Quantum statistics effects and fluctuations of particle numbers near the critical point of nuclear matter
aa r X i v : . [ nu c l - t h ] D ec Quantum statistics effects and fluctuations of particle numbers near the critical pointof nuclear matter
S.N. Fedotkin, A.G. Magner, and U.V. Grygoriev
Institute for Nuclear Research NASU, 03680 Kiev, Ukraine
Equation of state with quantum statistics corrections is derived for a multi-component gas ofparticles interacting through the repulsive and attractive van der Waals (vdW) forces up to firstfew orders over a small parameter δ ≈ ~ n ( mT ) − / [ g (1 − bn )] − , where n and T are the particlenumber density and temperature, m and g the particle mass and degeneracy factor. The parameter b corresponds to the vdW excluded volume. For interacting system of Fermi nucleon and Bose α particles, a small impurity of α particles to the nucleon system at leading first order in both α particle and nucleon small parameters δ does not change much the basic results for the symmetricnuclear matter. The particle number fluctuations ω determined by the isothermal in-compressibility K ( n, T ) can be obtained analytically at the same first order quantum-statistics approximation forsymmetric nucleon matter. Our approximate analytical results appear to be in good agreement withthe accurate numerical calculations. I. INTRODUCTION
A study of hadron matter, first of all, an interactingsystem of protons and neutrons, has a long history; see,e.g., Refs. [1–11]. Realistic versions of the nuclear mat-ter equation of state includes both the attractive and re-pulsive forces between particles. Thermodynamical be-havior of this matter leads to the liquid-gas first-orderphase transition which ends at the critical point. Exper-imentally, a presence of the liquid-gas phase transitionin nuclear matter was reported and then analyzed in nu-merous papers (see, e.g., Refs. [12–17]). Critical pointsin different systems of hadrons were studied in Refs. [18–21], see also references therein.Recently, the proposed van der Waals (vdW) equa-tion of state accounting for the quantum statistics(QS) [18, 22, 23] was used to describe the properties ofhadronic matter, also with many component extensionsand applications to the fluctuation calculations for dif-ferent thermodynamical averages [24–31]. The role andsize of the effects of QS was studied analytically for nu-clear matter, also for pure neutron and pure α -particlematter in Ref. [23]. Particularly, we investigated a de-pendence of the critical point parameters on the particlemass m , degeneracy factor g , and the vdW parameters a and b which describe particle interactions for each ofthese systems. Our consideration was restricted to smalltemperatures, T ∼ <
30 MeV, and not too large parti-cle densities. Within these restrictions, the number ofnucleons becomes a conserved number, and the chemi-cal potential of such systems regulates the number den-sity of particles. An extension to the fully relativistichadron resonances in a gas formulation with vdW inter-actions between baryons and between antibaryons wasconsidered in Ref. [32]. An application of this extendedmodel to net baryon number fluctuations in relativisticnucleus-nucleus collisions was developed in Ref. [33]. Wedo not include the Coulomb forces and make no differ-ences between protons and neutrons (both these parti-cles are named as nucleons). In addition, under theserestrictions the non-relativistic treatment becomes veryaccurate and is adopted in our studies. In the presentwork we are going to apply the same analytical methodas in Ref. [23] to the mixed two-component system of nucleons and α particles. Another attractive subject ofthis work is to apply our analytical results to analysis ofthe particle number fluctuations near the critical pointsof the nuclear matter.The paper is organized as the following. In Sec. IIwe recall some results of the ideal Bose and Fermi gasestaking an exemplary case of the two-component N − α system. In Sec. III the QS effects near the criticalpoint are studied for the system of symmetric-nuclearand α -particle matter. Our analytical results are used fornucleon number fluctuations in Sec. IV. These results arethen discussed in Sec. IV C and summarized in Sec. V. II. IDEAL QUANTUM GASES
The pressure P i ( T, µ ) for the i -system of particles(e.g., i = { N, α } ) plays the role of the thermodynamicalpotential in the grand canonical ensemble (GCE) wheretemperature T and chemical potential µ are independentvariables. The particle number density n i ( T, µ ), entropydensity s i ( T, µ ), and energy density ε i ( T, µ ) are given as n i = (cid:18) ∂P i ∂µ (cid:19) T , s i = (cid:18) ∂P i ∂T (cid:19) µ , ε i = T s i + µn i − P i . (1)In the thermodynamic limit V → ∞ considered in thepresent paper all intensive thermodynamical functions – P , n , s , and ε – depend on T and µ , rather than on thesystem volume V , see for instance Ref. [34]. We startwith the GCE expressions P i P id i ( T, µ ) for the pres-sure P id ( T, µ ) and particle number density n id ( T, µ ) = P i n id i ( T, µ ) for the ideal non-relativistic quantum gas[25, 35], P id i = g i R d p (2 π ~ ) p m i h exp (cid:16) p m i T − µT (cid:17) − θ i i − , (2) n id i = g i R d p (2 π ~ ) h exp (cid:16) p m i T − µT (cid:17) − θ i i − , (3)where m i and g i are, respectively, the particle mass anddegeneracy factor of the i component. The value of θ i = − θ i = 1 to the Bose gas,and θ i = 0 is the Boltzmann (classical) approximationwhen effects of the QS are neglected .Equations (2) and (3) can be expressed in terms of thepower series over fugacity, z ≡ exp( µ/T ), as: P i ( T, z ) ≡ g i Tθ i Λ i Li / ( θ i z ) = g i Tθ i Λ i P ∞ k =1 ( θ i z ) k k / , (4) n i ( T, z ) ≡ g i θ i Λ i Li / ( θz ) = g i θ i Λ i P ∞ k =1 ( θ i z ) k k / . (5)Here, Λ i ≡ ~ r πm i T (6)is the de Broglie thermal wavelength [25], and Li ν is thepolylogarithmic function [36, 37]. The values of µ > z >
1, are forbidden in the ideal Bose gas. The point µ = 0 corresponds to an onset of the Bose-Einstein con-densation in the system of bosons. For fermions, anyvalues of µ are possible, i.e., integrals (2) and (3) existfor θ i = − µ . The power series(4) and (5) are obviously convergent at z <
1. For theFermi statistics at z >
