symmetry and classes of transport equations

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Physica A 308 (2002) 292 – 300 www.elsevier.com/locate/physa Symmetry and classes of transport equations J.A. Cardeal a; b , A.E. Santana a ; , T.M. Rocha Filho a; c a Instituto de F sica, Universidade Federal da Bahia, Campus de Ondina, 40210-340 Salvador, Bahia, Brazil b Departamento de Ciˆ encias Exatas, Universidade Estadual, de Feira de Santana, 44031-460 Feira de Santana, Bahia, Brazil c Instituto de F sica, Universidade de Bras lia, 70919-970 Bras lia, DF, Brazil Received 4 June 2001; received in revised form 7 December 2001 Abstract A method for classication and determination of local transport equations is presented based on the use of Lie symmetries. The starting point is the choice of a specic symmetry group and the denition of a class of partial-dierential equations. Results associated with the Fokker–Planck equation in (1+1) and (2+1) dimensions are given. c 2002 Published by Elsevier Science B.V. 1. Introduction The existence of symmetries provides a powerful method to describe and solve several aspects in the study of physical systems. On one hand, symmetry transforma- tions can be used for classication purposes and for the determination of solutions of dierential equations, which describe the evolution of mechanical states (see for instance Refs. [1– 4]). On the other hand, since the seminal paper by Wigner [5], the analysis of the Lie-symmetry representation theory has provided solid foundations to describe and characterize the properties of elementary particles in the relativistic realm of the Poincar e group. In addition, this machinery of symmetry has also been developed to derive from the Galilei group the set of dierential equations describ- ing the non-relativistic quantum physics, as for example the Schr odinger equation and its generalizations for higher spin [6]. Given the intricate nature of many-body sys- tems, however, this program has not been fully developed to describe non-equilibrium statistical systems or to derive transport equations [7]. Corresponding author. Fax: +55-71-235-5592. E-mail address: [email protected] (A.E. Santana). 0378-4371/02/$ - see front matter c 2002 Published by Elsevier Science B.V. PII: S0378-4371(02)00616-7

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Page 1: Symmetry and classes of transport equations

Physica A 308 (2002) 292–300www.elsevier.com/locate/physa

Symmetry and classes of transport equationsJ.A. Cardeala;b, A.E. Santanaa ; ∗, T.M. Rocha Filhoa;c

aInstituto de F��sica, Universidade Federal da Bahia, Campus de Ondina,40210-340 Salvador, Bahia, Brazil

bDepartamento de Ciencias Exatas, Universidade Estadual, de Feira de Santana,44031-460 Feira de Santana, Bahia, Brazil

cInstituto de F��sica, Universidade de Bras��lia, 70919-970 Bras��lia, DF, Brazil

Received 4 June 2001; received in revised form 7 December 2001

Abstract

A method for classi0cation and determination of local transport equations is presented based onthe use of Lie symmetries. The starting point is the choice of a speci0c symmetry group and thede0nition of a class of partial-di4erential equations. Results associated with the Fokker–Planckequation in (1+1) and (2+1) dimensions are given. c© 2002 Published by Elsevier Science B.V.

1. Introduction

The existence of symmetries provides a powerful method to describe and solveseveral aspects in the study of physical systems. On one hand, symmetry transforma-tions can be used for classi0cation purposes and for the determination of solutionsof di4erential equations, which describe the evolution of mechanical states (see forinstance Refs. [1–4]). On the other hand, since the seminal paper by Wigner [5],the analysis of the Lie-symmetry representation theory has provided solid foundationsto describe and characterize the properties of elementary particles in the relativisticrealm of the Poincar@e group. In addition, this machinery of symmetry has also beendeveloped to derive from the Galilei group the set of di4erential equations describ-ing the non-relativistic quantum physics, as for example the SchrCodinger equation andits generalizations for higher spin [6]. Given the intricate nature of many-body sys-tems, however, this program has not been fully developed to describe non-equilibriumstatistical systems or to derive transport equations [7].

