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On growth of spinodal instabilities in nuclear matter

Early growth of density fluctuations of nuclear matter in spinodal region is investigated employing the stochastic mean-field approach. In contrast to the earlier treatments in which only collective modes were included in the calculations, in the present work non-collective modes are also included, thus providing a complete treatment of the density correlation functions. Calculations are carried out for symmetric matter in non-relativistic framework using a semi-classical approximation.


I. INTRODUCTION
Much work has been done to analyze multi-fragmentation processes observed in heavy-ion collisions. These processes exhibit many features compatible with liquid-gas phase transition of nuclear matter. Experimental measurements of charge correlations between produced fragments suggest that the induced phase transformation is due to the amplification of density fluctuations in the spinodal region. Microscopic description based on mean-field approximation can provide only a limited information about the spinodal dynamics. In the mean-field framework, it is only possible to determine the boundary of the dynamical instabilities and the growth rates of the most-unstable collective modes. Growth of density fluctuations and large amplitude dynamical evolution of the system require a microscopic framework beyond the mean-field that incorporates the dynamics of density fluctuations [1][2][3][4][5].
Considerable effort has been spent to improve the standard mean-field approximation by incorporating dynamics of density fluctuations [6][7][8][9]. In a recently proposed stochastic mean-field approach (SMF), the effect of quantal and thermal fluctuations in the initial state is incorporated into the description in a stochastic manner [10]. The standard meanfield approximation is a deterministic description in the sense that a well-defined initial state gives rise to a unique final state. On the other hand, in the SMF approach, starting from a proper distribution of initial states, specified by quantal and thermal fluctuations, an ensemble of mean-field trajectories are evolved each with its own self-consistent Hamiltonian.
Employing the SMF approach in non-relativistic framework [17], as well as in relativistic framework based on Walecka-type effective field theory [18], we recently carried out a number of investigations of the early development of spinodal instabilities and baryon density correlation functions in nuclear matter in the semi-classical and quantal descriptions [19][20][21][22].
In the linear response regime, linearizing the equation of motion around a suitable initial state inside the spinodal zone and employing the method of one-sided Fourier transform, it is possible to carry out nearly analytical treatment of the baryon density correlation functions.
In these investigations only unstable collective modes were included in the calculations. As pointed out in Ref. [23], for a complete treatment of initial growth of density fluctuations it is necessary to include non-collective modes as well. In the present investigation, we undertake calculations of correlation functions of baryon density fluctuations including collective as well as non-collective modes.
In Section 2, we present nearly analytical description of the density correlation functions, including collective and non-collective modes in the linear response framework, in the semi-classical limit. In Section 3, we present calculations at different initial densities and temperatures for symmetric matter. Conclusions are given in Section 4.

II. DENSITY CORRELATION FUNCTIONS IN SPINODAL REGION
The SMF approach goes beyond the standard mean-field theory by including quantal and thermal fluctuations in the initial state. An ensemble of single-particle density matrices are generated by incorporating the initial fluctuations. It is possible to formulate the approach in relativistic and non-relativistic frameworks. Here we consider the non-relativistic approach.
The single-particle density matrix of each event (indicated by the event label λ),ρ λ ( r, r , t), is evolved through to its own mean-field according to where the label a = p ↑, p ↓, n ↑, n ↓ indicate the isospin-spin quantum numbers, and h λ a (t) is the mean-field Hamiltonian for the event. For the description of early growth of density fluctuations around an initial stateρ a , it is sufficient to linearize the equation of motion around this initial state. Subsequently small fluctuations, δρ λ a (t) =ρ λ a (t) −ρ a , in the density matrix are determined by where h a denotes the mean field Hamiltonian at the initial state and δU λ a is the fluctuating part of the mean-field for the event. We are interested in investigating the development of spinodal instabilities in nuclear matter around a uniform initial stateρ a . In this case, by employing a plane wave representation, we can carry out an almost analytical treatment of the density correlation function. In the plane wave basis, the linear response equations here f a ( p) denotes Fermi-Dirac factors at finite temperature. In the current work, we consider symmetric nuclear matter and assume that the mean-field potential depends only on the local nucleon density. Consequently, small fluctuations in the mean-field can be written as δU λ a ( r, t) = (∂U/∂ρ) 0 δρ λ ( r, t), where δρ λ ( r, t) denotes the small fluctuations in total nucleon density. Also, because of relative simplicity, we carry out calculations in the semiclassical limit. The Fourier transform of nucleon density fluctuations, δρ λ ( k, t) is related to the fluctuations of the density matrix according to where summation a is over the spin-isospin quantum numbers. The method of one-sided Fourier transform is very useful for solving Eq. 3 [24,25]. We introduce the one-sided Fourier transform of the local density in time as This yield an algebraic equation for the Fourier transform of the local density, which can be solved to obtain where ε( k, ω) = 1 + F 0 χ( k, ω) is the susceptibility, with F 0 = (∂U/∂ρ) 0 as the zeroth order Landau parameter, and χ( k, ω) denotes the Lindhard function. The semi-classical expression of the Lindhard function is given by The quantity G λ ( k, ω) in Eq.6 is determined by the initial conditions which acts as a source for developing fluctuations. According to the basic postulate of the SMF approach, elements of the initial density matrix are uncorrelated Gaussian random numbers with zero mean values and with well defined variances. In the semi-classical limit their variances are given by Here and below the bar over the expressions indicates the ensemble average. If we ignore the Coulomb potential, this has the same form for neutrons and protons in symmetric nuclear matter, therefore we omit the label a in the Fermi-Dirac factors. The factor δ ab reflects the assumption that local density fluctuations in spin-isospin modes are uncorrelated in the initial state.
According to the method of one-sided Fourier transform, time evolution of the density fluctuations is determined by the inverse transformation in time, which is expressed as a contour integral in the complex ω-plane as where integration path passes above all singularities of the integrand, as shown by line C 1 in Fig.1. We can calculate this integral by employing the residue theorem and closing the contour in a suitable manner. For the calculation of this integral, we need to investigate analytical structure of the integrand. In the spinodal zone, there are collective poles determined by imaginary solutions of the dispersion relation ε( k, ω) = 0 → ω = ±iΓ k . The collective poles play an important role in early growth of density fluctuations. However, as pointed out by Bozek [23], collective poles alone do not give the full description of the growth of instabilities. In fact, collective pole contributions alone do not even satisfy the initial conditions. By calculating the angular integral in the Lindhard function given by Eq.7, it is possible to see that, there is a cut singularity of the integrand in Eq.10 along real axis in the complex ω-plane. In addition, G λ ( k, ω) may also have singularities on the real axis. These non-collective poles of the integrand in Eq.10 can be seen more clearly in discrete description of the plane wave representation. If the system is placed in a finite box, plane waves are characterized by discrete values of momentum. In this case as illustrated in [26], dispersion relation has both imaginary collective poles and non-collective poles on the real axis. In the continuous limit, these non-collective poles appear as a cut singularity.
Since the integrand in complex ω-plane is multivalued, the entire real ω-axis is a branch cut.
In order to calculate the integral in Eq.10, we choose the contour C, as shown in Fig. 1.
We exclude the real ω-axis by drawing the contour from +∞ to the origin just above the real ω-axis, and after jumping from the first Riemann surface to the second at the origin, we draw the contour just below the real ω-axis from origin to +∞ . Contour is continued with a large semi-circle and it is completed by jumping from the second Riemann surface to the first one at origin,as shown in Fig.1. As a result, the integral can be expressed as where the first term is the pole contribution and the cut contribution is given by Local nucleon density fluctuations δρ λ ( r , t) are determined by the Fourier transform of In order to investigate the development of the spinodal instabilities a very useful quantity is the equal time correlation function of local density fluctuations The spectral intensity σ( k, t) of the correlation function is determined in terms of the variance of the Fourier transform of density fluctuations according to We can evaluate the spectral intensity σ( k, t) by evaluating the ensemble average using the Eqs. (12) and (13) for pole δρ λ pole ( k, t) and cut δρ λ cut ( k, t) part of the Fourier transform of density fluctuations and the Eq. (9) for the initial fluctuations. As a result, the spectral intensity becomes where the first and last terms are due to pole and cut parts of the spectral intensity and the middle term denotes the cross contribution. The pole part is where The pole contributions are the same as what we found in our previous investigation [17].
Mixed term in the spectral intensity of Eq. (17) has the following form with andB