1, the integral representation ofthe corresponding polylogarithmic function can be used.Particularly, at z → ∞ one can use the asymptotic Som-merfeld expansion of the Li ν ( − z ) functions over 1 / ln | z | [38].For nucleon gas we take m N ∼ = 938 MeV neglectinga small difference between proton and neutron masses.The degeneracy factor is then g N = 4 which takes intoaccount two spin and two isospin states of nucleon. Forideal Bose gas of α -nuclei, one has g α = 1 and m α ∼ =3727 MeV.At z ≪
1, only one term k = 1 is enough in Eqs. (4)and (5) which leads to the classical ideal gas relation P = n T . (7)Note that the result (7) follows automatically fromEqs. (2) and (3) at θ i = 0. The classical Boltzmannapproximation at z ≪ T and/or small n region of the n - T plane. In fact, at very small n , oneobserves z < T too.Inverting the z k power series in Eq. (5), one transformsthe power expansion of z ( e i ) to the parameter e i (see,e.g., Ref. [26]), e i = − θ i n i Λ i √ g i ≡ − θ i ǫ i , (8)where ǫ i = ~ π / n i g i ( m i T ) / (9)Taking a given component i , e.g., for nucleon mat-ter ( θ i = − i The units with Boltzmann constant κ B = 1 are used. We keepthe Plank constant in the formulae to illustrate the effects of QS,but put ~ = h/ π = 1 in all numerical calculations. For simplic-ity, we omitted here and below the subscript id for the ideal gaseverywhere where it will not lead to a misunderstanding. in discussions of Fig. 1. The exact fugacity z ( ǫ ) canbe obtained by multiplying equation (5) by the factorΛ / (4 √ g ) to get ǫ = ǫ ( z ) and, then, inverting thisequation with respect to z . Different other curves inFig. 1 present the maximal power k max of the sum ofEq. (5) over k after the same multiplying and cut-off theseries for the polylogarithmic function Li( − z ) in powersof z (at the order k max ). As seen from this figure, onehas the asymptotic convergence over k max - the betterthe smaller ǫ . Even the first-order correction is lead-ing and good in the region of ǫ ≈ ǫ c = 0 . − . z ≈
1. The second ( k max = 2) correction improves theconvergence such that the cut-off sum for Li at the power k max practically coincides with the exact result (Fig.1).For larger ǫ , say, ǫ >
1, where the fugacity z is muchlarger than 1 (e.g., in the small temperature limit), weneed more and more terms and one has a divergence ofthe series in k max . In this region the series for Li fails,and one has to use another asymptotic expansion, forinstance, over 1 /z as suggested by Zommerfeld [38].Fig. 2 shows the contour graphics in the n − T planewhere black lines mean z ( n, T ) = const on left, and ǫ ( n, T ) = const on right with the values written in whitesquares. As seens from these plots, all values of z ∼ < ǫ ≪ ǫ , even when the fugacityis of the order of 1 and somewhat larger. In particular,the critical points obtained in Ref. [23] belong to such aregion.The expansion of z ( ǫ ) in powers of ǫ is inserted theninto Eq. (4). At small ǫ i < ǫ i is rapidly convergent asymp-totically, i.e., converges to the exact (polylogarithmic)function result (4) and (5), the faster the smaller ǫ i , suchthat a few first terms give already a good approximationof the QS effects. Notice that the fugacity values of z can be larger 1, however, for small ǫ and, similarly, forother corrections of a maximal power k max in the Li( z )polynomials. Taking the two terms, k = 1, and 2, inEqs. (4) and (5), one obtains a classical gas result (7)plus the leading first few-order corrections due to theeffects of QS: P i ( T, n i ) = n i T (cid:2) e i − c e i − c e i + O( e i ) (cid:3) , (10)where c = 4[16 / (9 √ − ∼ = 0 .
106 , c = 4(15 + 9 √ − √ / ∼ = 0 . ǫ i -terms in Eq. (10) as the first andsecond (order) quantum corrections.Equation (10) demonstrates explicitly a deviation ofthe quantum ideal gas pressure from the classical ideal-gas value (7): the Fermi statistics leads to an increasingof the classical pressure, while the Bose statistics to itsdecreasing. This is often interpreted [25] as the effec-tive Fermi ‘repulsion’ and Bose ‘attraction’ between QSparticles. k123 z ε exact max Figure 1. Fugacity z as function of the quantum statistics parameter ǫ for small values where one finds the critical points( ǫ c = 0 . − . z ( ǫ ), and k max is the maximal power of cut-offseries for polylogarithm Li . - T , M e V - T , M e V Figure 2. Contour plots for the first-order fugacity z ( n, T ) and parameter ǫ ( n, T ) for nucleon matter in the plane of density n and temperature T are shown in left and right panels, respectively. The red line (left) shows the zero entropy line, suchthat the white area is related to a nonphysical region where the entropy of the ideal gas is negative. The critical point for ourfirst-order and the zero-order (standard vdW) approximations for nuclear matter at the parameters a and b [18] are shownon right by the red and black points, relatively. The blue point in the same plot presents the numerical result for the criticalpoint (Ref. [23]). III. VDW MODEL WITHQUANTUM-STATISTICS CORRECTIONS
For the infinite system of a mixture of of different par-ticles, e.g., Fermi and Bose particles – nucleons and α particles, one can present the pressure function of the vdW model (vdWM) with the QS (QvdWM) [18] P ( T, n ) = P id N ( T, µ ∗ N ) + P id α ( T, µ ∗ α ) − a NN n N − a Nα n N n α − a αα n α , (11)where P id N ( T, µ ∗ N ) = g N T √ π Λ N R ∞ dη η / exp (cid:18) η − µ ∗ NT (cid:19) +1 ,P id α ( T, µ ∗ α ) = g α T √ π Λ α R ∞ dη η / exp (cid:16) η − µ ∗ αT (cid:17) − , (12)Here n is the baryon number density, n = n N + 4 n α , P id i is given by Eq. (2), and µ ∗ i are the solutions of transcen-dental equations: n ∗ N = n id N ( T, µ ∗ N ) ≡ g N √ π Λ N R ∞ dη η / exp (cid:18) η − µ ∗ NT (cid:19) +1 ,n ∗ α = n id α ( T, µ ∗ α ) ≡ g α √ π Λ α R ∞ dη η / exp (cid:16) η − µ ∗ αT (cid:17) − , (13)and n id i is defined by Eq. (3), see more details in Refs. [18,22]. The relationship between the densities n i of Eq. (11)and auxiliary ones n ∗ i can be written in the followingform [18]: n N = n ∗ N [1+( b αα − b αN ) n ∗ α ]1+ b NN n ∗ N + b αα n ∗ α + ( b NN b αα − b Nα b αN ) n ∗ N n ∗ α , (14) n α = n ∗ α [1+( b NN − b Nα ) n ∗ N ]1+ b NN n ∗ N + b αα n ∗ α + ( b NN b αα − b Nα b αN ) n ∗ N n ∗ α , (15)where b ij are the vdWM exclusion volume constants [18]: b NN = 3 .
