Tsallis Entropy and the Vlasov-Poisson Equations

Home , Tsallis distribution, Vlasov equation

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 79

1;2;3 3;4

A. R. Plastino and A. Plastino

Faculty of Astronomy and Geophysics,

National University La Plata,

C. C. 727, 1900 La Plata, Argentina

CentroBrasileiro de Pesquisas F sicas CBPF

Rua Xavier Sigaud 150, Rio de Janeiro, Brazil

Argentine Research Agency CONICET.

Physics Department,

National University La Plata,

C.C. 727, 1900 La Plata, Argentina

Received 07 Decemb er, 1998

We revisit Tsallis Maximum Entropy Solutions to the Vlasov-Poisson Equation describing gravita-

tional N -b o dy systems. We review their main characteristics and discuss their relationshi p with

other applicatio ns of Tsallis statistics to systems with long range interactions. In the following con-

siderations we shall b e dealing with a D -dimensional space so as to b e in a p osition to investigate

p ossible dimensional dep endences of Tsallis' parameter q . The particular and imp ortant case of the

Schuster solution is studied in detail, and the p ertinent Tsallis parameter q is given as a function of

the space dimension. In the sp ecial case of three dimensional space we recover the value q =7=9,

that has already app eared in many applications of Tsallis' formalism involving long range forces.

I Intro duction

1 [f x] dx

S = 1

q 1

where q is a real parameter characterizing the entropy

Foravarietyofphysical reasons, muchwork has b een

functional S , and f x is a probability distribution de-

devoted recently to the exploration of alternativeor q

ned for x 2 R . In the limiting case q ! 1 the stan-

generalized information measures, based up on entropy

dard logarithmic entropy, S = f xlnf x dx,is

functionals di erent from the standard Boltzmann- 1

recovered. From an Information Theory viewp oint, 1

Gibbs-Shannon-Jaynes entropy[1]. Despite its great

constitutes an information measure distinct from that

overall success, this ortho dox formalism is unable to

of Shannon.

deal with a varietyofinteresting physical problems

The Tsallis entropy preserves or suitable generalizes

such as the thermo dynamics of self-gravitating systems,

the relevant features of the Boltzamnn-Gibbs entropy

some anomalous di usion phenomena, L evy ights and

[5, 6]. It has b een shown to b e compatible with the In-

distributions, and turbulence, among others see [1] for

formation Theory foundations of Statistical Mechanics

a more detailed list. The approachofJaynes to Statis-

given byJaynes [4] and with the dynamical thermostat-

tical Mechanics [2, 3] is also compatible with the p os-

ing approach to statistical ensembles [7]. Increasing at-

sibility of building up a thermostatistics starting with

tention is b eing devoted to exploring the mathematical

a nonlogarithmicentropy, as demonstrated by Plastino

prop erties of Tsallis formalism [8-14,30].

and Plastino [4]. Tsallis has intro duced a family of gen-

eralized entropies [5], namely, The Tsallis formalism has b een already applied to

80 A. Plastino and A. R. Plastino

astrophysical self-gravitating systems [16,17, 18], cos- The rst physical application of Tsallis entropywas

mology [19,20], the solar neutrino problem [21], L evy concerned with self-gravitational systems [16]. It is well

ights and distributions [22, 23, 24], phonon-electron known that the standard Boltzmann-Gibbs formalism

thermalization in solids [25], turbulence phenomena is unable to provide a useful description of this typ e of

[26, 27], low-dimensional dissipative systems [28,29], systems [33,34,35, 36]. Moreover, the connection b e-

non linear Fokker-Planck equations [30], etc. The inter- tween Tsallis entropy and gravitation provided the rst

ested reader is referred to [1] for additional references. hint of just which are the kind of problems where the

generalized formalism might b e useful.

