Systems of particles with interaction and the cluster formation in condensed matter

We investigate the behaviour of a system of particles with the different character of interaction. The approach makes it possible to describe systems of interacting particles by statistical methods taking into account a spatial nonhomogeneous distribution of particles, i.e. cluster formation. For...

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Дата:	2003
Автори:	Krasnoholovets, V., Lev, B.
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Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2003
Назва видання:	Condensed Matter Physics
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Цитувати:	Systems of particles with interaction and the cluster formation in condensed matter / V. Krasnoholovets, B. Lev // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 67-83. — Бібліогр.: 11 назв. — англ.

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spelling	irk-123456789-1206932017-06-13T03:06:39Z Systems of particles with interaction and the cluster formation in condensed matter Krasnoholovets, V. Lev, B. We investigate the behaviour of a system of particles with the different character of interaction. The approach makes it possible to describe systems of interacting particles by statistical methods taking into account a spatial nonhomogeneous distribution of particles, i.e. cluster formation. For these clusters are evaluated: their size, the number of particles in a cluster, and the temperature of phase transition to the cluster state. Three systems are under consideration: electrons on the liquid helium surface, particles interacting by the shielding Coulomb potential, which are found under the effect of an elastic field (e.g. nucleons in a nucleus), and gravitating masses with the Hubble expansion. Досліджено поведінку системи частинок з різними типами взаємодії. Пропонований підхід дозволяє описати системи взаємодіючих частинок статистичними методами, беручи до уваги просторову неоднорідність розподілу частинок, тобто, формування кластерів. Для кластерів оцінено: їхній розмір, число частинок в кластері і температуру фазового переходу до стану кластера. Розглянуто три системи: електрони на поверхні рідкого гелію; частинки, що перебувають в пружньому полі і додатково взаємодіють за допомогою екранованого кулонівського потенціалу (наприклад, нуклони в ядрі); частинки з гравітаційним потенціалом при наявності Хаблівського розширення. 2003 Article Systems of particles with interaction and the cluster formation in condensed matter / V. Krasnoholovets, B. Lev // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 67-83. — Бібліогр.: 11 назв. — англ. 1607-324X PACS: 34.10.+x, 36.90.f DOI:10.5488/CMP.6.1.67 http://dspace.nbuv.gov.ua/handle/123456789/120693 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We investigate the behaviour of a system of particles with the different character of interaction. The approach makes it possible to describe systems of interacting particles by statistical methods taking into account a spatial nonhomogeneous distribution of particles, i.e. cluster formation. For these clusters are evaluated: their size, the number of particles in a cluster, and the temperature of phase transition to the cluster state. Three systems are under consideration: electrons on the liquid helium surface, particles interacting by the shielding Coulomb potential, which are found under the effect of an elastic field (e.g. nucleons in a nucleus), and gravitating masses with the Hubble expansion.
format	Article
author	Krasnoholovets, V. Lev, B.
spellingShingle	Krasnoholovets, V. Lev, B. Systems of particles with interaction and the cluster formation in condensed matter Condensed Matter Physics
author_facet	Krasnoholovets, V. Lev, B.
author_sort	Krasnoholovets, V.
title	Systems of particles with interaction and the cluster formation in condensed matter
title_short	Systems of particles with interaction and the cluster formation in condensed matter
title_full	Systems of particles with interaction and the cluster formation in condensed matter
title_fullStr	Systems of particles with interaction and the cluster formation in condensed matter
title_full_unstemmed	Systems of particles with interaction and the cluster formation in condensed matter
title_sort	systems of particles with interaction and the cluster formation in condensed matter
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2003
url	http://dspace.nbuv.gov.ua/handle/123456789/120693
citation_txt	Systems of particles with interaction and the cluster formation in condensed matter / V. Krasnoholovets, B. Lev // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 67-83. — Бібліогр.: 11 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT krasnoholovetsv systemsofparticleswithinteractionandtheclusterformationincondensedmatter AT levb systemsofparticleswithinteractionandtheclusterformationincondensedmatter
first_indexed	2025-07-08T18:25:04Z
last_indexed	2025-07-08T18:25:04Z
