Study of the higher-order correlation functions of number fluctuations in simple fluids with radial many-body interactions

OCTOBRE 1 1373J. 42 1373-1385 1981, Physique (1981)
Classification
AbstractsPhysics
05.40
of the correlation functions of number fluctuationsStudy higher-order
in fluids with radial interactions simple many-body (*)
Knast and W. ChmielowskiK.
Nonlinear Division, Institute of A. Mickiewicz PolandOptics Physics, University, Pozna0144,
le 22 décembre révisé le 1 er le 24 (Reçu 1980, juin 1981, accepté juin 1981)
2014 Résumé. En traitant les fluctuations de densité comme un on 0394N(K, t) processus étudiestochastique complexe,
d’ordre élevé du le de leurs fonctions de corrélation Ces fonctions desproblème dépendantes temps. apportent
informations sur la des et le de en statistique processus stochastiques prouvent que processus 0394N(K, t) est, général,
deux non Gaussien. Chacune de ces fonctions être dont l’une est liée aux atomespeut exprimée par composantes
libres et l’autre aux atomes corrélés dans le et Les fonctions de corrélation (sans interactions) temps l’espace. dépen-
dantes du nous les calculées dans le avons les mêmes castemps, que calculées, possèdent propriétés que
de la limite On montre les fonctions des fluctuations de densité étudiées sont cohérentesthermodynamique. que
avec l’ensemble de ce de les des fonctions de grand canonique Gibbs, qui permet exprimer par répartition d’équi-
libre les la fonctions de isother-m-atomiques g(m)(R1, R2, ..., Rm) ou, thermodynamiquement, par compressibilité
mique 03B2T.
2014 Abstract. with number fluctuations as a stochastic a discussion isDealing (NF) 0394N(K, t), complex process,
of the of correlation functions of NF. The functions informationgiven problem higher-order time-dependent convey
on of and show the to in non-Gaussian. Each of thethe statistics the stochastic be, process 0394N(K, t) process general,
correlation functions can be two the one related with free atoms andexpressed by components, (non-interacting)
the other with atoms correlated in and time. The correlation functions calculated in thisspace time-dependent
are shown to be in with the of the same calculated for the case of the thermo-paper agreement properties functions,
limit. The correlation functions of NF under are then shown to conform to the Gibbsdynamical investigation
in canonical their terms of m-atomic distribution functionsgrand ensemble, permitting expression equilibrium
in terms of the functions of isothermal or, g(m)(R1, R2, ..., Rm) thermodynamically, appropriate compressibility 03B2T.
- of the scattered is related with the two-1. Introduction. Number fluctuations light directly (NF) play
a role in all atom radial distribution function account-very g(2)(Rh R2) important physical phenomena
statistical of media. Smolu- for radial correlations of of as wellatoms, involving inhomogeneity ing pairs
as with chowski has shown on the distribution functions that, level, higher g(3)(Rb R3)[1] microscopic R2,
and In accordance with the Gibbsno medium is since chaotic g(4)(R1, R2, R3, R4). completely homogeneous
thermal motion of the atoms within canonical ensemble the two-atom distri-leads, elementary grand [9],
related with the meanto NF i.e. to sta- bution function is volumes, unceasing spontaneous g(2)(R1, R2)
number fluctuation of atoms.tistical all sorts square inhomogeneity. Significantly, of optical
statistical inhomo-are sensitive to The phenomena of highly techniquesdevelopment spectroscopic
of the in of enhanced interest in geneity médium, particular phenomena NF ; herein, especial importance
in material where NF cause localmedia, was to stochastic thelight scattering given approaches, permitting
tovariations of the refractive index fluctuations. The[2]. determination of According time-dependent
Smoluchowski the increase of NF in the stochastic treatment of fluctuations in[1], steep originating
of the critical is as critical the work of Smoluchowski was Chan-vicinity point apparent developed by
The of and has drasekhar and Kac beenopalescence. thermodynamical theory light [11], recently [10]
Brillouin on NF is due to Einstein Brenner et al. [3], [4],scattering analysed by [12].