Tsallis theory has also b een confronted with direct

Although the standard Boltzmann-Gibbs formalism

exp erimental and observational data. Tsallis MaxEnt

characterized by the value q = 1 of Tsallis parameter

distributions have b een shown to provide b etter de-

has great diculties in dealing with gravitation in three

scriptions than the ones given by the standard ther-

dimensional space, it shows no problems in the case of

mostatistics, for b oth the exp erimentally measured dis-

one or two dimensions. On the other hand, it is p ossible

tributions of pure electron plasmas [26], and the obser-

to obtain physically acceptable answers in D =3,ifwe

vational p eculiar velo city distribution of galaxy clusters

employ an appropriate value of q<7=9. These facts

[31]. Tsallis thermostatistics has also b een advanced as

suggest that dimensionality plays a decisive role within

a p ossible explanation for the solar neutrino problem

this context. The aim of the presentwork is to clarify

[21]. It is worth remarking that all these exp erimental

some asp ects of the application of Tsallis formalism to

evidences involve N -b o dy systems whose constituent

the Valsov-Poisson equations, and to generalize some

particles interact according to the r gravitational or

of the known results connecting Tsallis Entropy with

Coulomb p otential.

self-gravitating systems, in order to explore the p ossi-

Other alternative measures of entropy or informa-

ble dep endence of the Tsallis parameter q on the spatial

tion have also b een successfully applied in di erent

dimensionality D .

areas of theoretical physics. For instance, R enyi in-

formation is a very useful to ol in the study of chaotic

dynamical systems [32]. However, it lacks a de nite

II Gravitation and other Long

concavity, crucial for the discussion of thermo dynami-

Range Interactions.

cal stability. Tsallis entropy, on the other hand, do es

exhibit such an imp ortant mathematical prop erty, and

One of the main features characterizing the thermo dy-

was thus historically the rst one b esides the standard

namics of self-gravitating systems is that the total en-

Boltzmann-Gibbs employed to develop an entirely con-

ergy,aswell as other variables usually regarded as addi-

sistent Statistical Mechanics formalism [6, 4].

tive quantities, lose their extensivecharacter. This fact

suggests that a non-additiveentropic measure, like the Active research conducted during the last few years

one prop osed by Tsallis, may b e more useful in dealing is giving raise to a general picture of the physical sce-

with gravitation than the standard Boltzamnn-Gibbs narios where Tsallis theory provides a b etter thermo-

entropy. Indeed, non-extensivity is one of the main statistical description than the one given by the stan-

prop erties of Tsallis entropy. Given two non-correlated dard Boltzmann-Gibbs formalism. These physical sys-

systems A and B , with entropies S [A] and S [B ], the tems are characterized by the presence of long range

q q

Tsallis entropy S [A + B ] corresp onding two b oth sys- interactions, memory e ects, or a fractal phase space.

tems regarded as a single, unique, system is given by Summing up, they show nonextensive e ects.

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 81

1 =D 1

N = ; 3

S [A + B ]=S [A]+ S [B ]+1 q S [A]S [B ]: 2

q q q q q

1 =D

In the limit q ! 1, yielding Boltzmann-Gibbs thermo-

D b eing the spatial dimension. But this scaling with

statistics, the standard additive b ehaviour of entropyis

the numb er of particles is not applicable to the strict

recovered.

gravitational case, since the canonical ensemble for a

In the case of gravitational physics, the non-

given nite numb er of particles is meaningless. There

extensive b ehaviour is due to the long range charac-

is another fundamental feature of the thermo dynam-

ter of the gravitational interaction. The early applica-

ics of systems endowed with long range interactions

tions of Tsallis' Thermostatistics to gravitational sys-

that has not yet b een fully addressed by the simpli-

tems stimulated the exploration, within Tsallis formal-

ed mo dels discussed in [37, 38, 39]. This prop ertyis

ism, of other many b o dy problems with long range

the breaking of the translational symmetry of space that

interaction showing non-extensive e ects [37,38,39].

is implicitly assumed in standard thermo dynamics [40].

These studies considered systems with a Lennard-Jones

This prop erty of systems with long range interactions

likeinterparticle p otential, characterized by a repulsive

is illustrated by the p olytropic mo del discussed in [16]

b ehaviour at short distances together with an attrac-

and reviewed here.

tive long range comp onent falling like r . Itisim-

p ortant to realize that this kind of p otentials, while

III The Vlasov-Poisson Equa-

b eing useful in order to illuminate the thermo dynam-

tions.