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fulltext	Condensed Matter Physics, 2003, Vol. 6, No. 1(33), pp. 67–83 Systems of particles with interaction and the cluster formation in condensed matter V.Krasnoholovets∗, B.Lev† Institute of Physics of the National Academy of Sciences of Ukraine, 46 Nauky Ave., 03028 Kyiv, Ukraine Received December 21, 2000, in final form December 29, 2002 We investigate the behaviour of a system of particles with the different char- acter of interaction. The approach makes it possible to describe systems of interacting particles by statistical methods taking into account a spatial nonhomogeneous distribution of particles, i.e. cluster formation. For these clusters are evaluated: their size, the number of particles in a cluster, and the temperature of phase transition to the cluster state. Three systems are under consideration: electrons on the liquid helium surface, particles inter- acting by the shielding Coulomb potential, which are found under the effect of an elastic field (e.g. nucleons in a nucleus), and gravitating masses with the Hubble expansion. Key words: statistical mechanics, interparticle interactions, clusters PACS: 34.10.+x, 36.90.f 1. Introduction Basically, statistical description of many-particle systems concerns homogeneous states. In monograph [1], a general method based on quantum field theory was suc- cessfully applied to homogeneous states of condensed media. An approach that took into account spatial nonhomogeneous states of a system of particles was first pro- posed in paper [2]; the approach was resting on the employment of the methodology of quantum field theory as well. The most general approach to the description of a system of particles, which included the interparticle interaction was offered in paper [3] in which the procedure of discount of the number of states providing for the nonhomogeneous particle distribution was proposed. However, a correct solution of nonlinear equations in the saddle-point is an impossible task for the most part: the general approach does not permit one to obtain analytical solutions for a sys- ∗E-mail: krasnoh@iop.kiev.ua †E-mail: lev@iop.kiev.ua c© V.Krasnoholovets, B.Lev 67 V.Krasnoholovets, B.Lev tem of particles when the inverse operator of the interaction energy does not exist. In the interim we need to know a series of parameters which characterize possible clusters. The major parameters are the cluster size and the critical temperature of cluster stability. Because of that, one can raise a question about a more simplified approach, which, however, would permit the study of cluster formation in an initially homogeneous system of particles. In the present work we study a system of interacting particles having regard to the concept proposed in papers [2,4]. Such an approach reduces the number of variables, which determine the free energy and introduces a peculiar “combined variable” [5] playing a role of a typical variable that describes the thermodynamics of the system with a spatial nonhomogeneous particle order. For instance, as an analog of a certain combined variable one can consider the interaction of two waves on the water surface, which interfere in a limited range. However, after the interference, the waves depart and hence in this case the combined variable does not bring into existence any instability of the system studied. On the other hand, the interaction of an incident electromagnetic wave with a polar crystal results in the polariton, a stable formation, which is indeed described by a combined (canonical) variable – a mixture of the electromagnetic field and optical phonon. Can the combined variable give rise to the change in the homogeneous distri- bution of interacting particles? We analyze such an option considering concrete examples. Below we take a look at the specific systems with long-range repul- sion/attraction and short-range attraction/repulsion. In particular, the approach provides a way of estimating the cluster size, the number of particles in a cluster and the temperature of phase transition to the cluster state. The approach is exam- ined in three examples: i) electrons on the liquid helium surface, ii) particles with the shielding Coulomb interaction, which are found in an applied elastic potential, and iii) gravitating masses with the Hubble expansion. 2. Statistical mechanics for model systems with interaction. The principle of states selection Let us consider a model system of particles with arbitrary kinds of interactions. Nevertheless, the de Broglie thermal wavelength λ of a particle is taken to be greater than the mean distant l between particles. However on the other hand, the value of λ should be smaller than the mean amplitude of scattering. The type of the statistics introduces significant peculiarities to the interparticle interaction, which, however, can be rendered by the classical methods allowing one to do away with dynamic quantum correlations. In this case the macroscopic state of the system in question may be specified by filling numbers ns, where the symbol s describes the state of the system. In other words, particles can occupy only the knots that are still unoccupied. This means that the filling numbers ns can run only two meanings: 1 (the s-th knot is occupied) or 0 (the s-th knot is not occupied). Besides, the allocation of particles in the lattice invites the type of statistics, which set limits to the behaviour of particles, i.e. the statistics refers particles to either bosons or fermions, see below. 