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arXiv:1001.4477v1 [nucl-th] 25 Jan 2010

EXACTLY SOLVABLE MODELS

EXACTLY SOLVABLE MODELS: THE ROAD TOWARDS A

RIGOROUS TREATMENT OF PHASE TRANSITIONS IN

FINITE NUCLEAR SYSTEMS

K.A. Bugaev1,2, P. T. Reuter3

UDC 539.12

c

?2006

1Bogolyubov Institute for Theoretical Physics, Nat. Acad. Sci. of Ukraine

(14b, Metrologichna Str., Kyiv 03143, Ukraine; e-mail: KABugaev@th.physik.uni-frankfurt.de)

2Lawrence Berkeley National Laboratory

(1, Cyclotron Rd., Berkeley, CA 94720, USA )

Triumf, Canada

(4004 Wesbrook Mall, Vancouver, Canada V6T 2A3; e-mail: reuter@triumf.ca)

We discuss exact analytical solutions of a variety of statistical models recently obtained for finite systems by a novel powerful mathemati-

cal method, the Laplace-Fourier transform. Among them are a constrained version of the statistical multifragmentation model, the Gas

of Bags Model and the Hills and Dales Model of surface partition. Thus, the Laplace-Fourier transform allows one to study the nuclear

matter equation of state, the equation of state of hadronic and quark gluon matter and surface partitions on the same footing. A

complete analysis of the isobaric partition singularities of these models is done for finite systems. The developed formalism allows us, for

the first time, to exactly define the finite volume analogs of gaseous, liquid and mixed phases of these models from the first principles of

statistical mechanics and demonstrate the pitfalls of earlier works. The found solutions may be used for building up a new theoretical

apparatus to rigorously study phase transitions in finite systems. The strategic directions of future research opened by these exact

results are also discussed.

There is always a sufficient amount of

facts. Imagination is what we lack.

D. I. Blokhintsev

1.Theoretical Description of Phase Transitions in Finite Systems

A rigorous theory of critical phenomena in finite systems was not built up to now. However, the experimental

studies of phase transitions (PTs) in some systems demand the formulation of such a theory. In particular, the

investigations of the nuclear liquid-gas PT [1–3] require the development of theoretical approaches which would

allow us to study the critical phenomena without going into the thermodynamic limit V → ∞ (V is the volume of

the system) because such a limit does not exist due the long range Coulomb interaction. Therefore, there is a great

need in the theoretical approaches which may shed light on the “internal mechanism” of how the PTs happen in

finite systems.

The general situation in the theory of critical phenomena for finite (small) systems is not very optimistic at the

moment because theoretical progress in this field has been slow. It is well known that the mathematical theory of

phase transitions was worked out by T. D. Lee and C. N. Yang [4]. Unfortunately, there is no direct generic relation

between the physical observables and zeros of the grand canonical partition in a complex fugacity plane. Therefore,

we know very well what are the gaseous phase and liquid at infinite volumes: mixture of fragments of all sizes and

ocean, respectively. This is known both for pure phases and for their mixture, but, despite some limited success [5],

this general approach is not useful for the specific problems of critical phenomena in finite systems (see Sect. VIII

below).

The tremendous complexity of critical phenomena in finite systems prevented their systematic and rigorous

theoretical study. For instance, even the best formulation of the statistical mechanics and thermodynamics of finite

systems by Hill [6] is not rigorous while discussing PTs. As a result, the absence of a well established definition of

the liquid and mixed phase for finite volumes delays the progress of several related fields, including the theoretical

and experimental searches for the reliable signals of several PTs which are expected to exist in strongly interacting

matter. Therefore, the task of highest priority of the theory of critical phenomena is to define the finite volume

analogs of phases from first principles of statistical mechanics. At present it is unclear whether such definitions can

be made for a general case, but it turns out that such finite volume definitions can be formulated for a variety of

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K.A. BUGAEV, P. T. REUTER

realistic nonclassical (= non mean-field) statistical models which are successfully used in nuclear multifragmentation

and in relativistic heavy collsisions.