ical consequences of long range forces, do di er in an

Vlasov equation constitutes one of the most imp or-

essential way from the gravitational interaction. The

tant mathematical to ols employed to mo del astrophys-

Gibbs canonical ensemble for a system of a N particles

ical self-gravitating N -b o dy systems like galaxies and

interacting via a Lennard-Jones-like p otential, enclosed

galaxy clusters. Let us consider a system of N identi-

within a b oxofvolume V ,iswell de ned and has a

cal stars of mass m. Vlasov equation reads [33]

convergent partition function. On the other hand, and

due to the singularity at the origin of the gravitational

@f @ @f @f

p otential, the Gibbs canonical ensemble for a system of

+ v =0; 4

@t @ x @ x @ v

N gravitationally interacting particles even if they are

where f x; v dxdv denotes the numb er of stars in

enclosed within a nite b ox has a divergent partition

the 2D -dimensional volume element dxdv in p osition-

function and thus is not well de ned. The non-extensive

velo city space, and the gravitational p otential xis

prop erties of the Lennard-Jones like gases are more ap-

given by the D -dimensional Poisson equation

parent when we try to de ne the N !1 thermo dy-

namic limit. The main results obtained so far in con-

r = D D 2V G: 5

nection with this issue deal with the exotic scaling laws

that are to b e employed in orther to de ne the thermo-

In the last equation G is the universal gravitational

dynamic limit in a sensible way. Cogent evidence has

constant, V denotes the volume of a D -dimensional

already b een obtained, showing that the correct scaling

unitary sphere, and the mass density given by

with particle numb er of the thermo dynamic variables

suchasinternal and free energy, is the one prop osed by

x=m f x; v dv : 6

Tsallis, that go es with NN , where

82 A. Plastino and A. R. Plastino

In the particular case of a central p otential r de- IV MaxEnt Distributions in

p ending only on the radial co ordinate r , the Laplacian

Stellar Dynamics

op erator adopts the form [41],

Presentday galaxies are describ ed by stationary solu-

tions to the Vlasov-Poisson equations. These equilib-

1 d d

2 D 1

r = r 7

D 1

r dr dr

rium con gurations are usually assumed to b e the result

of appropriate relaxation pro cesses such as the ones de-

It is imp ortant to realize that Vlasov equation can b e

noted by the names \phase mixing" and \violent relax-

cast in the form

ation" [33]. These kind of pro cesses are characterized

by a loss of memory ab out the initial conditions of the

=0; 8

system. It has b een argued that the nal equilibrium

or bit

state should b e determined from a MaxEntvariational

where the total time derivativeisevaluated along the

principle [35]. The maximization of Boltzmann' loga-

orbit of any star moving in the p otential . In the case

rithmic entropy under the constraints imp osed by the

of stationary solutions i.e.; @ f =@ t = 0, equation 8

conservation of the total mass and energy, leads to the

implies that f x; v dep ends on the co ordinates and the

stellar isothermal sphere distribution,

velo city comp onents only through integration constants

C of the motion in the p otential . This result,

2 3=2 2

f =A 2 exp= ; 13

f = f C ;::: C ; 9

est: 1 N

where stands for the velo city disp ersion and A is an

appropriate normalization constant. Unfortunately, the

constitutes the well known Jeans' Theorem in Stellar

isothermal sphere is characterized byanin nite total

Dynamics [33]. The solutions describing spherically

mass [35]. The real meaning of this unphysical result

symmetric systems with isotropic velo city distributions

is that the p osed variational problem do es not havea

dep end only on the energy, and it is convenient to write

solution. For any given star distribution f x; v , it is

them in the form

always p ossible to obtain a new distribution f x; v

with the same mass and energy but showing a larger

f = f ; 10

Boltzmann entropy [33]. For given values of total mass

and energy,entropy is not b ounded from ab ove. An

where

illustration of this feature of self-gravitating systems is

provided by the so-called \red giant structure". This

=x v ; 11

con guration is characterized by a high density inner

core surrounded by a diluted extended \atmosphere".

and

The core accounts for almost all the energy of the sys-

tem. The outer envelop e makes the main contribution

= : 12

to the total entropy. By increasing the concentration

of the core while simultaneously expanding the \atmo-

The quantities and are usually called the rela-

sphere", the entropy can b e raised as muchaswanted

tive energy and p otential, resp ectively. The constant

without changing the total energy. [33].

is chosen in suchaway that the relative p otential

vanishes at the b oundary of the system. Summing up, a MaxEnt approach based on the

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 83

Gibbs-Boltzmann entropic measure seems to b e unable thermostatistical formalism, based on a non-extensive

to characterize relaxed self-gravitating systems. This entropy, is able to deal in a physically sensible way

fact has motivated the exploration of alternative en- with self-gravitating systems. Moreover, the present

tropy functionals [36] of the form formalism is relevant in connection with many of the

empirical veri cations of Tsallis theory that wehave

up to now. Bogoshian's application of Tsallis entropy

S = C [f ] dx dv ; 14

to the pure electron plasma [26] is based essential ly in

C [f ] b eing a convex function that vanishes for f =0.