68 Particles with interaction and cluster formation The Hamiltonian of the system of interacting particles can be written in the form H(n) = ∑ s Esns − 1 2 ∑ ss′ vss′nsns′ + 1 2 ∑ ss′ uss′nsns′ , (1) where Es is the additive part of the particle energy in the s-th state; vss′ and uss′ are, respectively, the paired energy of attraction and repulsion between particles in the states s and s′. Signs before the potentials in expression (1) reflect the proper signs of the attractive and repulsive paired energies and, therefore, both of the functions vss′ and uss′ are positive. Such writing of the Hamiltonian corresponds to the model in which particles are found in sites of the three-dimensional lattice. As will be evident from the further consideration, a specific form of the lattice will be unnecessary. However the medium will be supposed isotropic at the transition to the continual approximation. Note that the Hamiltonian of the model of replacement of solid solutions [6] amounts to the Hamiltonian (1) as well. This means that the proposed approach can be used to describe the behaviour of defects in solids, alloys, liquids, etc. (in these cases one particle will characterize an unfilled place and another particle will characterize a filled one). The reduced solution Hamiltonian can be amounted to a general Ising model, which takes into account an arbitrary interaction allowing for the spatial-nonhomogeneous distribution of particles. The partition function of the system is Z = ∑ {ns} exp(−H(ns)/kBT ), (2) where the summation provides for all possible values ns, i.e., all states of the system. Such summation can be formally performed, if one passes to field variables according to [2]. It is known from the theory of Gauss integrals that exp[ ν2 2 ∑ ss′ wss′nsns′ ] = Re ∞ ∫ −∞ Dχ exp[ν ∑ s nsχs − 1 2 ∑ ss′ w−1 ss′χsχs′], (3) where Dχ ≡ Πs √ det\|\|Wss′\|\| √ 2πdχs implies the functional integration with respect to the field χ and ν2 = ±1 in relation to the sign of interaction (+1 for attractive interaction and −1 for repulsion). Besides the obligatory condition ∑ s′′ w−1 ss′′ws′′s′ = δss′ (4) is fulfilled. In particular, condition (4), in principle, could allow one to determine the inverse operator w−1 ss′ (in the case when it exists) which in its turn would permit the construction of the Green function for interacting particles. Let us introduce the dimensionless energies w̃ss′ ≡ wss′/kBT and Ẽs ≡ Es/kBT . Note that hereinafter the tilde over a symbol means the division of the symbol by kBT . 69 V.Krasnoholovets, B.Lev So the partition function (2) can be presented in the form Z = Re ∞ ∫ −∞ Dφ ∞ ∫ −∞ Dψ ∑ {ns} exp [ − ∑ s Ẽsns + ∑ s (ψs + iφs)ns − 1 2 ∑ ss′ (ũ−1 ss′φsφs′ + ṽ−1 ss′ψsψs′)]. (5) We do not include the dependence on momentum of particles into the functional integrals in the partition function (5); the sum takes into consideration only the spatial distribution of particles and their energy. Now we can settle the quantity of particles in the system, ∑ s ns = N . The procedure is an equivalent to the consideration of the canonical ensemble. For this purpose one can use the formula known in the theory of complex variable 1 2πi ∮ dz zN−1− ∑ s ns = 1. (6) It makes it possible to introduce the sum of the canonical ensemble ZN = Re 1 2πi ∮ dz ∫ Dφ ∫ Dψ exp { − 1 2 ∑ ss′ (ũ−1 ss′φsφs′ + ṽ−1 ss′ψsψs′) + (N − 1) ln z} ∑ {ns} exp { ∑ s ns(ψs + iφs − Ẽs − ln z)}. (7) Summing over ns (note {ns} = 0, 1) we get Z = Re 1 2πi ∫ Dφ ∫ Dψ ∮ dz eS(φ,ψ,z), (8) where S = ∑ s {−1 2 ∑ s′ (ũ−1 ss′φsφs′ + ṽ−1 ss′ψsψs′) + η ln \|1 + η z e−Ẽseψs cosφs\|} + (N − 1) ln z. (9) Here, the function η is equal to ±1 (Fermi or Bose statistics), see e.g. [7–9]. This provides the means determining the necessary procedure of the states selection that introduces the most essential contribution to the partition function and defines the free energy of the system. Let us set z = ξ + iζ and consider the action S on the transit path passing through the saddle-point with a fixed imaginable variable Im z = ζ0. In this case, it stands to reason that the action S, similarly to quantum field theory, is the variational functional that depends on three variables: the fields φs and ψs, and the fugacity ξ (here ξ = e−µ/kBT where µ is the chemical potential). The extremum of the functional should be realized at solutions of the equations 70 Particles with interaction and cluster formation δS/δφs = 0, δS/δψs = 0, and δS/δξ = 0. These equations appear as follows: ∑ s′ ũ−1 ss′φs′ = − e−Ẽseψs sinφs ξ + ηe−Ẽeψs cosφs , (10) ∑ s′ ṽ−1 ss′ψs′ = e−Ẽseψs cosφs ξ + ηe−Ẽeψs cosφs , (11) ∑ s′ e−Ẽs′ eψs′ cosφs′ ξ + ηe−Ẽs′ eψs′ cosφs′ = N − 1. (12) Equations