35 fm , b αα = 16 .
76 fm ,b αN = 13 .
95 fm , b Nα = 2 .
85 fm . (16)Notice that for Bose particles, the restriction µ ∗ α ≤ µ α ≤ a ij > b ij > θ = 0 inEqs. (2) and (3), the QvdWM is reduced to the classical vdWM [25], P i = X j (cid:20) n i T − n j b ij − a ij n i n j (cid:21) . (17)Note that the classical vdWM (17) is further reduced tothe ideal classical gas (7) at a ij = 0 and b ij = 0. At a ij = 0 and b ij = 0 the QvdWM turns into the quantum ideal gas Eqs. (2) and (3).Following Ref. [18], one can fix the model parameters a ij and b ij using the ground state properties of the cor-responding system components, (see, e.g., Ref. [39]) by a NN = 329 . · fm , a Nα = a αN = a αα = 0 . (18)These values are very close to those found in Refs. [18,22]. Other constants are taken from Ref. [18] [seeEq. (16)]. Small differences appear because of the non-relativistic formulation used in the present studies. No-tice that the system of N + α was studied in Ref. [19] inthe Skyrme model, and the QvdW approach is criticized because the Bose condensation cannot be described inthe QvdW model.In what follows, a few first quantum corrections ofthe QvdWM will be considered. Expanding P id i ( T, µ ∗ i ),Eq. (12) used in Eq. (11), over small parameters ǫ ∗ i (Eq. (8) with i = N, n i = n ∗ N or i = α, n i = n ∗ α , andsuperscript in ǫ ∗ i corresponds to that of n ∗ i ), one obtains P id N ( T, n ∗ N ) = n ∗ N T [1 + ǫ ∗ N ] (19)and P id α ( T, n ∗ α ) = n ∗ α T [1 − ǫ ∗ α ] , (20)where ǫ ∗ i is given by Eq. (9) with replacing n i by n ∗ i .These expressions are similar to those of Eq. (10) at thefirst order in ǫ i . We proved that at small ǫ ∗ i the expan-sion of the pressure over powers of ǫ ∗ i becomes rapidlyconvergent to the exact results, and a few first termsgive already a good approximation. Our Eq. (19), incontrast to Eq. (10) discussed in Refs. [25, 26], takesinto account the particle interaction effects (cf. with theprevious section II). A new point of our considerationis the analytical estimates of the QS effects in a mixedsystem of interacting fermions and bosons. Similarly tothe ideal gases, the quantum corrections in Eq. (19) in-creases with the particle number density n i and decreaseswith the system temperature T , particle mass m i , anddegeneracy factor g i .As in Ref. [18], we introduce now the impurity con-tribution of the α - particles in the symmetric nuclearmatter as the ratio of the number of nucleons in the α particle impurity referred to the total number of nucle-ons, X α = 4 n α n N + 4 n α ≡ n α n , (21)where n is the baryon number density defined alreadyabove (below Eq. (12). According to the numerical so-lutions in Ref. [18], for the parameters of Eq. (18), thevalue of X α has been approximately obtained, X α ≈ . n ∗ N and n ∗ α from equations (14) and (15). Then,using Eqs. (21) and (16), one can present them in the fol-lowing approximate form: n ∗ N ≈ r n − ˜ b N n , n ∗ α ≈ r n − ˜ b α n , (22)where r = (1 − X α ) = 0 . , r = X α . . (23)Here, ˜ b i are coefficients related approximately to the in-teraction constants b ij , Eq. (16),˜ b N ≈ .
29 fm , ˜ b α ≈ .
81 fm . (24)For another interaction parameter a , one can use a = r a NN ≈ . · fm . (25) Critical points vdW ( k max = 0) 1 2 QvdW T c [MeV] 29.2 19.0 19.67 19.7 n c [fm − ] 0.100 0.065 0.072 0.072 P c [MeV · fm − ] 1.09 0.48 0.52 0.52Table I. Results for the CP parameters of the symmetric nu-clear matter ( g = 4 , m = 938 MeV); k max = 0 , , k max = 0 is the vdW ap-proach (Eq. (31)) (from Ref. [23]). Using also Eq. (11) with Eqs. (19) and (20), for the pa-rameter values of the order of mentioned above, one ar-rives at P ( T, n ) =
T r n [1 + δ N ]1 − ˜ b N n + T r n [1 − δ α ]1 − ˜ b α n − a n , (26)where δ i = ǫ i − ˜ b i n , n N = r n, n α = r n , (27)and i = N, α , r and r are given by Eq.(23). Note thatthe expression for the pressure, Eq. (26), in a case of r = 0 and r = 1 exactly the same as for a pure nuclearmatter in Ref. [23]. A new feature of the quantum ef-fects in the system of particle with the vdW interactionsis the additional factors (1 − ˜ b i n ) − in the quantum cor-rection δ i , i.e., the QS effects becomes stronger due tothe repulsive interactions between particles.The vdW, both in its classical form (17) and in itsQvdW extension (11) and (26), describes the first orderliquid-gas phase transition. As the value of X α , used inour derivations, is very small, the approximate criticalpoints in the considered approach will be determined bythe following equations: (cid:18) ∂P ( T, n ) ∂n (cid:19) T = 0 , (cid:18) ∂ P ( T, n ) ∂n (cid:19) T = 0 . (28)Using Eq. (26) in the first approximation in δ i , one de-rives from Eq. (28) the system of two equations for theCP parameters n c and T c at the same first order:2 na = T r (1+2 δ N ) ( − ˜ b n ) + T r (1 − δ α ) ( − ˜ b n ) , (29) a = T r ˜ b ( − ˜ b n ) h δ N (1+2˜ b n )˜ b n i + T r ˜ b ( − ˜ b n ) h − δ α (1+2˜ b n )˜ b n i . (30)Note that the equations (29) and Eq. (30) for the CP inthe case of r = 0 exactly the same as for a pure nucleonmatter in Ref. [23].For the CP parameters of the classical vdWM, whichare found from Eq. (28) for the equation (17), one has T (0) c = a b ∼ = 29 . , n (0) c = b ∼ = 0 .
