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In this chapter, we review some kinetic and geometric properties of two-species annihilation in which an encounter between two distinct species A and B leads to the formation of an inert product I, A + B → I [7.1]. Examples of this... more
In this chapter, we review some kinetic and geometric properties of two-species annihilation in which an encounter between two distinct species A and B leads to the formation of an inert product I, A + B → I [7.1]. Examples of this reaction include electron-hole recombination in irradiated semiconductors [7.2], catalytic reactions on surfaces [7.3], exciton dynamics [7.4], and annihilation of primordial monopoles in the early universe [7.5]. At a more abstract level, two-species annihilation is a realization of an interacting Brownian particle system, a connection which has been helpful in establishing rigorous results [7.6]. For detailed discussion of Brownian motion see Chap. 5. For other types of reactions see Chap. 8.
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A general qualitative theory for the mean-field rate equation describing aggregation processes is described. Various exponents describing the cluster-size distribution and its time evolution are derived. It is found that this theory... more
A general qualitative theory for the mean-field rate equation describing aggregation processes is described. Various exponents describing the cluster-size distribution and its time evolution are derived. It is found that this theory frequently gives a good account of systems with more realistic kinetics, such as cluster-cluster aggregation, but that considerable caution must be exercised to ensure being in the actual asymptotic regime, since very long crossovers are sometimes observed.
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A class of N-body problems is identified, characterized by second-order discrete-time evolution equations determining the motion in the complex z-plane of an arbitrary number N of points zn ≡ zn(ℓ), where... more
A class of N-body problems is identified, characterized by second-order discrete-time evolution equations determining the motion in the complex z-plane of an arbitrary number N of points zn ≡ zn(ℓ), where \documentclass[12pt]{minimal}\begin{document}$\ell =0,\pm 1,\pm 2,{\kern -2.1pt}...$\end{document}ℓ=0,±1,±2,... is the discrete-time independent variable. Both these equations of motion, and the solution of their initial-value problem, only involve algebraic operations: finding the zeros of explicitly known polynomials of degree N in z, finding the eigenvectors and eigenvalues of explicitly known N × N matrices. These models feature an arbitrarily large number of arbitrary parameters (“coupling constants”).
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ABSTRACTPreliminary results of an aggregation model that takes into account both the Brownian motion as well as the gravitational drift experienced by the colloidal particles and clusters is presented. It is shown that for high strengths... more
ABSTRACTPreliminary results of an aggregation model that takes into account both the Brownian motion as well as the gravitational drift experienced by the colloidal particles and clusters is presented. It is shown that for high strengths of the drift the system crosses over to a regime different from diffusion-limited colloid aggregation, for which there is an increase of the fractal dimension, a speeding up of the aggregation rate and a widening of the cluster size distribution, becoming algebraically decaying with an exponent τ.
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A Reply to the Comment by K. Michaelian, I. Santamaría-Holek, and A. Pérez-Madrid.
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A model is suggested for interpreting the experimental dielectric dispersion data in nematic liquid crystals. The high‐frequency dispersion is observed in both, parallel and perpendicular alignment of substances with only a molecular... more
A model is suggested for interpreting the experimental dielectric dispersion data in nematic liquid crystals. The high‐frequency dispersion is observed in both, parallel and perpendicular alignment of substances with only a molecular dipole moment parallel to the long axis of the molecule. In the absence of a perpendicular dipole moment component these high‐frequency absorptions cannot be associated with rotation about the molecular axis. The model is based on Kubo's linear response theory and describes the dipole relaxation processes in frames of rotational diffusion. Results are compared with experimental data obtained on 4‐cyano‐4′‐octylbiphenyl.
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ABSTRACT We study a system of kinetic equations describing irreversible aggregation—the so-called Smoluchowski equations—for a peculiar case in which even starting from initial conditions for which only a monomer is present, at... more
ABSTRACT We study a system of kinetic equations describing irreversible aggregation—the so-called Smoluchowski equations—for a peculiar case in which even starting from initial conditions for which only a monomer is present, at arbitrarily small times an infinite cluster appears and causes the total mass of the system to decrease. This phenomenon, known as instantaneous gelation, is known to arise in cases where the reaction rates between large clusters increase strongly as the cluster size goes to infinity. We resolve two puzzles linked to this behaviour: first, from general results it is well known that a power-law behaviour must form instantly at large masses for arbitrarily small times. We show that the small time behaviour can be analysed for these systems, but this must be done in an essentially different manner than in the regular case. The correct treatment provides a mechanism for the instantaneous appearance of a power-law tail at large masses in the aggregate size distribution. We additionally show that instantaneous gelation is connected to the presence of an essential singularity in the solution and also show how, from the numerical study of the formal power series describing the solution, the nature of the singularity can be guessed and the time dependence of the mass scale for the onset of the power-law behaviour conjectured.