III. RESULTS
In the numerical calculations we employ an effective Skyrme potential given in Ref. [6].
For symmetric nuclear matter it has the form  earlier studies [1]. We also observe that both pole and cut contributions have divergent behavior with opposite signs, as wave numbers approach their upper limit, k → k max . These divergent behaviors cancel each other out to produce a nice regular behavior of the spectral intensity as a function of wave number.  this mechanism leads to a mixture of various cluster sizes . As seen from Figs. We like to clarify that the linear response treatment provides a description for the early evolution of the condensation mechanism inside the spinodal region. Since the density fluc- However, we note that according to the constraint, as seen in Fig. (7), at this low initial density the linear response regime is already exceeded beyond time t = 25f m/c. We also note that, in the linear response treatment, even though density fluctuations and hence early  In previous studies [17,[19][20][21][22], in the calculations of the density correlation functions we included only the contributions due to the collective pole in the residue integral in Eq. (10).
As seen from dashed lines in Figs. (3)(4)(5)(6), we encounter the problem that both pole and cut contributions have divergent behavior as wave numbers approach its upper limit, k → k max .
To cure this problem in those calculations, we introduced a cut-off in the integral over the wave numbers in Eq. (18). At initial density of ρ = 0.2ρ 0 f m −3 the cut-off value was taken at the local minimum point of the spectral intensity.
In cut the integrations over the wave number at k cut = 0.8f m −1 . In Fig.(10), we compare the density correlation function calculated with the complete spectral intensity and with those calculated with pole contribution with cut-off at temperatures T = 1M eV and T = 5M eV .
We observe that the results obtained with pole approximations are rather sensitive to the cut-off wave number.

IV. CONCLUSIONS
We investigate early growth of density fluctuations in nuclear matter in using the SMF approach. In the linear response framework, employing the method of one-sided Fourier transform, it is possible to carry out a nearly analytical treatment of the correlation function of density fluctuations. The density correlation function provides a very useful information about the early condensation mechanism during the liquid-gas phase transformation of the system. The one-sided Fourier transform method involves a contour integration in the complex frequency plane. In earlier investigations, this contour integral was evaluated by keeping only effects of the unstable collective poles, and the effects of non-collective poles were ignored. In the spectral intensity of the density correlation function, the collective pole contribution has an important drawback that it leads to a divergent behavior as the wave numbers approaches its upper limit, k → k max . In previous studies to cure this problem, the integration over wave numbers was terminated at a cut-off wave number.
Since short wavelength fluctuations cannot grow in time, cut-off provides a relatively good approximation. However, the complete description is always desirable. In this work, we consider symmetric nuclear matter in non-relativistic framework of the SMF approach. We calculate the spectral intensity of the correlation function, including the effects of collective poles and non-collective poles in terms of the cut contribution in the complex frequency plane. The cut contribution also has a divergent behavior with opposite sign, as wave