100 fm − ,P (0) c = a b ∼ = 1 .
09 MeV · fm − . (31) The numerical calculations within the full QvdWM (11),(12), (15) and (14) give (see also Refs. [18, 22, 23]) T c ∼ = 19 . , n c ∼ = 0 . − ,P c ∼ = 0 .
562 MeV · fm − . (32)These our results (32) appear to be essentially the sameas those obtained in Ref. [18].A summary of the results for the CP parameters ispresented in Tables I (with Figs. 2 and 3) and II. Forsymmetrical nuclear matter ( X α = 0), Fig. 3 shows theisotherms of the pressure P as function of the reducedvolume v (left) and the particle number density n (rightpanel) with the first (and second) order corrections. Forthe same case, a difference of the results for the classi-cal vdWM (31) and QvdWM (32) demonstrates a roleof the effects of Fermi and Bose statistics at the CP ofthe symmetric nuclear particle matter. The size of theseeffects appears to be rather significant for the case of im-purity contributions X α ∼ = 1 of the α -particles into thenucleon matter. On the other hand, it is remarkable thatthe first order correction (Table I) reproduces these QSeffects with a high accuracy. The contribution of high or-der corrections in δ i , - second , third and fourth order ismuch smaller than the first-order correction that showsa fast convergence in δ i by first-order terms. Therefore,high-order corrections due to the QS effects can be ne-glected for evaluations of the critical points values.Table II shows that for the case of the mixed N − α system with X α , Eq. (21), even the first order correctionsare in good agreement with exact numerical QvdW re-sults (32), see Refs. [18, 22, 23]. As seen from TableII, the QS effects of the α - particle impurity can be ne-glected because, first of all, of too small relative con-centration X α of this impurity, according to Eq. (21) assuggested in Ref. [18]. By this reason, one can simplifyour calculations of the particle number fluctuations inthe next section IV, taking a pure symmetric nuclearmatter.Many other examples were recently considered inRef. [40]. All models investigated in that paper haverather different high-order virial-expansion coefficients.However, if the parameters of these different models arefixed by a requirement to reproduce properties of theground state, the obtained values of T c and n c appear tobe quite similar. For example, different T c values cometo the narrow region T c = 18 ± X α = 0). The effects of Fermi statisticsleads to much stronger changes of the T c values: about10 MeV in the nucleon matter. IV. PARTICLE NUMBER FLUCTUATIONS
From the Gibbs probability distribution for a gasof classical particles interacting through the repulsiveand attractive forces at large temperatures T and smallenough particle number density average h n i , one can usethe vdW equation of state (17) [see sec. II and Ref. [25])].In what follows, to simplify notations, we will omit an-gle brackets for statistical averages if it does not lead tomisunderstanding. Following, e.g., Ref. [31], the fluctu-ations of the particle number ω as the dispersion of the
10 20 30 40 50 60 7000.20.40.60.81 P ( M e V f m ) - v (fm ) T c cc T1.21.00.9 c TT c c cccccc P ( M e V f m ) - n (fm ) -31.0 T1.0 T Figure 3. Pressures P as functions of the reduced volume v (left) and particle number density n (right panel) at differenttemperatures T (in units of the critical value T c ) for the simplest case of the symmetric nucleon matter. The critical point isshown by the close circle found from the exact solution of equations (28). The dotted line shows the second order approximation[Ref. [23] and Eq. (26) ( r = 1 , r = 0) employing for nucleon matter]. The horizontal lines are plotted by using the Maxwellarea law in left and correspondingly in right panels. The unstable and metastable parts of the isothermal lines are presentedby dashed and dash-dotted lines, respectively. Other closed dots show schematically a binodal boundary for the two phasecoexistence curve in the transition from two- to one-phase range [23].Critical points Eq.(31) N N QvdWM N + α N + α QvdWM T c [MeV] 29.2 19.0 19.7 19.4 19.9 n c [fm − ] 0.100 0.065 0.072 0.072 0.073 P c [MeV · fm − ] 1.09 0.48 0.52 0.51 0.56Table II. Results for the CP parameters of the vdWM (2nd column), the symmetric nuclear matter ( N ) ( g = 4 , m = 938 MeV,3rd and 4th columns) and the mixed symmetric-nuclear and α -particle ( g = 1 , m = 3737 MeV) matter ( N + α , 5th and 6thcolumns). GCE Gibbs distribution function integrated over the ex-citation energy can be expressed in terms of the deriva-tive of particle number-density average.