whereas their treatment is duestatistical-microscopic The of on NFproblem dynamical light scattering
to Ornstein and Zernike Yvon Fixman Fischer [5], [6], [7] has been discussed Komarov and [13]by
and Kielich who have shown that the formalism of van[8], depolarization as well as Pecora [14], applying
Hove’s correlation whereasfunction, [15] space-time
Research MR. 1.9. the to the fluctuations isThis work was carried out under (*) Project hydrodynamical approach
- LE JOURNAL DE T. NO OCTOBRE 1981PHYSIQUE 42, I O,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01981004201001373001374
Mountain and Benedek et al. The of the stochastic are discussedapplied by [16] [17]. properties process
The recourse to in the for as well as for andhomodyne spectroscopy [18] statistically independent space
of in atomic without time correlated atoms. The results obtained shouldstudy light scattering systems
-+ interaction has enabled Schaefer and Berne to be in in the case [19] agreement, 0, t 0),limiting (K -
establish as a result the electric field of the with those derived with the ensemblethat, of NF, canonical grand
scattered is not a Gaussian stochastic the form of which for atomslight process. statistically independent
In the is case atoms a non-Gaussian that of the Poisson distribution general (perfect gas) of interacting [10].
correction arises for the inter- in the same the specifically Next, case, thermodynamicalaccounting limiting
actions. of the correlation functionsproperties higher-order
In the we discuss the of of NF are discussed the functions in termspresent paper problem expressing
of the corrélation functions of NF as well as isothermal of the medium.higher-order compressibility
their in as Problems related with the correlationThe NF are dealt with higher-order properties. general
a stochastic the are functions are essential in critical so complex AN(K, t) highly e.g. phenomena,process,
functions of a wave vector K and the time t. Van as well as usual[20, 21] multiple light scattering,
within the method Hove’s formalism is and of correlationscattering intensity applied generalized many-
atomic correlation functions are introduced. [22].body
- We 2. stochastic corrélation functions of number fluctuations. consider a Time-dependent macroscopic,
like or molecules.and of a of atoms numbers N > isotropic homogeneous isotropic system great optically
N with V- the volume of the TheThe mean number of atoms is ( n > = >IV, system. density expressed by
at a in V at the moment is defined as follows local number density R. of time tm [15] :microscopic point
the vector of the atom at The Fourier transform leads to the macros-being position 01531-th tm. spatial of (1) rCX1(tm)
copic quantity :
in V at the moment of time On the the number of atoms introducing followingdefining K.-dependent present tm.
the van Hove correlation function of the mth order definition of microscopic [23] :
correlation function of NFwe write the mth order
where occurs « is for odd m and « - » for even m.+ » ±
In use is made of total van Hove correlation functions defined in with the definition of(4), conformity
fluctuations as :
where Newton’s symbol.m is J
, , z
= the formulae of total van Hove3 and the For the values m 4, (5) following 2, general expression yields
functions of the third and fourth order :second, correlation-(’)
- Total van Hove correlation functions in case t 0 are in with cluster functions (1) case) agreement Ursell,limiting (equilibrium (see :
H. Proc. Philos. Soc. 23 D., Cambridge (1927) 685).1375
where the terms have to be written out in all right-hand symmetrically permutations R1, tl, ..., R4, t4.
In accordance with the definition
the the mth order van Hove correlation function introduced can be..., by (3) Gm(R1, ti, R2, t2, Rm, t.) expression
written in the form of a of three sum components :
= results from the irreducible of the m-fold summation in Equation ( 10) decomposition (9) : for a 1 a2 = ... = (Xm
« we have the o self » term G",sm(R1, 11’ R2, a2 # ... # a -thé distinct » term GmD(R1, tl,12’ ..., R"" lm); for (Xl =f:.