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Bonding of large clusters by surface reactions can be modelled by Smoluchowski's coagulation equation with coagulation rates Kij=(ij)omega with omega =(d-1)/d in d-dimensional systems. It is shown that the cluster size distribution... more
Bonding of large clusters by surface reactions can be modelled by Smoluchowski's coagulation equation with coagulation rates Kij=(ij)omega with omega =(d-1)/d in d-dimensional systems. It is shown that the cluster size distribution for large clusters well below the gelation transition has the form ck approximately k- theta xi k (k to infinity ) where theta =2 omega . Results are compared with those from lattice theories and from the Flory-Stockmayer theory. For the models Kij=1/2(imu jnu +inu jmu ) with mu , nu <1 one finds theta = mu + nu ; for Kij=ijomega +jiomega with -1<or= omega <1 one finds theta =1/2(1+ omega ) and for Kij=ij one has theta =5/2.
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The authors develop a Smoluchowski-type mean-field treatment for a recently introduced model of ballistic agglomeration. The predictions of this mean-field theory for the exponent characterizing typical cluster size are in agreement with... more
The authors develop a Smoluchowski-type mean-field treatment for a recently introduced model of ballistic agglomeration. The predictions of this mean-field theory for the exponent characterizing typical cluster size are in agreement with earlier results for all dimensions. Nevertheless, the predicted monomer decay and particle size distribution are totally at variance with the numerical observations in one dimension. The reason for this discrepancy is found to be the fact that high velocity particles coalesce rapidly independently of their mass, which introduces correlations not taken into account by the mean-field treatment. This is likely to persist in all dimensions, so that the model has no upper critical dimension. The case where the initial velocity distribution function of the particles has a power-law tail is also examined. It is found that, at least in one dimension, the typical cluster size behaves in a way that depends on the specific velocity distribution function, whereas the monomer decay does not.
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ABSTRACT An exact relation between one of the recently introduced models for kinetic gelation and random percolation is displayed. This explains the frequently observed identity between the exponents of the two models. Moreover, it... more
ABSTRACT An exact relation between one of the recently introduced models for kinetic gelation and random percolation is displayed. This explains the frequently observed identity between the exponents of the two models. Moreover, it strongly suggests that any further universal properties of percolation are also shared by kinetic gelation. In particular, a rigorous inequality due to van den Berg and Kesten (1985) can be used to display an inconsistency between the observed values of the exponents beta and nu and the observed value of the fractal dimension of the backbone in kinetic gelation. The more complex issue of the observed variation of the amplitude ratio cannot be easily resolved, however. Should the observed discrepancies be real, they would, according to the author&#39;s results, indicate that the amplitude ratio can indeed vary depending upon the underlying lattice structure.
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The authors consider j-mers Ai reacting irreversibly according to the scheme Aj+Ak to Aj+k. The kinetic equations for the concentration of Aj are examined, and particularly their behaviour near gelation. Only the case Rjk=jalpha kalpha... more
The authors consider j-mers Ai reacting irreversibly according to the scheme Aj+Ak to Aj+k. The kinetic equations for the concentration of Aj are examined, and particularly their behaviour near gelation. Only the case Rjk=jalpha kalpha (0<or= alpha <or=1) is considered; this is a variation of the usual Flory-Stockmayer model to take excluded volume and cyclisation effects roughly into account. The effect on certain critical exponents is estimated.
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ABSTRACT A reaction kernel, K(j,k) = K(k,j), is studied, for which the Smoluchowski equations of aggregation ċj = (1/2)∑k,l = 1∞K(k,l)ckcl[δk + l,j-δk,j -δl,j] can be solved. It takes only three values: K(j,k) = K if j and k are both odd,... more
ABSTRACT A reaction kernel, K(j,k) = K(k,j), is studied, for which the Smoluchowski equations of aggregation ċj = (1/2)∑k,l = 1∞K(k,l)ckcl[δk + l,j-δk,j -δl,j] can be solved. It takes only three values: K(j,k) = K if j and k are both odd, K(j,k) = L if j and k are both even and K(j,k) = M if j and k have different parities. A considerable simplification over previous treatments is presented for the general case (K, L and M are three arbitrary positive numbers), and the time evolution of the concentrations is exhibited in completely explicit form for the (new) special case L = 4M. In another special case, K = M, the equation for the generating function of the concentrations can be reduced to quadratures; the analysis of this case, and of the general case, is postponed to a future paper.
