 l i l t i n g cases.

To f i n d the master equation describing the t r a n s p o r t of a s i n g l e p a r t i c l e ~e f i r s t l i m i t ourselves t o the one dimensional case. C) Master equation of the oriented l a t t i c e gas model in one dilmnsion. In ~hat can be c a l l e d an oriented l a t t i c e gas model i t i s necessary t o p a r t i t i o n the p r o b a b i l i t y d i s t r i b u t i o n on a given s i t e i i n t o pO the p r o b a b i l i t y t o f i n d a p a r t i c l e thermalized in the well and the z components P~i indicating from ~hich nearest neighbouring s i t e the p a r t i c l e has j u s t made a jump (and has p r o b a b i l i t y a t o continue). For a cubic l a t t i c e in d dimensions z = 2d. In one dimension the components of the p r o b a b i l i t y d i s t r i b u t i o n are

Two main response f u n c t i o n s a r e of g r e a t general i n t e r e s t : t h e c o n d u c t i v i t ~ o ( q , ~ ) and the dynseical structure S(q,u). C2. Eigenvalues and eigenvectors of the master equation : AsT~I+¢~ equation (lb) i s equivalent at t>>~ t o the continuous aaster equation :

at

=

~:_I~

= up

1171

The eigenvalues of thisequation are the c h a r a c t e r i s t i c r e l a x a t i o n r a t e s of the problem. The associated eigenvectors w~ and eigenvalues kF(q) are given by

T~w

= ~Lm = - k (q)s

1181

The d e t a i l l e d c a l c u l a t i o n of the eigenvalues w i l l be presented l a t e r . The Iomest one ko(q) showing the long time behaviour i s

P~, P~ and P~ (+ :~ t o t h e r i g h t , - : ~ t o t h e l e f t ) or as f u n c t i o n of p o s i t i o n x and t i m e t : P + ( x , t ) , P ° ( x , t ) and P - ( x , t ) .

the most important. I f me l i m i t ourselves t o the f i r s t order in #o i t can be shoNn t h a t :

The evolution of the d i s t r i b u t i o n i s completely defined by i t s master equation and, at the same t i e s , a l l desired c o r r e l a t i o n functions.Let X be the space t r a n s l a t i o n operator such t h a t X f ( x , t ) = f ( x + a , t ) and l e t T be the time t r a n s l a t i o n operator such t h a t T f ( x , t ) = f ( x , t + ~ ) , • = I/(2~ A) being the

ko(q ) = B#° (1+¢)(1-cosq_a) + 0(#:) ¢2-2¢cosqa + 1

elecmntary time of our nx)del.

so that except f o r ¢=I, q-'O the corresponding eigenvecturs r e l a x e r a p i d l y . From ko(q) i t i s

I t i s easy t o shou t h a t t h e master e q u a t i o n can be u r i t t e n ( T = T x I )

~#o x

Or i n a more compact f o r I orlB =lle

The two other eigenvalues are approxi matel y k+lq) -" 1-sexp(+iqa) + 01#o)

easy t o v e r i f y t h a t the equilibrium d i s t r i b u t i o n corresponds t o ko(q=O), and that

ko(q) ~ 9q2~

Nhere

l+s a 2 2 6 D = 1 ~ 2-~ #o= vAa #oC°th(2T )

aX

DEyring = vAa=#o = a2#abs

Then ~P = ]P =~>eq independent on time. I f the Bean occupation of a s i t e i s

p~q=p~q.~o/12+2#ol ;

P~q=/( 1+# o) Or approximately as #o < and P~q ---(1--~o)

1211

idhen a = 0 the d i f f u s i o n c o e f f i c i e n t i s the usual Eyring r a t e :

T pIA = I ~ v pV

C1. Stationary d i s t r i b u t i o n

+ o +Peq= ~hile Peq+Peq

(20)

f o r small q,

x- °°x-1 l/i:/ P-(x,t)]

1191

116)

(21a)

while ~hen ¢41 then 612T<<1 and the Kramers regime i s reached: 2T 2. DKramer s = ~-a 9ab s

(21b)

The model c o r r e c t l y d e s c r i b e s t h e tNo l i m i t i n g regimes. F i g u r e 2 r e p r e s e n t s t h e e t g e n v a l u e ko(q) f o r d i f f e r e n t v a l u e s of a. The v a l u e a=0.8, f o r i n s t a n c e , corresponds t o a aean hop over 5 b a r r i e r s . We a l s o see t h a t t h e r e l a x a t i o n time f o r t h e h i g h e s t s p a t i a l f r e q u e n c i e s becomes l a r g e r and l a r g e r as ¢

Y. Boughaleb, J.F. Gouyet / Lattice gas model generalized to Kramers regime increases. We t r i e d moreover t o compare t h i s generalized hopping model with exact calculation. The ÷ i t between our model and rigorous r e s u l t s [13] f o r the halfwidth of the quasielastic l i n e i s shown on f i g 3. This leads t o the ÷ollowing remarks :

a) Formula (19) r e p r e s e n t s v e r y closely the exact r e s u l t s . The halfwidth & i s related t o ko(q) and A

ko(q)

uA

UA~

1405

code1. a',b"

: Rigorous r e s u l t s

fr=

a',b"

: The usual hopping model.

r e f . [ 13].

