Discrete-continuum process

Discrete-Continuum Process. For Case 3, the basic operator-differential equation (Equation 11) involves continuum states. A(t) will be a continuously infinite row-column matrix. If excitation of A is optically allowed, the matrix components corresponding to discrete-continuum components of A(t) are predominant—i.e., A(t) can be regarded as... 

Fick s (continuum) laws of diffusion can be related to the discrete atomic processes of the random walk, and the diffusion coefficient defined in terms of Fick s law can be equated to the random-walk displacement of the atoms. Again it is easiest to use a one-dimensional random walk in which an atom is constrained to jump from one... 

Judging from the calculation of the coupled equation (without the rotating atom approximation) in the resonant case and the discrete-continuum case, the error from the rotating atom approximation would not change the order of the magnitude (67, 68, 70). With these approximations, the process can be described in terms of the resonance defect [o> = (Et — Ef)/h] and the transition matrix dipole moments of A and B fiA and /xB. By Nakamura s calculation (50), the transfer probability is written as... 

Zhu HP, Zhou ZY, Hou QF, Yu AB Linking discrete particle simulation to continuum process modelling for granular matter theory and application, Particuology 9 342-357, 2011. 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

Warneck et al.38S have presented evidence that absorption in the 1800-2400 A range consists of discrete structure superimposed on a continuum which they attribute to the dissociation to ground state SO and O. The overlying bands are predissociated at X < 1900 A, but the nature and states of the resulting fragments are not established. On the basis of their measured quantum yield, relative importance of the possible primary processes. There is as yet, however, insufficient information for the evaluation of quantitative rate data for the photolysis in this region. 

Chemical and biological analyses of trace organic mixtures in aqueous environmental samples typically require that some type of isolation-concentration method be used prior to testing these residues the inclusion of bioassay in a testing scheme often dictates that large sample volumes (20-500 L) be processed. Discrete chemical analysis only requires demonstration that the isolation technique yields the desired compounds with known precision. However, chemical and/or toxicological characterization of the chemical continuum of molecular properties represented by the unknown mixtures of organics in environmental samples adds an extra dimension of the ideal isolation technique ... 

More recently, Mies and Kraus have presented a quantum mechanical theory of the unimolecular decay of activated molecules.13 Because of the similarity between this process and autoionization they used the Fano theory of resonant scattering.2 Their theory provides a detailed description of the relationships between level widths, matrix elements coupling discrete levels to the translational continuum, and the rate of fragmentation of the molecule. 

Defect diffusion traditionally is treated as a process in continuum medium. However, discreteness of the crystalline lattice becomes important in particular situations, e.g., when defect recombination occurs in several hops (nearest neighbour recombination) [3, 4] or even for nearest-site hops of defects if their recombination is controlled by the tunnelling whose probability greatly changes on a scale of lattice constant [45, 46],... 

In studying processes of accumulation of the Frenkel defects, one uses three different types of simple models the box, continuum, and discrete (lattice) models. In the simplest, box model, which was proposed first in [22], one studies the accumulation of complementary particles in boxes having a certain capacity, with walls impenetrable for diffusion of particles among the boxes. The continuum model treats respectively a continuous medium the intrinsic volume of similar defects at any point of the space is not bounded. In the model of a discrete medium a single cell (e.g., crystalline lattice site) cannot contain more than one defect (v or i). 

The analysis of the transient fluorescence spectra of polar molecules in polar solvents that was outlined in Section I.A assumes that the specific probe molecule has certain ideal properties. The probe should not be strongly polarizable. Probe/solvent interactions involving specific effects, such as hydrogen-bonding should be avoided because specific solute/solvent effects may lead to photophysically discrete probe/solvent complexes. Discrete probe/solvent interactions are inconsistent with the continuum picture inherent in the theoretical formalism. Probes should not possess low lying, upper excited states which could interact with the first-excited state during the solvation processes. In addition, the probe should not possess more than one thermally accessible isomer of the excited state. 

These are produced by autoionization transitions from highly excited atoms with an inner vacancy. In many cases it is the main process of spontaneous de-excitation of atoms with a vacancy. Let us recall that the wave function of the autoionizing state (33.1) is the superposition of wave functions of discrete and continuous spectra. Mixing of discrete state with continuum is conditioned by the matrix element of the Hamiltonian (actually, of electrostatic interaction between electrons) with respect to these functions. One electron fills in the vacancy, whereas the energy (in the form of a virtual photon) of its transition is transferred by the above mentioned interaction to the other electron, which leaves the atom as a free Auger electron. Its energy a equals the difference in the energies of the ion in initial and final states ... 

Summarizing the individual decay branches of the 4d5/2 -> 6p resonance, one finds that all final ionic states can also be reached by outer-shell photoionization, in (a) and (b) by main processes, and in (c)-(i) by discrete and continuous satellite processes. The effect of the resonance decay will then be a modification of these otherwise undisturbed direct outer-shell photoionization processes which turns out to be an enhancement in the present case. Therefore, these outer-shell satellites are called resonantly enhanced satellites. In this context it is important to note that outer-shell photoionization also populates other satellites, attached, for example, to electron configurations 5s25p4ns and 5s25p4nd. However, the parity of these satellites is even, while the decay branches (c)-(/) lead to odd parity. Therefore, both groups of final ionic states can be treated independently of each other (if configuration interaction in the continuum is neglected). 

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