Renormalization of quasi-hamiltonians under heterophase averaging

Volume 125, number 2,3

PHYSICS LETTERS A

2 November 1987

RENORMALIZATION OF QUASI-HAMILTONIANS UNDER HETEROPHASE AVERAGING V.I. Y U K A L O V

Laboratory of TheoreticalPhysics, Joint Institutefor NuclearResearch, P.O. Box 79, Dubna, USSR Received 30 March 1987;accepted for publication 28 August 1987 Communicated by A.R. Bishop

A representation of the algebra of local observables for a heterophase system is constructed. Averagingover phase configurations is defined as a functional integration over characteristic functions of submanifolds. Averagesfor the operators of observables are given with the use of an ensemble of the quasi-equilibrium Gibbs ensembles. Expressions for these averages, transformed as a result of the heterophase averaging, are found. This allows us to obtain an explicit form for a renormalized quasi-hamiltonian. A generalization of the approach to the case of a continuous many of phases is made.

The theory of heterophase fluctuations developed by the author [ 1-3 ] is essentially based on the notion o f an effective hamiltonian. The latter appears after a summation over heterophase fluctuations [ 2-4 ] in the partition function like a renormalized hamiltonian appears after summing a part o f variables in the renormalization group method [ 5,6]. The system with heterophase fluctuations is generally nonequilibrium, it is quasi-equilibrium. Its most logical description presupposes the use o f the quasiequilibrium Gibbs ensemble whose statistical operator contains a quasi-hamiltonian in place o f a hamiltonian. In ref. [7] a heterophase ensemble consisting of a set o f quasi-equilibrium ensembles with various phase configurations has been constructed, and it has been shown how to calculate the corresponding thermodynamic potential. However, solely one question is yet undetermined - how to define in a correct way mathematical expectations for the operators of observables when averaging over this heterophase quasi-equilibrium ensemble. An answer to this question is given in the present paper. The succession of actions is formulated in the abstract. Consider the system of particles on the Lebesgue measurable manifold V = { x l m e s V = f v d x = V}. A Hilbert space Yf o f microscopic states is given on the manifold V. The algebra o f local observables ~¢(A) = {A(A) } (A c V) is defined in the space ~ ; this algebra being composed o f operators o f the form

A(A) = E A k( Xl , X 2 k

....

Xk ) dXl ...dXk ,

where Ak is an operator distribution and A o - c o n s t × ]. Constructing an ordered manifold {Ai[i= 1,2,...} of bounded open regions At c At+ ~and an isotonic sequence of algebras ~¢(Al ) c d ( A 2 ) c .... in which Aj c A2 c .... one obtains a net of algebras {~(Ai)}. For a net o f algebras an inductive limit can be defined [8], called a quasi-local algebra. Suppose the considered system consists o f several thermodynamic phases, enumerated by the index a = 1, 2, ... s. The separation of phases in the real space is characterized by a family o f submanifolds {V~} forming a covering o f the manifold V, 0V~=V, c¢=l

~ V,~=V

(V~-mesV~).

(1)

ot=l

In its turn, in the space ~ of microscopic states one is able [7,9] to separate subspaces ~ , c ~ff (o~= 1,2, ... s ) , such that a conditional probability measure corresponding to the thermodynamic phase a is concentrated on the subspace ~ . Each o f the spaces ~,~ is a set o f vectors that are typical [ 10] for the phase a. The representation 1G[d(A,,)] o f the algebra o f local observables for the regions A , c V~ is defined on the space ~ , . Writing down the representations o f the operators from this algebra one can define the operator distributions Ak,~( ) by the equality 95

Volume 125, number 2,3

PHYSICS LETTERSA

u ~ [ A ( A , ) ] = ~ fAk~(X,,X:,...Xk)dXl dx; ... dXk .

2 November 1987

pings ~ . ~ : , ( a = 1, 2 .... s) should be called the standard fiber space

A~

(2) To divide the manifold V into a set of submanifolds {V~}, one may use the Gibbs method of separating surfaces [ 11 ] in his theory of heterogeneous systems. Mathematically, it is convenient [ 3,4,7] to produce such a division by fixing a set of characteristic functions of submanifolds, ~(x)=l, =0,

+~=®,~++

(map,d . ~ - ~ , ~ ) •

Fiber bases corresponding to different thermodynamic phases are not necessarily mutually orthogonal, although in many cases they are [9]. Thus, the global algebra d ( V ) is to be interpreted as a direct sum of quasi-local algebras d ( V . ~ ) , and its representation n[ d ( V ) ] =- d ( ~ ) , where - {~,~(x) l a = 1,2,...s;.w V', .

x~V~, (3)

xCV~.