About 25 years ago, when the theoretical foundations of nuclear multifragmentation were established, there was

an illusion that the theoretical basis is simple and clear and, therefore, we need only the data and models which

will describe them. The analysis of finite volume systems has proven to be very difficult. However, there was a clear

way out of troubles by making numerical codes that are able to describe the data. This is, of course, a common way

to handle such problems and there were many successes achieved in this way [1–3,7]. However, there is another side

of the coin which tells us that our understanding did not change much since then. This is so because the numerical

simulations of this level do not provide us with any proof. At best they just demonstrate something. With time

the number of codes increased, but the common theoretical approach was not developed. This led to a bitter result

- there are many good guesses in the nuclear multifragmentation community, but, unfortunately, little analytical

work to back up these expectations. As a result the absence of a firm theoretical ground led to formulation of such

highly speculative “signals” of the nuclear liquid-vapor PT as negative heat capacity [8,9], bimodality [10], which

later on were disproved, in Refs [11] and [12], respectively.

Thus, there is a paradoxic situation: there are many experimental data and facts, but there is no a single

theoretical approach which is able to describe them. Similar to the searches for quark-gluon plasma (QGP) [13]

there is lack of a firm and rigorous theoretical approach to describe phase transitions in finite systems.

However, our understanding of the multifragmentation phenomenon [1–3] was improved recently, when an exact

analytical solution of a simplified version of the statistical multifragmentation model (SMM) [14,15] was found in

Refs. [16,17]. These analytical results not only allowed us to understand the important role of the Fisher exponent

τ on the phase structure of the nuclear liquid-gas PT and the properties of its (tri)critical point, but to calculate

the critical indices α′,β,γ′,δ of the SMM [18] as functions of index τ. The determination of the simplified SMM

exponents allowed us to show explicitly [18] that, in contrast to expectations, the scaling relations for critical indices

of the SMM differ from the corresponding relations of a well known Fisher droplet model (FDM) [19]. This exact

analytical solution allowed us to predict a narrow range of values, 1.799 < τ < 1.846, which, in contrast to FDM

value τFDM≈ 2.16, is consistent with ISiS Collaboration data [20] and EOS Collaboration data [21]. This finding

is not only of a principal theoretical importance, since it allows one to find out the universality class of the nuclear

liquid-gas phase transition, if τ index can be determined from experimental mass distribution of fragments, but also

it enhanced a great activity in extracting the value of τ exponent from the data [22].

It is necessary to stress that such results in principle cannot be obtained either within the widely used mean-

filed approach or numerically. This is the reason why exactly solvable models with phase transitions play a special

role in statistical mechanics - they are the benchmarks of our understanding of critical phenomena that occur in

more complicated substances. They are our theoretical laboratories, where we can study the most fundamental

problems of critical phenomena which cannot be studied elsewhere. Their great advantage compared to other

methods is that they provide us with the information obtained directly from the first principles of statistical

mechanics being unspoiled by mean-field or other simplifying approximations without which the analytical analysis

is usually impossible. On the other hand an exact analytical solution gives the physical picture of PT, which cannot

be obtained by numerical evaluation. Therefore, one can expect that an extension of the exact analytical solutions

to finite systems may provide us with the ultimate and reliable experimental signals of the nuclear liquid-vapor PT

which are established on a firm theoretical ground of statistical mechanics. This, however, is a very difficult general

task of the critical phenomena theory in finite systems.

Fortunately, we do not need to solve this very general task, but to find its solution for a specific problem of

nuclear liquid-gas PT, which is less complicated and more realistic. In this case the straightforward way is to

start from a few statistical models, like FDM and/or SMM, which are successful in describing the most of the

experimental data. A systematic study of the various modifications of the FDM for finite volumes was performed by

Moretto and collaborators [23] and it led to a discovery of thermal reducibility of the fragment charge spectra [3],

to a determination of a quantitative liquid-vapor phase diagram containing the coexistence line up to critical

temperature for small systems [24,25], to the generalization of the FDM for finite systems and to a formulation

of the complement concept [26,27] which allows one to account for finite size effects of (small) liquid drop on the

properties of its vapor. However, such a systematic analysis for the SMM was not possible until recently, when its

finite volume analytical solution was found in [28].