the same mathematical formalism as the one involved

Here enters Tsallis generalized entropy S . It has b een

in the discussion of stel lar polytropes. Furthermore, it

shown that adopting appropriate values for Tsallis pa-

has b een recently shown [31] that a Tsallis distribution

rameter q , the extremization of Tsallis functional S

ts the observed velo city distribution of galaxy clusters

under the constraints imp osed by the total mass and

inamuch b etter way than other mo dels based on the

energy leads to sensible stellar distributions. Indeed,

standard entropy. The reason for the success of Tsallis

the so-called stellar p olytrop es are obtained [16]. These

theory within this context, is presumably due to the

distributions have b een widely employed in the mo d-

long-range gravitational forces involved. And, so far,

elization of astrophysical ob jects like galaxies [33], al-

the stellar p olytrop es are the only detailed theoretical

though their relationship with Tsallis entropywas not

mo del establishing a link b etween Tsallis distributions

known. The mass density of a stellar p olytrop e b ehaves

and N -b o dy self-gravitational systems. For all these

in the same fashion as the density of a self-gravitating

reasons we b elieve that a further study of the relation-

sphere constituted by a gas with a p olytropic equation

ship b etween stellar p olytrop es and Tsallis statistics is

of state,

worthwhile, and will contribute to the understanding of

non-extensive systems.

p = const: : 15

V Tsallis MaxEnt Solutions.

The exp onent in the equation of state is related to

the p olytropic index n by

Nowwe will extremize the Tsallis entropy of the star

distribution function f x; v ,

=1+ : 16

f f dx dv ; 17 S =

Stellar p olytrop es constitute the most simple, but

q 1

still physically acceptable, mo dels for equilibrium stel-

under the constraints imp osed by the total mass

lar systems. In the earlier days of stellar dynamics,

p olytrop es where considered as p ossible realistic mo dels M

= f x; v dx dv ; 18

for galaxies and globular clusters. Nowadays it is known

and the total energy E ,

that stellar p olytrop es do not t prop erly the concomi-

tant observational data. However, they still play an im-

p ortant role in theoretical astrophysics, and are widely E 1

= f x; v v + x dx dv : 19

m 2

employed as a rst theoretical approach, b oth in numer-

ical simulations and analytical studies, for the descrip- In order to avoid some misunderstandings that arose in

tion of stellar systems. From our p oint of view they the literature in connection with this last formula [26],

are relevant b ecause they illustrate how a generalized wemust stress that the ab ove expression deals with the

84 A. Plastino and A. R. Plastino

system's gravitational p otential in a self consistentway. the MaxEnt distribution adopts the form of a stellar

The p otential app earing in 19 is generated by the p olytrop e,

mass distribution asso ciated with the distribution f

itself. We are not considering a given external p oten-

f = F 25

tial. In such a case, the appropriate expression would

with

; 26 =

q 1

E 1

= f x; v v + x dx dv : 20

F b eing constant. So far wehave used, as the con-

m 2

straints in our variational approach, the ordinary kind

Intro ducing now appropriate Lagrange multipliers

of mean values. However, within Tsallis formalism,a

and , our variational problem can b e put in the form

generalization of the concept of mean value has b een

E M

considered [6, 26]. If the ab ove calculations are re-done

S =0; 21

m m

using Tsallis mean values, a p olitropic distribution is

yielding the Tsallis MaxEnt solution

again obtained, but with a di erentvalue of the pa-

rameter q . This imp ortant p oint will b e discussed in

q 1

more detail in section IV.

f = [1 + 1 q q 1e]

The cut-o condition on the generalized MaxEnt

q 1

1 q +1

distribution was originally intro duced by Tsallis in a

e = const ; 22

q 1

rather ad-ho c way.However, within the present appli-

where

cation of Tsallis generalized thermostatistics, the cut-

o prescription admits of a clear physical interpreta-

v + 23 e =

tion. Within the p olytrop e stellar distributions, the

Tsallis cut-o corresp onds to the escap e velo city from is the energy p er unit mass of an individual star.

the system [33].

