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Study of the higher-order correlation functions of number fluctuations in simple fluids with radial
many-body interactions
K. Knast, W. Chmielowski
To cite this version:
K. Knast, W. Chmielowski. Study of the higher-order correlation functions of number fluctuations in simple fluids with radial many-body interactions. Journal de Physique, 1981, 42 (10), pp.1373-1385.
�10.1051/jphys:0198100420100137300�. �jpa-00209329�
Study of the higher-order correlation functions of number fluctuations in simple fluids with radial many-body interactions (*)
K. Knast and W. Chmielowski
Nonlinear Optics Division, Institute of Physics, A. Mickiewicz University, Pozna0144, Poland
(Reçu le 22 décembre 1980, révisé le 1 er juin 1981, accepté le 24 juin 1981)
Résumé. 2014 En traitant les fluctuations de densité 0394N(K, t) comme un processus stochastique complexe, on étudie
le problème de leurs fonctions de corrélation d’ordre élevé dépendantes du temps. Ces fonctions apportent des informations sur la statistique des processus stochastiques et prouvent que le processus de 0394N(K, t) est, en général,
non Gaussien. Chacune de ces fonctions peut être exprimée par deux composantes dont l’une est liée aux atomes libres (sans interactions) et l’autre aux atomes corrélés dans le temps et l’espace. Les fonctions de corrélation dépen-
dantes du temps, que nous avons calculées, possèdent les mêmes propriétés que les fonctions calculées dans le cas de la limite thermodynamique. On montre que les fonctions des fluctuations de densité étudiées sont cohérentes
avec l’ensemble grand canonique de Gibbs, ce qui permet de les exprimer par des fonctions de répartition d’équi-
libre m-atomiques g(m)(R1, R2, ..., Rm) ou, thermodynamiquement, par les fonctions de la compressibilité isother- mique 03B2T.
Abstract. 2014 Dealing with number fluctuations (NF) 0394N(K, t), as a complex stochastic process, a discussion is
given of the problem of higher-order time-dependent correlation functions of NF. The functions convey information
on the statistics of the stochastic process and show the 0394N(K, t) process to be, in general, non-Gaussian. Each of the correlation functions can be expressed by two components, the one related with free (non-interacting) atoms and
the other with atoms correlated in space and time. The time-dependent correlation functions calculated in this paper are shown to be in agreement with the properties of the same functions, calculated for the case of the thermo-
dynamical limit. The correlation functions of NF under investigation are then shown to conform to the Gibbs grand canonical ensemble, permitting their expression in terms of equilibrium m-atomic distribution functions
g(m)(R1, R2, ..., Rm) or, thermodynamically, in terms of the appropriate functions of isothermal compressibility 03B2T.
Classification
Physics Abstracts
05.40
1. Introduction. - Number fluctuations (NF) play
a very important role in all physical phenomena involving statistical inhomogeneity of media. Smolu- chowski [1] has shown that, on the microscopic level,
no medium is completely homogeneous since chaotic thermal motion of the atoms leads, within elementary volumes, to unceasing spontaneous NF i.e. to sta- tistical inhomogeneity. Significantly, all sorts of optical phenomena are highly sensitive to statistical inhomo-
geneity of the médium, in particular phenomena of light scattering in material media, where NF cause local
variations of the refractive index [2]. According to
Smoluchowski [1], the steep increase of NF in the
vicinity of the critical point is apparent as critical opalescence. The thermodynamical theory of light scattering on NF is due to Einstein [3], Brillouin [4],
whereas their statistical-microscopic treatment is due
to Ornstein and Zernike [5], Yvon [6], Fixman [7]
and Kielich [8], who have shown that depolarization
(*) This work was carried out under Research Project MR. 1.9.
LE JOURNAL DE PHYSIQUE - T. 42, NO I O, OCTOBRE 1981
of the scattered light is directly related with the two- atom radial distribution function g(2)(Rh R2) account- ing for radial correlations of pairs of atoms, as well
as with higher distribution functions g(3)(Rb R2, R3)
and g(4)(R1, R2, R3, R4). In accordance with the Gibbs
grand canonical ensemble [9], the two-atom distri-
bution function g(2)(R1, R2) is related with the mean
square number fluctuation of atoms.
The development of spectroscopic techniques
enhanced interest in NF ; herein, especial importance
was given to stochastic approaches, permitting the
determination of time-dependent fluctuations. The stochastic treatment of fluctuations originating in
the work of Smoluchowski was developed by Chan-
drasekhar [10] and Kac [11], and has recently been analysed by Brenner et al. [12].