A. Fluctuations and susceptibility in the GCE
For calculations of classical fluctuations of the particlenumbers, ω , within the grand canonical ensemble (GCE)one can start with the particle number average [25, 31] h N i = Z ρ N ( q , p ; µ, T, V )dΓ , (33)where ρ N ( q , p ; µ, T, V ) is the GCE distribution functionof the phase space variables q , p , dΓ = d q d p (normal-ized as usually for a classical system), µ is the chemicalpotential, T the temperature, and V the volume of theclassical system. The Gibbs probability distribution canbe written as ρ N ( q , p ; µ, T, V ) = Z − exp [ − ( H N ( q , p ) − µN ) /T ] . (34)Here H N ( q , p ) is the classical Hamiltonian, Z the nor-malization factor which is the partition function, Z = X N Z dΓ exp [ − ( H N ( q , p ) − µN ) /T ] . (35) Taking variations of both sides of Eq. (33) over µ withthe help of Eqs. (34) and (35) and changing the orderof the integral over the phase space Γ and derivativeover the chemical potential µ , at first order variations,i.e., the second order in fluctuations one obtains (seeRefs. [25, 31]) ω ( n, T ) = h (∆ N ) ih N i = T ( δn/δµ ) T n , (36)where ∆ N = N − h N i is the fluctuation of N aroundits average h N i , n = n ( µ, T ) is the particle number-density average in the GCE. In Eq. (39), the variationalderivative, χ = ( δn/δµ ) T , (37)is the isothermal susceptibility. For the linear (first or-der) variations, χ = ( ∂n/∂µ ) T , (38)one has explicitly, ω ( n, T ) ≡ S = h N i − h N i h N i = T ( ∂n/∂µ ) T n . (39)The particle number density n ( µ, T ), entropy density s ( µ, T )), and energy density ε ( T, µ ) in the GCE are givenby Eq. (1).
12 345 6 7 89 (cid:144) n c T (cid:144) T c
123 456 789 (cid:144) n c T (cid:144) T c
23 45 67 89 (cid:144) n c T (cid:144) T c
23 45 67 89 (cid:144) n c T (cid:144) T c Figure 4. Contour plots for the vdW [zeroth, upper ] and first (lower plots) orders in the QS expansion over a smallparameter δ for the particle number fluctuations as functions of the density n and temperature T (in units of n c and T c ) withfull in-compressibility K T ( n, T ) (left) and the main derivative approximation (MDA) (right panels). Let us consider variations of the relationship (33) overthe chemical potential µ taking into account high ordervariations, for instance, second-order ones. For simplic-ity, we shall still take these variations at constant tem-perature, i.e. consider non-linear (second-order) isother-mal susceptibility. Eq. (36) is correct for any order of thevariational derivative (non-linear susceptibility, Eq. (37))but now we can specify it at the 2nd order. Taking im-mediately the variations over µ up to the second order at T = const in Eq. (33), one obtains the next (2nd) ordercorrections to Eqs. (39) and (38), which were consideredat first order. These corrections are proportional to theso called kurtosis, defined in Ref. [22] in a slightly differ-ent way. Fluctuations accounting for the third cumulantmoment of the Gibbs distribution, take the form: T h N i δ h N i = S ( δµ ) + 12 T S ( δµ ) + . . . , (40)where S is the kurtosis which can be normalized by the h N i as S , Eq. (39), S = h N i − h N i h N i . (41)Similarly, one can obtain the 4th order moment (or 4th-order cumulant moment) of the Gibbs distribution, i.e.,from the third order variations of the average (33) overchemical potential µ , and so on. This allows us to go be-yond the restrictions of the 2nd order cumulant momentfluctuations ω , shown explicitly in Eq. (39), i.e. beyondthe first variational derivative for the susceptibility χ , –linear susceptibility χ , Eq. (38).The expression (36) for the fluctuation ω of the par-ticle number is more general though it is still singularexactly at the CP where the linear susceptibility χ (38)is ∞ in the sum (40).The integral traces of cumulants as given by C N = Z d q d p exp {− β [ H N ( q , p ) − µN ] } , (42)can be calculated by the saddle point method (SDM).We may try to introduce the entropy S = − X N P N log P N , (43)where P N = ρ N ( q , p , µ, T, V ) is the probability distri-bution (34) P N = exp {− β [ H N ( q , p ) − E ] − µN } , (44)Writing the SPM condition δS = 0, i.e., (cid:18) ∂S∂ q (cid:19) ∗ = 0 , (cid:18) ∂S∂ p (cid:19) ∗ = 0 , (45)one obtains the classical trajectories q ∗ ( t ) , p ∗ ( t ) fromthese Hamiltotian equations (45). We have also to spec-ify the mean field in the Hamilton function, H N ( q , p ) = P κ (cid:2) p κ / m + V κ ( q κ ) (cid:3) ). If we are far from the bifur-cations (CPs), one can use the standard SPM, and thenon-zero 2nd order terms of the entropy expansion. Suchderivations lead to the results of the standard thermody-namics but near the CP. Near the bifurcation, where thesecond order terms of the entropy expansion is zero, wemay employ the improved SPM (ISPM) [41, 42] trans-forming the ISPM for the action phase integral of thePOT [42–46] to the real exponent argument – the en-tropy S , Eq. (43). The simplest ISPM is the second -order expansion of the entropy (43), but with finite in-tegration limits. Now, one can take the path integralanalytically in terms of the erf functions of the real ar-gument. Here is the place where we can apply for thecatastrophe theory of Fedoryuk [47, 48] by expanding theentropy to the third order terms. In this way, we arriveat the Airy-kind integrals with the finite contributionsof the two SPM points which turn into one bifurcationpoint at the limit to the CP. Note that in order to re-move singularity of the fluctuations ω near at the criticalpoint with generalization to the QS description, one cancalculate ω through the moments of the statistical leveldensity ρ ( E, A ) [49], also with using the ISPM.Thus, for the first simplest classical dynamic case,the integral for the fluctuation (36) can be presentedas the Feynman path integral over the formal trajec-tories q ( t ) and p ( t ) with the SDM condition (45) forthe main contributions from the classical trajectories atlarge excitation energy with the system temperature, T = ( ∂S/∂E ) ∗ , where the standard thermodynamics butwith critical points is working well.In the next sections, we will study more a popularformula (see Appendix A, Ref. [25, 26, 30]) used forcalculations of the fluctuations ω , which is expressedin terms of the isothermal in-compressibility K T , andcompare the results obtained by different approxima-tions. Our purpose of the next sections is to find theranges of good agreement between the approximate ex-pansion near the critical point and accurate analyticalresult for the vdWM to check a validness of both expres-sions through the non-linear susceptibility χ and non-linear in-compressibility K T . B. Fluctuations and in-compressibility