from (10) to (12) completely solve the problem of the statistical descrip- tion of the systems with any type of interaction. One or another state of the system is realized in accordance with the solution of the nonlinear equations. Among the solutions there are solutions which correspond to the spatial nonhomogeneous dis- tribution of particles. So the Bose condensation is realized in the real space because we can say that an aggregation of particles is the Bose condensation of a sort. One of the possibilities of cluster formation was demonstrated in [2]; the nonhomogeneous distribution of particles which were characterized by the shielded Coulomb interac- tion for both the attraction and the repulsion just yielded stable clusters. Such type of the interaction was chosen owing to the existence of the inverse operator of the interaction energy in an explicit form. Nevertheless, it is a very difficult problem to find the inverse operator among the most widespread types of interactions. That is why, there is a necessity to develop methods which are capable of describing the particle distribution at any interaction. In [4,5] the approach based on the selection of states in the form of a “combined variable” has been proposed. The combined variable has given us the chance to find connections between different fields (such as φs and ψs in the present paper) along the extremal path that passes through the saddle-point. 3. Combined variable and the selection of states Let us introduce the designation Γs = e−Ẽseψs cosφs ξ + ηe−Ẽseψs cosφs . (13) The value Γs includes the two field variables, ψs and φs, and the fugacity ξ. Inserting Γs into the left hand side of equation (12) we will see that the total sum ∑ s Γs is equal to the number of particles in the system studied ∑ s Γs = N − 1. (14) This means that Γs directly specifies the quantity of particles in the s-th state. Hence, the value Γs might be considered as a characteristic of the number of particles 71 V.Krasnoholovets, B.Lev contained in the sth cluster. So, it is quite possible to consider Γs to be the variable of particle number and, on the other hand, it may be called the combined variable owing to the fact that it combines the variables ψs, φs, and ξ. Probably the introduction of the combined variable is a rough approximation, but it permits to determine the clusterization conditions in cases when the inverse operator of the interaction potential cannot be found. Utilizing the variable (13) we can present the action (9) as a function of only two variables, Γs and ξ, and it makes it possible to do away with the inverse operators ṽ−1 ss′ and ũ−1 ss′ in the final expression. If we multiply two sides of equation (11) by the function ṽss′′ and then summing equation (11) over s, we obtain ψs′ = ∑ s ṽss′Γs . (15) Now let us multiply the same equation (11) by ψs; then summing it over s we acquire ∑ ss′ ṽ−1 ss′ψs′ψs = ∑ ss′ ṽss′Γs′Γs . (16) Multiplying equation (10) by ũs′′s and summing it over s we have ∑ ss′ ũ−1 ss′ ũs′′sφs′ = − ∑ s ũss′′ e−Ẽseψs sinφs ξ + ηe−Ẽseψs cosφs . (17) From relationship (13) one obtains 1 ξ e−Ẽseψs cosφs = Γs 1 − ηΓs . (18) Let us substitute (18) into (17) taking into account equation (15) and using condition (4) for the left-hand side of equation (17). We gain φs = − ∑ s′ ũs′s 1 ξ e−Ẽs e ∑ s′′ ṽs′′sΓs′′ (1 − ηΓs′) sinφs′ . (19) With the formula sinφs′ = √ 1 − cos2 φs′, cosφs can be substituted from (18) into (19). As a result, instead of (19) we have φs = − ∑ s′ ũs′s √ 1 ξ2 e−2Ẽs′ e2 ∑ s′′ ṽs′′sΓs′′ (1 − ηΓs′)2 − Γ2 s′ . (20) Consequently using equations (10) and (13) we get: ∑ ss′ ũ−1 ss′φs′φs = − ∑ s Γsφs tanφs , 72 Particles with interaction and cluster formation or in the explicit form: ∑ ss′ ũ−1 ss′φs′φs = ∑ ss′ ũss′ √ 1 ξ2 e−2Ẽs′ e2 ∑ s′′ ṽs′s′′Γs′′ (1 − ηΓs′)2 − Γ2 s′ × √ 1 ξ2 e−2Ẽse2 ∑ s′′ ṽss′′Γs′′ (1 − ηΓs)2 − Γ2 s , (21) here, allowance is made for the negative sign of the field φs (see (19)). Now using relationships (16) and (21) we can rewrite the action (9) in the point of extremum as follows: S = −1 2 ∑ ss′ ṽss′Γs′Γs − η ∑ s ln \|1 − ηΓs\| + (N − 1) ln ξ − 1 2 ∑ ss′ ũss′ √ 1 ξ2 e−2Ẽs′ e2 ∑ s′′ ṽs′s′′Γs′′ (1 − ηΓs′)2 − Γ2 s′ × √ 1 ξ2 e−2Ẽse2 ∑ s′′ ṽss′′Γs′′ (1 − ηΓs)2 − Γ2 s . (22) It has been shown in [5] that the minimum of the free energy based on the expression (22) is lower than that obtained in the framework of the mean field approximation. Indeed, let us assume following [7] that Nu and Nv are numbers of particles getting under the effect of the repulsive and attractive force, respectively. The number of particles Γs in a cluster can be put constant equals Γ. Let the interaction energies in the range of the effect of these forces be changed to their average values. In this case expression (22) becomes: S = −1 2 (N − 1){NvΓŪΓ +NvΓV̄ Γ + 2η ln \|1 − ηΓ\|} − 1 2 (N − 1)[NuŪ(1 − ηΓ)e−2Ēe2Nv V̄ Γ2 ξ−2 − 2 ln ξ], (23) where Ū and V̄ are average values of the repulsive and attractive potential, respec- tively. Here, in the mean field approximation the expression in the curly brackets coincides with the