Concerning the excited states k_+(q) i t has been shown,and t h i s w i l l be published in a more d e t a i l l e d paper that in the v i c i n i t y of q = 0 or • t small corrections of order 0o s p l i t the real parts of )~+ and X_. These

(22)

Comparing ~ f i t t e d t o ~=1/(2v A) we o b t a i n f o r t h e two c u r v e s a. and b. :

corrections remain always saall compared with the values Re k_+(q) Nhich are at least of order B = I - ¢ . I f we suppose B>>~o these corrections

By ~ f i t = 1.25~ (23) which i s very reasonable as the real t r a n s i t time between two b a r r i e r s i s c e r t a i n l y larger than half a period in a Nell. b)The ÷ i t of ¢ gives s f i t = 0.30 f u r curve a. and 0.37 f o r curve b . , that i s t o say 6/T would be respectively 1.2 and 0.99 using (12), while (8), gives 0.53 and 0.32 respectSvely. We can conclude either than f o r ~ l a (8) underestimates 6 or more probably that ¢ = exp(-6/T) i s not the best expression when 6/T approaches unity.

be n e g l e c t e d . N e v e r t h e l e s s a problem

remains as in t h i s case the corresponding eigenvectors w+ and w_ become degenerated when q=Oor

w. A p a r t i c u l a r drawing of the eigenvalues X+(q) i s given on figures 4 and 5. The d e t a i l s of t h e i r behaviour in the v i c i n i t y of 0 and w i s also shown.

I ......

1+~ I

Xo(q)

2'o

o(;0

=OZ 0

~:o~

q0

TT

qFig.2 : Lowest eigenvalue f o r the generalized hopping model in one dimension.

~-~0

¸

i

/lib"

0.003 L /

<

b

,,' 0.001

// /,"

0.0

/

/

".. Ira{k+)

//[ / / /

.''"

a a

/ / /

0.1

02

03

0.4

0.5

qa/(2n) Fig.3 : Halfwidth of the quasielastic l i n e f o r ¥/UA=l/(2w) and T/(VM-VR) equal 0.3 (curves a) and 0.5 (curves b). a ,b : The generalized hopping

F i g . 4 : Real and i m a g i n a r y p a r t s of X+ f o r ¢=0.9 and ~o=0.01.

1406

Y. Boughaleb, J.F. Gouyet / Lattice gas model generalized to Kramers regime

•

.....

.....

;,e I~'+/

O.

~ Ix

C3. Propagator and mean values

. .", *_4 1-o~

=

I t

The propagator ~ i s defined 10

0.5

by

F,,~(x,t) =~oJdxo~.o(XtlXoto)~'°~Xoto)(27)

~.q (X 10 -3 } (xlO -3}

(~ or more exactly F.~/x plays in f a c t the r o l e of the vel~:ity) The boundary conditions f o r P~g are

1.0

O

0.5

p~ olXtolXoto) = ~(X-Xol~Pl=o 0.0

-1£

lim PF~o(XtlXoto ) = ~ t~ I

;

t

I

I ~ I 0.5

I

t

I

Fig.5 : D e t a i l s of the v i c i n i t y of q:O f o r k_+. Similar behaviour appears around q=•.

Now, from the evolution equation 1171 we can w r i t e , ~ithout d i f { i c u l t i e s f o r the one dimension case the general s o l u t i o n in ( q , t ) space :

=~-1 exp (-~t/x) a~(~,0)

12~)

ao :.

aObO and

b-

a -1

P~ ( x t l x ) I~°

O

: (eLf) F

I~°

~(x-x ).

(31)

o

definition:

" =Peq " J%(x) dx (321 : Jdx AF(x) Peq "-

=/bo b+ b_

/ ; o c+ c

then one has e x p l i c i t e l y a~=b~[1-¢ e x p ( - i q a ) - X ~ ] - l x 6

(25)

c~=bF[1-¢ exp( i q a ) - k F ] - I x~ ~. =(B../bP)(4~/2)

and L i s in f a c t a forward Kolmcx~urov operator leading t o an exponential operator acting on x, such t h a t ,

Let Ap(x) be a dynamical v a r i a b l e , by

~here exp ( a t / x ) = II Xl=(q,t) ~ v II ~=O~+~and xF(q,t) =exp(-Xlx(q)t/x )

b+

(30)

From these r e l a t i o n s i t i s easy t o calculate any mean value.