∗ Corresponding author. Fax: +55-71-235-5592.E-mail address: [email protected] (A.E. Santana).

0378-4371/02/$ - see front matter c© 2002 Published by Elsevier Science B.V.PII: S 0378 -4371(02)00616 -7

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J.A. Cardeal et al. / Physica A 308 (2002) 292–300 293

Symmetry properties of transport equations has raised interest in the literature indi4erent ways [8–11]. In particular Chen and Guo [12] obtained SL(1 + 1;R)⊗ U (1)as the symmetry group for the Fokker–Planck equation with non-linear drift term,describing the Rayleigh process. One diHcult aspect in this context is to derive 0nite-dimensional symmetries for Fokker–Planck like equations with arbitrary non-linear driftterms. In this paper this kind of problem is addressed by systematizing a method toderive and classify transport equations based on the notion of invariance of a partialdi4erential equation (PDE) under a symmetry group. In this way, as an application, weanalyze classes of partial PDE’s assuming the SL(1+1;R)⊗U (I) symmetry group. Thatis, as a starting point we use the sl(1 + 1;R)⊕ I Lie algebra to derive Fokker–Planckequations (in 1+1 and 2+1 dimensions) with Rayleigh drift terms as a particular caseof non-linear drift terms. We also consider the same problem with a six-dimensionalLie algebra which has as subalgebras the sl(1 + 1; R) algebra and the Weyl algebra.The investigation is then carried out for the subalgebras and the whole six-dimensionalalgebra. Once again linear and non-linear drift terms are derived.The structure of the paper is the following. In the next section we present an outline

of a standard method to calculate symmetries of PDE’s, adapting it to show how todetermine classes of equations imposing a symmetry algebra. In Section 3 the methodis applied to (1+1)-dimensional transport equations. Section 4 is devoted to the lesstrivial situation of considering transport equations in (2+ 1) dimensions. We close thepaper with some concluding remarks in Section 5.

2. Symmetries of PDE’s and transport equations

Let us consider a group G which is also a di4erential manifold (essentially a setwith coordinate systems such that the coordinate transformations are di4erentiable).Thus, each element p∈G is in correspondence to the coordinates pi = �i(p), withi = 1; : : : ; m; in Rm where m is the group dimension. The composition law is denotedby the dot “·”, and it can be expressed by using the group coordinate system. That isfor p; q; r ∈G and r = pq; we have

ri = fi(p; q); i = 1; : : : ; m (1)

for some given functions fi. If the functions fi are analytical, then G is said to bea Lie group. The analyticity of fi implies that these functions are C∞; but usuallythis condition can be relaxed to functions of the class Ck for some 0nite k. Forconvenience the identity element of the group will be associated to the origin of thecoordinate system.An important property of Lie Groups is that the neighborhood of the origin (the

identity element) uniquely determines the subgroup MG ⊂ G formed by all element ofG which can be connected continuously to the origin [1]. We can sum the coordinatesof two in0nitesimal elements of G and the result gives the coordinates of a thirdin0nitesimal element of G.Consider that the elements of G are the transformations in the set F of

functions of class C∞ on n variables. The action of an element g∈G on f∈F is

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294 J.A. Cardeal et al. / Physica A 308 (2002) 292–300

given by

f′(x) = f(x′) = g(1; : : : ; m)f(x) ; (2)

where x ≡ (x1; : : : ; xn). For an in0nitesimal transformation we have

x′i − xi = �m∑

k=1

k�ik(x) (3)

for some functions �ik . Then Eq. (2) can be written as

g(1; : : : ; m)f(x) =

1 +

∑i; k

kTk

f(x) ; (4)

where

Tk =∑i

�ik@@xi

; k = 1; : : : ; m (5)

are the generators of the elements of MG. The set of operators Tk expands a vector spaceto be denoted by g; which is equipped with an associative algebra structure, where theproduct is the usual subsequent applications of two operators, say TjTi.Given a set of di4erential operators of the form Eq. (5) expanding a m-dimensional

vector space, there is an associated Lie group provided that the following conditionsare satis0ed [1]:

[Ti; Tj] ≡ TiTj − TjTi =∑k

CkijTk ; (6)

where [Ti; Tj] is the commutator of Ti and Tj. The vector space g is thus also equippedwith an algebra with respect to the anti-symmetric product de0ned by the commutator,given in Eq. (6). The vector space g is the so-called Lie algebra of G and the constantsCkij are called the structure constants of the Lie group. A generic element G of the Lie

group G connected to the identity can then be written as

G = exp

(m∑

k=1

kTk

)(7)

with the coordinates k as 0nite numbers. One interesting aspect of the Lie theory isits association with PDE.Consider a PDE given by

�(x)�(x) = 0 (8)

de0ned in Rm with coordinates x=(x1; x2; : : : ; xm), where �(x) is a function in Rm and�(x) is a partial di4erential operator. A symmetry group of Eq. (8) is a local group oftransformations, say G, acting on Rm. Considering a symmetry transformation generatorL(x) ∈ g, where g is the corresponding Lie algebra of G, then G is symmetry groupof Eq. (8) if [1]

L(x)�(x)�(x) = 0 : (9)

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J.A. Cardeal et al. / Physica A 308 (2002) 292–300 295

The operator L(x) can be written as an expansion of k-linearly independent symmetrygenerators Ti ∈ g, that is

L(x) = �iTi(x) (10)

with �i constant (the sum rule for repeated indices is assumed). The Lie algebra of thegenerators Ti is speci0ed by Eq. (6). The invariance-condition Eq. (9) is equivalentlywritten as the following set of algebraic equations:

[Ti(x); �(x)] = ri(x)�(x); i = 1; : : : ; k ; (11)

where [ ; ] is the commutator and ri(x) are functions in Rm. This approach has beenused to analyze symmetry of di4erential equations in di4erent situations [2,12,13]. Ourproposal here in to apply this result based on Eq. (11) for the derivation of transportequations, proceeding in the following way:

(i) A class of DPE is given in a generic form as �(x)=∑

n an(x) f(@nx); where f(@nx)

is a function of the vector 0elds in Rm and an(x) are arbitrary functions.(ii) The generators Ti(x) are given in a speci0c realization.(iii) The determining Eqs. (11) are solved for �(x) following recipe (i).

The generators Ti will be selected as those ones associated with the Fokker–Planckequation. In this sense one can expect to derive Fokker–Planck equations for di4erentdrift terms, and so to obtain a better understanding about 0nding 0nite-dimensionalsymmetries for equations with non-linear drift terms. This procedure can in additionbe used as a guide for classifying transport equations from a systematic table of Liealgebras. In order to carry out the calculations, throughout the paper we have used apackage written for this purposes in the symbolic computation system MAPLE as ageneralization of the method presented in Ref. [4].

3. Fokker–Planck equation in (1+1) dimension

Let us consider a (1+1)-dimensional manifold with x1 → x; x2 → t, and �(x) givenby

�(x; t) = @t + a1(x; t)fx + a1(x; t)f(x)@x + a2(x; t)@xx ; (12)

where fx = @f(x)=@x. This Eq. (12) has the content of a Fokker–Planck equation butfor an unknown drift term (a1(x; t)fx + a1(x; t)f(x)@x = a1(x; t)@x · f(x)) as well asa di4usion term (a2(x; t)@xx). The coeHcients a1(x; t); a2(x; t) and the function f(x)will be determined using the symmetry condition given by recipes (i)–(iii) in lastsection, assuming Ti as the generators of the sl(1 + 1;R)⊕ I algebra in the followingrepresentation [12]:

T1 =1#@t +

$2�

+12; (13)