For the relative fluctuations of the particle numbers ω , one has [24–31] ω ( n, T ) = T K T , (46)where K T is the isothermal in-compressibility, K T = (cid:18) δPδn (cid:19) T , (47)and P is given by equation of state which is given inthe one-component QvdW (symmetric nucleon matter)by Eq. (26) (with r = 1, r = 0, ˜ b = b NN ). Thein-compressibility K T , Eq. (47), in Eq. (46) as functionof the density n and temperature T , can be expandedin power series near the critical point n c , T c over bothvariables n and T but taking derivatives at the currentpoint n, T , K T = (cid:0) ∂P∂n (cid:1) T + (cid:16) ∂ P∂n (cid:17) T ( n − n c )+ ∂ P∂n∂T ( T − T c ) + (cid:16) ∂ P∂n (cid:17) T ( n − n c ) + . . . . (48)Using approximately the definition (28) valid at the crit-ical point , and assuming that the linear in tempera-ture and quadratic in density variations are dominatingabove other high order variations, one can define themain derivative approximation (MDA): K MDA T ≈ ∂ P∂n∂T ( T − T c ) + 12 (cid:18) ∂ P∂n (cid:19) T ( n − n c ) . (49)We may compare their contributions K MDA T into the fullexpansion (48) with the definition K T ( n, T ), Eq. (47) interms of the first order derivative in the expansion (48)as function of n, T , K T ≈ K (1) T = (cid:18) ∂P∂n (cid:19) T . (50)Notice that Eq. (46) can be derived from Eq. (36) byusing linear variations for the chemical potential µ asfunction of the particle number density n (see, e.g., Ap-pendix A).Approximating Eq. (47) for in-compressibility K T ( n, T ) by the first derivative of the pressure,Eq. (50), at the first order in a small quantum-statisticsparameter δ , Eq. (27) ( r = 1, r = 0, ˜ b = b NN = b ,Eq. (16)), one obtains ω ( n, T ) = TT [1 + 2 δ ] / (1 − nb ) − na . (51)Studying now a behavior of ω ( T, n ), Eq. (51), nearthe critical point n c , T c , see 3rd column in Table I, within The CP is assumed to be of the simplest second order, in contrastto a high order CP when high order derivatives become also zero. n/n c T=T c ω Figure 5. Fluctuations of the particle numbers ω for nucleon system as function of the particle number density n (in units of n c ) at the critical value of the temperature T = T c with zeroth (vdW), k max = 0, and first corrections of the QS expansion, k max = 1. Solids show Eq. (36) for the full (without expansion near the CP) fluctuations ω and dashed curves present thetested main-derivative approximation (MDA), Eq. (46) with (49). the QvdW model, we will use now the expansion of thesefluctuations in powers of the distance from the CP n c , T c taking derivatives at the CP. With the help of the newvariables, τ ≡ T /T c − , ν ≡ n/n c − , (52)one can fix first n = n c and find the behavior of ω ( n, T )as function of temperature T near the critical point. Forthis purpose, it is convenient to present δ ( n, T ) in thefollowing form: δ ( n, T ) ≈ δ [(1 + ν ) n c , (1 + τ ) T c ] . (53)We will find now the limit of this expression at ν = 0and very small τ and, then, at τ = 0 and very small ν .In the first case, ν = 0, one can approximate Eq. (53) by δ ( n c , (1 + τ ) T c ) ≈ ~ π / ˜ n g ( mT c ) / (1 − ˜ n ) (cid:18) − τ (cid:19) . (54)Using also Eq. (51), one finds ω ( n c , (1 + τ ) T c ) ≈ (1 − ˜ n ) − δ τ − = T c n c P c G τ τ − , ν = 0 , (55)where ˜ n = bn c , δ = δ ( T c , n c ), and G τ ≈ P c T c n c (1 − ˜ n ) − δ . (56)Taking b from Eq. (16) , one finally obtains G τ ≈ . G τ ≈ . G τ = 1 /
6. Similarly, using Eq. (54), for the fluctuations ω ( n, T ),Eq. (51), at the constant T = T c one finds δ ((1 + ν ) n c , T c ) ≈ ~ π / ˜ n g ( mT c ) / (1 − ˜ n ) × (cid:16) ν − ˜ n + ν ˜ n (1 − ˜ n ) (cid:17) . (57)Finally, for the fluctuations ω , Eq. (51), one arrives at ω ((1 + ν ) n c , T c ) ≈ T c n c P c G ν ν − , τ = 0 , (58)where G ν ≈ P c T c n c (1 − ˜ n ) n [2 δ (1 + ˜ n ) + ˜ n ] ≈ . . (59)For the case of the classical vdWM ( δ = 0), fromEq. (59) one obtains G ν = 2 / ω ( n, T ), Eqs. (46) or(51), is function of the two variables n and T , one needsto introduce the two-dimensional critical index, with thefirst component being along the n and second one alongthe T axis. Another characteristics of the critical point( n c , T c ) in the n − T plane is the two-dimensional fluc-tuation slope coefficient { G τ , G ν } ≈ { . , . } . Noticethat the temperature, Eq. (55), and the density, Eq. (58),dependence near the CP can be seen also from Eq. (49)of the MDA. C. Discussion of the results
Fig. 4 shows the particle number fluctuations ω ( n, T )in units of the critical values n c and T c for symmetric nu-clear matter at the zeroth [vdW, upper] and first (lowerpanels) order in the quantum statistics expansion. Left0and right contour plots of Fig. 4 present the calcula-tions using respectively the standard in-compressibility K (1) T ( n, T ), Eq. (50), and its MDA, Eq. (49). For theMDA calculations we assume the dominance of thederivative contributions of Eq. (48) above high ordervariations in the in-compressibility, see Eq. (49), andneglect first- and second-derivative terms by using ap-proximately Eq. (28). As seen from Fig. 4 (cf. lowerwith upper plots), the quantum statistics effects is signif-icant for the fluctuations ω even after exclusion of a largeshift of the critical point by choosing the scaling unitsto a lower critical values due to the quantum statisticseffect, in agreement with the accurate numerical result[Eq.