- Email: [email protected]
Extended TDHF with a coherent collision term
9 November 2000
Physics Letters B 493 (2000) 47–53 www.elsevier.nl/locate/npe
Extended TDHF with a coherent collision term Sakir Ayik a,b a Tennessee Technological University, Cookeville, TN 38505, USA b Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606, Japan
Received 1 May 2000; received in revised form 20 September 2000; accepted 27 September 2000 Editor: W. Haxton
Abstract It is shown that the stochastic one-body transport description contains, in addition to incoherent binary collisions, a coherent damping mechanism due to coupling between mean-field fluctuations and single-particle motion. An extended TDHF model is proposed by incorporating the coherent mechanism into the equation of motion. In the limit of small fluctuations around equilibrium, the collective and single-particle self-energies due to the coherent mechanism are deduced. 2000 Published by Elsevier Science B.V. PACS: 24.30.Cz; 25.70.Ef; 25.70.Lm Keywords: Transport theory; Dissipation-extended TDHF
Over last two decades, much work have been done to improve the time-dependent Hartree–Fock (TDHF) theory beyond the mean-field approximation [1]. In so called extended TDHF theory, two-body correlations are incorporated into the equation of motion in the Born approximation [2–6]. The resultant collision term describes the coupling of the single-particle motion to the incoherent 2p–2h excitations. Such an incoherent damping mechanism is very important at relatively high energy heavy-ion collisions to convert the collective energy of the relative motion into incoherent excitations and thermalize the system. However, at low energies including giant resonance excitations, incoherent damping mechanism is not effective due to long nucleon mean-free-path. Therefore, for a proper description of damping mechanism at low energies, coherence between the p–h pairs should be taken into account [7,8]. For this reason, it is highly desirable to
improve the TDHF theory by incorporating a coherent collision term into the equation of motion. The one-body density matrix is coupled to two-body and higher order correlations through the BBGKY hierarchy. According to the generalized Langevin description of reduced variables developed by Mori, correlations due to coupling with degrees of freedom which are not considered explicitly, have two different but intimately connected effects: (i) dissipation of energy associated with the reduced variables leading to thermalization of the entire system, which is described by the friction or the collision term in the equation of motion, and (ii) dynamical fluctuations of the reduced variables, which are described by the random force term originating from the initial correlations [9]. Consequently, temporal evolution of the reduced one-body density matrix should have a stochastic nature. The associated “random force” in the equation of motion should originate from the initial correlations, with statistical properties specified in ac-
E-mail address: [email protected] (S. Ayik). 0370-2693/00/$ – see front matter 2000 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 0 ) 0 1 1 4 4 - 8
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S. Ayik / Physics Letters B 493 (2000) 47–53
cordance with the fluctuation–dissipation relation. The stochastic transport theories, both quantal and semiclassical forms, provide a one-body framework to describe dynamics of density fluctuations in a manner that is consistent with the dissipation–fluctuation relation of non-equilibrium statistical mechanics [10– 13]. Furthermore, the coherent damping mechanism is naturally included in the stochastic transport description. Density fluctuations excited by the stochastic source in the equation of motion are propagated by the mean-field, that gives rise to nonlinear fluctuations of the mean-field with random amplitudes on the top of its average evolution. The coherent damping mechanism arises from coupling of the single-particle motion with the mean-field fluctuations. We illustrate this mechanism by considering the average evolution of the single-particle density matrix, and propose an extended TDHF description by including the coherent mechanism. We investigate small amplitude vibrations around equilibrium and derive expressions for damping widths of the collective and single-particle excitations due to coherent damping mechanism. For a sufficiently dilute system, retaining the lowest order terms in the residual interactions, the BBGKY hierarchy can be truncated at the second level. As a result, temporal evolution of the single-particle density matrix ρ(t) ˆ is governed by [14], ∂ ρ(t) ˆ − h(ρ), ˆ ρ(t) ˆ = K(ρ) ˆ + δK(t), (1) ∂t where h(ρ) ˆ is the effective mean-field Hamiltonian, K(ρ) ˆ is an incoherent binary collision term and the second term on the left-hand-side represents the initial correlation term, δK(t) = Tr2 v, δσ12 (t) . (2)
i
The expression on the right-hand-side in Eq. (1) is obtained by determining two-body correlations from the solution of the second equation of the BBGKY hierarchy over a time interval from an initial time t0 to time t. The solution for two-body correlations, in general, involves two different contributions: (i) correlations developed by the effective residual interactions v during the time interval, which give rise to the binary collision term, and (ii) propagation of the initial correlations represented by δσ12 (t), that gives rise to the initial correlation term. The time interval t– t0 cannot be taken arbitrary large, but should be suf-
ficiently small to justify the neglect of the explicit coupling to higher order terms in the equation of the motion during the time interval. However, the correlations developed from the remote past until time t0 are contained the initial correlation term. If we consider an ensemble of identical systems that are prepared with slightly different initial conditions at the remote past, the exact initial correlations exhibit nearly random fluctuations. In the extended TDHF theory, a statistical averaging is carried out over such an ensemble and the average value of the initial correlations is assumed to vanish, δσ12 (t) = 0 [15]. This assumption in the semi-classical context is known as the “molecular chaos assumption”, and it corresponds to the factorization of the two-particle phase-space density before each binary collisions [16]. In the stochastic transport description, the initial correlations δσ12 (t) are retained, but treated as a Gaussian random quantity. Consequently, the initial correlation term δK(t) has a Gaussian random distribution characterized by a zero mean and a second moment which can be determined in accordance with the fluctuation–dissipation