The problem of dynamical light scattering on NF
has been discussed by Komarov and Fischer [13]
as well as Pecora [14], applying the formalism of van
Hove’s [15] space-time correlation function, whereas
the hydrodynamical approach to the fluctuations is
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420100137300
applied by Mountain [16] and Benedek et al. [17].
The recourse to homodyne spectroscopy [18] in the study of light scattering in atomic systems without interaction has enabled Schaefer and Berne [19] to
establish that, as a result of NF, the electric field of the
light scattered is not a Gaussian stochastic process.
In the general case of interacting atoms a non-Gaussian correction arises specifically accounting for the inter- actions.
In the present paper we discuss the problem of higher-order corrélation functions of NF as well as
their properties. The NF are dealt with in general as
a complex stochastic process, so the AN(K, t) are
functions of a wave vector K and the time t. Van Hove’s formalism is applied and generalized many-
body atomic correlation functions are introduced.
The properties of the stochastic process are discussed for statistically independent as well as for space and time correlated atoms. The results obtained should be in agreement, in the limiting case (K - 0, t -+ 0),
with those derived with the grand canonical ensemble the form of which for statistically independent atoms (perfect gas) is that of the Poisson distribution [10].
Next, in the same limiting case, the thermodynamical properties of the higher-order correlation functions of NF are discussed expressing the functions in terms of the isothermal compressibility of the medium.
Problems related with the higher-order correlation functions are highly essential e.g. in critical phenomena, multiple [20, 21] light scattering, as well as usual scattering within the method of intensity correlation
[22].
2. Time-dependent stochastic corrélation functions of number fluctuations. - We consider a macroscopic, homogeneous and isotropic system of a great numbers N > of optically isotropic like atoms or molecules.
The mean number density of atoms is expressed by ( n > = N >IV, with V- the volume of the system. The local microscopic number density at a point R. in V at the moment of time tm is defined as follows [15] :
rCX1(tm) being the position vector of the 01531-th atom at tm. The spatial Fourier transform of (1) leads to the macros- copic quantity :
defining the K.-dependent number of atoms present in V at the moment of time tm. On introducing the following
definition of the microscopic van Hove correlation function of the mth order [23] :
we write the mth order correlation function of NF
where ± occurs « + » is for odd m and « - » for even m.
In (4), use is made of total van Hove correlation functions defined in conformity with the definition of fluctuations as :
where m
J
is Newton’s symbol. , , zFor the values m = 2, 3 and 4, the general expression (5) yields the following formulae of total van Hove
correlation-(’) functions of the second, third and fourth order :
(1) Total van Hove correlation functions in limiting case t - 0 (equilibrium case) are in agreement with cluster functions (see : Ursell, H. D., Proc. Cambridge Philos. Soc. 23 (1927) 685).
where the right-hand terms have to be written out symmetrically in all permutations R1, tl, ..., R4, t4.
In accordance with the definition
the mth order van Hove correlation function Gm(R1, ti, R2, t2, ..., Rm, t.) introduced by the expression (3) can be
written in the form of a sum of three components :
Equation ( 10) results from the irreducible decomposition of the m-fold summation in (9) : for a 1 = a2 = ... = (Xm
we have the o self » term G",sm(R1, 11’ R2, 12’ ..., R"" lm); for (Xl =f:. a2 # ... # a -thé « distinct » term GmD(R1, tl,
R2, 12’ ..., Rm, lm) whereas for all the others (Xl #- a2 = ... = (Xm, (Xi = (a2 #- ... = am, (Xl = a2 = ... #- am,
(X #- (a2 #- ... = (Xm etc. - the « mixed » term Gm(M)(R 1, tb R2, t2, ..., Rm, tm). For m = 2, the irreducible decom-
position of the double sum leads to the well known expression
The van Hove correlation function of mth order (9) is related with some arbitrary group of m atoms, selected from all the ( N > atoms of the system, and gives the probability of finding these m atoms at the space points R1, R2, ..., Rm in the moments of time ti, t2, ..., tm respectively, this probability being of a conditional type.
For m = 2, the « self » term
gives the probability of finding an atom at ti in the point RI under the condition that, at t2, the same atom had been in the point R2 ; whereas the « distinct » term
gives the probability of finding, at tl, an atom in the point Ri under the condition that, at the moment of time t2,
another atom had been at R2.
In the papers by Egelstaff et al. [23, 24], Groome et al. [15] and Gubbins et al. [26], the van Hove correlation function of 3rd order G3(R1, tl, R2, t2, R3, t3) has been studied both theoretically and experimentally.
