Hans A. Weidenmüller

Random matrix theories in quantum physics: Common concepts

We review the development of random-matrix theory (RMT) during the last fifteen years. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new “statistical mechanics”: Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Grassmann integration in stochastic quantum physics: The case of compound-nucleus scattering

Using astochasticmodel for N compound—nucleusresonances coupledto thechannels,we calculatein the limit N—* theensembleaverageof theS-matrix (the “one-point function”),andof theproductof an S-matrixelementwith thecomplexconjugateof another,both takenat different energies(the “two-point function”). Using a generatingfunction involving both commutingand anticommutingintegrationvariables,we evaluate the ensembleaveragestrivially. The problem of carrying Out the remaining integrationsis solved with the help of theHubbard—Stratonovitch transformation.We put specialemphasison theconvergencepropertiesof this transformation,andon the underlyingsymmetriesof thestochastic model for thecompoundnucleus.Thesetwo featurestogethercompletelydefinetheparametrizationof thecompositevariablesin termsof agroup of transformations.Thisgroupis compactin the“Fermion—Fermionblock” andnon-compactin the“Boson—Bosonblock”. The limit N-. x is taken with thehelp of thesaddle-pointapproximation.After integrationover the“massive modes”,we showthat thetwo-point function can be expressed in terms of thetransmissioncoefficients. In this way we prove that thefluctuation propertiesof thenuclearS-matrix arethesameover theentire spectrumof the random Hamiltonian describingthe compoundnucleus.The integration over the saddle-pointmanifold is carried out using symmetry propertiesof the randomHamiltonian. We finally obtain a closed-formexpressionfor the two-point function in terms of a threefold integralover realvariables.This expressioncan be easilyevaluatednumerically.

The effective interaction in nuclei and its perturbation expansion: An algebraic approach

Abstract We consider a finite-dimensional model for the Hilbert space of the A -nucleon problem. In the frame of this model we explicitly construct the energy-independent effective interaction W first introduced by Des Cloiseaux and Brandow. We demonstrate the connection between this explicit form and the implicit equation for W given by these authors. Using the explicit form for W , we investigate the perturbation expansion of W in powers of the interaction. (When used in the nuclear problem, this expansion leads to the folded diagrams.) We show that this expansion is likely to diverge in most cases of practical interest. Several methods are given which may yield a convergent expansion for W . The implications of our results for practical calculations are discussed.

The statistical theory of nuclear reactions for strongly overlapping resonances as a theory of transport phenomena

Abstract We present a unified microscopic statistical theory of preequilibrium and equilibrium processes of the compound nucleus, valid for mass numbers A ⪆ 40, light incident projectiles ( A ′4), and for excitation energies a few MeV above neutron threshold or larger. The theory is based on a two-body random matrix model for the nuclear Hamiltonian, and on the idea of a chain of statistical doorway [hallway] states, populated from the entrance channel in the direction of increasing complexity through a series of two- body collisions. Averages of fluctuating cross sections, and of other observables, are evaluated by taking ensemble averages, and by a method of calculation which is tailored to the dissipative character of the reaction processes under study. An expansion in terms of a small parametry y is introduced. This parameter is defined as a function of the mean level spacing, the spreading width, the decay width, and the rate of change with energy of these quantities, for each group of statistical doorway states of given complexity. Average cross sections, channel correlations due to direct reactions and/or isolated doorway states (isobaric analogue resonances), the probability distribution function for the elements of the scattering matrix, the correlation length of Ericson fluctuations, and the mean nuclear lifetime are evaluated to leading order in y . We have checked in special cases that the expansion in powers of y is consistent with the consraints imposed by unitarity. Average fluctuating cross sections are given in terms of transmission coefficients and a probability matrix. The latter obeys a probability balance equation, which is shown to be closely related to the Pauli master equation. In case the system equilibraates before it undergoes decay, the average fluctuation cross sections factorize, and we recover the well-known Hauser-Feshbach formula with its various modifications. Next-order correction terms to this formula are also evaluated. The connection between our results, and direct reaction theories on the one hand, and preequilibrium and equilibrium models on the other, is established. These two latter types of models emerge as special cases of the general theory, each with its own well-defined domain of validity, while direct reaction theories specify some input parameters from which channel correlations can be calculated.