Making the identi cation

The mass density distribution asso ciated with the

1 q +1

p olitropic distribution is = 24

q 1

D 1 2

x=DV F v dv 27 v

This expression can b e simpli ed making the change of variables

2 2

v = 2 cos ; 28

wich yields

D=2 +D=2 2 +1 D 1

=2 DV F sin cos d : 29

Intro ducing now the p olitropic index

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 85

n = + ; 30

we obtain

2nD +1 D 1 D=2 n

sin cos d : 31 =2 DV F

This last equation can b e recast in the simpler form

= C ; 32

where C is now a constant given by

2nD +1 D 1 D=2

sin cos d : 33 C =2 DV F

Inserting this expression in the D -dimensional Poisson equation 5 we obtain a non-linear ordinary di erential

equation for recall that = ,

1 d d

D 1 n

r + D D 2V GC =0: 34

D 1

r dr dr

The expression for the mass density also allows us to obtain the radial derivative of the total mass M r contained

within radious r ,

D 1 D 1 n

= DV r = DV r C : 35

1 1

Inserting this last result in the di erential equation veri ed by , it is easy to see that

D 1

+ GD 2M r = const; 36 r

whichevaluated in r = 0 leads to

D 1

r + GD 2M r = 0: 37

From this last equation it follows that r is the limiting asimptotic b ehaviour still yielding a nite total

mass.

the di erential equation for reads VI D -dimensional Schuster

Spheres.

1 dh d

D 1 n

y + h =0: 40

D 1

In terms of the dimensionless variables

y dy dy

If we adopt the p olitropic index

h == ; 38

and D +2

; 41 n =

D 2

1=2 n1

the ab ove equation admits the solution ] r; 39 y =[D D 2V GC

1 0

86 A. Plastino and A. R. Plastino

The physical interpretation of the q -generalized mean

values p oses a dicult problem. For instance, wehave

hy = 1+ ; 42

D D 2

to deal with the baing fact that the mean value of

which, in term of the physical variables reads,

1 is not equal to 1. The original motivation for in-

2 2

tro ducing the generalized mean values was of a rather

j r

1+ = ; 43

D D 2

formal nature. Tsallis and Curado proved that, if we re-

where

place the standard mean values by the generalized ones,

all the usual thermo dynamical relations are preserved

1=2

j =[D D 2V GC ] : 44

within Tsallis formalism [6]. Wenow know, however,

that the connection with thermo dynamics still holds

From equation 37 it follows that the D -

true if we employ Tsallis entropy along with the stan-

dimensional Schuster Sphere shows the limiting asimp-

dard mean values [43, 44]. This last result somewhat

totic b ehaviour for a nite total mass.

weakened the case for the q -generalized mean values.

The exp onent app earing in the stellar p olitrop e

On the other hand, the generalized mean values show

distribution is now

some desirable prop erties in connection with thermo-

D 8 D 2

dynamic stability. Another advantage of the q -values

= n = : 45

2 2D 2

is that they are useful in order to characterize distribu-

Notice that for a space dimension D>D , where

tions with divergent ordinary moments. For instance,

hx i is divergent for L evy distributions while, employ-

2; 46 D = 2+2

ing an appropriate Tsallis parameter q di erent from

the D -dimensional Schuster p olitrop e is a monotonic

unity, hx i converges.

increasing function of the energy.

The rst discussion of Vlasov-Poisson dynamics in

connection with Tsallis statistics was done employ-

VI I Tsallis generalized Mean

ing the usual linear mean values [16]. Afterwards,

Values

Boghosian [26] reformulated those results in terms of

Tsallis generalized mean values constitute a remarck-

the q -generalized mean values. In order to appreciate

able comp onent of the generalized thermostatistical for-

Boghosian' contribution, it is useful to consider the fol-

malism that has not app eared yet in the present consid-

lowing expression for the kinetical and gravitational p o-

erations. Given a probability distribution f , the gener-

tential energies of the system

alized mean value of a quantity A is de ned as

hAi = Af d ; 47

T = m v f r; v dr dv 48

where d stands for the volume element on the appro-

priate phase space where the distribution f is de ned. and

r r

1 2

W = G dr dr

1 2

D 2

j r r j

1 2

f r ; v f r ; v

1 1 2 2

= Gm dr dv dr dv : 49

1 1 2 2

D 2

j r r j

1 2

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 87

In order to intro duce Tsallis generalized mean values, Boghosian de ned the following generalizations for the kinetic