(4)

~Y

(5)

has to be defined on the standard fiber space (4) in the form

Then, invoking the identity

f Ak~(X~ ,,..X~) dx~ ... dXk The representations of operators have the structure

-- f Ak~( x~ ,...Xk ; ~ ) dx~ ... dxk , the representation of the quasi-local algebra, g~[W(V~)], can be extended to the representation of a quasi-local algebra, zr~[d(V; ~ ) ] = d ~ ( ~ ) with the operator distributions

Ak~(X, ,X2,...X~;~) k

=akMx, ,x2,...x~) F[ ~ ( x , ) •

A(~) = @ A , ( ~ , ) .

A~,(x,,xz,...x~.)H~,(x,) cL'c,.

A,~(~,)=~

(7)

i=l

%

The many of all possible collections of ~ form the topological space {~}, on which a functional measure ~ can be given [7]. The statistical operator of a quasi-equilibrium heterophase ensemble is presentable as

j=l

As is clear, the function ~ ( x ) plays the role of an additional functional variable. In order to define a representation of the quasi-local algebra d ( V ) , that could be called a global algebra as distinct from the quasi-local algebras, consider .~-~. Suppose that there exists a topological space .~, on which a mapping map~: ~ ~,~ is given. The three ( ~ , map~, .~-~) is called the fiber space, ,~ is the total space, ~,~ is the fiber base [ 12]. The procedure of obtaining ~ from ~ by means of map,~ is called fibering, and the inverse process of reconstructing .~ out of ~ is a fiber section. When the total spaces of different fiberings are homeomorphic and their bases are the same, then such fiberings are equivalent. For our purpose any of the equivalent fiberings may be used. It is convenient to choose the so-called standard fibering with the total space as a tensor product ®~ ~,~. This total space under a fixed set of map96

p(~)=e

;I~ Tr

e

~

(8)

where F(4) is the quasi-hamiltonian of the system. Mathematical expectations corresponding to observable quantities are defined by the formula

The functional integrals in eqs. (8) and (9) describe the averaging over phase configurations. Introducing the functional measure ~ one notices that the averaging over phase configurations contains two kinds of actions. The first one deals with all possible configrations under a fixed set p=- {p,~I c~= 1,2,...s} of geometrical probabilities

p~=V,~/V

(0<~p~
(10)

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PHYSICS LETTERSA

The second action is the variation of each p~ from zero to unity taking account of their normalization. In correspondence to these actions s

s

.

~,-~.~p, dp, dp~. ( ~a=lPCt-1)I'=I=idPot

(11)

2 November 1987

characteristic functions of submanifolds if to implicate, as it is usually supposed in physical problems, that summation and integration can be interchanged. Then formula (9) for the mathematical expectation leads to 1

The functional differential ~p~ is defined in the following manner. One divides each of the submanifolds Va by means of subcoverings {V~,} so that net

= Tr fp(p)A (p) dp,

where the differential dp is defined in eq. (11 ), and

nct

Uv~,=v~, E v.~=V. i=l

(16)

0

1

i=1

--1

p(p):e-mP)(Trfe -rtp) dp)

(17)

0

(~=n~=n,V~-mesV~,i).

Theorem 2. Let the function The characteristic function (3) is presentable as the sum

1 y(p) = - ~ l n T r e -rap) ,

(18)

n~

~ ( x ) = 2 ~.~(x-a~) ,

in which N-N(V) is a number such that

i=l

N~oo, V-.oo, N/V-.const ,

~i(x-a~i)= 1, xeV~i , =0,

(19)

have an absolute minimum

xCV.i,

in which a~V,~i. Implying the limiting transition

y(w) = abs min y(p) P

n o ~ , n . ~ o o , V.i~0 (p,~ =const) ,

(12)

one can write the asymptotic expression i~= ~" da.i =1

(n--.oo).

(w-= {w. l a = 1,2,-=s}) .

(20)

Then (13)

i=1

1

l/m 1 Tr(~p(p)A(p)dp-p(w)A(w))=O,

(21)

0

Finally, the averaging of a functional F(O over phase configurations under a fixed set of geometric probabilities (10) is defined as the functional integral

where the limit is understood in the sense ofeq. (19), and

f F(~)~p~=,~lim I F(~) ~ =~:=~ f~[daai V '

p(w) = e-r~ w)/Tr e-r(w) .