= = ... = = = whereas for all the others = ... #- R2, 12’ ..., Rm, lm) (Xl #- a2 (Xm, (Xi (a2 #- ... am, (Xl a2 am,
- = etc. the « mixed » term For m the irreducible decom-2, (X #- (a2 #- ... = (Xm Gm(M)(R 1, tb R2, t2, ..., Rm, tm).
of the double sum leads to the well known position expression
selectedvan Hove correlation function of mth order is related with some of m The atoms, (9) arbitrary group
of these m atoms at the from all the atoms of the and the finding space points( N > system, gives probability
of time this of a conditional in the moments ti, t2, ..., tm respectively, probability being type.R1, R2, ..., Rm
= For m the « self » term2,
hadthe an atom in the under the condition at the same atom that, gives probability of finding at ti point RI t2,
« been in the whereas the distinct » termpoint R2 ;
an atom in the under the condition at the moment of the at that, of finding, tl, point Ri time t2,gives probability
another atom had been at R2.
and al. the van Hove correlationIn the et al. Groome et al. Gubbins et 24], [15] [26], papers by Egelstaff [23,
has been studied both and function of 3rd order R2, t3) theoretically experimentally.G3(R1, tl, t2, R3,
- The mth order correlation function of NF relevant to our2.1 STATISTICALLY INDEPENDENT ATOMS.
the sum of two considerations can be as components :expressed
The first defines the contribution from free whereas(stochastically atoms, component ... >FA independent)
all contributions in atoms correlated in and time.the other contains originating space component, ( ... > cA
there the total functionsTo the functions (14) correspond microscopic
the first in to a lack of interatomic correlations and the other to the component corresponding (as (14)) presence
= cases 3 and we can of mutual correlations. For several with to particular (m 2, 4) write, regard (6)-equations
(10),1376
orders of the functions can be calculated. On insertion into higher tl, R2, ..., (4),Similarly, H m FI (Rl, t2, Rm, t.)
m B
the determination of correlation functions valid for atomic ofthey permit AN( + K j, systems of NF fl t j)
1 j=1 /
atoms. Table 1 contains the correlation functions thus calculated for several values of the index m.non-interacting
In table we have introduced the notation :I, following
is Kronecker’s Formula is a result of the of the medium whensymbol. (20) macroscopic isotropicity where ô,,,o
is a function relative of the atoms whereas the factorG:;> pf the positions (Ri - R2, R2 - R3, ...) exponential
is on the sum of the atomic dependent positions.
Table 1 renders certain of the stochastic described aapparent properties process by AN(K, t). Namely,
is stochastic of this kind it to the Gaussian and non-complex non-Gaussian, process being possible separate
Gaussian the correlation functions of NF for of free atoms can be written in the parts. Thus, systems following
form :
- Table I. Gaussian and non-Gaussian thecomponents of
a atoms.correlation function for system of statistically independent of number fuctuations
The terms of the above should be in (*) right-hand (K1, K2, (1’ t3, expressions expanded permitted permutations K3, K4, K5, K6, t2, t4l
are : These t il t6). pairs -(1, 2), (1, 4), (1, 6), (2, 3), (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),
- 5, 6), (2, 3, 2, 4, 4, 3, 2, 4, 4, 3, 2, 3, (2, 6), (1, 5, 4), (1, 6), (3, 4, 5), (2, 3, 6), (1, 5), (1, 6), (2, 5), (1, 5), (3, 6), (2, 5), (1, 6) ; quartets (1, 5),
(l, 3, 4, 5), (1, 2, 4, 6), (1, 3, 5, 6), (2, 3, 4, 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, 4, 5, 6),
(2,3,4,5),(1,4,5,6).1377
NG are the Gaussian and non-Gaussian of the correlation function, where...> ’and ( ... ) parts respectively.
From table the Gaussian has the I, part following properties :
where we = have introduced the notation and n ! 1.3.5... 1.2.3 ... n. (2 n - 1) ! ! = (2 n - 1) Obviously,
= in the case for 0 well known theorem we obtain the of Reed general (for j 1, 2,..., 2 n) Kj =1= [27]
the Gaussian of the function on certain combinations of the functionscorrelation part being dependent
in contain information on the behaviour of the atoms in two momentswhich, turn, P2(Kl, K2, 11’ t2) N>
of time and the atomic motions translational diffusion we can writeti t2. Approximating by [10],
D 1 where is the translational diffusion table we moreover the non-Gaussian give partscoefficient..In
NG NG
of the correlation function of NF in the DN( ± originating essentially functions pm(K 1, K2, ... ,Il m K j, t j) ( j =1 /FA
for m > 2 and information the behaviour of the atoms in etc...., three, four, Km’ t1, t2, lm) conveying regarding
moments of time. These functions are in the caselimiting
= 4 with to the definition Certain forms of these functions for m have been3, regard general (19). approximate
basis of the construed Brenner et al. on the short-time [12] by approximation.