and p otential energies,

2 q

T = m v f r; v dr dv ; 50

and

q q

f r ; v f r ; v

1 1 2 2

dr dv dr dv : 51 W = Gm

1 1 2 2

D 2

j r r j

1 2

where d is the volume element in the relevant phase Moreover, he prop osed to identify the mass density with

space where the distribution f \lives". These general-

ized mean values still do exhibit the convenient prop- r=m f r; v dv ; 52

erties of Tsallis' original q -moments. However, the new

Boghosian' approachistantamount to identify the ac-

normalized mean values are free of many of the trou-

tual stellar distribution with f instead of making the

blesome features of the original q -values. For instance,

identi cation with f itself. Consequently, a Tsallis dis-

wenow recover the desirable relation

tribution with parameter q , obtained by recourse to the

usual mean values, is equivalent to a Tsallis distribution

a la Boghosian with a new parameter q given by

hh1ii =1: 56

1 q

: 53 =

It would b e interesting to reformulate the q -

1 q q 1

nonextensive approach to stellar dynamics in terms

Hence, making use of the complete Tsallis formalism,

of the normalized q -mean values 55. When the

the Tsallis parameter characterizing the D -dimensional

normalized q -values are used, the nonextensive q -

Schuster distributions is given by

thermostatistics can b e reformulated in terms of ordi-

8 D 2

nary linear mean values [46]. This fact suggests that

: 54 q =

8 D 2 +2D 2

this new formulation of Tsallis formalism may b e par-

It is interesting to notice that in the limit D ! 2, we

ticularly useful in order to develop e the q -nonextensive

obtein q ! 1. This is consistent with the known fact

version of the BBGKY hierarchy [33]. This line of work

that the standard Boltzmann-Gibbs Thermostatistics

p oses formidable diculties when the unnormalized q -

is able to deal in a physically acceptable way with self-

mean values are used.

gravitating systems with spatial dimension less than 2

[34, 42]. In the case of D =3,we recover the known

value q =7=9 [26] corresp onding to the limit value of

VIII The q = 1 MaxEnt Time

q providing a p olytropic distribution with nit total

Dep endent Solutions to

mass.

the Vlasov- Poisson Equa-

A di erent kind of generalized mean values have

tions

b een recently intro duced by Tsallis and coworkers

[45, 46]. These mean values are de ned by

The Tsallis nonextensive thermostatistics is also useful

in order to obtain approximate time dependent solutions

f Ad

hhAii = ; 55

to the Poisson-Vlasov equations [17]. The q -MaxEnt

f d

88 A. Plastino and A. R. Plastino

scheme develop ed in [17] is based on the idea of follow- of M relevant dynamical quantities A x; v ,

ing the time evolution of the mean values of a small set

hA i = f x; v A x; v dx dv i =1;:::;M; 57

i q i

where h is just a constant with the dimensions of the elementofvolume in the p ertinentx; v space. If the

distribution function f x; v evolves according to the Vlasov equation the time derivative of the relevant mean

values is given by

1 d

hA i = f x; v v r A r r A dx dv i =1;:::;M: 58

i q x i v i

dt h

Unfortunatelly, this last set of equations do es not constitute, in general, a closed set of M ordinary di erential

equations for the mean values hA i . However, the MaxEnt approach allows us to build up, at each time t,a

i q

q -MaxEnt distribution

1=1q

f x; v ;t= ; 59 1 1 q tA x; v

q i i

i=1

taking as constraints the instantaneous values adopted tipliers turn out to b e canonicaly conjugate dynamical

by the relevant moments hA i and intro ducing appro- variables. Due to this Hamiltonian structure the asso-

i q

priate Lagrange multipliers i =1;:::;M. The ciated ow in phase space do es not have sinks, nor do es

right hand sides of equations 58 can then b e evaluated it have sources. Moreover, the Tsallis entropy S eval-

by recourse to this q -MaxEnt approximation. In such uated up on the MaxEnt approximate solution f x; v

a manner a closed system of equations is formally ob- is a constant of the motion. This means that the evo-

tained. The ensuing system, however, b ecomes highly lution of the system do es not exhibit any relaxation

non-linear. As a counterpart, the Vlasov equation, a pro cess. This is an imp ortant feature of the exact so-

partial di erential equation, b ecomes now a system of lutions that is shared by our MaxEnt approximations.

