Canonical Formalism For Description Of Critical Phenomena In Systems, Isomorphic To Simple Liquids

The basic elements of the canonical formalism for description of the critical phenomena in systems isomorphic to Ising model are stated. The reduction of the eeective Hamiltonian to the canonical form and numerous consequences of such a transformation are thoroughly investigated. The shift of xed point and the critical exponents of the system are considered.


Introduction
At the present time there exists the well-developed approach to the critical phenomena in di erent systems 1].All most essential features of the critical behaviour were explained qualitatively and to a certain extent quantitatively 1,2].Especially important role in the investigation of the uctuation properties belongs to the Renormalization Group (RG) method 3,4].It allows to describe in an adequate way the symmetry properties of uctuations as well as to develop the analytical method for calculation of the critical exponents, the universal ratios of critical amplitudes etc..But in some aspects RG method is not quite consecutive.The critical exponents can be obtained with accuracy o( 2 ) basing on the Landau-Ginsburg Hamiltonian (\ 4 "-model) To calculate the higher terms in the -expansions the certain extension of the Hamiltonian should be done.The Hamiltonian must be completed with contributions g 6 6 for calculation of 3 -order and with g 8 8   for 4 -order and so on 4,1].However, this circumstance is not consistent with the assumption that the model \ 4 " is su cient for the description of the long-range uctuations near critical point.Add, that the Quantum Field Theory analysis of critical anomalies is namely based on \ 4 "-model 6].This fact can be considered as a property of su ciency and completeness of that model for that kind of phenomena.The certain di culty of the standard variant of RG approach is its phenomenological character.The dependence of the Landau-Ginsburg Hamiltonian on the details of microscopic interactions does not ascertained.As a consequence the ne details of the critical behaviour of systems which belong to the same class of universality can not be taken into account.To a certain extent this de ciency can be made up with help of Hubbard-Stratonovich identity for lattice systems 7] or collective variables approach 9,10].But the problems raised by the 30 V.L.Koulinskii, N.P.Malomuzh operating with in nite series have the same acuity as the mentioned above.In addition the inclusion of isomorphism principle for the critical phenomena in the multicomponent mixtures into general Hamiltonian formalism remains unsolved problem.
The solution of these problems and other questions can be found in the framework of the canonical formalism.This approach is a synthesis of ideas and methods of the scale-invariant uctuation theory 1], the catastrophe theory 5] and the statistical theory of condensed matter 11].Some elements of the canonical formalism were discussed in 14{16].
The central part of such approach is a nonlinear analytical transformation { the canonical transformation of an order parameter that reduces the initial Hamiltonian to polynomial form, so called, canonical form.Its form depends on only the structure of the equations which de ne the critical point.The list of canonical forms is given in 5].
In this paper the basic ideas of canonical formalism for description of the critical phenomena are formulated.In the rst two sections the details of transformation of the e ective Hamiltonian to the canonical form in the thermodynamic and the uctuation regions are discussed.In the third section the renormalization of the parameters of the system caused by the canonical transformation is discussed in the cases of 3-D Ising model and liquid Ar and Xe as an example.Section four includes the results obtained in the framework of the canonical RG.The fth section is devoted to the problem of rectilinear diameter singularity and the analysis of the manifestation of individual features of the system in its critical behaviour.Some new problems and possible applications of the canonical formalism are enumerated in Conclusion.