- 2.2 ATOMS MUTUALLY CORRELATED IN SPACE AND TIME. We with our considerations of theproceed
correlation related with mutual correlations of the latter in the medium. The function of NF space-time function,
namely :
van Hove function the mutualis related with the total HmA(R1, tl, R2, t2, ..., Rm, tm), directly defining microscopic
functions incorrelations of the atoms. For several values table II us to write the of m, gives expressions enabling
« « and « mixed » terms of van Hove functions of the self » distinct » type Gm(M).type G(s), type G2»,
which holds for free we can write the to the (21), atoms, following expression :By analogy expression
for the case of atoms correlated in and time. Its is valid space microscopic counterpart given by :
van Hove function.where and the Gaussian and non-Gaussian of the are, H,c,A, HÀkG respectively, components
The Gaussian has the component properties 1378
- several values Table II. Total van correlated HmcA(R1, ..., of m.Hove functions tl, R2, t2, Rm, tm) for atoms, for
In the terms of these all of the variables to of the individual(*) right-hand expressions, possible permutations leading symmetrization
should be taken into account.components
The total van Hove functions informa-..., microscopic H;A(Rl’ tl, R2, t2, Rm, tm) convey highly important
tion the in four and m correlated atoms. in accordance with of two, concerning motions, liquids, three, Thus, (29),
the Gaussian inform us of the and time correlation of a of..., components HmG(R1, ti, R2, t2, Rm, tm) space pair
whereas information on correlations of orders is contained in the non-Gaussian atoms, componentshigher
The form of these functions is not obtained and so aHcA MN CA(R1, tl, R2, t2, ..., Rm, tm). analytical easily poses
for the sake various model are introduced on based the separate problem ; of simplicity, approximations equili-
brium of the functions in has the convolution forproperties question. Vineyard [28] proposed approximation
the of whereas Nelkin and have discussed a differentthem, simplest H2A(Rl, tl, R2, t2), Ranganathan [29]
to it based a Vlasov on linearized equation.approximation
The has also been dealt with in a Kerr A to the of time-problem paper by [30J. general approach problems
correlation certain stochastic is due to Mori functions, dependent characterizing processes, [31].
- of 3. The of the corrélation functions number fluctuations. We with ourequilibrium properties proceed
considerations for correlation functions of NF calculated in the caselimiting
to mean values of AN. we can writeleading of higher powers By equation (4)
where the are total van Hove the case of the ..., functions, being Rm) equilibrium limiting dynamicalHm(R1, R2,
...
= On over to the 0 in the and left-functions ..., going limit t1 = t2 = tm -+ right- Hm(R1, t1, R2, t2, Rm, tm).
we obtainhand terms of (5), 1379
the van Hove functions which are the case of the functions ..., dynamical ti,Gm(R1, R2, Rm) limiting Gm(R1,
of the van Hove functions are to be obtained fromFurther ..., properties Hm(P-1, R2, ..., Rm R2, t2, Rm, tm).
the fact that a of NF conforms to the canonical Gibbs which the follow-ensemble, system admitting grand gives
function normalization condition for the m-atomic distribution ing (correlation) [32]
of The term can be re-written first-order numbers i.e. the(32) right-hand equation introducing Stirling Sk(m)
of has the formmicroscopic counterpart (33)
where is Dirac’s delta function.b(Rm-k - R. - k 1 )
The numbers of the first-order fulfil the relations [33] :Stirling following
the Bernoulli numbers of order m. and the definition the van Hove functions(3), B(m)m-k being By (33)
can be written in recurrential formGm(R1, R2, ..., Rm)
the values of m. With to and calculation of concrete forms of the functions for various regard (32) (36),permitting
the total van Hove are related with the distribution functionsfunctions many-atomic Hm(R1 R2, ..., Rm)
the formulag(m)(R1 R2, ..., Rm) by
2 now on atomsAs done us in section we to consider’the two cases of independent by go particular statistically
- correlations and atoms with mutual corrélations. (no gas), spatial perfect 1380
- In this of the of number3.1 INDEPENDENT ATOMS. case the mean values STATISTICALLY higher powers
are obtained in table 1 the condition (25))fluctuations (AN)"" ) immediately applying (perfect gas) (Eq.