2.1 STATISTICALLY INDEPENDENT ATOMS. - The mth order correlation function of NF relevant to our
considerations can be expressed as the sum of two components :
The first component ... >FA defines the contribution from free (stochastically independent) atoms, whereas the other component, ( ... > cA contains all contributions originating in atoms correlated in space and time.
To the functions (14) there correspond the total microscopic functions
the first component corresponding (as in (14)) to a lack of interatomic correlations and the other to the presence of mutual correlations. For several particular cases (m = 2, 3 and 4) we can write, with regard to equations (6)- (10),
Similarly, higher orders of the functions H m FI (Rl, tl, R2, t2, ..., Rm, t.) can be calculated. On insertion into (4),
m B
they permit the determination of correlation functions of NF fl
j=1
1 AN( + K j, t j)/
valid for atomic systems ofnon-interacting atoms. Table 1 contains the correlation functions thus calculated for several values of the index m.
In table I, we have introduced the following notation :
where ô,,,o is Kronecker’s symbol. Formula (20) is a result of the macroscopic isotropicity of the medium when
G:;> is a function pf the relative positions (Ri - R2, R2 - R3, ...) of the atoms whereas the exponential factor
is dependent on the sum of the atomic positions.
Table 1 renders apparent certain properties of the stochastic process described by AN(K, t). Namely, a complex stochastic process of this kind is non-Gaussian, it being possible to separate the Gaussian and non-
Gaussian parts. Thus, the correlation functions of NF for systems of free atoms can be written in the following
form :
Table I. - Gaussian and non-Gaussian components of the
correlation function of number fuctuations for a system of statistically independent atoms.
(*) The right-hand terms of the above expressions should be expanded in permitted permutations (K1, K2, K3, K4, K5, K6, (1’ t2, t3, t4l t il t6). These are : pairs -(1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (3, 4), (3, 6), (4, 5), (5, 6) ; triplets -(1, 2, 3), (4, 5, 6), (1, 2, 4), (3, 5, 6), (1, 3, 4), (2, 5, 6), (1, 5, 6), (2, 3, 4), (1, 2, 6), (3, 4, 5), (2, 3, 6), (1, 4, 5), (1, 4, 6), (2, 3, 5), (1, 2, 5), (3, 4, 6), (2, 4, 5), (1, 3, 6) ; quartets - (1, 2, 3, 5), (l, 3, 4, 5), (1, 2, 4, 6), (1, 3, 5, 6), (2, 3, 4, 6), (2, 4, 5, 6), (1, 2, 3, 4), (l, 2, 3, 6), (1, 2, 5, 6), (1, 3, 4, 6), (1, 2, 4, 5), (2, 3, 5, 6), (3, 4, 5, 6), (2,3,4,5),(1,4,5,6).
where...> ’and ( ... ) NG are the Gaussian and non-Gaussian parts of the correlation function, respectively.
From table I, the Gaussian part has the following properties :
where we have introduced the notation (2 n - 1) ! ! = 1.3.5... (2 n - 1) and n ! = 1.2.3 ... n. Obviously,
in the general case for Kj =1= 0 (for j = 1, 2,..., 2 n) we obtain the well known theorem of Reed [27]
the Gaussian part of the correlation function being dependent on certain combinations of the functions
P2(Kl, K2, 11’ t2) which, in turn, contain information on the behaviour of the N> atoms in two moments
of time ti and t2. Approximating the atomic motions by translational diffusion [10], we can write
where D is the translational diffusion coefficient..In table 1 we moreover give the non-Gaussian parts
(
Il m DN( ± K j, t j) NGNG of the correlation function of NF originating essentially in the functions pm(K 1, K2, ... ,j =1 /FA
Km’ t1, t2, ..., lm) for m > 2 and conveying information regarding the behaviour of the atoms in three, four, etc.
moments of time. These functions are in the limiting case
with regard to the general definition (19). Certain approximate forms of these functions for m = 3, 4 have been
construed by Brenner et al. [12] on the basis of the short-time approximation.
2.2 ATOMS MUTUALLY CORRELATED IN SPACE AND TIME. - We proceed with our considerations of the correlation function of NF related with mutual space-time correlations of the latter in the medium. The function, namely :
is related with the microscopic total van Hove function HmA(R1, tl, R2, t2, ..., Rm, tm), directly defining the mutual
correlations of the atoms. For several values of m, table II gives expressions enabling us to write the functions in terms of van Hove functions of the « self » type G(s), « distinct » type G2», and « mixed » type Gm(M).
By analogy to the expression (21), which holds for free atoms, we can write the following expression :
valid for the case of atoms correlated in space and time. Its microscopic counterpart is given by :
where H,c,A, and HÀkG are, respectively, the Gaussian and non-Gaussian components of the van Hove function.
