Coupled channel T-operator equations with connected kernels: (II). N coupled two-body channels

A time-independent theory of rearrangement collisions involving transitions between two-body states is presented. It is assumed that the system of interest consists of particles that may be partitioned into two-body systems in N ways, including interchanges of particle labels without changing the kind of channel. An infinite family of sets of N coupled T-operator equations is derived by use of the channel coupling array, as in previous work on the three-body problem. Specialization to the channel-permuting arrays guaranteeing connected (N?1)th iterates of the kernel of the coupled equations is made in the N-channel case (N > 3) and the nature of the solutions to the coupled equations is discussed. Various approximation schemes to be used with numerical calculations are suggested. Since the transition operators for all rearrangement channels are coupled together, no problems concerning non-orthogonality of the eigenstates of different channel Hamiltonians are encountered; also the presence of the outgoing wave boundary condition in all channels is made explicit. The close resemblance of the equations in matrix form to those of one-channel scattering is exploited by introducing Møller wave operators and associated channel scattering states, an optical potential formalism that leads to rearrangement channel optical potential operators, and a variational formulation of the coupled equations using a Schwinger-like variational principle. A brief comparison with other many-body formalisms is also given.     

Integral equations for the scattering of N identical particles

Integral equations are obtained for the scattering of N identical particles using a form of the N-particle scattering equations derived previously. The equations couple together only transition operators between physical two cluster channels, the breakup amplitudes being expressed in terms of quadratures over two-cluster amplitudes. The kernel of the equations becomes connected after a single iteration. The number of coupled equations for identical particles is $\text{1}\text{2}\text{N}\text{or}\text{1}\text{2}\text{(N?1)}$ when N is even or odd respectively.     

Connected kernel methods in nuclear reactions: (III). Effective interactions

The recently derived connected kernel equation (CKE) for N-body scattering operators is applied to direct nuclear reactions. A spectral representation is derived for the kernel of the CKE in order to obtain manageable approximations. This allows the kernel to be split into orders corresponding to the propagation of different numbers of bound clusters. By formally solving one part of the kernel at a time, the CKE is written as a hierarchy of nested equations in increasingly many variables. The first equation of this hierarchy is a set of coupled channel Lippmann-Schwinger equations coupling together all two-cluster channels. These equations reduce to the usual coupled channel equations for inelastic scattering and to the coupled channel Born approximation for rearrangement reactions when weak coupling assumptions are made. The second equation of the hierarchy is a two-variable integral equation for the effective interactions appearing in the coupled channel equations. The driving terms and kernel of this integral equation are obtained from the third equation of the hierarchy which is a three-variable integral equation and so forth. The use of the spectral expansion results in a renormalized theory in the sense that the bound state and reaction problems are separated. This permits the inclusion of nuclear models in the theory in a straightforward manner. The hierarchy is applied to a particular example, that of nucleon-nucleus scattering. For this case the hierarchy is truncated at the level allowing no more than three clusters in the continuum. By suppressing exchange and keeping only one-particle transfer and single-nucléon knockout channels, a set of equations for the optical potentials and transfer operators is obtained. These equations provide a three-body treatment of the single scattering approximation to the optical potential. Iteration of the equations yields the usual single scattering approximation in first order including three-body off-shell effects. After suppression of Fermi motion and off-shell effects, the standard impulse approximation is recovered. Modifications of the method for other cases are discussed and other possible applications suggested.     

Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit N→+∞, where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.     

Complete reduction of fermion-antifermion bethe-salpeter equation with static kernel

The Bethe-Salpeter (BS) equation for a $\text{spin}\text{-}\text{1}\text{2}$ fermion-antifermion bound system is considered for the case in which the kernel is static and is the fourth component (i.e., three-scalar part) of a vector potential. Relative time (or relative energy) dependence can be eliminated easily. The 16 BS bispinor amplitudes are reexpressed in the usual way in terms of corresponding tensor amplitudes which satisfy 16 coupled integrodifferential equations. If Lorentz-, parity-, and charge-conjugation invariance are used, these equations can be reduced through a sequence of transformations to single eigenvalue equations, involving scalar and three-vector wavefunctions for singlet and triplet states, respectively. The effective Hamiltonians obtained in these equations are correct to all orders in the coupling constant and have a simple structure, consisting in general of a scalar, a spin-orbit, and a tensor part, which are explicitly exhibited.Although the equations could well be used for consideration of a general particleantiparticle system (e.g., quark-antiquark), for the present only positronium with a Coulomb interaction kernel is treated as an illustrative example. There exists a singlettriplet splitting in leading order mα⁶ ln α even though no spin-spin forces are directly introduced in the kernel. The splitting is calculated in detail in perturbation theory to order mα⁶ ln α and mα⁶.     