relation of the non-equilibrium statistical mechanics [10,14]. In analogy with the generalized Langevin description of the reduced dynamical variables, Eq. (1) is regarded as a stochastic transport equation for the fluctuating density matrix in which the stochastic part of the collision term δK(t) acts as a random noise. In the stochastic transport description, dynamical evolution is characterized by constructing an ensemble of solutions of the stochastic transport Eq. (1). In this manner, the theory provides a basis for describing the average evolution, as well as, dynamics of density fluctuations. Furthermore, the stochastic evolution involves a coherent dissipation mechanism arising from the coupling of the single-particle motion with randomly excited non-linear mean-field fluctuations. When the amplitude of the fluctuations is small, this mechanism appears as a coupling between the single-particle motion and the time-dependent RPA phonons around the mean trajectory. We consider the ˆ average evolution of the density matrix, ρ(t) = ρ(t), and calculate the ensemble average of Eq. (1) by expressing the mean-field and the density matrix as ˆ h(ρ) ˆ = h(ρ) + δ h(t) and ρ(t) ˆ = ρ(t) + δ ρ(t), ˆ where ˆ = (∂h/∂ρ) · δ ρ(t) δ h(t) ˆ and δ ρ(t) ˆ represent the fluctuating parts of the mean-field and the density matrix, respectively. Noting that the ensemble average of the
S. Ayik / Physics Letters B 493 (2000) 47–53
49
noise δK(t) vanishes, we obtain a transport equation for evolution of the average density matrix,
time evolutions are determined by
∂ i ρ(t) − h(ρ), ρ(t) = Kc (ρ) + K(ρ), (3) ∂t where K(ρ) represents the incoherent collision term and the additional term arises from the correlations of the mean-field fluctuations and the density fluctuations,
i
ˆ δ ρ(t)], ˆ Kc (ρ) = [δ h(t),
(4)
and it is referred to as the coherent collision term. This collision term has been investigated in previous publications in quantal [17] and semi-classical frameworks [18] for spatially uniform systems near equilibrium. Here, we carry out a quantal treatment of the collision term in non-equilibrium for finite systems. For details of the treatment we refer to [14]. In order calculate the coherent collision term, we consider that the fluctuations are small, and can be described by the linearized transport equation around the average evolution ρ(t), i
∂ ˆ ρ − h(ρ), δ ρˆ δ ρˆ − δ h, ∂t Zt b12 + δK(t). = −i dt 0 Tr2 v, δ C
(5)
t0
b12 can In the linearized collision term, the quantity δ C approximately be expressed as, b12 = 1 − ρ1 (t) 1 − ρ2 (t) G(t, t 0 ) δC b 0 ) G† (t, t 0 )ρ1^ (t)ρ2 (t) − h.c., (6) × v, δ Φ(t b describes the density fluctuations accordwhere δ Φ(t) b ing to δ ρ(t) ˆ = [δ Φ(t), ρ(t)]. We restrict our treatment to the stable regime and analyze linearized transport equation in a time-dependent RPA approach by expanding the small amplitude density fluctuations in terms of the time-dependent RPA functions, X δ ρ(t) ˆ = δzλ (t)ρλ† (t) + δzλ∗ (t)ρλ (t). (7) ρλ† (t)
and ρλ (t) denote the time-dependent RPA Here functions, and δzλ (t) and δzλ∗ (t) are the stochastic amplitudes associated with these modes. The timedependent RPA functions describe the correlated p–h excitations around the average trajectory, and their
∂ † ρ (t) − h(ρ), ρλ† (t) − h†λ (t), ρ(t) = −iηρλ† (t), ∂t λ (8)
and its hermitian conjugate, where the amplitude of the mean-field fluctuations is indicated by h†λ (t) = (∂h/∂ρ) · ρλ† (t) and small damping term is included in the equation of motion. The initial conditions are supplied by the static RPA functions, and their positive and negative frequency components are evolved according to Eq. (8) and its hermitian conjugate, respectively. We, also, introduce the dual wavefunctions, Qλ (t) and Q†λ (t) associated with the RPA modes [14, 19], i
∂ Qλ (t) − h(ρ), Qλ (t) − Qλ , ρ(t) (∂h/∂ρ) ∂t = −iηQλ (t) (9)
and its hermitian conjugate. It can be shown that, if the RPA functions and their dual functions form a bi-orthonormal system at the initial instant, they remain orthonormal in time according to the definition, Tr Qλ (t)ρµ† (t) = δλµ . When the occupation numbers of the natural states do not change in time, two types of wavefunctions are related by ρµ† (t) = [Q†λ (t), ρ(t)] and ρµ (t) = −[Qλ (t), ρ(t)]. In the following, we use these relations which provide a good approximation in the collision term even when the occupation numbers are changing in time. Projecting Eq. (5) on the collective RPA modes and noting that the collision term can be expanded usP b = δzλ (t)Q† (t) − δz∗ (t)Qλ (t), we can ing δ Φ(t) λ λ deduce stochastic equations for the random amplitudes, d i δzλ (t) = dt
Zt
dt 0 Σλ (t, t 0 )δzλ (t 0 ) + Fλ (t),
(10)
t0
where Fλ (t) = Tr Qλ (t) δK(t) is the projected noise and Σλ (t, t 0 ) denotes the collisional self-energy of the RPA mode. Following this result, the variances of the random amplitudes are determined by d |δzλ |2 = −Γλ (t)|δzλ |2 + 2Dλ (t), dt
(11)
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S. Ayik / Physics Letters B 493 (2000) 47–53
η describing the damping of the p–h states into more complex configurations. We calculate the ensemble average in the coherent collision term (4) by expanding the mean-field fluctuations and the density fluctuations in terms of the time-dependent RPA functions, X ˆ = δ h(t) δzλ (t)h†λ (t) + δzλ∗ (t)hλ (t) (15)
where Zt Γλ (t) = −2 Im
dt 0 Σλ (t, t 0 )
denotes the collisional damping width and Zt Dλ (t) = Re
dt 0 Fλ (t)Fλ∗ (t 0 )
is the diffusion coefficient associated with the mode. In the case of small fluctuations around equilibrium, the RPA modes become harmonic ρλ† (t) = ρλ† exp(−iωλ t) and ρλ (t) = ρλ exp(+iωλ t), and diffusion coefficient is related to damping width in accordance with the quantal dissipation–fluctuation relation [14,20], ωλ 1 coth . (12) 2 2T As a result, the thermal equilibrium value of the variance becomes |δzλ |2eq = Nλ0 + 1/2, where Nλ0 = 1/[exp(ωλ /T ) − 1] is the phonon occupation factor. Following this property, we define Dλ = Γλ ·
1 (13) 2 and regard Nλ (t) as the time-dependent occupation factors of the RPA functions. We note that, besides the collisional damping, there are other mechanisms involved in damping of the mean-field fluctuations, that are not incorporated in Eq. (10) for the random amplitudes. In particular, the low-frequency fluctuations are mainly damped by the one-body dissipation mechanism. In order to account for the one-body dissipation mechanisms in an approximate manner, we can evaluate the variances of the random amplitudes directly in terms of the uncorrelated density fluctuations and obtain an approximate expression for the phonon occupation factors |δzλ (t)|2 = Nλ (t) +
1 2 1 = Tr Qλ (t)ρ(t)Q†λ (t) 1 − ρ(t) 2 + Qλ (t) 1 − ρ(t) Q†λ (t)ρ(t) .