T2 =12e2#t(x@x +

1#@t + 1

); (14)

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296 J.A. Cardeal et al. / Physica A 308 (2002) 292–300

T3 =12e−2#t

(x@x − 1

#@t − #

�x2 − $

); (15)

T4 = 1 (16)

ful0lling the following Lie algebra:

[T1; T2] = 2T2 ; (17)

[T1; T3] =−2T3 ; (18)

[T2; T3] = T1 ; (19)

[Ti; T4] = 0; i = 1; : : : ; 4 : (20)

Using Eqs. (11) and Eqs. (13)–(16) we 0nd

a1(x; t) =1c(#x2 + $)($−b)=2$x(b−$)$ ;

a2(x; t) =−�

and

f(x; t) = cx−b=$(#x2 + $)($+b)=2$ ; (21)

where b; c and � are arbitrary constant. The 0nal form of the operator �(x; t) is then

�(x; t) = @t + #− bx2

+(#x +

$x

)@x − �@xx : (22)

Observe that the drift term in Eq. (22)

#− bx2

+(#x +

$x

)@x (23)

associated with the 0nite-dimensional algebra given in Eqs. (17)–(20), describes theusual Rayleigh process when b = $. The result for �(x; t) given by Eq. (22) is alsoan example of using Eqs. (11), with the arbitrary �(x; t) given in Eq. (12), to deriveprototypes for the Fokker–Planck equation with a given symmetry. In this case thefocus is the nature of the drift term.Consider then the following symmetry generators:

T1 =1#@t +

12; (24)

T2 =12e2#t(x@x +

1#@t + 1

); (25)

T3 =12e−2#t

(x@x − 1

#@t − #

�x2)

; (26)

T4 = 1 ; (27)

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J.A. Cardeal et al. / Physica A 308 (2002) 292–300 297

T5 = e#t@x ; (28)

T6 = e−#t(x2− �

2#@x

)(29)

with the associated algebra

[T1; T2] = 2T2; [T1; T3] =−2T3 ; (30)

[T2; T3] = T1; [T5; T6] = 12T4 ; (31)

[T1; T5] = T5; [T1; T6] =−T6 ; (32)

[T2; T5] = 0; [T2; T6] =�2#

T5 ; (33)

[T3; T5] =2#�T6; [T3; T6] = 0 ; (34)

[Ti; T4] = 0; i = 1; : : : ; 4 : (35)

We assume Eq. (12) as a generic form for �(x; t); but before to calculate the transportequation for the algebra given in Eqs. (30)–(35), we 0rst analyze its subalgebras.Observe that we have a subalgebra with the operators T1; T2; T3 and T4; satisfying

the commutation given in Eqs. (17)–(20), and realized in Eqs. (24)–(27) (which isbut a slightly di4erent realization for the generators given in Eqs. (13)–(16)). UsingEq. (12) we derive

a1(x; t) =1c# exp

(− b2#x2

);

a2(x; t) =−� ;

f(x; t) = cx exp(

b2#x2

);

where c and b are arbitrary constants. Therefore the operator �(x; t) reads

�(x; t) = @t + #− bx2

+ #x@x − �@xx : (36)

The operator term #− b=x2 + #x@x in this Eq. (36) is the drift term given by Eq. (23)but with $ = 0. This result shows that the realization of the algebra is also importantfor the explicit form of the drift term.Another subalgebra of Eqs. (30)–(35) is provided by the generators T1; T4; T5 and

T6. In this case, using Eq. (12) for �(x; t) and Eqs. (24), (27)–(29) we obtain

a1(x; t) =1c#x(#+b)=# ;

a2(x; t) =−� ;

f(x; t) = cx−b=#

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298 J.A. Cardeal et al. / Physica A 308 (2002) 292–300

resulting in

�(x; t) = @t − b+ #x@x − �@xx : (37)

As a consequence, we derive the usual linear drift term when b=−#.Finally let us study Eq. (12) under the whole symmetry given by the algebra