E ective Hamiltonian in thermodynamic region
Let h = h( ; T) and F = F( ; T) are the equation of state and the free energy of a system, respectively: h = @F @ T ; ( where T { the temperature, h { the external eld conjugated to the order parameter .In the vicinity of the critical point de ned by the equations: @h @ = 0; @ 2 h @ 2 = 0; (2) the free energy of a system can be represented as following: To determine coe cients k it is expedient to use the equation of state: k = 1 (k ?1)! @ k?1 h @ k?1 =0 ; k = 1; 2; : : : For example, in the case of Curie-Weiss equation for magnetics: h = ?J(0)m + atanh m; m; (5) where m and h are magnetization and external eld strength, respectively, is Fourier transform of the \spin" interaction potential, the coe cients k have the form: 1 = h; 2 = J(0) ?1; 2k?1 = 0; 2k = ?
Reducing of the in nite series (3) to the polynomial canonical form is the fact of great importance.The mathematical for such a procedure are discussed in the catastrophe theory 5].With the help of the canonical transformation of the initial order parameter ?! Ĉg = ?0 + + 1 2 2 2 + 1 3 3 3 + : : : ; equation ( 3) can be reduced to the canonical form which is the same as Landau free energy: F( ) = V ?h + 1 2 a c 2 2 + 1 4 a c 4 4 : (15) One can show that h = 1 + 1 Taking into account that parameters x i = (h ; A 2 ; A 3 ) are small near the critical point the coe cients can be represented in a form: n = (0) n + (i) n x i + (i;j) n x i x j + : : : ; (0) 3 = ?etc.
The formulas (16) can be considered as a result of canonical transformation in the space of the Hamiltonian coe cients.From the physical point of view the canonical transformation is connected with the renormalization of the initial order parameter.This transformation generalizes the notation of the temperature and the external eld conjugated to the order parameter.Due to canonical transformation the order parameter acquires rigorous physical meaning.It is important to mark that the roots of the equations a c 1 = 0; a c 2 = 0 (21) and those of ( 11) or ( 13) are the same.That is because of spatial homogeneity of both the initial and canonical order parameters.At the same time it is clear that only the change in the weight of short scale uctuations in uences the critical point position.
At the end of this section we touch upon the question of the convergence of used expansions.The convergence radius R for the series of the free energy (3) depends on both the choice of the order parameter and the equation of state.In the examples mentioned above: R = The radii of convergency for the expansions ( 14), ( 18) and ( 19) as it follows from the results of 5], are less than those given in (22).
?4 ? 5 (i) + ? 4 (i)? 2 (i) + 1 2 ? 2 3 (i) ??? 3 (i)? 2 2 (i) ? 1 4 ? 4 2 (i) : Here ?n (i) are the coe cients of the inverse canonical transformation on the i-th step of the recursive procedure: i?1 (r) = i (r) + + 1 2 ? 2 (i) 2 i (r) + 1 3 ?3 (i) 3 i (r) + : : : They can be calculated by formulas (19) and (20) in which instead of coefcients A n the coe cients Ãn (i ? 1) of expansions of H (i?1) l i (r)] in series of i (r) should be substituted.In accordance with (31) we get: A n !Ãn (i ? 1) = A n (i ? 1) + A J n (i ?1); p=2 p : Here P and P 0 denote the summation over all sets of n which satisfy the conditions X :; It should be pointed out that canonical form of the uctuation Hamiltonian coincides completely with Landau-Ginsburg Hamiltonian.But the order parameter of the system and the \temperature" a c 2 conjugated to it as well as the \external" eld a c 1 and also intermodal interaction coe cient a c 4 are connected with their initial values by nonlinear relations.It is im- portant that all odd terms in canonical form of the Hamiltonian are absent.Note, that the choice of the initial order parameter essentially in uence the convergence (or asymptotic character) of the recursive procedure described above.The closer the one is to the canonical order parameter the best convergency to the canonical form can be achieved.

Operator of the canonical RG-transformation
Let R( ) be the operator of RG-transformation.The action of R( ) on the Landau-Ginsburg Hamiltonian H LG changes its form: where ~ (r) = Z( ) (r), Z( ) is the coe cient of the scaling factor, H nl denotes the sum of nonlocal contributions which are not included in H LG .
To remove the \super uous" local and nonlocal terms Wilson 3] completed the de nition of RG-operator with projector operator P onto the space of Landau-Ginsburg Hamiltonians.In accordance with this R( ) ! RW ( ) = P R( ); RW ( ) H LG (r)] !H LG ~ (r)]: (42) One can improve the de nition of the RG-operator substituting P by the product P Ĉ1 , where the operator of local canonical transformation Ĉinfty is given by(32).The new RG-operator is the canonical RG-operator.We suppose also that operator P kills the contributions of small scale with wave vectors j q j> , where is the cut o parameter.Unlike R W the operator R c not only conserves the form of Landau-Ginsburg Hamiltonian: Rc ( ) H GL (r)] = H LG ~ (r)]; (44) but also takes into account all local contributions.This circumstance in uences essentially on the parameters of RG-transformation near xed point.
To analyse the properties of RG-transformations we pass from functional equation (44) to the di erential equations for the coe cients of Landau-Ginsburg Hamiltonian.In the limit t = ln !0 at a c 1 = 0 and xed value of b we obtain the equations:

The values of the parameters in the critical point
In this section we give the values of the several important parameters for 3-D Ising model and liquid Ar and Xe obtained in the framework of the canonical formalism.First of all we should determine the critical point position.In accordance with previous sections the values of critical temperature and the external eld are connected with their initial values by the following relations: T c = T 0 (1 + ); h c = h 0 + h ; (61) where and h are determined by the equations: a c 1 ( ; h ) = 0; a c 2 ( ; h ) = 0: (62) This is the system of nonlinear equations which has in general multiple solutions.The physical ones are those ( ; h ); i = 1; : : : ; n which belong to the convergence region and satisfy the stability conditions: a c 4 ( ; h ) < 0: (63) If still i > 1 then the solution which is our \analytic prolongation" of the initial critical point should be chosen.The equations (62) have the simplest form in the case of Ising model.The coe cient a c 1 does not depend on and the condition of its vanishing yields to h = 0 which agrees with the symmetry of the Ising model.The second one of the equations (62) expanded in series on is as following: 2? (0) 3 + (1 + 2? (2) ) + ? (2;2) 3 2 + : : : = 0: (64) Using the expressions (6) for coe cients k after rst iteration (see Section 3) we get: ?
3 ' ?0:3; ? (2)3 ' ?0:03; : : : is not satisfactory because they converge in a small vicinity of ; h .That is why the solutions were obtained by numerical method.In order to calculate the coe cients a; b; c in ( 7), ( 8) and ( 11) the standard formulas 8] were used: where (r) is an intermolecular potential, is hard core diameter.In (69) the following testing potentials: were substituted instead of (r).The calculations of di erent parameters and its comparison with experimental ones were made for Ar and Xe.The parameters and were put respectively: =k B = 119:8K; = 3:405 A; =k B = 225:3K; = 4:04 A: The obtained data are presented in table 2. Initial values of critical temperature and pressure were calculated with the help of ( 11) and (12).The roots of (62) were found by the dichotomy method in the vicinity of the initial critical point.There was used one iteration for calculating ?n like in the case of Ising model.It appeared that the root satisfying all required conditions exists only for mvdW equation with Lenard-Jones and 6-9 potentials.In other cases the poor convergency properties of the initial equation of state is the cause of the nonexistence of the root.It is important that prime values of the critical point coordinates were close to the experimental ones.In this case the applying of the canonical formalism might improve the theoretical results.The uctuation terms should be also taken into account to reduce the di erence in the coordinates of the critical point.The same scheme can be applied to other equations of state including the empiric ones which describe the properties of liquids.The respective results will be the subject of other paper.The results in table 2 obtained for 4 (T )j T =T0 , where T 0 is prime value of the critical temperature and the shift of the order parameter ( = ?a3 3a4 ) has been done.
Using the canonical formalism the di erent attening of closed lamination curves in binary mixtures with hydrogen bonds near upper (T u ) and lower (T l ) critical points can be explained 22].In this case the coe cients a 21 (T l ) and a 21 (T u ) responsible for attening can be estimated with the help of some model 23] and are quite di erent.
In the same way one can consider the connection between the asymmetry of equation of state and the choice of an order parameter both in simple liquids and in multicomponent mixtures.

Conclusion
In this paper the main elements of the canonical formalism and it's applications to Ising model and simple liquids were presented.We would like to accentuate that the proposed approach is not only formal procedure leading to more clear mathematical consideration but also gives new physical results.Among them we note: the rigorous de nition of the initial order parameter; generalized temperature and external eld conjugated to it, the nontrivial renormalization of initial values for many critical parameters, more correct formulation of RG-theory, the consecutive explanation of equation of state asymmetry etc.These additional possibilities of the theory are connected with the more accurate account of small-range uctuations.Due to the using of the canonical formalism we are able to separate the e ects raised by the long-wave uctuations which have general character and the individual properties of the system.As a result one can evaluate the theoretical constants like g s (0) and l s (0) and so on.Here we only considered the consequences of local canonical transformation.We think that wider possibilities are connected with further development of the canonical formalism in which the quasilocal canonical transformations reducing the e ective Hamiltonian to the canonical forms like Ginsburg-Landau Hamiltonian will play the key role.New applications of the proposed approach to electrolytes, polymer and micellar solutions as well as to conductor-insulator transition in the vicinity of the critical point are very attractive too.
?1; T = T=T c ; P = P=P c .The values of the critical temperature, pressure and density are: ); due to the Jacobian term in (26) the in nite series of local contributions appears again.The quasilocal part H ql (r)] of the uctuation Hamiltonian remains unchanged if we restrict ourselves only by quadratic term: arguments used can be completed by nonperturbative approach.If we start from the partition function for the local Hamiltonian the existence of the canonical form is equivalent to the existence of such a transformation ! for which the following conditions hold +1; ] = U +1; a c 1 ; a c 2 ] ?1; ] = U ?1; a c 1 ; a c 2 ]: ?H c ]):The equations (40) determines uniquely the canonical coe cients if the value of a c 4 is xed.

c 4 .
The equations (45) correspond to the Gell-Mann-Low equations in quantum eld theory 17].Below we consider their solutions near saddle-like xed point for local (b = 0) and quasilocal Hamiltonians.
) k = a(a+1) : : : (a+k ?1) is Pochgammer symbol.As a consequence of (49)-(53) we get the equations (45) with root of (64) equation is approximately equal to: = 0:53: (66) Since T 0 = zJ=k B , where z = z(D) is the coordination number, the equations (62) and (66) yield the following value of the critical temperature for \cubic" Ising model: for D = 3 case gives the value k B T c = J=0:22 and Onzager's exact solution for D = 2 Ising model leads to k B T c = J=0:44.The numerical values of initial and canonical coe cients of Landau-Ginsburg Hamiltonian are equal to: a 4 = 1=3; b = 1; = 1 { initial; a 4 = 0:98; b = 1; = 0:68 { canonical; lattice parameter.The situation becomes more complex for simple liquids which are described with help of mvdW equation.In this case the coe cients k are the functions of both the temperature and the external eld.The use of expansions

Table 1 .
The contributions appeared due to the canonical transformation are given in braces.The values of coordinates of the xed point and the correlation radius critical exponent are presented in table1.The comparison of the Correlation radius exponent and xed point coordinates.

Table 2 .
Some critical point parameters.