Nλ (t) +
(14)
In using this expression, damping term in Eq. (9) for dual functions should be included with a finite value of
and δ ρ(t) ˆ as given by Eq. (7). The collision term involves, in addition to the diagonal terms δzλ δzλ∗ , the ∗ arising from the coupling off-diagonal terms δzλ δzµ between different RPA modes through the incoherent collision term and its stochastic part. This coupling is neglected in Eq. (10) for the random amplitudes, which may not be very important between collective RPA modes. However, as a result of the collisional coupling, the non-collective RPA modes are strongly mixed up and loose their bosonic character [21]. Therefore, the non-collective modes should be excluded from the coherent collision term, since their effects can be included into the incoherent collision term by renormalizing the residual interactions. The diagonal contributions of the collective modes to the coherent collision term is given by ˆ [δ h(t), δ ρ(t)] ˆ coll 0 X 1 † hλ , ρQλ (1 − ρ) = Nλ + 2 1 † hλ , (1 − ρ)Qλ ρ − h.c., − Nλ + 2
(16)
where prime indicates the sum over the collective modes. By inspection, it can be seen that the first and second terms in this expression correspond to absorption and excitation of RPA phonons. These rates should be proportional to Nλ and Nλ + 1, respectively, but the average value Nλ + 12 appears in both rates. There are other contributions in Kc (ρ) arising from the cross-correlations between the collective and the non-collective modes. In schematic models, it is possible to show that these cross-correlations give rise to an additional contribution to the collision term, so that the excitation and absorption rates become proportional to Nλ + 1 and Nλ , as it should be. However, in the RPA analysis, it is difficult to extract such a contribution. For the time being, we replace the excitation and absorption factors in Eq. (16) by Nλ + 1 and Nλ , and
S. Ayik / Physics Letters B 493 (2000) 47–53
express the coherent collision term as, Kc (ρ) 0 X † hλ (t), Nλ (t) + 1 1 − ρ(t) Qλ (t)ρ(t) = − Nλ (t)ρ(t)Qλ (t) 1 − ρ(t) − h.c. (17) At low energies, since the dissipation is dominated by the coherent mechanism, we can omit the incoherent mechanism due to binary binary collisions in Eq. (3). As a result, we obtain an extended TDHF description by including the coherent collision term which describes coupling of the single-particle motion with the coherent 2p–2h excitations. The amplitudes of the mean-field fluctuations in the coherent collision term are self-consistently determined by the time-dependent RPA equation (9) and its conjugate. In general, it is difficult to determine the time evolution of the occupation factors Nλ (t) of the RPA functions, since they depend on rather complex damping mechanism of the mean-field fluctuations. Since, the high frequency fluctuations are mainly damped by the collisional effects, the corresponding occupation factors are determined by Eq. (11). However, for low frequency fluctuations it is more appropriate to use the expression (14) for an approximate description of these factors. As an illustration of the proposed model, we consider the nuclear collective vibrations around a finite temperature equilibrium and deduce expressions for the damping widths of collective vibrations and singleparticle excitations. Near equilibrium, since the small amplitude density fluctuations are harmonic, equilibrium mean-field fluctuations can be expressed in terms of the static RPA modes and the equilibrium occupation factors Nλ0 of these modes. For describing the damping of the single-particle excitations, we replace the mean-field Hamiltonian in Eq. (3) by its static value, h(ρ) → h0 , and express the density matrix in the P Hartree–Fock representation according to ρ(t) = |iini (t)hi|. Then, from transport Eq. (3), omitting the incoherent collision term, we can deduce a master equation for the occupation numbers d ni (t) = ni (t) 2 Im Σi> (t) dt − 1 − ni (t) 2 Im Σi< (t),
(18)
51
where the first and second terms represents loss and gain terms, respectively. The loss term is determined by the imaginary part of the self-energy X Σi> (t) = 1 − nj (t) |hi|h†λ |j i|2 Nλ0 + 1 × i − j − ωλ + iη |hi|hλ |j i|2 Nλ0 , + (19) i − j + ωλ + iη where, the first and second contributions describe the decay of the single-particle state |ii by excitation and absorption of a phonon, respectively. The expression of Σi< (t) in the gain term is obtained from Σi> (t) by making the substitutions, Nλ0 + 1 → Nλ0 , Nλ0 → Nλ0 + 1, and (1 − nj (t)) → nj (t). Similar expressions for the single-particle self-energies have been derived in the literature by following different approaches [19, 22,24]. In particular, we note that the retarded singleparticle self-energy Σiret (t) = Σi> (t) + Σi> (t), Σiret (t) =
X
|hi|h†λ |j i|2 Nλ0 + 1 − nj (t) i − j − ωλ + iη |hi|hλ |j i|2 Nλ0 + nj (t) + (20) i − j + ωλ + iη
has the same form as the one obtained by employing Matsubara formalism in [24]. In order to describe the collective vibrations, we linearize the transport Eq. (3) around a finite temperature equilibrium state ρ0 . Retaining only the coherent mechanism, the small amplitude vibrations δρ(t) = ρ(t) − ρ0 are determined by ∂ δρ(t) − h0 , δρ(t) − δh(t), ρ0 = δKc (t), (21) ∂t where the quantity δKc (t) represents the linearized form of the coherent collision term (17). An expression for this collision term can be obtained by following a treatment similar the one used in linearizing the collision term in Eq. (5). We analyze the linearized transport Eq. (21) by expanding the small amplitude density vibrations in terms of the static RPA functions as, X ∗ (t) Qµ , ρ0 , δρ(t) = zµ (t) Q†µ , ρ0 − zµ (22) i
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S. Ayik / Physics Letters B 493 (2000) 47–53
∗ (t) are the deterministic amplitudes where zµ (t) and zµ associated with the modes. By projecting the transport Eq. (21) on the RPA modes, we find that the corresponding amplitude are determined by
d i zµ (t) − ωµ zµ (t) = dt
Zt
dt 0 Σµ (t − t 0 )zµ (t 0 ).