(14)

in which the limit means eq. (12). Theorem 1. If the functional F(~) is a polynomial in the characteristic functions (3), then

fF(~) ~ , ~ = F ( p ) ,

(15)

where F(p) follows from F(~) as a result of the replacement ~ ) -,p,. Proof with all details has been given in ref. [ 7 ]. Corollary. The theorem can be extended to arbitrary functionals presentable as series in powers of

(22)

Proof. Introducing the notation .4(p) - T r e-r(p) A(p)/Tr e -rap) and using eq. (18), one can write 1

--1

Tr p(p)A(p)=e-N~(P)A(p) ( ofe-:vy(p) dp) Applying the Laplace method as N--,oo, we find 97

Volume 125, number 2,3

PHYSICS LETTERSA

tations of the algebra of field operators could be construced in a perfect analogy with the construction of the representation of the algebra of local observables. For the field operator ~ ( J ) = f ~ . f ( x ) ~ ( x ) d x . in which f ( x ) is any square-integrable function, one sets the representation n~[ ~z(f)] on the space .~,~. In its turn, this representation defines the operator distribution ~,~(x) with the help of the relations

I

fe-'~'Y~P~A(p) dp 0

.~

I

" 2~:

2 November 1987

" J/2

where y~; _=O~y(w)/Ow~ ( N - - , ~ ) . Therefore, I

n~[~,(f)] = t f ( x ) v / ~ ( x ) ctv 0

= f f ( x ) ~ , d x ) ~ , ~ ( x ) &v.

(27)

Remembering the notation for A(p) and definition (22), we obtain (21). Corollary. The mathematical expectation (16) becomes

The representation of the operator H ( V ) on the fiber space (4) according to (7), has the form

...Trp(w)A(w)

H(¢) = @H~(¢~),

(V-,oo) .

(23)

~j

(v

The representation of operators (7) of observables on the fiber space (4) takes the structure

a(w) = ~A,(w,) , + f H:~(x,x'){~(x)¢.(x') K

-V

(24) The set w_= {w~} defines the probabilities of thermodynamic phases and is to be found from the minimization of he thermodynamic potential 1

y(w)=_~lnTre

r,,,)

(25)

under the normalization condition X;= ~w~ = 1 following from eq. (10). To concretize the approach developed above, consider a system with the hamiltonian in the Heisenberg representation

H(A)= f H,(x) A

fI-l (x,x')

,

A

H, (x) = ~u+ (x)[ - V 2/2m + U(x)] ~u(x),

H2(x,x') = ½~u+ (x)~u + (x')~(x,x')~u(x')~,(x)

. (26)

Operator distributions Hl~(X) and H2,,(x,x') can be defined by rule (2) when introducing the representation of operator (26) on the space ~ . Represen98

dx dx' .

(28)

.

=

The partition of the manifold V into submanifolds V~ occupied by different thermodynamic phases by no means presupposes the uniformity of these phases. These phases as a whole are already nonuniform if only due to the existence ofinterpbase transition layers. The Gibbs dividing surface [ 11 ] presents a conditional geometric boundary placed somewhere inside a transition layer. The measured thermodynamic quantities can be defined so that they do not depend on the position of the dividing surface. Imposing some additional limitations, e.g. the equimolecularity condition [ 11 ], one may fix the dividing surface in a macroscopically unique manner. In our case an ambiguity of choosing separating surfaces is not at all important as far as we average over all their possible positions. A nonuniform system consisting of several thermodynamic phases is quasi-equilibrium [7]. Consequently, in the same fashion as for any locally equilibrium system [13], the local quantities must have a meaning, such as the local energy density

Volume 125, number2,3

PHYSICSLETTERSA

2 November 1987

e~(x; ~,~)=Trp(~)H~(x; G ) ,

in which the renormalized quantities

Ha(x; ~ ) =H,,~(x)~,~(x)

fl,~(x) = (fl,~(x,~,~)),

+ f H z ~ ( X , X ' ) ~ ( x ) ~ ( x ' ) dx' ,

(29)

V

and the local number-of-particle density

n,~(x; ~,~) =Yrp(~)N,~(x; ~,~) , N~(x; ~)=N,~(x)~,~(x) , Nc~(x) = ~+ (x)q/~(x) .

(30)

The statistical operator is given by formula (8) from which the normalization is evident

T:

(3,)

Expression (8) contains a yet unknown quasi-hamiltonian F(~). An explicit form of the quasi-hamiltonian can be found by demanding that the entropy S ' = - T r fp(e) lnp(e) ~ e

(32)

be maximal with respect to variations ofp(~) under conditions ( 29)- ( 31 ). This yields e(~) = ~ r ~ ( ~ ) , ot

M

- U . ( x , ~ . ) N . ( x ; ~,~)] dx,

(33)

where the inverse temperature p ~ ( x , ~ ) and the chemical potential/z~(x,~) play the role of the Lagrange multipliers, ensuring the nonuniformity of a system corresponding to a given choice of separating surfaces. After averaging over phase configurations the renormalized quasi-hamiltonian F ( w ) entering the statistical operator (22) assumes the form