in table III. is to the conclusion which leads to the concrete Equation (40) equivalent of Ziff [39],expressions given
fluctuations the method of cumulants and showed for the thewho calculated the number that, by perfect gas,
cumulant mth order is N Table III reveals another of a of arbitrary equal to >. property system of statistically
atoms the the Poisson statistics [10]independent (perfect gas) : system obeys
is in accordance with the canonical Gibbs ensemble for the of Poisson form.which, grand perfect gas,
Gaussian and non-Gaussian the former In table III we have the of the parts, performed separation resulting
from the distribution
is well Poisson distribution does not lead to a Gaussian distribution and can be It known that the (41) approxi-
mated such a distribution in a first by approximation only.
- Table III. Mean values of higher powers of number
(0394N )m = + (AN)m fluctuations >FA (0394N )m>GFA )FA
the and where (AN)m >GFA (AN )"’ )FAfor perfect gas,
Gaussian and non-Gaussian are, respectively, parts.
A closer is one of the form :approximation following
where The its distribution function is with to rather ck not, (43) regard compli-
cated form, readily applicable.
- If the atoms interact with one the van Hove function charac-3.2 SPACE-CORRELATED ATOMS. another,
becomes a rather function of the fluctuations many-atomicterizing HmA(R1, R2, ..., Rm) highly complicated
the mutual radial correlations of selected ofdistribution functions R2, ..., Rm), characterizing groups g(m)(Rl,
atoms. Table IV shows how the functions are related with the functions m ..., ..., HmA(R1, R2, Rm) g(m)(R1, R2, Rm),
which characterize the of the of m atoms we have the Gaussiangroup too, equilibrium properties (here, separated
from the non-Gaussian part HmG part HMNUI*
the mean values of the fourth of the NF areThe show that and second, third, higher powers expressions
the radial correlations and numbers of atoms defined the on three, four, dependent of two, greater by many-atom
The radial distribution functions for and..., g(m)(R1, R2, Rm). expressions simplify partly systems isotropic
= for which the 1. in the one-atomic distribution function Thehomogeneous macroscopic meaning g(1)(R1) 1381
- Table IV. Gaussian and non-Gaussian van ..., HmG(R1, R2, Rm) HmN;(R1, R2, ..., Rm) parts of the Hove functions
some values for of m.
= for m 2 is well known from classical statistical it relates the mean of the ANexpression mechanics ; square
to the bi-atomic radial distribution function. The detailed discussion of the fluctuations ordersfurther, of higher
the of the of Cole Certain Kirkwood and requires application superpositional approximations [34] [35]. pro-
of the correlation functions in have been discussed Schofield perties simple liquids by [36, 37].
- 4. to the correlation functions of number fluctuations. The Thermodynamical approach equilibrium pre-
sections dealt with the of the mean the caseceding properties value (AN )"’ >, being limiting
thermodynamical limit)
of the function Those resulted from the canonical Gibbs ensemble properties grand which,1382
that of the of the fluc-beside the discussion of the atomic correlations, permits thermodynamical properties
the function tuations. On introducing grand partition [32]
one can show thateasily
to the forleading following expression
In stands for Boltzmann’s T for the absolute for the chemical constant, (44)-(47), kB temperature andy potential,
whereas is the Hamiltonian of the of N and h Planck’s constant.atoms, HN system
can be re-written in the form :Equation (47) following
where
and the due to Leibnitz but the function is obtained (47) following expression, [38] :Fm by equating
that of the definition of in the canonical Gibbstwo well known namely pressure grand Applying expressions,
ensemble In and the mean the isothermal and ZN density introducing compressibility
we obtainof the medium
which is result The functions are cumulants of the defined the of Ziff [39]. Cm function ( Nm > by
the functions the difference between the mean value of the mth of NF and the cumulantsThus, Fm represent power
of Nm For some values of we have ;n4 ( >.