(23)
t0
Here, Σµ (t − t 0 ) denotes the self-energy of the mode due to the coherent coupling mechanism, and its Fourier transform is given by [23] Σµ (ω) =
X
|hi|[Qµ , h†λ ]|j i|2 ω − ωλ − j + i + iη 0 × Nλ + 1 1 − n0j n0i − Nλ0 n0j 1 − n0i X
|hi|[Qµ , hλ ]|j i|2 ω + ωλ − j + i + iη 0 × Nλ 1 − n0j n0i − Nλ0 + 1 n0j 1 − n0i .
+
(24) The first term in the self-energy, describes the damping of the collective vibrations by exciting a phonon and a p–h pair. At a finite temperature, the reverse process with a weight Nλ0 n0j (1 − n0i ) is also possible, which decreases the damping. There is another contribution to the self-energy represented by the second term in this expression. It describes absorption of a phonon accompanied by p–h excitations that is possible only a finite temperatures. The self-energy of collective modes has been investigated by employing the Matsubara formalism in [24]. The expression (24) of the collective self-energy has essentially the same form as the one derived within the Matsubara formalism. The commutator structure in (24) gives rise to two direct and two cross terms, hi|[Qµ , h† ]|j i 2 = Fp (ij ; λµ) − Fv (ij ; λµ), (25) λ where 2 2 Fp (ij ; λµ) = hi|Qµ h†λ |j i + hi|h†λ Qµ |j i
(26)
and Fv (ij ; λµ) = hi|Qµ h†λ |j ihi|h†λ Qµ |j i∗ + hi|h†λ Qµ |j ihi|Qµ h†λ |j i∗
(27)
and similarly for |hi|[Qµ , h†λ ]|j i|2 . The parts of selfenergy determined by Fp (ij ; λµ) and Fv (ij ; λµ) correspond to the propagator and vertex corrections terms in Matsubara treatment, respectively. In the expression presented in [24], there are terms which do not involve the propagator ω ± ωλ − j + i + iη. These terms may be neglected, since they do not lead to damping of collective modes due to mixing with the particle–hole plus phonon states. Furthermore, it can be shown that in the pole approximation, i.e., ω ± ωλ − j + i = 0, the other terms presented in [24] may be combined together to give the same expression for the self-energy as the one given by (24), except an intermediate summation is missing in the propagator correction terms. In damping of high frequency collective modes and damping of single-particle excitations at low energies, the dominant contributions to the self-energies arises from the low-frequency density fluctuations that can be well approximated by surface vibrations. In previous publications, using this approximation, the Matsubara description have been applied to describe damping of the single-particle and giant resonance excitations at finite temperature [7,24]. In the extended TDHF, the effect of two body correlations are incorporated into the equation of motion in the form of an incoherent binary collision term. Such an incoherent mechanism is very important for damping of collective motion and thermalization of the system at intermediate energies around Fermi energy. However, at low energy nuclear dynamics including giant resonance excitations, the coherence in twobody correlations should be incorporated for a realistic description of the damping mechanism of the collective motion. In the stochastic theory, the transport description is improved by incorporating higher order correlations in an approximate manner by a stochastic mechanism. As a result, it is possible to address the average evolution as well as the dynamics of density fluctuations, in a manner similar to the generalized Langevin description of the reduced variables. Furthermore, the stochastic dynamics contains, in addition to incoherent binary collisions, a coherent damping mechanism resulting from the coupling between the mean-field fluctuations and the single-particle motion. We illustrate this damping mechanism by investigating the average dynamical evolution and propose an extended TDHF description by including the coherent collision term into the equation of motion. Consider-
S. Ayik / Physics Letters B 493 (2000) 47–53
ing small amplitude fluctuations around equilibrium, we deduce expressions for the self-energies of singleparticle and collective excitations due to the coherent coupling mechanism.
Acknowledgements Author gratefully acknowledges the Yukawa Institute for Theoretical Physics and Kyoto University for support and warm hospitality extended to him by the theory group and the supporting stuff. Also, Author thanks to Y. Abe for many useful discussions. This work is supported in part by the US DOE grant DEFG05-89ER40530.
References [1] K. Goeke, P.-G. Reinhard, in: Time-Dependent Hartree–Fock and Beyond, Proceedings, Bad Honnef, Germany, 1982. [2] C.Y. Wong, H.H.K. Tang, Phys. Rev. Lett. 40 (1978) 1070; C.Y. Wong, H.H.K. Tang, Phys. Rev. C 20 (1979) 1419. [3] S. Ayik, Z. Phys. A 298 (1980) 83. [4] R. Balian, M. Veneroni, Ann. Phys. 135 (1981) 270.