(32)] (see also Ref. [18]). Contour plots for fluctu-ations ω at a few next high orders (e.g., k max = 2 − ω , Eqs. (46) and (49),with those calculated through the the equation (50) forin-compressibility K T takes place, except for small whiteranges near the CP. (see Fig. 4).Fig. 5 presents more details in the comparison be-tween fluctuations (46) with the in-compressibility K T ,Eq. (50), using the pressure (26) ( r = 1 , r = 0) fornucleons at the zeroth and first orders of the QS ex-pansion and their MDA calculations by Eq. (49) for K , at T = T c . The derivatives of the MDA are cal-culated analytically at the ( n, T ) point on a small butfinite distance from the critical point ( n c , T c ) by assum-ing that the third derivative term over the density n isleading at the temperature T = T c (Eq. (49)) for vari-ations of the pressure of equation of state, which is de-termined by Eq. (26) for symmetric nuclear matter. Asshown in Fig. 5, the fluctuations (46) calculated throughthe in-compressibility, Eq. (50) (solids), and the MDA,Eq. (49)(dashed lines), within a given order k max = 0 or1 of the QS expansion shows a huge bump in the densitydependence, largely in agreement with the approximatesimple analytical asymptotic expression (58), and moreaccurate analytical formula, Eq. (51), also with numeri-cal calculations.Concerning the MDA, one finds a good agreement withthe expression (50) for the vdW ( k max = 0), and first(”1”) approximations in the QS expansion over δ nearthe critical point up to a small distance from the CP.As this distance decreases, one can see a divergence ofthe MDA as for the vdW approximation. For larger dis-tances from the CP in the range n ∼ > . n ∼ < . ω with QS corrections, it was convenient to usethe expression (39) for the fluctuation ω in terms of thesusceptibility using the fugacity variable z instead of theparticle density variable n . Similarly, one can considerthe fluctuations ω ( T, n ) at n = n c with analogous prop-erties.A validity of the expressions (46) and (39) for fluctu-ations ω and their MDAs can be evaluated from these calculations (Figs. 4 and 5) by the ranges where onefinds good agreement between the approximations (49)and (50) for the in-compressibility K ( n, T ). Their roughrelative estimates, about 1.5%, are found approximatelyfor both the cases, T = T c (Fig. 5) in the dependence ondensity n and in dependence on temperature at n = n c (see also Fig. 4). The fluctuation based on the MDAEq. (49), converges to that with the standard Eq. (50) forthe in-compressibility of the expression (39) (or Eq. (46))on much smaller relative distances, 0.05% for T = T c andfrom -0.02% to 0.005% for n = n c , with respect to thecritical values of Table I.Notice that it is difficult (impossible) to realize prac-tically the conditions for the application of the MDAin the limit to the CP, in particular, if we introducethe restrictions T = T c or n = n c . In the way to theCP, one has to stop at small but finite distance fromthe CP when a huge bump appear : the MDA varia-tions fail because it becomes smaller or of the order ofnext derivatives contributions in expansion (48) of thein-compressibility K T in the denominator of the fluctu-ations ω , Eq. (46), see Refs. [27–29]. The derivations ofEqs. (36) and (46) become invalid on enough small butfinite distances from the CP because, probably, we usethe mean field approach (in particular, the vdWM) asthe basis of the QS perturbation expansion. As shownin Ref. [25], in this case the correlation length of thecorrelation function, or the two-body amplitude of scat-tering in the quasi-particle Landau theory [50], infinitelydiverges by increasing relatively, in the considered limitto the CP, with respect to the mean distance betweenparticles. In this case, the arguments of validness forthe derivations of Eqs. (36) and (46) for the fluctua-tions ω through the derivatives of the thermodynamicaverages (pressure or particle number density) contradict[27, 28] with the background of the statistical physics forwhich we should have an opposite tendency such that theconsidered relative fluctuations must be small; see, e.g.,Refs. [25, 27, 28, 31]. V. SUMMARY
The QvdWM equation of state has been derived an-alytically and used to study the quantum statistics ef-fects in a vicinity of the critical point of two-componentsystem of nucleon and α -particle matter. The expres-sions for the pressure were obtained by using the quan-tum statistics expansion , over the small parameters δ i ( i = { N, α } ) near the vdW approach. A simple andexplicit dependence on the system parameters, such asthe particle mass m i and degeneracy factor g i , is demon-strated at the first order of this expansion. Such a depen-dence is absent within the classical vdWM. The quan-tum corrections to the CP parameters of the symmetric-nuclear and α -particle matter appear to be quite signif-icant. For example, the value of T (0) c = 29 . decreases dramatically to the value T (1) c = 19 . T c = 19 . α particles into thenucleon system occurs in the correct direction, namelythe CPs are somewhat increased in the critical point ascompared to those for pure nucleon system, and theseanalytical results are in good agreement with more ex-act numerical calculations.The particle number fluctuations for symmetric nu-cleon matter have been derived within the same ana-lytical QvdW approach near the critical point. Theirbehavior near the critical point in standard calculationsthrough the in-compressibility is in good agreement withmore exact numerical calculations. Main features, as ahuge bump near the CP, for the same QvdWM equa-tion of state was found as similar to the approximateanalytical and full numerical results obtained with andwithout using the expansion of the in-compressibilitynear the CP at zero (vdW) and first order over a smallparameter of quantum statistics. The convergence ofthe main derivative approximation for the isothermalin-compressibility near the CP by accounting for con-tributions of the mixed second density-temperature andthird density derivative terms to the corresponding fullexpansion of the in-compressibility was studied and therough estimates for ranges of validness of these QvdWapproximations was obtained.As perspectives, we will study the fluctuations nearthe critical point by using the improved saddle pointmethod similarly as applied for the oscillating compo-nents of the single-particle density of states within thesemiclassical periodic orbit theory of critical points (bi-furcations) [41, 42, 48] and in terms of the moments ofthe statistical level density. Note also that our consid-eration made for the QvdWM can be straightforwardlyextended to other types of inter-particle interactions. ACKNOWLEDGMENTS
We thank A.I. Sanzhur for many fruitful dis-cussions and suggestions, as well D.V. Anchishkin,M.I. Gorenstein, A. Motornenko, R.V. Poberezhnyuk,and V. Vovchenko for many useful discussions. The workof S.N.F. and A.G.M. on the project “Nuclear collec-tive dynamics for high temperatures and neutron-protonasymmetries” was supported in part by the Program“Fundamental researches in high energy physics and nu-clear physics (international collaboration)” at the De-partment of Nuclear Physics and Energy of the NationalAcademy of Sciences of Ukraine. S.N.F., A.G.M. andU.V.G. thank the support in part by the budget pro-gram “Support for the development of priority areas ofscientific reseraches”, the project of the Academy of Sci-ences of Ukraine, Code 6541230.