Eqs. (30)–(35), realized in Eqs. (24)–(29). The result is

a1(x; t) =#c;

a2(x; t) =−� ;

f(x; t) = cx ;

such that

�(x; t) = @t + #+ #x@x − �@xx : (38)

This is a Fokker–Planck equation with a (usual) linear drift term. ComparingEqs. (36)–(38) we see that the symmetry de0ned by the generators T1; T4; T5 and T6has a predominant e4ect over the symmetry de0ned by the generators T1; T2; T3 andT4 in determining �(x; t) under the whole symmetry given by Eqs. (30)–(35).

4. Fokker–Planck equation in (2+1) dimensions

In the previous section we have shown how to derive Fokker–Planck equationsin (1+1) dimensions considering a given Lie algebra. Here we consider a (2 + 1)-dimensional case with the symmetry satisfying the sl(2;R)⊕ I Lie algebra as given inEqs. (17)–(20), but now realized by [12]:

T1 =@t#+

12

($1�1

+$2�2

)+ 1 ; (39)

T2 =12e2#t(2 + x@x + y@y +

1#@t

); (40)

T3 =12e−2#t

[x@x + y@y − 1

#@t − #

(x2

�1+

y2

�2

)− $1

�1− $2

�2

]; (41)

T4 = 1 : (42)

The operator �(x; t) is written as

�= @t + a1(x; y; t)@xf(1)(x; y; t) + a2(x; y; t)@yf(2)(x; y; t)

+ a3(x; y; t)@xx + a4(x; y; t)@yy : (43)

With the generators given in Eqs. (39)–(42), following prescriptions (i)–(iii) inSection 2, the operator �(x; t), Eq. (43), reads

�=[#x +

$1x

+1�2x

F(yx

)]@x +

[#y +

$2y

− 1�1y

F(yx

)]@y

+2#− 1xy

G(yx

)− �1@xx − �2@yy + @t ; (44)

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J.A. Cardeal et al. / Physica A 308 (2002) 292–300 299

where G is an arbitrary function and F is given by

F(yx

)=

xy(I(y=x) + c1)�2x2 + �1y2 (45)

with c1 constant and

I(yx

)=−�1�2

∫ [$1 +

$2u2

− G(u)u

]du; u ≡ y

x: (46)

This is a Fokker–Planck equation with non-linear drift term without isotropy in space.Notice that the operator in Eq. (44) is not invariant under a permutation of x andy (and of indices 1 and 2), although the symmetry generators posses this property.Imposing this additional symmetry implies

F(yx

)=−F

(xy

)(47)

with solution

F(yx

)= c2 log

(yx

)(48)

with c2 constant and G given by

G(yx

)=[c2�2

(1 + log

(yx

))+ $1

]yx+[c2�1

(1− log

(yx

))+ $2

]xy: (49)

5. Concluding remarks

In this work we have systematized a method to derive transport equations when aspeci0c symmetry is given. The method is based on Eq. (11) and recipes (i)–(iii) givenin Section 2. As an application we have analyzed the Fokker–Planck equation underdi4erent symmetry Lie algebras, as the sl(1+1;R)⊕ I and the Weyl algebra, in (1+1)and (2+1) dimensions. This has provided a procedure to analyze di4erent linear andnon-linear drift terms, taking into account the e4ect of di4erent Lie symmetries. In thiscontext, the non-trivial problem about the existence of 0nite-dimensional symmetriesfor nonlinear drift terms has been addressed.Using this approach an interesting aspect is then to consider more general situations

in which non-linear or non-local collision terms are included. This aspect will beconsidered otherwise.

Acknowledgements

The authors would like to thank Prof. A. Ribeiro Filho for stimulating discussionsand the Referee for the suggestions. This work was supported by CNPq and CAPES(two Brazilian Government Agency).

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300 J.A. Cardeal et al. / Physica A 308 (2002) 292–300

References

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