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[5] P. Grange, H.A. Weidenmuller, G. Wolschin, Ann. Phys. 136 (1981) 190. [6] W. Cassing, S.J. Wang, Z. Phys A 337 (1990) 1. [7] G.F. Bertsch, P.F. Bortignon, R.A. Broglia, Rev. Mod. Phys. 55 (1983) 287. [8] G.F. Bertsch, R.A. Broglia, Oscillations in Finite Quantum Systems, Cambridge, 1994. [9] H. Mori, Theor. Phys. 33 (1965) 423. [10] S. Ayik, C. Gregoire, Phys. Lett. B 212 (1988) 269; S. Ayik, C. Gregoire, Nucl. Phys. A 513 (1990) 187. [11] J. Randrup, B. Remaud, Nucl. Phys. A 514 (1990) 339. [12] Y. Abe, S. Ayik, P.-G. Reinhard, E. Suraud, Phys. Rep. 275 (1996) 49. [13] G.F. Burgio, Ph. Chomaz, J. Randrup, Phys. Rev. Lett. 69 (1992) 885. [14] S. Ayik, Y. Abe, in preparation. [15] L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics, Benjamin, New York, 1962. [16] K. Huang, Statistical Mechanics, Wiley, New York, 1962. [17] S. Ayik, Z. Phys. A 350 (1994) 45. [18] Y.B. Ivanov, S. Ayik, Nucl. Phys. A 593 (1995) 233. [19] P. Danielewicz, Ann. Phys. 197 (1990) 154. [20] C.W. Gardiner, Quantum Noise, Springer, Berlin, 1991. [21] Ph. Chomaz, D. Lacroix, S. Ayik, M. Colonna, Phys. Rev. C (2000), in press. [22] P. Danielewicz, P. Schuck, Nucl. Phys. A 567 (1994) 78. [23] S. Ayik, Phys. Rev. Lett. 56 (1986) 38. [24] P. Bortignon, R.A. Broglia, G.F. Bertsch, J. Pacheco, Nucl. Phys. A 460 (1986) 149.
Physics Letters B 493 (2000) 47–53 www.elsevier.nl/locate/npe
Extended TDHF with a coherent collision term Sakir Ayik a,b a Tennessee Technological University, Cookeville, TN 38505, USA b Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606, Japan
Received 1 May 2000; received in revised form 20 September 2000; accepted 27 September 2000 Editor: W. Haxton
Abstract It is shown that the stochastic one-body transport description contains, in addition to incoherent binary collisions, a coherent damping mechanism due to coupling between mean-field fluctuations and single-particle motion. An extended TDHF model is proposed by incorporating the coherent mechanism into the equation of motion. In the limit of small fluctuations around equilibrium, the collective and single-particle self-energies due to the coherent mechanism are deduced. 2000 Published by Elsevier Science B.V. PACS: 24.30.Cz; 25.70.Ef; 25.70.Lm Keywords: Transport theory; Dissipation-extended TDHF
Over last two decades, much work have been done to improve the time-dependent Hartree–Fock (TDHF) theory beyond the mean-field approximation [1]. In so called extended TDHF theory, two-body correlations are incorporated into the equation of motion in the Born approximation [2–6]. The resultant collision term describes the coupling of the single-particle motion to the incoherent 2p–2h excitations. Such an incoherent damping mechanism is very important at relatively high energy heavy-ion collisions to convert the collective energy of the relative motion into incoherent excitations and thermalize the system. However, at low energies including giant resonance excitations, incoherent damping mechanism is not effective due to long nucleon mean-free-path. Therefore, for a proper description of damping mechanism at low energies, coherence between the p–h pairs should be taken into account [7,8]. For this reason, it is highly desirable to
improve the TDHF theory by incorporating a coherent collision term into the equation of motion. The one-body density matrix is coupled to two-body and higher order correlations through the BBGKY hierarchy. According to the generalized Langevin description of reduced variables developed by Mori, correlations due to coupling with degrees of freedom which are not considered explicitly, have two different but intimately connected effects: (i) dissipation of energy associated with the reduced variables leading to thermalization of the entire system, which is described by the friction or the collision term in the equation of motion, and (ii) dynamical fluctuations of the reduced variables, which are described by the random force term originating from the initial correlations [9]. Consequently, temporal evolution of the reduced one-body density matrix should have a stochastic nature. The associated “random force” in the equation of motion should originate from the initial correlations, with statistical properties specified in ac-
E-mail address: [email protected] (S. Ayik). 0370-2693/00/$ – see front matter 2000 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 2 6 9 3 ( 0 0 ) 0 1 1 4 4 - 8
48
S. Ayik / Physics Letters B 493 (2000) 47–53
cordance with the fluctuation–dissipation relation. The stochastic transport theories, both quantal and semiclassical forms, provide a one-body framework to describe dynamics of density fluctuations in a manner that is consistent with the dissipation–fluctuation relation of non-equilibrium statistical mechanics [10– 13]. Furthermore, the coherent damping mechanism is naturally included in the stochastic transport description. Density fluctuations excited by the stochastic source in the equation of motion are propagated by the mean-field, that gives rise to nonlinear fluctuations of the mean-field with random amplitudes on the top of its average evolution. The coherent damping mechanism arises from coupling of the single-particle motion with the mean-field fluctuations. We illustrate this mechanism by considering the average evolution of the single-particle density matrix, and propose an extended TDHF description by including the coherent mechanism. We investigate small amplitude vibrations around equilibrium and derive expressions for damping widths of the collective and single-particle excitations due to coherent damping mechanism. For a sufficiently dilute system, retaining the lowest order terms in the residual interactions, the BBGKY hierarchy can be truncated at the second level. As a result, temporal evolution of the single-particle density matrix ρ(t) ˆ is governed by [14], ∂ ρ(t) ˆ − h(ρ), ˆ ρ(t) ˆ = K(ρ) ˆ + δK(t), (1) ∂t where h(ρ) ˆ is the effective mean-field Hamiltonian, K(ρ) ˆ is an incoherent binary collision term and the second term on the left-hand-side represents the initial correlation term, δK(t) = Tr2 v, δσ12 (t) . (2)
i