Appendix A: Derivations of the classicalparticle-number fluctuactions
Within the canonical ensemble (CE), one can use thefree energy F ( V, T ) as a characteristic thermodynamicfunction of the volume V and temperature T for a fixed particle number N . Assuming the thermodynamic limitcondition for our infinite system, one can express F interms of that per particle [25], F ( V, T ) =
N f (˜ v, T ) , (A1)where ˜ v = 1 n , n = N/V . (A2)For the pressure P and chemical potential µ , one has P = − (cid:18) ∂F∂V (cid:19) T = − (cid:18) ∂f∂ ˜ v (cid:19) T , (A3)and µ = (cid:18) ∂F∂N (cid:19) T = f − n (cid:18) ∂f∂ ˜ v (cid:19) T , (A4)where the volume per particle ˜ v is given by Eq. (A2).Taking the first variation of Eq. (A4) over particlenumber density n through the relationship (A2), one ob-tains δµ = 1 n (cid:18) ∂ f∂ ˜ v (cid:19) T δn . (A5)Therefore, one finds (cid:18) ∂n∂µ (cid:19) T = n ( ∂ f /∂ ˜ v ) T . (A6)According to Eq. (39) and Eqs. (A6), (A3) and (A2),one arrives at Eq. (46).Note that the same result can be obtained muchshortly by using the Jacobian transformations within theGCE [25], (cid:18) ∂n∂µ (cid:19) T = D ( n, T ) D ( µ, T ) = 1 D ( µ, T ) /D ( n, T ) (A7)and n = (cid:18) ∂P∂µ (cid:19) T = D ( P, T ) D ( µ, T ) , (A8)see Eq.(1). Therefore, substituting these equations (A7)and (A8) into Eq. (39) for the particle number fluctua-tions ω , one can do cancellation in ratios of the denom-inator by using the Jacobian properties. Finally, oneobtains Eq. (46).Note that these derivations based on the first deriva-tive transformations fail near the critical point becauseof the divergence of fluctuations due to zeros in the de-nominators and, therefore, strictly speaking, cannot beused in enough a small vicinity of the critical point, seeEq. (28), in contrast to the fluctuation formula as theGibbs distribution dispersion (sec. III) and Eq. (36) interms of the susceptibility (37).As stated in the paper, our analysis can be appliedbeyond the vdW approach. In fact, similar estimatesof the quantum statistic effects can be straightforwardlydone also for the mean-field models. Concerning thesemodels see, e.g., [51] and references therein.2 [1] B. K. Jennings, S. Das Gupta, and N. Mobed, Phys.Rev. C , 278 (1982).[2] G. R¨opke, L. M´unchow, and H. Schulz, Nucl. Phys. A , 536 (1982).[3] G. F´ai and J. Randrup, Nucl. Phys. A , 557 (1982).[4] T. Biro, H. W. Barz, B. Lukacs, and J. Zimanyi, Phys.Rev. C , 2695 (1983).[5] L. P. Csernai, H. St¨ocker, P. R. Subramanian, G. Buch-wald, G. Graebner, A. Rosenhauer, J. A. Maruhn, andW. Greiner, Phys. Rev. C , 2001 (1983).[6] L. P. Csernai and J. I. Kapusta, Phys. Rept. , 223(1986).[7] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. , 1(1986).[8] J. Zimanyi and S.A. Moszkowski, Phys. Rev. C , 1416(1990).[9] R. Brockmann and R. Machleidt, Phys. Rev. C , 1965(1990).[10] H. Mueller and B. D. Serot, Nucl. Phys. A , 508(1996).[11] M. Bender, P. H. Heenen and P. G. Reinhard, Rev. Mod.Phys. , 121 (2003).[12] J. E. Finn et al., Phys. Rev. Lett. , 1321 (1982).[13] R. W. Minich et al., Phys. Lett. B , 458 (1982).[14] A. S. Hirsch et al., Phys. Rev. C , 508 (1984).[15] J. Pochodzalla et al., Phys. Rev. Lett. , 1040 (1995).[16] J. B. Natowitz, K. Hagel, Y. Ma, M. Murray, L. Qin, R.Wada, and J. Wang, Phys. Rev. Lett. , 212701 (2002).[17] V. A. Karnaukhov et al., Phys. Rev. C , 011601(2003).[18] V. Vovchenko, A. Motornenko, P. Alba, M.I. Gorenstein,L.M. Satarov, and H. Stoecker, Phys. Rev. C , 045202(2017).[19] L.M. Satarov, I.N. Mishustin, A. Motornenko,V. Vovchenko, M.I. Gorenstein, and H. Stocker,Phys. Rev. C , 024909 (2019).[20] R. V. Poberezhnyuk, V. Vovchenko, M. I. Gorenstein,and H. Stoecker Phys. Rev. C 99, 024907 (2019).[21] R.V. Poberezhnyuk, V. Vovchenko, D.V. Anchishkin,M.I. Gorenstein, J. Mod. Phys. E, , 1750061 (2017).[22] V. Vovchenko, D. Anchishkin, and M. Gorenstein, Phys.Rev. C, , 0.64314 (2015).[23] S.N. Fedotkin, A.G. Magner, and M.I. Gorenstein, Phys.Rev. C, , 054334 (2019).[24] M Anisimov and V. Sychev, Thermodynamics of crit-ical state for individual sustances , (Energoatomizdat,Moscow, 1990)(in Russian).[25] L.D. Landau and E.M. Lifshitz,
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