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In a previous paper, a new reaction kernel for the Smoluchowski equations of aggregation was solved exactly. This kernel, K(j,k) = 2-qj-qk, for 0<q<1 a real positive quantity, interpolates between two well understood exactly solved... more
In a previous paper, a new reaction kernel for the Smoluchowski equations of aggregation was solved exactly. This kernel, K(j,k) = 2-qj-qk, for 0<q<1 a real positive quantity, interpolates between two well understood exactly solved cases, namely that of K(j,k) = 2 and that of K(j,k) = j+k. This new model, however, shows a number of unexpected features, not found in either of the two limiting cases. It is shown that this model has a remarkable behaviour with respect to the commonly accepted scaling theory. On the one hand, it satisfies a rigorous form of the scaling hypothesis, but, on the other hand, it clearly violates some relations which are ordinarily assumed to follow from it. These issues are discussed, as well as the nature of the singular limit in which q is very close to one, for which our kernel becomes close to the sum kernel mentioned above. In particular, the form of the crossover between two kernels with different degrees of homogeneity can be discussed here in an exact way.
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Two-species diffusion-limited annihilation in less than four dimensions is well known to have a concentration decaying as t-d4/, as opposed to the rate equation result of t-1. This result had not, however, been demonstrated numerically in... more
Two-species diffusion-limited annihilation in less than four dimensions is well known to have a concentration decaying as t-d4/, as opposed to the rate equation result of t-1. This result had not, however, been demonstrated numerically in three dimensions, due to very strong transient effects. A variant of the model is proposed here, and is shown to reach the exactly known asymptotic behaviour in numerically accessible times. This also allows one to investigate open questions in this system with some certainty that the results are indeed asymptotic. In particular, it is shown that the distance between two particles of the same species scales identically to the distance between particles of different species. This is in contradistinction to recent (numerical and scaling) results in one and two dimensions. Some evidence is also presented that, contrary to previous suggestions, the domains formed in three-dimensional two-species annihilation are indeed regular objects with smooth interfaces.
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A new solvable discrete-time many-body problem is identified. It extends a model treated in a previous paper by introducing in its equations of motion an additional free parameter. Hence, it features 6 parameters, 2 of which can be... more
A new solvable discrete-time many-body problem is identified. It extends a model treated in a previous paper by introducing in its equations of motion an additional free parameter. Hence, it features 6 parameters, 2 of which can be eliminated (say, replaced by unity) by appropriate rescalings. Assignments of these parameters are identified which entail that the many-body model is asymptotically isochronous, namely, that its generic solution—when the discrete-time variable ℓ diverges, ℓ → ∞—becomes completely periodic up to exponentially vanishing corrections, with a fixed period independent of the initial data.
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Imperfections of Bunimovich mushroom Billiards are analyzed. Any experiment will be affected by such imperfections, and it will be necessary to estimate their influence. In particular some of the corners will be rounded and small... more
Imperfections of Bunimovich mushroom Billiards are analyzed. Any experiment will be affected by such imperfections, and it will be necessary to estimate their influence. In particular some of the corners will be rounded and small deviations of the angle of the underside of the ...
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A correspondence between the orbits of a system of two complex, homogeneous, polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed. This correspondence is general in the sense... more
A correspondence between the orbits of a system of two complex, homogeneous, polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed. This correspondence is general in the sense that it applies to an open set of systems of ordinary differential equations of the specified kind. This allows us to transfer results well known from the theory of polygonal billiards, such as ergodicity, the existence of periodic orbits, the absence of exponential divergence, the existence of additional conservation laws, and the presence of discontinuities in the dynamics, to the corresponding systems of ordinary differential equations. It also shows that the considerable intricacy known to exist for polygonal billiards also attends these apparently simpler systems of ordinary differential equations.
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Recently, a general technique has been introduced to Omega-modify a Hamiltonian so that the Hamiltonian thereby produced is, in the classical context, isochronous. In this paper, we introduce and discuss simple examples of isochronous... more
Recently, a general technique has been introduced to Omega-modify a Hamiltonian so that the Hamiltonian thereby produced is, in the classical context, isochronous. In this paper, we introduce and discuss simple examples of isochronous Hamiltonians manufactured in this manner. We also outline their quantal treatment, yielding equispaced spectra.
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A new class of isochronous dynamical systems is introduced and briefly discussed. These systems feature in their phase space a fully dimensional region (part of which can be explicitly identified) where all their solutions are completely... more
A new class of isochronous dynamical systems is introduced and briefly discussed. These systems feature in their phase space a fully dimensional region (part of which can be explicitly identified) where all their solutions are completely periodic (periodic in all their degrees of freedom) with the same period. But in other regions of their phase space their evolution might be quite complicated.