The expression on the right-hand-side in Eq. (1) is obtained by determining two-body correlations from the solution of the second equation of the BBGKY hierarchy over a time interval from an initial time t0 to time t. The solution for two-body correlations, in general, involves two different contributions: (i) correlations developed by the effective residual interactions v during the time interval, which give rise to the binary collision term, and (ii) propagation of the initial correlations represented by δσ12 (t), that gives rise to the initial correlation term. The time interval t– t0 cannot be taken arbitrary large, but should be suf-
ficiently small to justify the neglect of the explicit coupling to higher order terms in the equation of the motion during the time interval. However, the correlations developed from the remote past until time t0 are contained the initial correlation term. If we consider an ensemble of identical systems that are prepared with slightly different initial conditions at the remote past, the exact initial correlations exhibit nearly random fluctuations. In the extended TDHF theory, a statistical averaging is carried out over such an ensemble and the average value of the initial correlations is assumed to vanish, δσ12 (t) = 0 [15]. This assumption in the semi-classical context is known as the “molecular chaos assumption”, and it corresponds to the factorization of the two-particle phase-space density before each binary collisions [16]. In the stochastic transport description, the initial correlations δσ12 (t) are retained, but treated as a Gaussian random quantity. Consequently, the initial correlation term δK(t) has a Gaussian random distribution characterized by a zero mean and a second moment which can be determined in accordance with the fluctuation–dissipation relation of the non-equilibrium statistical mechanics [10,14]. In analogy with the generalized Langevin description of the reduced dynamical variables, Eq. (1) is regarded as a stochastic transport equation for the fluctuating density matrix in which the stochastic part of the collision term δK(t) acts as a random noise. In the stochastic transport description, dynamical evolution is characterized by constructing an ensemble of solutions of the stochastic transport Eq. (1). In this manner, the theory provides a basis for describing the average evolution, as well as, dynamics of density fluctuations. Furthermore, the stochastic evolution involves a coherent dissipation mechanism arising from the coupling of the single-particle motion with randomly excited non-linear mean-field fluctuations. When the amplitude of the fluctuations is small, this mechanism appears as a coupling between the single-particle motion and the time-dependent RPA phonons around the mean trajectory. We consider the ˆ average evolution of the density matrix, ρ(t) = ρ(t), and calculate the ensemble average of Eq. (1) by expressing the mean-field and the density matrix as ˆ h(ρ) ˆ = h(ρ) + δ h(t) and ρ(t) ˆ = ρ(t) + δ ρ(t), ˆ where ˆ = (∂h/∂ρ) · δ ρ(t) δ h(t) ˆ and δ ρ(t) ˆ represent the fluctuating parts of the mean-field and the density matrix, respectively. Noting that the ensemble average of the
S. Ayik / Physics Letters B 493 (2000) 47–53
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noise δK(t) vanishes, we obtain a transport equation for evolution of the average density matrix,
time evolutions are determined by
∂ i ρ(t) − h(ρ), ρ(t) = Kc (ρ) + K(ρ), (3) ∂t where K(ρ) represents the incoherent collision term and the additional term arises from the correlations of the mean-field fluctuations and the density fluctuations,
i
ˆ δ ρ(t)], ˆ Kc (ρ) = [δ h(t),
(4)
and it is referred to as the coherent collision term. This collision term has been investigated in previous publications in quantal [17] and semi-classical frameworks [18] for spatially uniform systems near equilibrium. Here, we carry out a quantal treatment of the collision term in non-equilibrium for finite systems. For details of the treatment we refer to [14]. In order calculate the coherent collision term, we consider that the fluctuations are small, and can be described by the linearized transport equation around the average evolution ρ(t), i
∂ ˆ ρ − h(ρ), δ ρˆ δ ρˆ − δ h, ∂t Zt b12 + δK(t). = −i dt 0 Tr2 v, δ C
(5)
t0
b12 can In the linearized collision term, the quantity δ C approximately be expressed as, b12 = 1 − ρ1 (t) 1 − ρ2 (t) G(t, t 0 ) δC b 0 ) G† (t, t 0 )ρ1^ (t)ρ2 (t) − h.c., (6) × v, δ Φ(t b describes the density fluctuations accordwhere δ Φ(t) b ing to δ ρ(t) ˆ = [δ Φ(t), ρ(t)]. We restrict our treatment to the stable regime and analyze linearized transport equation in a time-dependent RPA approach by expanding the small amplitude density fluctuations in terms of the time-dependent RPA functions, X δ ρ(t) ˆ = δzλ (t)ρλ† (t) + δzλ∗ (t)ρλ (t). (7) ρλ† (t)
and ρλ (t) denote the time-dependent RPA Here functions, and δzλ (t) and δzλ∗ (t) are the stochastic amplitudes associated with these modes. The timedependent RPA functions describe the correlated p–h excitations around the average trajectory, and their
∂ † ρ (t) − h(ρ), ρλ† (t) − h†λ (t), ρ(t) = −iηρλ† (t), ∂t λ (8)
and its hermitian conjugate, where the amplitude of the mean-field fluctuations is indicated by h†λ (t) = (∂h/∂ρ) · ρλ† (t) and small damping term is included in the equation of motion. The initial conditions are supplied by the static RPA functions, and their positive and negative frequency components are evolved according to Eq. (8) and its hermitian conjugate, respectively. We, also, introduce the dual wavefunctions, Qλ (t) and Q†λ (t) associated with the RPA modes [14, 19], i
∂ Qλ (t) − h(ρ), Qλ (t) − Qλ , ρ(t) (∂h/∂ρ) ∂t = −iηQλ (t) (9)
and its hermitian conjugate. It can be shown that, if the RPA functions and their dual functions form a bi-orthonormal system at the initial instant, they remain orthonormal in time according to the definition, Tr Qλ (t)ρµ† (t) = δλµ . When the occupation numbers of the natural states do not change in time, two types of wavefunctions are related by ρµ† (t) = [Q†λ (t), ρ(t)] and ρµ (t) = −[Qλ (t), ρ(t)]. In the following, we use these relations which provide a good approximation in the collision term even when the occupation numbers are changing in time. Projecting Eq. (5) on the collective RPA modes and noting that the collision term can be expanded usP b = δzλ (t)Q† (t) − δz∗ (t)Qλ (t), we can ing δ Φ(t) λ λ deduce stochastic equations for the random amplitudes, d i δzλ (t) = dt