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An invention of a new powerful mathematical method [28], the Laplace-Fourier transform, is a major theoretical

breakthrough in the statistical mechanics of finite systems of the last decade because it allowed us to solve exactly

not only the simplified SMM for finite volumes [28], but also a variety of statistical surface partitions for finite

clusters [29] and to find out their surface entropy and to shed light on a source of the Fisher exponent τ. It was

shown [28] that for finite volumes the analysis of the grand canonical partition (GCP) of the simplified SMM is

reduced to the analysis of the simple poles of the corresponding isobaric partition, obtained as a Laplace-Fourier

transform of the GCP. Such a representation of the GCP allows one not only to show from first principles that

for finite systems there exist the complex values of the effective chemical potential, but to define the finite volume

analogs of phases straightforwardly. Moreover, this method allows one to include into consideration all complicated

features of the interaction (including the Coulomb one) which have been neglected in the simplified SMM because

it was originally formulated for infinite nuclear matter. Consequently, the Laplace-Fourier transform method opens

a principally new possibility to study the nuclear liquid-gas phase transition directly from the partition of finite

system without taking its thermodynamic limit. Now this method is also applied [30] to the finite volume formulation

of the Gas of Bags Model (GBM) [31] which is used to describe the PT between the hadronic matter and QGP.

Thus, the Laplace-Fourier transform method not only gives an analytical solution for a variety of statistical models

with PTs in finite volumes, but provides us with a common framework for several critical phenomena in strongly

interacting matter. Therefore, it turns out that further applications and developments of this method are very

promising and important not only for the nuclear multifragmentation community, but for several communities

studying PTs in finite systems because this method may provide them with the firm theoretical foundations and a

common theoretical language.

It is necessary to remember that further progress of this approach and its extension to other communities

cannot be successfully achieved without new theoretical ideas about formalism it-self and its applications to the

data measured in low and high energy nuclear collisions. Both of these require essential and coherent efforts of two

or three theoretical groups working on the theory of PTs in finite systems, which, according to our best knowledge,

do not exist at the moment either in multifragmentation community or elsewhere. Therefore, the second task of

highest priority is to attract young and promising theoretical students to these theoretical problems and create

the necessary manpower to solve the up coming problems. Otherwise the negative consequences of a complete

dominance of experimental groups and numerical codes will never be overcome and a good chance to build up a

common theoretical apparatus for a few PTs will be lost forever. If this will be the case, then an essential part of

the nuclear physics associated with nuclear multifragmentation will have no chance to survive in the next years.

Therefore, the first necessary step to resolve these two tasks of highest priority is to formulate the up to

day achievements of the exactly solvable models and to discuss the strategy for their further developments and

improvements along with their possible impact on transport and hydrodynamic approaches. For these reasons the

paper is organized as follows: in Sect. II we formulate the simplified SMM and present its analytical solution in

thermodynamic limit; in Sect. III we discuss the necessary conditions for PT of given order and their relation to

the singularities of the isobaric partition and apply these findings to the simplified SMM; Sect. IV is devoted to

the SMM critical indices as the functions of Fisher exponent τ and their scaling relations; the Laplace-Fourier

transform method is presented in Sect. V along with an exact analytical solution of the simplified SMM which

has a constraint on the size of largest fragment, whereas the analysis of its isobaric partition singularities and the

meaning of the complex values of free energy are given in Sect. VI; Sect. VII and VIII are devoted to the discussion

of the case without PT and with it, respectively; at the end of Sect. VIII there is a discussion of the Chomaz and

Gulminelli’s approach to bimodality [5]; in Sect. IX we discuss the finite volume modifications of the Gas of Bags,

i.e. the statistical model describing the PT between hadrons and QGP, whereas in Sect. X we formulate the Hills

and Dales Model for the surface partition and present the limit of the vanishing amplitudes of deformations; and,

finally, in Sect. XI we discuss the strategy of future research which is necessary to build up a truly microscopic

kinetics of phase transitions in finite systems.