ordinary di erential equations for the evolution of the All these general prop erties of the q -MaxEntscheme do

relevant mean values. In [17] it has b een shown that not dep end on the dimensionality of the one-particle

such an approach is indeed useful b ecause it do es pre- con guration space. They hold true for anynumber of

serve some imp ortant prop erties of the exact evolution spatial dimensions. However, in the particular case of

equation. q = 1 and D =1 or D = 3, our MaxEnt approxima-

tion generates, if the relevant mean values are prop erly

The q -MaxEnt approxiamte solution f x; v ;t pre-

chosen, exact solutions to the Vlasov-Poisson equations

serves, by construction, the exact equations of motion

[17]. Indeed, various of the known exact solutions of

of our M relevant moments. These moments, along

the Valsov-Poisson equations turn out to b e particular

with the concomitant Lagrange multipliers, evolve ac-

instances of our general q -MaxEnt approach [17].

cording to a set of 2M ordinary di erential equations

that can b e given a Hamiltonian form [17]: the relevant

mean values hA i and their asso ciated Lagrange mul- It should b e stressed that these time dep endent q -

i q

Brazilian Journal of Physics, vol. 29, no. 1, March, 1999 89

[3] J. Aliaga, D. Otero, A. Plastino and A. N. Proto, Phys. MaxEnt solutions of the Vlasov equation are not re-

Rev. A37, 918 1988; E. Duering, D. Otero, A. Plas-

lated in a direct fashion to the previously discussed sta-

tino and A. N. Proto, Phys. Rev. A32, 2455 1985; E.

Duering, D. Otero, A. Plastino and A. N. Proto, Phys.

tionary MaxEnt solutions. The total energy do es not

Rev. A35, 2314 1987; A. R. Plastino and A. Plastino,

app ear as a constraint within the time dep endent sce-

Physica A232, 458 1996.

nario, as happ ens in the stationary case. This means

[4] A. R. Plastino and A. Plastino, Phys. Lett. A177, 177

1993.

that the q = 1value of the Tsallis parameter lead-

[5] C. Tsallis, J. Stat. Phys. 52, 479 1988.

ing to exact time dep endent solutions should not b e

[6] E. M. F. Curado and C. Tsallis, J. Phys. A24, L69

regarded as b elonging to the \allowed" q<7=9 range

1991; Corrigenda: 24, 3187 1991 and 25, 1019

1992.

of q -values yielding physically acceptable stationary so-

[7] A. R. Plastino and C. Anteneo do, Ann. Phys., 255,

lutions. These two sp ecial q -values, 7=9 and 1, arise

250 1997.

from quite di erent circumstances and, as far as we

[8] G. A. Raggio, J. Math. Phys. 36, 4785 1995.

know, are completely unrelated.

[9] G. R. Guerb ero and G. A. Raggio, J. Math. Phys. 37,

1776 1996.

[10] G. R. Guerb ero , P. A. Pury and G. A. Raggio, J.

IX Conclusions

Math. Phys. 37, 1790 1996.

[11] A. M. Meson and F. Vericat, J. Math. Phys. 37, 4480

We have considered MaxEnt Tsallis distributions

1996.

in connection with stationary solutions of the D -

[12] A. K. Ra jagopal, Phys. Rev. Lett. 76, 3469 1996.

dimensional Vlasov-Poisson equations. In particular,

[13] E. K. Lenzi, L. C. Malacarne and R. S. Mendez, Phys.

Rev. Lett. 80, 218 1997.

we obtained an analitical relation b etween Tsallis pa-

[14] S. Ab e, Phys. Lett. A 224, 326 1997.

rameter q and the dimension D of physical space for

[15] E. P. Borges, J. Phys. A: Math. Gen. 31, 5281 1998.

the sp ecial case of D -dimensional Schuster solutions.

[16] A. R. Plastino and A. Plastino, Phys. Lett. A174, 384

1993.

Although these solutions have in nite spacial extent,

[17] A. R. Plastino and A. Plastino, Phys. Lett. A193, 251

the density falls rapidly enough so as to have a nite

1994.

total mass. Furthermore, they show, within the larger

[18] J.J. Aly, in \N-Bo dy Problems and Gravitational Dy-

set of Tsallis MaxEnt solutions, the limiting b ehaviour

namics", Pro c. of the Meeting held at Aussois-France

21-25 March 1993, eds. F. Comb es and E. Athanas-

between sensible solutions with nite mass and unphys-

soula Publicatio ns de l'Observatoire de Paris, 1993.

ical solutions with divergent mass.