free energy. The next term, i.e. the expression in the square brack- ets decreases the free energy at ξ < 1. This inequality, ξ < 1, corresponds to the coherent state of a system of interacting particles. Thus, the approach that is be- ing developed can be considered as a general methodology of the mean field theory applying to the nonhomogeneous particle distribution. Expression (22) is a function of only one variable Γ and the fugacity ξ and it is applicable to any kind of interaction, even though the inverse operator is unknown. The most beneficial advantage of expression (22) lies in the fact that it provides a way of finding the characteristics of the system such as its size, the number of particles in a cluster and the temperature of phase transition. Now, let us pass to continual variables into the action (22). Inasmuch as we are interested in the nonhomogeneous distribution of particles in an indeterminate 73 V.Krasnoholovets, B.Lev volume, let a radius R be the fitting parameter of the system studied. Assume that the density of particles is distinguished from zero only in the cluster volume. Then equation (14) can be written as ΓK = N − 1, (24) where K is the number of clusters in the system and Γ defined in expression (13) is the combined variable of the fields φs and ψs, and the fugacity ξ in a cluster. As is seen from expression (24), the variable Γ can be interpreted as the mean quantity of particles in a cluster. Then the passage to the continual presentation is realized by the substitution: ∑ s fs = K 1 V ∫ cluster f(~r ) d~r, (25) here, the integration extends to the volume of a cluster and V = 4π 3 g3 is the effective volume occupied by one particle where g is the distance between particles, i.e., the lattice constant. Thus, in the continual presentation and with allowing for the assumption (14), we can transform the action (22) to the form: S = − 1 2V 2 ∫ d~r ∫ d~r ′ṽ(~r − ~r ′)Γ(~r ′)Γ(~r ) − 1 2V 2 ∫ d~r ∫ d~r ′ũ(~r − ~r ′)Γ(~r ′)Γ(~r ) × √ e−2Ẽ(~r ) e2 ∫ ṽ(~r−~r ′′)Γ(~r ′′)d~r ′′ 1 ξ2 ( 1 Γ(~r ) − η)2 − 1 × √ e−2Ẽ(~r ) e2 ∫ ṽ(~r−~r ′′)Γ(~r ′′)d~r ′′ 1 ξ2 ( 1 Γ(~r ) − η)2 − 1 + η 1 V ∫ ln \|1 − ηΓ(~r )\|d~r + (N − 1) ln ξ, (26) where the integrals are the same as in equation (25). The integration is effected in accord with the rule: 1 V ∫ cluster f(~r )d~r = 1 4π 3 g3 2π ∫ 0 dϕ π ∫ 0 sin θdθ R ∫ g f(r)r2dr = 1 4π 3 g3 4π R ∫ g f(r)r2dr. (27) Here we shift the limits of integration from 0 and R to g and R, respectively, as it allows one to eliminate the singularity of the integrand (the same procedure is made 74 Particles with interaction and cluster formation below in expression (29)). Thus, we can now normalize the integrals to the number of particles Γ in a cluster, that is, 1 V ∫ d~r = 1 (4π/3)g3 4π R ∫ g r2dr = R3 − g3 g3 = Γ − 1 ∼= Γ, (28) the last approximation is, in fact, correct as we assume that Γ satisfies the inequality Γ � 1. Relation (28) allows us to introduce the dimensionless variable x = r/g in the integral (27). Thereby, the rule of transformation from summation to integration, i.e. (25), becomes 1 K ∑ s fs = 3 Γ1/3 ∫ 1 f(gx)x2dx (29) (once again, in expressions (27) we shift the limits of integration from 0 and R to g and R, respectively, as it allows one to eliminate the singularity of the integrand; the same shift is made in expression (29)). Having integrated the action (26), we should exploit the following relationships 1 V 2 ∫ d~r ∫ d~r ′ ṽ(~r − ~r ′)Γ(~r )Γ(~r ′) = 1 V 2 ∫ d~r ṽ(~r )Γ(~r ) · ∫ d~r ′ Γ(~r ′) = 1 V 2 ∫ d~r ṽ(~r )Γ · Γ ∫ d~r ′ = Γ2 V · ∫ d~r ṽ(~r ) = 3Γ2 Γ1/3 ∫ 1 dx x2ṽ(gx), (30) 1 V 2 ∫ d~r ∫ d~r ′′ ũ(~r − ~r ′′)Γ(~r )Γ(~r ′′) = 1 V 2 ∫ d~r ũ(~r ) Γ2 = 3Γ2 Γ1/3 ∫ 1 dxx2ũ(gx).(31) With the transformations, we have used the step function: Γ(r) = Γϑ(Γ − r3/g3) where ϑ(Γ − r3/g3) = 0 if r3/g3 > Γ and ϑ(Γ − r3/g3) = 1 if r3/g3 < Γ. Besides, 1 V 2 ∫ d~r ∫ d~r ′ ũ(~r, ~r ′)f(~r )f(~r ′) = 1 V 2 ∫ d~r ũ(~r )f(Γ) · ∫ d~r ′f(Γ) = 3f 2 Γ1/3 ∫ 1 dx x2ũ(gx), (32) where f = √ exp[−2Ẽ + 2Γ ∫ d~r ṽ(~r )]. Now, if we introduce the designations a = 3 Γ1/3 ∫ 1 dx x2ũ(gx), b = 3 Γ1/3 ∫ 1 dx x2ṽ(gx), (33) 75 V.Krasnoholovets, B.Lev we will present the action (26) in the following final form S = K · {1 2 (a− b)Γ2 − 1 2 1 ξ2 ( 1− ηΓ )2 e−2Ẽ+2bΓ + η ln \|1− ηΓ\| } + (N − 1) ln ξ. (34) If we minimize the action (34) by the variable Γ, we will be able to define the number of particles in a cluster, the cluster size, and the temperature of phase transition that triggers the nonhomogeneous distribution of particles. It should be emphasized that the definition of the parameters of nonhomogeneous formations requires only explicit forms of potentials of interparticle interactions. Then, the system itself will select the realization which will provide for the minimum of the free energy. Nonetheless, first of all we assume the availability of clusters and only then the conditions and parameters of their existence are defined. Studying the phase transition of clusters regarding the spatial homogeneous dis- tribution of particles, we should take into account the