A multi-cluster,n-particle generalization of the three-particle faddeev wavefunction equations

Recent work applying certain forms of many-body scattering theory to problems such as molecular potential energy surfaces and equations for nonequilibrium statistical mechanics indicates that a formulation of the theory based directly on multi-cluster, n-particle, wave function components could be of some utility. Such a formulation is derived in this paper using techniques from the Baer-Kouri-Levin-Tobocman and Bencze-Redish-Sloan-Polyzou theories of multi-particle scattering. It is based on components corresponding to the various multi-cluster partitions of an n-particle scattering system and is a generalization of the three-body Faddeev wave function formalism, to which it reduces when n = 3. Except for the full breakup partition, which does not enter the equations, the new components are defined for all possible m-cluster partitions of the n-particles, 2 ≤ m ≤ n ? 1. The sum of all the components yields the solution to the Schrödinger equation for scattering and either the Schrödinger equation solution or an easily identified spurious solution in the case of bound states. Both the two-cluster components and two-cluster transition operators are shown to be solutions of equations involving quantities carrying only two-cluster partition labels. Discussions of the Born term and a multiple scattering representation for the non-rearrangement transition operator and the inclusion of distortion operators in the formalism are also included.     

Coupled channels treatment of transfer reactions

Consistent coupled channels calculations, using folded optical potentials, are performed by iteration for the reactions ¹⁶O(d, d), ¹⁶O(d, p), and ¹⁷O(p, p). The kernel is greatly changes when the nonorthogonality of the channel wave functions is included. The non-orthogonality corrections tend to cancel the effects from higher-order processes except for ¹⁷O(p, p) where only higher-order processes are important. The use of folded potentials reduces both effects.     

Slow neutron scattering parameters for stable nuclei with nucleon numbers betweenA=79 andA=96

The relation between the method of coupled channels for rearrangement reactions (CRC) and the bound state approximation to the channel coupling array formalism (BSCCA), advocated in recent years, is investigated in detail for a simple 3-body system expressed in terms of truncated component wavefunctions of Faddeev type. The system is described by coupled differential or integral equations that are truncated into a model space of strongly-coupled channels. It is shown that CRC can be derived from the truncated coupled equations, either in differential or integral form, provided care is taken to use the entire model space. The corresponding BSCCA in this model space can be obtained from a restrictive condition on the integral from of the coupled equations, while it cannot be obtained consistently from the differential form of the coupled equations. The boundary conditions for the component functions are discussed in detail.     

Multi three-cluster coupling model of nuclear reactions

Two alternative versions of practicable connected kernel theories of nuclear reactions are proposed. The basic assumption is that the state of a many-body system can be approximated by a superposition of two- and three-cluster states corresponding to various possible reaction processes. The approach is based on the concept of transitions due to particle exchange as in the AmadoLovelace (AL) formalism. All important three-cluster partitions can be incorporated via multichannel couplings in two-cluster subsystems. The simpler of two models, which is called the multi three-cluster coupling (MTCC) model, is a direct extension of the AL formalism. Using the separable representation of two-cluster potentials, the AL type coupled equations among reaction amplitudes are postulated. The MTCC equation is shown to be reduced to the equation with the same structure as the Faddeev equation. Thus, while the MTCC model can treat a much wider class of nuclear reactions than the Faddeev approach, the calculation is no more complicated than that. The basic assumption in this model, as well as in the AL and AGS theories, is shown to contain some degree of inconsistency regarding the treatment of bound state pole parts of interacting pairs. This is remedied in the other model, which we call the multi two- and three-cluster coupling (MTTC) model. This is the most extended MTCC model that is possible within the two- and three-cluster coupling model. In the MTTC treatment, it is shown that sequential (multi-step) transfers can also be accommodated in addition to all various processes that are possible in the MTCC approach.     