2.Statistical Multifragmentation in Thermodynamic Limit

The system states in the SMM are specified by the multiplicity sets {nk} (nk= 0,1,2,...) of k-nucleon fragments.

The partition function of a single fragment with k nucleons is [1]: V φk(T) = V (mTk/2π)3/2zk, where k = 1,2,...,A

(A is the total number of nucleons in the system), V and T are, respectively, the volume and the temperature of the

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system, m is the nucleon mass. The first two factors on the right hand side (r.h.s.) of the single fragment partition

originate from the non-relativistic thermal motion and the last factor, zk, represents the intrinsic partition function

of the k-nucleon fragment. Therefore, the function φk(T) is a phase space density of the k-nucleon fragment. For

k = 1 (nucleon) we take z1= 4 (4 internal spin-isospin states) and for fragments with k > 1 we use the expression

motivated by the liquid drop model (see details in Ref. [1]): zk= exp(−fk/T), with fragment free energy

fk= −W(T) k + σ(T) k2/3+ (τ + 3/2)T lnk ,

(1)

with W(T) = Wo+ T2/ǫo. Here Wo= 16 MeV is the bulk binding energy per nucleon. T2/ǫois the contribution

of the excited states taken in the Fermi-gas approximation (ǫo = 16 MeV). σ(T) is the temperature dependent

surface tension parameterized in the following relation: σ(T) = σ(T)|SMM ≡ σo[(T2

σo= 18 MeV and Tc= 18 MeV (σ = 0 at T ≥ Tc). The last contribution in Eq. (1) involves the famous Fisher’s

term with dimensionless parameter τ. As we will show later, at the critical (tricritical) point the fragment mass

distribution will lose it exponential form and will become a power law k−τ.

It is necessary to stress that the SMM parametrization of the surface tension coefficient is not a unique one. For

instance, the FDM successfully employs another one σ(T)|FDM= σo[1 − T/Tc]. As we shall see in Sect. IV the

temperature dependence of the surface tension coefficient in the vicinity of the critical point will define the critical

indices of the model, but the following mathematical analysis of the SMM is general and is valid for an arbitrary

σ(T) function.

The canonical partition function (CPF) of nuclear fragments in the SMM has the following form:

c − T2)/(T2

c + T2)]5/4, with

Zid

A(V,T) =

?

{nk}

?A

k=1

?

[V φk(T)]nk

nk!

?

δ(A −?

kknk).

(2)

In Eq. (2) the nuclear fragments are treated as point-like objects. However, these fragments have non-zero proper

volumes and they should not overlap in the coordinate space. In the excluded volume (Van der Waals) approximation

this is achieved by substituting the total volume V in Eq. (2) by the free (available) volume Vf ≡ V − b?

Zid

A(V − bA,T). The SMM defined by Eq. (2) was studied numerically in Refs. [14,15]. This is a simplified version

of the SMM, since the symmetry and Coulomb contributions are neglected. However, its investigation appears to

be of principal importance for studies of the nuclear liquid-gas phase transition.

The calculation of ZA(V,T) is difficult due to the constraint?

∞

?

where µ denotes a chemical potential. The calculation of Z is still rather difficult. The summation over {nk} sets in

ZAcannot be performed analytically because of additional A-dependence in the free volume Vf and the restriction

Vf> 0. The presence of the theta-function in the GCP (3) guarantees that only configurations with positive value

of the free volume are counted. However, similarly to the delta function restriction in Eq. (2), it makes again the

calculation of Z(V,T,µ) (3) to be rather difficult. This problem was resolved [16,17] by performing the Laplace

transformation of Z(V,T,µ). This introduces the so-called isobaric partition function (IP) [31]:

kknk,

where b = 1/ρo(ρo= 0.16 fm−3is the normal nuclear density). Therefore, the corrected CPF becomes: ZA(V,T) =

kknk= A. This difficulty can be partly avoided

by evaluating the grand canonical partition (GCP)

Z(V,T,µ) ≡

A=0

exp

?