Zt
dt 0 Σλ (t, t 0 )δzλ (t 0 ) + Fλ (t),
(10)
t0
where Fλ (t) = Tr Qλ (t) δK(t) is the projected noise and Σλ (t, t 0 ) denotes the collisional self-energy of the RPA mode. Following this result, the variances of the random amplitudes are determined by d |δzλ |2 = −Γλ (t)|δzλ |2 + 2Dλ (t), dt
(11)
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S. Ayik / Physics Letters B 493 (2000) 47–53
η describing the damping of the p–h states into more complex configurations. We calculate the ensemble average in the coherent collision term (4) by expanding the mean-field fluctuations and the density fluctuations in terms of the time-dependent RPA functions, X ˆ = δ h(t) δzλ (t)h†λ (t) + δzλ∗ (t)hλ (t) (15)
where Zt Γλ (t) = −2 Im
dt 0 Σλ (t, t 0 )
denotes the collisional damping width and Zt Dλ (t) = Re
dt 0 Fλ (t)Fλ∗ (t 0 )
is the diffusion coefficient associated with the mode. In the case of small fluctuations around equilibrium, the RPA modes become harmonic ρλ† (t) = ρλ† exp(−iωλ t) and ρλ (t) = ρλ exp(+iωλ t), and diffusion coefficient is related to damping width in accordance with the quantal dissipation–fluctuation relation [14,20], ωλ 1 coth . (12) 2 2T As a result, the thermal equilibrium value of the variance becomes |δzλ |2eq = Nλ0 + 1/2, where Nλ0 = 1/[exp(ωλ /T ) − 1] is the phonon occupation factor. Following this property, we define Dλ = Γλ ·
1 (13) 2 and regard Nλ (t) as the time-dependent occupation factors of the RPA functions. We note that, besides the collisional damping, there are other mechanisms involved in damping of the mean-field fluctuations, that are not incorporated in Eq. (10) for the random amplitudes. In particular, the low-frequency fluctuations are mainly damped by the one-body dissipation mechanism. In order to account for the one-body dissipation mechanisms in an approximate manner, we can evaluate the variances of the random amplitudes directly in terms of the uncorrelated density fluctuations and obtain an approximate expression for the phonon occupation factors |δzλ (t)|2 = Nλ (t) +
1 2 1 = Tr Qλ (t)ρ(t)Q†λ (t) 1 − ρ(t) 2 + Qλ (t) 1 − ρ(t) Q†λ (t)ρ(t) .
Nλ (t) +
(14)
In using this expression, damping term in Eq. (9) for dual functions should be included with a finite value of
and δ ρ(t) ˆ as given by Eq. (7). The collision term involves, in addition to the diagonal terms δzλ δzλ∗ , the ∗ arising from the coupling off-diagonal terms δzλ δzµ between different RPA modes through the incoherent collision term and its stochastic part. This coupling is neglected in Eq. (10) for the random amplitudes, which may not be very important between collective RPA modes. However, as a result of the collisional coupling, the non-collective RPA modes are strongly mixed up and loose their bosonic character [21]. Therefore, the non-collective modes should be excluded from the coherent collision term, since their effects can be included into the incoherent collision term by renormalizing the residual interactions. The diagonal contributions of the collective modes to the coherent collision term is given by ˆ [δ h(t), δ ρ(t)] ˆ coll 0 X 1 † hλ , ρQλ (1 − ρ) = Nλ + 2 1 † hλ , (1 − ρ)Qλ ρ − h.c., − Nλ + 2
(16)
where prime indicates the sum over the collective modes. By inspection, it can be seen that the first and second terms in this expression correspond to absorption and excitation of RPA phonons. These rates should be proportional to Nλ and Nλ + 1, respectively, but the average value Nλ + 12 appears in both rates. There are other contributions in Kc (ρ) arising from the cross-correlations between the collective and the non-collective modes. In schematic models, it is possible to show that these cross-correlations give rise to an additional contribution to the collision term, so that the excitation and absorption rates become proportional to Nλ + 1 and Nλ , as it should be. However, in the RPA analysis, it is difficult to extract such a contribution. For the time being, we replace the excitation and absorption factors in Eq. (16) by Nλ + 1 and Nλ , and
S. Ayik / Physics Letters B 493 (2000) 47–53
express the coherent collision term as, Kc (ρ) 0 X † hλ (t), Nλ (t) + 1 1 − ρ(t) Qλ (t)ρ(t) = − Nλ (t)ρ(t)Qλ (t) 1 − ρ(t) − h.c. (17) At low energies, since the dissipation is dominated by the coherent mechanism, we can omit the incoherent mechanism due to binary binary collisions in Eq. (3). As a result, we obtain an extended TDHF description by including the coherent collision term which describes coupling of the single-particle motion with the coherent 2p–2h excitations. The amplitudes of the mean-field fluctuations in the coherent collision term are self-consistently determined by the time-dependent RPA equation (9) and its conjugate. In general, it is difficult to determine the time evolution of the occupation factors Nλ (t) of the RPA functions, since they depend on rather complex damping mechanism of the mean-field fluctuations. Since, the high frequency fluctuations are mainly damped by the collisional effects, the corresponding occupation factors are determined by Eq. (11). However, for low frequency fluctuations it is more appropriate to use the expression (14) for an approximate description of these factors. As an illustration of the proposed model, we consider the nuclear collective vibrations around a finite temperature equilibrium and deduce expressions for the damping widths of collective vibrations and singleparticle excitations. Near equilibrium, since the small amplitude density fluctuations are harmonic, equilibrium mean-field fluctuations can be expressed in terms of the static RPA modes and the equilibrium occupation factors Nλ0 of these modes. For describing the damping of the single-particle excitations, we replace the mean-field Hamiltonian in Eq. (3) by its static value, h(ρ) → h0 , and express the density matrix in the P Hartree–Fock representation according to ρ(t) = |iini (t)hi|. Then, from transport Eq. (3), omitting the incoherent collision term, we can deduce a master equation for the occupation numbers d ni (t) = ni (t) 2 Im Σi> (t) dt − 1 − ni (t) 2 Im Σi< (t),
(18)
51