Connected kernel methods in nuclear reactions: (I). Derivation of the connected kernel equations

A set of connected kernel equations for the scattering operator are derived. The equations connect only the two cluster channels and generalize the equations of Lovelace (for N = 3) and Sloan (for N = 4) to arbitrary N. This is done by summing all disconnected diagrams explicity by induction. Methods for handling multiple summations over partitions are developed and presented. The resulting equations are similar to those given by Bencze but have a different Born term. An error in Bencze's derivation is pointed out but we show that only the two cluster connected part of the Born term contributes on-shell so his final equation is correct and equivalent to ours.     

Appearance of gauge potentials in atomic collision physics

The dynamical equations obtained from the method of perturbed stationary states (PSS) are shown to be formally equivalent to the dynamical equations of a particle with N internal degrees of freedom minimally coupled to U (N) static gauge potentials. Several examples are given that illustrate the appearance of non-abelian and magnetic monopole gauge potentials in simple systems. Advantages of expressing the PSS equations as a gauge theory are discussed.     

Symmetry conserving generalisation of Hartree Fock Bogoliubov theory

We derive a generalisation of the HFB equations which conserve particle number. This is achieved in using the equation of motion method or alternatively the Green's function technique. The price we have to pay is that there is not only one mean field for the particle numberN but a set of coupled mean field equations for the whole bandN, N±2,N±4... Nevertheless we think that our theory is a quite interesting variant in comparison with the conventional projection technique. We apply our theory to simple models and find that the results are excellent.     

Integral equations for N-particle scattering

Integral equations are derived for the N-particle transition operators. The equations couple together only transition operators between two-body channels. The kernel of the equations becomes connected after a single iteration. Transition operators involving channels with three or more particles can be obtained by quadratures from the solution of the equations. It is also shown that the N-particle equations can be reduced to multichannel two-body equations by the use of the quasiparticle method.     

Calculations of inelastic cross sections for rotational excitation

Cross sections for scattering of N₂ (j=0) molecules on He atoms are calculated for relative energies below the excitation threshold for the N₂ (j=2) rotational state. The close coupling method is used and the coupled differential equations are solved numerically. Very sharp resonances (corresponding to a lifetime of 10^?10 sec) caused by quasistable states of the N₂He system are found in the calculated cross section, when the closed channels corresponding to the N₂ (j=2) states were included in the coupled equations. The position of the resonances is compared with the calculated energy eigenvalues of the corresponding two body potential. Furthermore the equilibrium concentration of the N₂He quasistable and orbiting states is calculated at 80 °K andp _{N 2}=1 Atm. Both concentrations are found to be 1%.     

Evidence of projectile polarisation effects on the reaction 208Pb(17O, 16O)209Pb

Coupled reaction channels calculations (CRC) of the reactions ²⁰⁸Pb(¹⁷O, ¹⁶O)²⁰⁹Pb leading to different states of ²⁰⁹Pb are compared with DWBA predictions at projectile energies of 78, 86 and 102 MeV. The calculations exhibit strong effects of multistep processes on Q-value and angular-momentum-mismatched transfer reactions. It is shown that the contribution to the transfer through the inelastic excitation of ¹⁷O contains a major part of the multistep effect. A simple three-channel model comprising the elastic, inelastic and transfer channels is constructed which simulates the CRC effects on the transfer cross sections. The polarization effects of the eliminated channels give rise to effective potentials which are mainly imaginary.     

Kinetic and hydrodynamic models of chemotactic aggregation

We derive general kinetic and hydrodynamic models of chemotactic aggregation that describe certain features of the morphogenesis of biological colonies (like bacteria, amoebae, endothelial cells or social insects). Starting from a stochastic model defined in terms of N coupled Langevin equations, we derive a nonlinear mean-field Fokker-Planck equation governing the evolution of the distribution function of the system in phase space. By taking the successive moments of this kinetic equation and using a local thermodynamic equilibrium condition, we derive a set of hydrodynamic equations involving a damping term. In the limit of small frictions, we obtain a hyperbolic model describing the formation of network patterns (filaments) and in the limit of strong frictions we obtain a parabolic model which is a generalization of the standard Keller-Segel model describing the formation of clusters (clumps). Our approach connects and generalizes several models introduced in the chemotactic literature. We discuss the analogy between bacterial colonies and self-gravitating systems and between the chemotactic collapse and the gravitational collapse (Jeans instability). We also show that the basic equations of chemotaxis are similar to nonlinear mean-field Fokker-Planck equations so that a notion of effective generalized thermodynamics can be developed.