The Gaussian component has the properties
Table II. - Total van Hove functions HmcA(R1, tl, R2, t2, ..., Rm, tm) for correlated atoms, for several values of m.
(*) In the right-hand terms of these expressions, all possible permutations of the variables leading to symmetrization of the individual components should be taken into account.
The microscopic total van Hove functions H;A(Rl’ tl, R2, t2, ..., Rm, tm) convey highly important informa-
tion concerning the motions, in liquids, of two, three, four and m correlated atoms. Thus, in accordance with (29),
the Gaussian components HmG(R1, ti, R2, t2, ..., Rm, tm) inform us of the space and time correlation of a pair of
atoms, whereas information on correlations of higher orders is contained in the non-Gaussian components
HcA MN CA(R1, tl, R2, t2, ..., Rm, tm). The analytical form of these functions is not easily obtained and so poses a separate problem ; for the sake of simplicity, various model approximations are introduced based on the equili-
brium properties of the functions in question. Vineyard [28] has proposed the convolution approximation for
the simplest of them, H2A(Rl, tl, R2, t2), whereas Nelkin and Ranganathan [29] have discussed a different
approximation to it based on a linearized Vlasov equation.
The problem has also been dealt with in a paper by Kerr [30J. A general approach to the problems of time- dependent correlation functions, characterizing certain stochastic processes, is due to Mori [31].
3. The equilibrium properties of the corrélation functions of number fluctuations. - We proceed with our
considerations for correlation functions of NF calculated in the limiting case
leading to mean values of higher powers of AN. By equation (4) we can write
where the Hm(R1, R2, ..., Rm) are total equilibrium van Hove functions, being the limiting case of the dynamical
functions Hm(R1, t1, R2, t2, ..., Rm, tm). On going over to the limit t1 = t2 = ... = tm -+ 0 in the right- and left-
hand terms of (5), we obtain
the van Hove functions Gm(R1, R2, ..., Rm) which are the limiting case of the dynamical functions Gm(R1, ti,
R2, t2, ..., Rm, tm). Further properties of the van Hove functions Hm(P-1, R2, ..., Rm are to be obtained from
the fact that a system admitting of NF conforms to the grand canonical Gibbs ensemble, which gives the follow- ing normalization condition for the m-atomic distribution (correlation) function [32]
The right-hand term of equation (32) can be re-written introducing first-order Stirling numbers Sk(m) i.e. the microscopic counterpart of (33) has the form
where b(Rm-k - R. - k 1 ) is Dirac’s delta function.
The Stirling numbers of the first-order fulfil the following relations [33] :
the B(m)m-k being Bernoulli numbers of order m. By (33) and the definition (3), the van Hove functions
Gm(R1, R2, ..., Rm) can be written in recurrential form
permitting the calculation of concrete forms of the functions for various values of m. With regard to (32) and (36),
the total van Hove functions Hm(R1 R2, ..., Rm) are related with the many-atomic distribution functions
g(m)(R1 R2, ..., Rm) by the formula
As done by us in section 2 we now go on to consider’the two particular cases of statistically independent atoms (no correlations - perfect gas), and atoms with mutual spatial corrélations.
3.1 STATISTICALLY INDEPENDENT ATOMS. - In this case the mean values of the higher powers of number
fluctuations (AN)"" ) are obtained immediately applying in table 1 the (perfect gas) condition (Eq. (25))
which leads to the concrete expressions given in table III. Equation (40) is equivalent to the conclusion of Ziff [39],
who calculated the number fluctuations by the method of cumulants and showed that, for the perfect gas, the cumulant of arbitrary mth order is equal to N >. Table III reveals another property of a system of statistically independent atoms (perfect gas) : the system obeys the Poisson statistics [10]
in accordance with the grand canonical Gibbs ensemble which, for the perfect gas, is of Poisson form.
In table III we have performed the separation of the Gaussian and non-Gaussian parts, the former resulting
from the distribution
It is well known that the Poisson distribution (41) does not lead to a Gaussian distribution and can be approxi-
mated by such a distribution in a first approximation only.
Table III. - Mean values of higher powers of number fluctuations (0394N )m >FA = (0394N )m>GFA + (AN)m )FA for the perfect gas, where (AN)m >GFA and (AN )"’ )FA
are, respectively, Gaussian and non-Gaussian parts.
A closer approximation is one of the following form :
where ck The distribution function (43) is not, with regard to its rather compli-
cated form, readily applicable.
