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Physics Letters A 373 ( 2009 ) 4012-4016

Contents lists available at ScienceDirect

Physics Letters A


Statistical theory of self-similarly distributed fields

Alexander Olemskoia[1], Irina Shudab

a Institute of Applied Physics, National Academy of Sciences of Ukraine, 58, Petropavlovskaya St., 40030 Sumy, Ukraine b Sumy State University, 2, Rimskii-Korsakov St., 40007 Sumy, Ukraine



Article history: Received 10 July 2009 Accepted 19 August 2009 Available online 19 September 2009 Communicated by C.R. Doering

PACS: 02.20.Uw 05.30.Pr


A field theory is built for self-similar statistical systems with both generating functional being the Mellin transform of the Tsallis exponential and generator of the scale transformation that is reduced to the Jackson derivative. With such a choice, the role of a fluctuating order parameter is shown to play deformed logarithm of the amplitude of a hydrodynamic mode. Within the harmonic approach, deformed partition function and moments of the order parameter of lower powers are found. A set of equations for the generating functional is obtained to take into account constraints and symmetry of the statistical system.

© 2009 Elsevier B.V. All rights reserved.

Keywords: Field theory Generating functional Jackson derivative Deformation

1. Introduction

A formal basis of the statistical theory, using quantum field methods, is known to be a generating functional which presents the Fourier-Laplace transform of the partition function from the dependence on the fluctuating distribution of an order parameter to an auxiliary field [1]. Due to the exponential character of this transform, determination of correlators of the order parameter is provided by ordinary differentiation of the generating functional over auxiliary field.

Above scheme becomes inconsistent with passage from simple systems to complex ones because the phase space gets forbidden regions and the phase flow does not ensure statistical mixing [2]. As is known from the theory of critical phenomena, analytical de­scription of complex systems is achieved in the presence of the scaling invariance only [3]. Because in this case the role of a basic function plays the power-law function instead of the exponential one, we need in use of the Mellin transform at constructing of the generating functional. Moreover, one should introduce the Jackson derivative as a generator of the scaling transformation instead of the ordinary derivation operator.

This Letter is devoted to building a field-theoretical scheme based on the use of both Mellin transform and Jackson deriva­tive. The work is organized as follows. In Section 2 we adduce necessary information from the theory of quantum calculus. Sec­tion 3 is devoted to construction of the generating functional and finding its connection with related correlators. As the simplest ex­ample, the harmonic approach is studied in Section 4 to obtain the partition function and the order parameter moments of the first and second powers in dependence of the deformation param­eter. In Section 5 we introduce pair of additive functional whose expansion into deformed series yields both Green functions and proper vertices. Moreover, we find here formal equations govern­ing by the generating functional of systems possessing a symmetry with respect to a field variation and being subjected to an arbitrary constrain. Section 6 concludes our consideration.

2. Preliminaries

We begin by citing an information from the quantum calculus [4,5] that will be needed below. A basis of this calculus is the di­latation operator D% := Xxdx being determined by the deformation parameter X and the differentiation operator dx = д/dx. Expanding formally the operator DX into the Taylor series, it is easily to define its action onto the power-law function: DXxn = (Xx)n. Similarly, the expansion of an analytical function f (x) shows that, in correspon-

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A. Olemskoi, I. Shuda / Physics Letters A 373 (2009) 4012-4016 4013

dence with the denomination, the operator DX arrives at the di­latation X of the argument of this function: DX f (x) = f (Xx).Aset of eigen-functions of the dilatation operator is reduced to a homo­geneous functions h(x) defined by the equality DXh(x) = Xqh(x) with the self-similarity degree q. In general case, this function takes the form h(x) = AX(x)xq where a factor AX(x) is obeyed the invariance condition AX(Xx) = AX(x).Itisconvenienttopassfrom the dilatation operator to the Jackson derivative

DX - 1

\= -

x     (X - 1)x


in accordance with the commutation rule \Р\, x] = DX. The action the Jackson derivative onto the homogeneous function is given by the relations:

(xV^h(x) = [q]xh(x),   [q]x :

Xq - 1 X- 1 ' (2)

Basic deformed number [q]X represents a generalization of the ex­ponent of the homogeneous function.

Principle peculiarity of self-similar statistical systems is that their consideration is based on the use of the deformed logarithm and exponential [2]

lnq (x) :=

x1-q - 1 ~1

expq(x) := [1 + (1 - q)x]




where [y]+ = max(0, y). Moreover, one needs to deform the sum and product as follows:

x ®qy = x + y + (1 - q)xy,

x ®qy = [x1-q + y1-q - 1]_



here, in the last equality x, y > 0. Respectively, the deformed prod­uct of n > 1 identical multipliers gives the expression of the de­formed power-law function:

x ®qx ®q ■■■®qx = [nx1  q - (n - 1)]


Making use of the rules (4) shows that functions (3) are obeyed the conditions

lnq(x ®q x) = lnqx + lnq y,        lnq(xy) = lnqx ®q lnq y; expq(x + y) = expq(x) ®q expq(y),

expq(x ®qy) = expq(x) expq(y). (6)

3. Generating functional

As mentioned above, the generating functional of self-similar systems is defined by the Mellin transform

Zq{ J} := j Zq{ф}фJ N



where index i runs over lattice sites with number N -^oo.[2] Ac­cording to Ref. [2], the partition functional Zq{ф} = expq(-S{ф}) is reduced to the deformed exponential with the exponent being

inverse action S = S^} determined by the order parameter distri­bution. Functional Zq{ J} canbepresentedbythedeformedseries [5]

n=0 i1...in


x (Ji2 - X) ■■■{ Jin - X


where basic deformed factorial [n]X! = [1]X[2]X... [n]X is deter­mined by the product of related numbers (2), while the coefficients

Ztn = On Vl)( Ji2 V

■■■ in

lJi1 ^ Jin =[3]


are given by the Jackson derivative (1).Bydefinition,

(Ji vJ) Ф j-1\

_ фх - Фі

Ji=[4]   (X - 1)Фі

-1 = а(фі)^lk-1, (10)

where one denotes о(фі) = 1п2-х(Фі). Then, the Jackson derivation of the functional (7)

(Ji V). )Zq{ J }\h =1 =[ Za {ф}Оі і) dфi]_[ [5]k

= j оі (ФіЩ (Фі) dфi, (11)

arriving at the function Zq(фі) := f Zq^}Y\k= [6] dфk,yields the first moment of the deformed order parameter:

S = (о(Фі)) := Z- f 2-х(Фі)Zq(Фі) dфi,

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