[19] V.H. Hamity and D.E. Barraco, Phys. Rev. Lett. 76,

4664 1996.

[20] D.F. Torres, H. Vucetich, and A. Plastino, Phys. Rev.

Lett. 79, 1588 1997.

Acknowledgments

[21] G. Kaniadakis, A. Lavagno and P. Quarati, Phys. Lett.

B 369, 308 1996.

A. R. Plastino gratefully acknowledges the hospi-

[22] P. A. Alemany and D. H. Zanette, Phys. Rev. E49, 956

tality of the Centro Brasileiro de Pesquisas F sicas

1994.

CBPF, where part of this researchwas done.

[23] D. H. Zanette and P. A. Alemany,Phys. Rev. Lett. 75,

366 1995.

[24] C. Tsallis, S. V. F. Levy, A. M. C. Souza and R. May-

References

nard, Phys. Rev. Lett. 75, 3589 1995.

[1] C. Tsallis, Physica A221, 277 1995; C. Tsallis,

[25] I. Kop onen, Phys. Rev. E55, 7759 1997.

Physics World 10, 42 July 1997.

[26] B. M. Boghosian, Phys. Rev. E53, 4754 1995.

See http://tsallis.cat.cbpf.br/bi bl io .htm for a complete

[27] C. Anteneo do and C. Tsallis, J. Mol. Liq. 71, 255

and up dated bibliography.

1997.

[2] E. T. Jaynes in Statistical Physics, ed. W. K. Ford

Benjamin, New York, 1963; A. Katz, Statistical Me- [28] C. Tsallis, A. R. Plastino and W.-M. Zheng, Chaos,

chanics,Freeman, San Francisco, 1967. Solitons, and Fractals 8, 885 1997.

90 A. Plastino and A. R. Plastino

[29] M. L. Lyra and C. Tsallis, Phys. Rev. Lett., 80,53 [38] L. S. Lucena, L. R. da Silva and C. Tsallis Phys. Rev.

1998. E51, 1995 6247.

[39] J. R. Grigera, Phys. Lett. A217, 1996 47.

[30] L. Borland, Phys. Rev. E57, 6634 1998.

[40] J. D. Brown and J. W. York Jr., in Physiscal Origins

[31] A. Lavagno, G. Kaniadakis, M. Rego-Monteiro, P.

of Time Assymetry, J. J. Halliwell , J. Perez-Mercader,

Quarati and C. Tsallis, Nonextensive Thermostatistical

and W. H. Zurek Editors, Cambridge Univ. Press.,

Approach to the Peculiar Velocity Function of Galaxy

Cambridge, 1996 pp. 465-474.

Clusters, Astrophysical Letters and Communications,

35, 449 1998.

[41] P. M. Morse and H. Feshbach, Methods of Theoretical

Physics, McGraw-Hilll, New York, 1953.

[32] C. Beck and F. Schlo egl, Thermodynamics of chaotic

systems, Cambridge UP, Cambridge, 1993.

[42] G. Rybicki, Astrophys. and Space Sci. 14 1971 56.

[33] J. Binney and S. Tremaine, Galactic Dynamics, [43] A. Plastino and A. R. Plastino, Phys. Lett. A 226, 257

Princeton U.P., Princeton, 1987.

1997.

[44] R. S. Mendes, Physica A 242, 299 1997.

[34] W. C. Saslaw, Gravitational Physics of Stel lar and

Galactic Systems, Cambridge University Press, Cam-

[45] C. Tsallis, R. S. Mendes and A. R. Plastino, Role of the

bridge, 1985.

Constraints Within Generalized Nonextensive Statis-

tics,Physica A 261, 534 1998.

[35] D. Lynden-Bell, Mon. Not. R. Astron. So c. 124 1967

279.

[46] R. P. Di Sisto, S. Mart nez, R. B. Orellana, A. R. Plas-

tino, and A. Plastino, General ThermostatisticalFor-

[36] S. Tremaine, M. Henon and D. Lynden-Bell, Mon. Not.

malisms, Invariance Under Uniform Spectrum Trans-

R. Astron. So c. 219 1986 285.

lations, and Tsal lis q-Additivity, 1998 Physica A,in

[37] P. Jund, S.G. Kim and C. Tsallis, Phys. Rev. B52

press. 1995 50.