3.2 SPACE-CORRELATED ATOMS. - If the atoms interact with one another, the van Hove function charac-
terizing the fluctuations HmA(R1, R2, ..., Rm) becomes a rather highly complicated function of many-atomic
distribution functions g(m)(Rl, R2, ..., Rm), characterizing the mutual radial correlations of selected groups of
m atoms. Table IV shows how the functions HmA(R1, R2, ..., Rm) are related with the functions g(m)(R1, R2, ..., Rm),
which characterize the equilibrium properties of the group of m atoms (here, too, we have separated the Gaussian part HmG from the non-Gaussian part HMNUI*
The expressions show that the mean values of the second, third, fourth and higher powers of the NF are
dependent on the radial correlations of two, three, four, and greater numbers of atoms defined by the many-atom radial distribution functions g(m)(R1, R2, ..., Rm). The expressions simplify partly for systems isotropic and homogeneous in the macroscopic meaning for which the one-atomic distribution function g(1)(R1) = 1. The
Table IV. - Gaussian HmG(R1, R2, ..., Rm) and non-Gaussian HmN;(R1, R2, ..., Rm) parts of the van Hove functions for some values of m.
expression for m = 2 is well known from classical statistical mechanics ; it relates the mean square of the AN
to the bi-atomic radial distribution function. The further, detailed discussion of the fluctuations of higher orders requires the application of the superpositional approximations of Kirkwood [34] and Cole [35]. Certain pro-
perties of the correlation functions in simple liquids have been discussed by Schofield [36, 37].
4. Thermodynamical approach to the equilibrium correlation functions of number fluctuations. - The pre-
ceding sections dealt with the properties of the mean value (AN )"’ >, being the limiting case thermodynamical limit)
of the function Those properties resulted from the grand canonical Gibbs ensemble which,
beside the discussion of the atomic correlations, permits that of the thermodynamical properties of the fluc- tuations. On introducing the grand partition function [32]
one can show easily that
leading to the following expression for
In (44)-(47), kB stands for Boltzmann’s constant, T for the absolute temperature andy for the chemical potential,
whereas HN is the Hamiltonian of the system of N atoms, and h Planck’s constant.
Equation (47) can be re-written in the following form :
where
but the function Fm is obtained by equating (47) and the following expression, due to Leibnitz [38] :
Applying two well known expressions, namely that of the definition of pressure in the grand canonical Gibbs ensemble In ZN and the mean density and introducing the isothermal compressibility
of the medium we obtain
which is the result of Ziff [39]. The functions Cm are cumulants of the function ( Nm > defined by
Thus, the functions Fm represent the difference between the mean value of the mth power of NF and the cumulants of ( Nm >. For some values of n4 we have ;
Table V. - Thermodynamical properties of the function ( (AN)m > for some values of m ;
(AN)m ) = ( (ON)m >G + (AN)m >NG where ( (AN)"’ >G and (AN)m >NG are, respectively, the Gaussian and non-Gaussian part.
In table V we give the basic thermodynamical properties of (AN )"’ ). As previously, we have separated the
Gaussian part ( (AN)m >G and non-Gaussian part ( (AN)’ >NG, the former conforming to the distribution
and possessing the properties :
One notes from table V that the mean values of
higher powers of the NF involve, in addition to higher
powers of the isothermal compressibility of the medium, terms dependent on the derivatives of the
compressibility with respect to pressure. The expres- sion for m = 2 is the well known Einstein-Smolu- chowski [1, 3] formula. Hemmer [40] has discussed
the relationship between the higher order derivatives of the isothermal compressibility and the m-atomic
distribution function g(m)(R1, R2, ..., Rm) generalizing
the Ornstein-Zernike formula for m = 2.
For the perfect gas, which fulfils the equation of
state pV = N > kB T we have :
and the expressions of table V reduce to those of table III for the perfect gas (Poisson distribution).
5. Summary and conclusions. - We have shown that the AN(K, t), considered as a complex stochastic
process, are non-Gaussian even in the simplest case,
namely that of the perfect gas. The process is of
generality higher than Gaussian and comprises certain properties of the Gaussian process. This permits the separation of the Gaussian and non-Gaussian parts in all cases : in the dynamical case of the perfect gas
(Section 2. 1), in the dynamical case of a real gas or
liquid (Section 2.2), as well as in the static case
(Section 3). The above results from our analvsis of the
correlation function obviously
r - 1
from m = 2 and m = 3, no complete information is obtained concerning the statistics of the stochastic process, the former providing information on the Gaussian properties only, and the latter - on the non-Gaussian properties. It is only from m = 4 that
the correlation function provides more general infor-