where the first and second terms represents loss and gain terms, respectively. The loss term is determined by the imaginary part of the self-energy X Σi> (t) = 1 − nj (t) |hi|h†λ |j i|2 Nλ0 + 1 × i − j − ωλ + iη |hi|hλ |j i|2 Nλ0 , + (19) i − j + ωλ + iη where, the first and second contributions describe the decay of the single-particle state |ii by excitation and absorption of a phonon, respectively. The expression of Σi< (t) in the gain term is obtained from Σi> (t) by making the substitutions, Nλ0 + 1 → Nλ0 , Nλ0 → Nλ0 + 1, and (1 − nj (t)) → nj (t). Similar expressions for the single-particle self-energies have been derived in the literature by following different approaches [19, 22,24]. In particular, we note that the retarded singleparticle self-energy Σiret (t) = Σi> (t) + Σi> (t), Σiret (t) =
X
|hi|h†λ |j i|2 Nλ0 + 1 − nj (t) i − j − ωλ + iη |hi|hλ |j i|2 Nλ0 + nj (t) + (20) i − j + ωλ + iη
has the same form as the one obtained by employing Matsubara formalism in [24]. In order to describe the collective vibrations, we linearize the transport Eq. (3) around a finite temperature equilibrium state ρ0 . Retaining only the coherent mechanism, the small amplitude vibrations δρ(t) = ρ(t) − ρ0 are determined by ∂ δρ(t) − h0 , δρ(t) − δh(t), ρ0 = δKc (t), (21) ∂t where the quantity δKc (t) represents the linearized form of the coherent collision term (17). An expression for this collision term can be obtained by following a treatment similar the one used in linearizing the collision term in Eq. (5). We analyze the linearized transport Eq. (21) by expanding the small amplitude density vibrations in terms of the static RPA functions as, X ∗ (t) Qµ , ρ0 , δρ(t) = zµ (t) Q†µ , ρ0 − zµ (22) i
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∗ (t) are the deterministic amplitudes where zµ (t) and zµ associated with the modes. By projecting the transport Eq. (21) on the RPA modes, we find that the corresponding amplitude are determined by
d i zµ (t) − ωµ zµ (t) = dt
Zt
dt 0 Σµ (t − t 0 )zµ (t 0 ).
(23)
t0
Here, Σµ (t − t 0 ) denotes the self-energy of the mode due to the coherent coupling mechanism, and its Fourier transform is given by [23] Σµ (ω) =
X
|hi|[Qµ , h†λ ]|j i|2 ω − ωλ − j + i + iη 0 × Nλ + 1 1 − n0j n0i − Nλ0 n0j 1 − n0i X
|hi|[Qµ , hλ ]|j i|2 ω + ωλ − j + i + iη 0 × Nλ 1 − n0j n0i − Nλ0 + 1 n0j 1 − n0i .
+
(24) The first term in the self-energy, describes the damping of the collective vibrations by exciting a phonon and a p–h pair. At a finite temperature, the reverse process with a weight Nλ0 n0j (1 − n0i ) is also possible, which decreases the damping. There is another contribution to the self-energy represented by the second term in this expression. It describes absorption of a phonon accompanied by p–h excitations that is possible only a finite temperatures. The self-energy of collective modes has been investigated by employing the Matsubara formalism in [24]. The expression (24) of the collective self-energy has essentially the same form as the one derived within the Matsubara formalism. The commutator structure in (24) gives rise to two direct and two cross terms, hi|[Qµ , h† ]|j i 2 = Fp (ij ; λµ) − Fv (ij ; λµ), (25) λ where 2 2 Fp (ij ; λµ) = hi|Qµ h†λ |j i + hi|h†λ Qµ |j i
(26)
and Fv (ij ; λµ) = hi|Qµ h†λ |j ihi|h†λ Qµ |j i∗ + hi|h†λ Qµ |j ihi|Qµ h†λ |j i∗
(27)
and similarly for |hi|[Qµ , h†λ ]|j i|2 . The parts of selfenergy determined by Fp (ij ; λµ) and Fv (ij ; λµ) correspond to the propagator and vertex corrections terms in Matsubara treatment, respectively. In the expression presented in [24], there are terms which do not involve the propagator ω ± ωλ − j + i + iη. These terms may be neglected, since they do not lead to damping of collective modes due to mixing with the particle–hole plus phonon states. Furthermore, it can be shown that in the pole approximation, i.e., ω ± ωλ − j + i = 0, the other terms presented in [24] may be combined together to give the same expression for the self-energy as the one given by (24), except an intermediate summation is missing in the propagator correction terms. In damping of high frequency collective modes and damping of single-particle excitations at low energies, the dominant contributions to the self-energies arises from the low-frequency density fluctuations that can be well approximated by surface vibrations. In previous publications, using this approximation, the Matsubara description have been applied to describe damping of the single-particle and giant resonance excitations at finite temperature [7,24]. In the extended TDHF, the effect of two body correlations are incorporated into the equation of motion in the form of an incoherent binary collision term. Such an incoherent mechanism is very important for damping of collective motion and thermalization of the system at intermediate energies around Fermi energy. However, at low energy nuclear dynamics including giant resonance excitations, the coherence in twobody correlations should be incorporated for a realistic description of the damping mechanism of the collective motion. In the stochastic theory, the transport description is improved by incorporating higher order correlations in an approximate manner by a stochastic mechanism. As a result, it is possible to address the average evolution as well as the dynamics of density fluctuations, in a manner similar to the generalized Langevin description of the reduced variables. Furthermore, the stochastic dynamics contains, in addition to incoherent binary collisions, a coherent damping mechanism resulting from the coupling between the mean-field fluctuations and the single-particle motion. We illustrate this damping mechanism by investigating the average dynamical evolution and propose an extended TDHF description by including the coherent collision term into the equation of motion. Consider-
S. Ayik / Physics Letters B 493 (2000) 47–53
ing small amplitude fluctuations around equilibrium, we deduce expressions for the self-energies of singleparticle and collective excitations due to the coherent coupling mechanism.
Acknowledgements Author gratefully acknowledges the Yukawa Institute for Theoretical Physics and Kyoto University for support and warm hospitality extended to him by the theory group and the supporting stuff. Also, Author thanks to Y. Abe for many useful discussions. This work is supported in part by the US DOE grant DEFG05-89ER40530.
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