behaviour of the fugacity ξ = e−µ/kBT . Indeed, in the classical system the chemical potential: µ = kBT ln(λ3 Tn), (35) where n is the particle concentration and the de Broglie thermal wavelength of a particle with a mass m: λT = h/ √ 3mkBT . (36) Thus, in the classical case the fugacity ξ = λ3 T n� 1. (37) In the case of a quantum system, the strong inequality does not hold; in this case ξ < 1 and ξ is only slightly smaller than the unit. Thereby, when we investigate the action (34) searching for clusters at the temperature T < Tc, we may neglect the last term as \| ln ξ\| is rather small in comparison with the terms which include the highest orders of Γ. However, if we consider the action (34) scanning for the critical temperature T = Tc, we should retain the last term because in this case ξ tends to a very small magnitude such that \| ln ξ\| � 1 and yet other terms decrease owing to the diminution of the value of Γ. 4. Clusters in condensed media In this section we apply the methodology developed above to the systems of par- ticles with long-range repulsion (attraction) and short range attraction (repulsion). 4.1. Electrons on liquid helium surface The energy of electrostatic repulsion between electrons can be written as uss′ = 1/4πε0 ·Q2/(rs − rs′) where Q is the effective charge of the electron in the helium film. The attraction between electrons caused by the deformation of the helium film can be taken as vss′ = 1 2 γfilm(rs − rs′) 2 where the coefficient γfilm characterizes 76 Particles with interaction and cluster formation the elasticity of the helium film in respect to the deformation caused by imbedded electrons (see, e.g. review [10]). In this case the parameters a and b (33) of the action (34) are transformed to the following a = 3 Q2/(4πε0g) kBT Γ1/3 ∫ 1 1 x x2dx ∼= 3 8πε0 Q2 gkBT Γ2/3, (38) b = 3 γfilmg 2 2kBT Γ1/3 ∫ 1 x2 x2dx ∼= 3 10 γfilmg 2 kBT Γ5/3 (39) or a = α̃Γ2/3, b = β̃Γ5/3, (40) where α̃ = 3 8πε0 Q2 gkBT , β̃ = 3 10 γfilmg 2 kBT (41) and α � β (hereafter the tilde over a symbol means the division by kBT ). Since for electrons η = +1 and the inequality Ẽ ≡ E/kBT � 1 is legitimate (i.e., exp(−2E/kBT ) � 1 where E is the kinetic energy of an electron), the action (34) can be rewritten as S ∼= K · {1 2 α̃Γ8/3 − 1 2 β̃Γ11/3 + ln Γ + Γ ln ξ } . (42) The extremum of the free energy is achieved at the solution of the equation S ′(Γ) = 0, or in the explicit form: 4 3 α̃Γ5/3 − 11 6 β̃Γ8/3 + 1 Γ + ln ξ = 0. (43) Retaining leading terms in equation (43), i.e. the highest powers to Γ, the equation is reduced to 4 3 α̃Γ5/3 − 11 6 β̃Γ8/3 ≈ 0 (44) and hence the solution to it is equal to Γ = 8 11 α β ≡ 40 33 nQ2 ε0γfilm , (45) where n = (4π 3 g3)−1 is the concentration of electrons in a cluster. The sufficiency is also satisfied, i.e., S ′′(Γ) > 0 with the solution (45) and, therefore, it actually determines the minimum of the action (42). The critical temperature Tc is defined from the condition Γ = 2, i.e. when the temperature is so high that no more than two particles are capable of interconnect- ing. In this case equation (43) turns into kBTc = −4α · 28/3 + 11 2 β · 211/3 3 ln ξ ≈ −12 π (2π 3 )1/3 Q2n1/3 ε0 ln ξ . (46) 77 V.Krasnoholovets, B.Lev In the first approximation we may allow that in the critical point the fugacity ξ ≡ e−µ/kBT reaches its classical meaning λ3 T n (see expressions (35) and (36)). Expanding function ln ξ in a Taylor series, we obtain lnλ3 T n ≈ 3(λT n 1/3 − 1). Substituting the expansion in equation (46) and taking into account expression (36) for λT , we arrive at the following equation for Tc kBTc ≈ 4 π (2π 3 )1/3 e2n1/3 ε0 . (47) Now, let us assign the concrete numerical values to the parameters. Let the charge Q be the elementary charge e. Put n = 2.4 · 1019 m−3 and γfilm = 8.4 · 10−14 N/m. Then, setting these values into expression (45), we get the number of electrons in a cluster: Γ ≈ 108. For the radius of the cluster we have R = (3Γ/4πn)1/3 ≈ 10−4 m. The critical temperature Tc estimated from expression (47) is of the order of 100 K that is far beyond the critical temperature (4 K) of the liquid helium film. Thus, the thermodynamic condition of the liquid helium stability sets limits to the real temperature Tc of the cluster existence. The behaviour of electrons in the liquid helium film has been studied by many researchers [10]. An experimental appraisal of the mean number of electrons in a near-surface bubblon was approximately equal to 108. The radius of the bubblon was estimated as 10−2 cm. In such a manner our qualitative evaluation of the values Γ and R is in agreement with the experimental data. 4.2. Shielding Coulomb potential Let particles be repulsed by the shielding Coulomb potential uss′ = 1/4πε0 × Q2e−κ\|rs−rs′ \| × (rs − rs′) −1, where κ is the effective radius of screening of nucleons in a nucleus, and attracted by the potential vss′ = 1 2 γ(rs− rs′) 2 that can be created for an external reason (so γ is the force constant of an off-site elastic field). In this case, the parameters a and b (33) of the action (34) become: a = 3 Q2 4πε0gkBT