~ ( q , t ) = exp(Et) P(q,O)

~=

(to=O)

PF ( x t | x ) = (eit)Fp, plZ~.o(XOIx o ) ~ ;~o o

tLq (×10-3}

Let

(29)

The evolution equation i s again 1.0

1281

exp(-iqa) a~

(26)

cF=(BF/b )(#o12) e x p ( i q a ) and the normalization condition a~+b~Bl=+c~l=

In a s i m i l a r manner, a c o r r e l a t i o ~ function i s given by F Po Po =~IdXodxA~(x) P l=o(Xt Ixo)A (xo) Peq

III.

Conductivity and s t r u c t u [ e

f actorUsing the above r e s u l t s the a.c. conductivity has been evaluated. A current operator aF can be defined such t h a t the current j ( x , t ) w i l l be f o r an e l e c t r i c f i e l d E :

j(x,t)

= a;iPt~(x,t)

(34)

v ,the eigenvectors being n a t u r a l l y defined = 6;= by

Nhere the operator J = 3 (°) + ~(1) Nith

(35)

Y. Boughaleb, J.F. Gouyet / Lattice gas model generalized to Kramers regime j ( O ) = e a , t - t ( e X -1/~'

-i]~o(Xt/2_X-t/2)/2

_¢tXt/'-')

2 2 ,](1)=e._~a_(ciX-,/2 Tt "

B,o(Xt/'S'+X-t/c')/2

=X,/~')E

I t was shoNn that the conductivity i s , taking q = ~ / c -~ O)

o(u)

1 l+¢fe-iut~

= o(-)+ .... T

Explicitely 2

=

/J

J

(o)

(t)

J

(o)

(O)>dt

(36)

Ne f o u n d a c o n d u c t i v i t y

e a o(u)

a

--T

2

~ ~ 11+0.)(1+ ~ ) Ao • i~+B

(37)

1407

IV) Conclusion This approach i s a f i r s t step in introducing the underdamped regime in a l a t t i c e gas model. The second one which i s in progress consists in defining c o l l i s i o n s of pairs of p a r t i c l e s in agreement as closed as possible with classical mechanics. For instance a jump of a p a r t i c l e on s i t e i towards an occupied s i t e j giving the sequence: At time t : n i = l , pi=+ and n j = l , ~j=Oj While at time t+At : n i = l , l~i--O and n j = l , IAj=+, in a Nay s i m i l a r t o that ÷ollowed by Kadanoff and Swift in t h e i r non hermitian l a t t i c e gas model t o simulate liquid-gas t r a n s i t i o n [14]. The l a s t step w i l l be then the building up of a Monte-Carlo method t o describe the dynamic of a many body system of underdaaped p a r t i c l e s .

and the d i f f u s i o n c o e f f i c i e n t , REFERENCES 2

D(w) = v ~ a (I+G)(I+-i~-----~--~Z'~-~L~p-~ Ao

(38)

When u=O we r e c o v e r f o r m u l a ( 2 1 ) , D(O)=D i • m r e n v e r the correlation f a c t o r i s a d i r e c t measurement of the number of jumps in a hop : fc

D(O) 1 = --- = -

D(-)

B

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The structure f a c t o r can be equivalently treated and gives:

S(q,u)=Re ~ Mp(q)/(iu~+kF(q)) (40) where M~ = < n > / ~ (~1~+~v+~v)(bV+~o[a~+cV-2bP]12) This cenTirms using the results described above (see figure 3 ) the correct description of the width of the structure factor by t h i s very simpliTied model.

[ I ] R.M. FOWLER and E.A. 8 1 ~ I M , S t a t i s t i c a l Theraodynamics, Cambridge University Press (1939) p. 541. [2] T.D. LEE and C.N. YP~8, Phys. Rev 87, 410 (1952). [3] C. DOMBand M.S. GREEN, Phase t r a n s i t i o n s and C r i t i c a l Phenomena, AcademicP r e s s (1972). Vol 2. [4] M. DIETERICH, P. FULDE and I . PESCHEL~ Adv. in Physics 29, 527 (1980) [5] K. KAMASAKI in Phase Transitions and C r i t i c a l Phenoaena. R~F [ 3 ] p. 443 [6] H. DIETERICH, I . PESCHEL and #.R. SCHNEIDER, Z. Phys. B 27, 177 (1977) [7] K. BINDER in Phase transitionm and C r i t i c a l Phenomena~ C. IX]MBand N . B . 8REENEds, Academic Press (1976) Vol 5B p. 1. [8] H. SATO and R. KIKUCHI~ J. Chem. Phys. 55~ (677 ou 702) ( 1 9 7 1 ) . [10] H.A. KRAHERS, Physica 7, 284 (1940) [11] H. EYRING, J. Chem. Phys. 3_, 107 (1935) [12] P. NOZIERES and S. ICHE9 J. Physique 40, 225 (1979) [13] ~. DIETERICH, T. 6EISEL, I . PESCHEL, Z. Physik B, 29, 5 (1978) [14] L.P. KAI)RNOFF and J.SMIFT0 Phys. Rev. 165 ,310 (1968).