µA

T

?

ZA(V,T) Θ(V − bA) ,

(3)

ˆ Z(s,T,µ) ≡

?∞

0

dV e−sVZ(V,T,µ)=

?∞

?∞

0

dV′e−sV′?

{nk}

?

?

k

1

nk!

?

V′φk(T) e

(µ−sbT)k

T

?nk

=

0

dV′e−sV′expV′

∞

?

k=1

φke

(µ−sbT)k

T

?

.

(4)

After changing the integration variable V → V′, the constraint of Θ-function has disappeared. Then all nk were

summed independently leading to the exponential function. Now the integration over V′in Eq. (4) can be done

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resulting in

ˆ Z(s,T,µ) =

1

s − F(s,T,µ),

(5)

where

F(s,T,µ) =

∞

?

k=1

φk exp

?(µ − sbT)k

T

?

=

?mT

2π

?3

2?

z1exp

?µ − sbT

T

?

+

∞

?

k=2

k−τexp

?(˜ µ − sbT)k − σk2/3

T

??

.

(6)

Here we have introduced the shifted chemical potential ˜ µ ≡ µ + W(T). From the definition of pressure in the

grand canonical ensemble it follows that, in the thermodynamic limit, the GCP of the system behaves as

p(T,µ) ≡ T lim

V →∞

ln Z(V,T,µ)

V

⇒Z(V,T,µ)

????

V →∞

∼ exp

?p(T,µ)V

T

?

.

(7)

An exponentially over V increasing part of Z(V,T,µ) in the right-hand side of Eq. (7) generates the rightmost

singularity s∗of the functionˆ Z(s,T,µ), because for s < p(T,µ)/T the V -integral forˆ Z(s,T,µ) (4) diverges at

its upper limit. Therefore, in the thermodynamic limit, V → ∞ the system pressure is defined by this rightmost

singularity, s∗(T,µ), of IPˆ Z(s,T,µ) (4):

p(T,µ) = T s∗(T,µ) .

(8)

Note that this simple connection of the rightmost s-singularity ofˆ Z, Eq. (4), to the asymptotic, V → ∞, behavior

of Z, Eq. (7), is a general mathematical property of the Laplace transform. Due to this property the study of the

system behavior in the thermodynamic limit V → ∞ can be reduced to the investigation of the singularities ofˆ Z.

3.Singularities of Isobaric Partition and Phase Transitions

The IP, Eq. (4), has two types of singularities: 1) the simple pole singularity defined by the equation

sg(T,µ) = F(sg,T,µ) ,

(9)

2) the singularity of the function F(s,T,µ) it-self at the point slwhere the coefficient in linear over k terms in the

exponent is equal to zero,

sl(T,µ) =

˜ µ

Tb.

(10)

The simple pole singularity corresponds to the gaseous phase where pressure is determined by the equation

pg(T,µ)=

?mT

2π

?3/2

T

?

z1exp

?µ − bpg

T

?

+

∞

?

k=2

k−τexp

?(˜ µ − bpg)k − σk2/3

T

??

.

(11)

The singularity sl(T,µ) of the function F(s,T,µ) (6) defines the liquid pressure

pl(T,µ) ≡ T sl(T,µ) =

˜ µ

b.

(12)

In the considered model the liquid phase is represented by an infinite fragment, i.e. it corresponds to the

macroscopic population of the single mode k = ∞. Here one can see the analogy with the Bose condensation where

the macroscopic population of a single mode occurs in the momentum space.

In the (T,µ)-regions where ˜ µ < bpg(T,µ) the gas phase dominates (pg> pl), while the liquid phase corresponds

to ˜ µ > bpg(T,µ). The liquid-gas phase transition occurs when two singularities coincide, i.e. sg(T,µ) = sl(T,µ). A

schematic view of singular points is shown in Fig. 1a for T < Tc, i.e. when σ > 0. The two-phase coexistence region

is therefore defined by the equation

pl(T,µ) = pg(T,µ) ,i.e., ˜ µ = b pg(T,µ) .

(13)

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