Γ1/3 ∫ 1 1 x e−κgxx2dx = 3Q2 4πε0κ2g3kBT [ − ( κgΓ1/3 + 1 ) e−κgΓ 1/3 + (κg + 1)e−κg ] . (48) We will consider weight nuclei (Γ � 1) and, therefore, Γ1/3 should be several times as large as the unit. Therefore, at the first approximation we may neglect the terms with e−κgΓ 1/3 and then expression (48) becomes: a ' 3 4πε0 Q2 κ2g3kBT (κg + 1)e−κg. (49) Now b = 3 γg2 2kBT Γ1/3 ∫ 1 x2x2dx ∼= 3 10 γg2 kBT Γ5/3. (50) 78 Particles with interaction and cluster formation Let us rewrite these expressions in the form: a = α̃, b = β̃Γ5/3, (51) where α̃ = 3 4πε0 Q2 κ2g3kBT (κg + 1)e−κg, β̃ = 3 10 γg2 kBT (52) and α� β. Let us now substitute expressions (49) and (50) for a and b in the action (34). With regard to inequality Ẽ � 1 and with allowance for η = +1 we will get: S ∼= K · {1 2 α̃Γ2 − 1 2 β̃Γ11/3 + ln Γ + Γ ln ξ } . (53) The extremum of the action S is reached with resolving the equation S ′(Γ) = 0, or explicitly α̃Γ − 11 3 β̃Γ8/3 + 1 Γ + ln ξ = 0. (54) Retaining two highest order terms in equation (54), we obtain: Γ ≈ ( 6 11 α β )3/5 = 4 3 π2/5n {15 11 Q2 ε0κ2γ [( 3 4π )1/3 κ n1/3 + 1 ]}3/5 exp [ − 3 5 ( 3 4π )1/3 κ n1/3 ] . (55) The critical temperature Tc of the cluster destruction is defined from equa- tion (54) if one puts Γ = 2: kBTc ≈ 4α − ln ξ . (56) However, having found the fugacity ξ, we now cannot follow the procedure de- scribed in the previous subsection. In a nucleus, the temperature and the thermal de Broglie wavelength are different. In other words, we should exploit the laws of high energy physics, i.e. that the total energy of a nucleon is linked with the momentum of the nucleon by the relation E = √ p 2c2 +m2c4. Hence, in this case 3 2 kBT = c √ p 2 +m2c2 (57) and, therefore, p ' 3kBT/2c. (58) Thus, the de Broglie thermal wavelength of the nucleon in a nucleus is determined as λT = 2h 3c kBT . (59) Expanding function ln ξ = ln(λ3 Tn) in equation (56) in a Taylor series we obtain ln ξ ' −3 and, therefore, equation (53) becomes kBTc ≈ 4 3 α ≡ 4 3 nQ2 ε0κ2 {( 3 4π )1/3 κ n1/3 + 1 } exp [ − ( 3 4π )1/3 κ n1/3 ] . (60) 79 V.Krasnoholovets, B.Lev The results obtained in this section may account for the reasons of the atomic nucleus stability from the microscopic viewpoint. The shielding Coulomb potential u = 1/4πε0 · Q2e−κr/r is the typical nuclear (Yukawa) potential that provides for repulsion between protons. The potential v = 1 2 γr2, which is applied to nucleons, ensures their mutual attraction. For example, setting γ = 4 · 1017 N/m, we get from expression (55) Γ ≈ 30, which corresponds to the number of nucleons in a nucle- us of zinc. The critical temperature of fission of a proton-neutron pair calculated from expression (60) results in Tc ∼ 1011 K. Note that this value is several times of magnitude greater than the typical temperature of a weight nucleus (the nucle- us temperature is of the order of the Coulomb repulsion between protons in the nucleus). 4.3. Gravitating masses with Hubble expansion Gravitational physics may also be assigned to the condensed matter [11]. The gravitational attraction between particles, i.e. big masses, is beyond question. How- ever, the uniform character of the Hubble expansion is doubtful. We will use only the fact that an additional kinetic energy of particles Ess′ = 1 2 m(ws − ws′) 2 is asso- ciated with such an expansion. The relative velocity ws − ws′ of particles which are found in points s and s′, respectively, pertains to the relative distance between the particles, since ws − ws′ = H(rs − rs′), where H is the Hubble constant. In this connection, we will consider a model system of identical gravitating masses, i.e. stars, with the attraction vss′ = Gm2/(rs − rs′) and the effective repulsion uss′ = 1 2 H2m(rs − rs′) 2. For such kinds of interactions one has for the parameters a and b (33): a = 3 g2H2m 2kBT Γ1/3 ∫ 1 x2x2dx ∼= 3 10 g2H2m kBT Γ5/3, (61) b = 3 Gm2 gkBT Γ1/3 ∫ 1 1 x x2dx ∼= 3 2 Gm2 gkBT Γ2/3 (62) or a = α̃Γ5/3, b = β̃Γ2/3, (63) where α̃ = 3 10 g2H2m kBT , β̃ = 3 2 Gm2 gkBT . (64) Inasmuch as gravitating masses are associated with the Bose statistics, i.e. η = −1, the action (34) is rewritten as follows S ' K · {1 2 α̃Γ11/3 − 1 2 β̃Γ8/3 − ln Γ + Γ ln ξ } . (65) 80 Particles with interaction and cluster formation In expression (65) we omit the exponent term because of the assumption that the temperature T of the universe is very close to the absolute zero (at small T the exponent in expression (34) tends to zero). Then the equation S ′(Γ) = 0 becomes: 11 6 α̃Γ8/3 − 4 3 β̃Γ5/3 − 1 Γ + ln ξ = 0. (66) Here, α � β. Retaining the highest order terms in equation (66) we immediately obtain the solution Γ ' 8 11 β α = 120π 33 Gmn H2 . (67) where n = (4π 3 g3)−1 is the concentration of particles, i.e. stars. The equation for the critical temperature Tc that determines the break-up of two coupled masses is obtained from equation (66) in which we set Γ = 2 (and the exponent term is neglected). With the inequality α� β, we get kBTc ≈ 22/3 3 β ln ξ ≡ (2π 3 )1/3 Gm2n1/3 ln ξ . (68) By definition, the fugacity ξ = e−µ/kBT . At the transition from