F(w)=(BFo,(wo,) ,

/t,~(x) = (/~(x,~,~))

(35)

figure as functions defining the taken heterophase ensemble. Each of the renormalized quasi-hamiltonians F~(w,~) corresponds not to a sole part of a real system, occupied by the phase a, but to an abstract system representing an averaged infinite many of spatially nonuniform subsystems taking arbitrary shapes and sizes, and having the properties of the thermodynamic phase a. Such an averaged abstract many can be called the phase replica [ 7 ]. Note that the renormalized quasi-hamiltonian (34) retains information about the presence of transition layers and the corresponding surface energy [ 7 ]. Let there be no external fields acting on the considered system so that a stationary separation of phases could occur. That is the appearance of nuclei of different competing phases is a purely fluctuational process. The quasi-equilibrium system with such heterophase fluctuations serves as an example of self-optimizing systems [ 14 ]. When all phases and all parts of the system are in equal external conditions, then the average quantities (35) characterizing these conditions have to be constant: fl,,(x) =fl, /z,~(x)=/x.

(36)

Equalities (36) showing that in the system there is a heterophase equilibrium on the average can be called the equilibrium condition for phase replicas [7]. Eqs. (36) being true, the renormalized hamiltonian (34) becomes

F(w)=flB,

B=~H~, ot

H , ~ = w ~ [ g / + ( x ) [ - V 2/2m + V ( x ) - U ] q/,~(x) dx V

1 ,,,2 jv,,~ f,,,+ + :.,,~

(x)~'+(x')O(x,x ')

V

× V ~ ( x ' ) g ~ ( x ) dx dx' .

(37)

Now the mathematical expectation (23) is

ot

( A ) -.-Try5.4 (V-,oo)

r,~ (w~) = wa f f l , ( x ) [ H , a -lt,~(x)Na(x)] dx V

+ wZa[fl.(x)H2,~(x,x ') dx dr' , V

(34)

(38)

which formally corresponds to an equilibrium case with the statistical operator ~ = e - P n / T r e -pn

(39) 99

Volume 125, number 2,3

PHYSICS LETTERS A

,4=-A(w)

a n d the operator representation on the fiber space (4). The renormalized chemical potential /~ can be expressed in the usual way through the renormalized inverse temperature fl and the average n u m b e r of particles

N= ~N~, N,~=w~f<~+~(x)~,~(x))dx. 0~=1

Here a n d in what follows f dx means the integration over the whole m a n i f o l d V. M i n i m i z i n g the t h e r m o d y n a m i c potential (25) u n d e r the n o r m a l i z a t i o n condition E~)~w,~ = 1 or finding an absolute m i n i m u m of the potential +//2 E ,~ 3 = ~w,~ we get the equations for phase probabilities

y=y(w)

w,~=(p.R,-K~-2)/2¢,~

(c~= 1,2,...s),

U(x)]~u,~(x)>dx,

¢~=2~f <~,+(x)~u+(x')¢(x,x ') ×~,~(x')~,~(x) > dxdx', 1 + R~=~rf(g]~ (x)~(x)

> dx.

The developed theory is applicable to heterophase systems of arbitrary nature a n d any n u m b e r of therm o d y n a m i c phases. It can be even generalized to the case of a c o n t i n u o u s phase mixture, when the phase index a runs over a c o n t i n u o u s m a n y {c~}. For example, this can have to do with magnetic phases with different local values or directions of magnetizations. Such a situation can arise in disordered matter with r a n d o m interactions [ 15] or in r a n d o m external fields [ 16 ] in the presence of a frustration [ 17 ]. It might be relevant to spin glasses [ 18] having a cluster structure [ 19,20]. The generalization to a

i00

I should like to thank D. ter Haar, the n u m e r o u s discussions with whom on the properties of heterophase systems have given me an essential help in writing this final and, as I think, in principle total foundation for the theory of heterophase fluctuations.

References

a n d the n o t a t i o n -VZ/2m+

c o n t i n u o u s phase mixture can be done quite simply. There the fiber space (4) becomes a c o n t i n u o u s product whose definition has been given in ref. [9]. A measure dm(c~) on the m a n y {~} lets us to represent the global algebra on the fiber space (4) as a direct integral on the field of representations {A,~( ~ } , , ~ / ( ~ ) = f ® , ~ , ~ ( ~ ) d m ( ~ ) . All subsequent expressions retain their sense when changing the sums over ce by the corresponding integrals over dm(c~).

(40)

with the Lagrange multiplier

1f K~=~J (~t~+ (x)[

2 November 1987

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