Direct reactions and Hauser--Feshbach theory

In the presence of direct reactions, the calculation of energy-averaged cross sections, polarizations, analyzing powers etc. in terms of the energy-averaged S-matrix is possible with the help of a unitary transformation U, defined as the matrix which diagonalizes Satchlers transmission matrix. For a fixed number of channels and in the limit of a very large number N of compound nucleus resonances, we show analytically that the transition from the origially given statistical scattering matrix S to the matrix S = USUT reduces the Hauser-Feshbach problem with direct reactions to the one without. The analytical proof is supported by numerical calculations which show that the results are valid on the 2 to 3% level of statistical accuracy of the numerical calculations for values of N as small as 100. We also show analytically that in the absence of direct reactions, energy-averaged cross sections depend only upon the transmission coefficients. The form of this dependence as well as the dependence on the transmission coefficients of other quantities needed for the evaluation of cross sections in the presence of direct reactions, is parametrized by simple fit formulas which, while lacking theoretical support, give a very satisfactory overall representation of our numerical results.

Generalized Exciton Model for the Description of Preequilibrium Angular Distributions

A generalization of the exciton model for preequilibrium decay of the compound nucleus is presented. This model successfully describes both spectra and angular distributions of neutrons and protons emitted in reactions induced by light (A≦4) projectiles. Limitations of the model due to the finite nuclear geometry are discussed. The connection with a statistical model for the nuclear Hamiltonian is established.

Transport theory of deeply inelastic heavy-ion collisions based on a random-matrix model. i. Derivation of the transport equation

A random-matrix model is used to describe the transformation of kinetic energy of relative motion into intrinsic excitation energy typical of a deeply inelastic heavy-ion collision. The random-matrix model is based upon statistical assumptions regarding the form factors coupling relative motion with intrinsic excitation of either fragment. Average cross sections are calculated by means of an ensemble average over the random matrix model. Summations over intermediate and final intrinsic spin values are performed. As a result, average cross sections are given by the asymptotic behavior of a probability density which in turn obeys a transport equation. In the transport equation there is no further reference to intrinsic spins. The physical and mathematical properties of this equation are exhibited.

Introduction to the theory of heavy-ion collisions

Some basic tools of theoretical heavy-ion physics are presented. Topics covered include: classical theory of heavy ion collisions; gross properties of heavy ion reactions, compound nucleus formation; some elements of nuclear scattering theory; elastic scattering; coulomb excitation; inelastic scattering and transfer reactions; statistical theory; and atomic effects in ions atom collisions. (GHT)

Statistical theory of precompound reactions: The multistep compound process

Abstract Using a quantum-statistical framework, the method of the generating function involving both commuting and anticommuting variables, and the saddle-point approximation followed by the loop expansion, we derive a theoretical framework for multistep-compound reactions. Our statistical input distinguishes between several classes of states of increasing complexity; this distinction is possible only at the expense of relinquishing the orthogonal invariance of the distribution of Hamiltonian matrix elements usually required in compound-nucleus theories. Our result contains both the compound-nucleus scattering cross section and the theory of Agassi Weidenmuller, and Montzouranis ( Phys. Lett. C 22 (1975), 145.) developed earlier as special cases. It goes beyond this theory, and extends the framework of precompound theories in general, by allowing the couplings between classes, and to the channels, to be reasonably strong. A self-consistency condition embodied in the saddle-point equation implies in this case that the level densities used in precompound calculations must be modified. We investigate the modification in simple model cases. Our results suggest that the modification may be relevant for the high-energy tail of the spectrum of precompound particles.

Beyond the TDHF: A collision term from a random-matrix model

Abstract The mean-field description of low-energy heavy-ion scattering is extended to include the residual nucleon-nucleon interaction. The collision term is derived from a random-matrix model for this interaction in the weak-coupling limit. Particular attention is paid to justifying the approximations made in terms of the time scales typical of nuclei, and to the conservation of energy and particle number.