the Bose statistics to the classical one, the chemical potential gradually decreases with temperature starting from a value µ ∼ 0 (but still µ < 0) to µ � 0. So, the function ln ξ in equation (68) is strongly negative: ln ξ = −\|µ\|/kBT . This signifies that the critical temperature in the system of gravitating masses is absent. In other words, in the universe, masses tend to aggregation at any conditions. Thus, the results obtained demonstrate that the availability of the Hubble ex- pansion is the decisive factor in the formation of galaxies in the universe (note that without H, masses in the universe will merely tend to the total mergence and any spatial distribution will not be observed). Actually, expression (67) shows that Hub- ble’s constant, H ≈ 50 km/(s ·Mpsc) = 1.6 ·10−18 s−1, is one of the main parameters that determines the steady state of masses in a cluster, i.e., galaxy. So the greater the mean mass m, the greater the number Γ of stars in a galaxy. 5. Conclusion In the present paper the statistical description of a system of interacting parti- cles, which enables the study of their spatial nonhomogeneous distribution, has been proposed. One of the most important properties of our model is the presentation of the total potential of interacting particles in the form of the paired energy of attrac- tion and repulsion. We have started from the Hamiltonian that describes interacting particles located in the knots of the three-dimensional lattice and applied the quan- tum field theory methodology based on the functional integrals for the description of the partition function of the systems studied. We have shown that the solutions of equations characterizing the behaviour of the field variables yield a complete resolving of the initial statistical task. The solutions 81 V.Krasnoholovets, B.Lev may correspond to both the homogeneous and nonhomogeneous distributions of particles and only the character of interparticle interaction and the temperature define the type of the solution. The free energy represented through field variables in the saddle-point has al- lowed us to investigate special conditions, which provide for the formation of clusters in a system with the initially homogeneous particle distribution. The conditions are also responsible for the size of clusters and for the temperature of phase transition to the cluster state. The approach proposed does not need fitting parameters: the cluster size and the temperature of phase transition to the cluster state are completely determined by the value and character of paired potentials in the systems of interacting particles. Besides, the introduction of a peculiar type of the combined variable has simplified the investigation procedure. Specifically, the procedure has provided a way of ob- taining the necessity criterion of the transition of a system of separate particles into the spatial nonhomogeneous state. The authors are very thankful to the referees for the concrete constructive re- marks that made significant improvements of the contents of the paper possible. References 1. Kleinert H. Gauge Fields in Condensed Matter. Singapore, World Scientific, 1989. 2. Belotsky E.D., Lev B.I. Formation of clusters in condensed media. // Theor. Math. Phys., 1984, vol. 60, No. 1, p. 120–132 (in Russian). 3. Lev B.I., Zhugaevich A.Ya. Statistical description of model systems of interacting particles and phase transitions accompanied by cluster formation. // Phys. Rev. E, 1998, vol. 57, No. 6, p. 6460–6469. 4. Krasnoholovets V., Lev B.I. On mechanism of H-associate polymerazation in alkyl- and alkoxybenzene acids. // Ukrain. Fiz. Zhurn., 1994, vol. 33, No. 3, p. 296–300 (in Ukrainian). 5. Belotsky E.D. Nonlinear and nonequilibrium spartial-limited structures (questions of theory). Thesis, Inst. of Phys., Kyiv, 1992, 242 p. (in Ukrainian). 6. Khachaturian A.G. The Theory of Phase Transitions and the Structure of Solid Mix- tures. Moscow, Nauka, 1974 (in Russian). 7. Magalinsky V.B. Thermodynamics of a one-dimensional model of particles with a Coulomb interaction. // JETP, 1965, vol. 48, No. 1, p. 167–174 (in Russian). 8. Edwards S.F. The statistical thermodynamics of a gas with long and short-range forces. // Phil. Mag., 1959, vol. 4, No. 46, p. 1171–1182. 9. Edwards S.F., Lennard A.J. Exact statistical mechanics of a one dimensional system with Coulomb forces. II. The method of functional integration. // J. Math. Phys., 1962, vol. 3, No. 4, p. 778–792. 10. Edelman P.S. Levitating electrons. // Uspekhi Fiz. Nauk, 1980, vol. 130, No. 4, p. 675– 706 (in Russian). 11. Saslaw W.C. Gravitational Physics of Stellar and Galactic Systems. Cambridge, N.Y., Melbourne, Sydney, Cambridge University Press, 1989. 82 Particles with interaction and cluster formation �� !�#"$��%�� &��' �)(�� % �+��",�-��.�/� ��$�&�� 0"�� %��/��12� 3/465�7�8.9�:�;&<=;.>�;&?A@�B�C.D-EF4 G�@�? 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Systems of particles with interaction and the cluster formation in condensed matter

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