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ABSTRACT We introduce a new technique—characterized by an arbitrary positive constant Ω, with which we associate the period T = 2π/Ω—to &#39;Ω-modify&#39; a Hamiltonian so that the new Hamiltonian thereby obtained is entirely... more
ABSTRACT We introduce a new technique—characterized by an arbitrary positive constant Ω, with which we associate the period T = 2π/Ω—to &#39;Ω-modify&#39; a Hamiltonian so that the new Hamiltonian thereby obtained is entirely isochronous, namely it yields motions all of which (except possibly for a lower dimensional set of singular motions) are periodic with the same fixed period T in all their degrees of freedom. This technique transforms real autonomous Hamiltonians into Ω-modified Hamiltonians which are also real and autonomous, and it is widely applicable, for instance, to the most general many-body problem characterized by Newtonian equations of motion (&#39;acceleration equal force&#39;) provided it is translation invariant. The Ω-modified Hamiltonians are of course not translation invariant, but for Ω = 0 they reduce (up to marginal changes) to the unmodified Hamiltonians they were obtained from. Hence, when this technique is applied to translation-invariant Hamiltonians yielding, in their center-of-mass systems, chaotic motions with a natural time scale much smaller than T, the corresponding Ω-modified Hamiltonians shall display a chaotic behavior for quite some time before the isochronous character of the motions takes over. We moreover show that the quantized versions of these Ω-modified Hamiltonians feature equispaced spectra.
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ABSTRACT Hamiltonians linear in the momenta and yielding (in the classical context) trajectories isochronous in configuration space are considered. It is shown that their motions are completely periodic in phase space as well, with the... more
ABSTRACT Hamiltonians linear in the momenta and yielding (in the classical context) trajectories isochronous in configuration space are considered. It is shown that their motions are completely periodic in phase space as well, with the same common period as the orbits in configuration space. Moreover, it is shown that for this particular class of Hamiltonians the semiclassical quantization prescription is exact, so that to the isochronous character of their classical dynamics there corresponds in the quantized context an equidistant spectrum, for a broad range of ordering prescriptions, including non-symmetrical ones. Examples illustrating these findings are presented.
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ABSTRACT A new reaction kernel, K(j,k) = 2-qj-qk with 0&lt;q&lt;1, is introduced, for which the Smoluchowski equations of aggregation can be solved. The time evolution of the concentrations cj(t) and of their moments is analysed.... more
ABSTRACT A new reaction kernel, K(j,k) = 2-qj-qk with 0&lt;q&lt;1, is introduced, for which the Smoluchowski equations of aggregation can be solved. The time evolution of the concentrations cj(t) and of their moments is analysed. The cj(t) decay at large times as t-(2-qj) in striking contrast to the behaviour of the constant kernel K(j,k) = 2, for which cj(t) behaves as t-2 at large times. On the other hand, the moments behave in leading order at large times exactly like the moments of the constant kernel, though differences appear at higher orders.
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We modify (in two different manners) the Hamiltonian describing motions in the Poincaré half-plane so that the modified Hamiltonians thereby obtained are entirely isochronous: indeed, in the classical context, all the motions they entail... more
We modify (in two different manners) the Hamiltonian describing motions in the Poincaré half-plane so that the modified Hamiltonians thereby obtained are entirely isochronous: indeed, in the classical context, all the motions they entail are periodic with the same period. We then investigate suitably quantized versions of these systems and show that their spectra are equispaced.
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A technique is provided that allows to associate to a Hamiltonian another, ω-modified, Hamiltonian, which reduces to the original one when the parameter ω vanishes, and for ω>0 features an open, hence fully dimensional, region in its... more
A technique is provided that allows to associate to a Hamiltonian another, ω-modified, Hamiltonian, which reduces to the original one when the parameter ω vanishes, and for ω>0 features an open, hence fully dimensional, region in its phase space where all its solutions are isochronous, i.e., completely periodic with the same period. The class of Hamiltonians to which this technique is applicable is large: it includes for instance the Hamiltonian characterizing the classical many-body problem with potentials that are translation-invariant but otherwise completely arbitrary, which is largely used in this paper to illustrate these findings. We also discuss variants of this technique that yield partially isochronous Hamiltonians, which also feature a region in their phase space where all solutions are isochronous, that region having however a bit less than full dimensionality (for instance codimension one or two) in phase space.
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We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides and the solvable character of these dynamical... more
We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides and the solvable character of these dynamical systems amounting to the possibility to obtain the solution of their initial value problem via algebraic operations. Equivalently, by considering the analytic continuation of these systems to complex time, their algebraically solvable character corresponds to the fact that their general solution either is single-valued or features only a finite number of algebraic branch points as functions of complex time (the independent variable). Thus, our results provide a major enlargement of the class of solvable systems beyond those with a single-valued general solution identified by Garnier about 60 years ago. An interesting property of several of these new dynamical systems is the elementary character of their general solution, identifiable as the roots of a pol...
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ABSTRACT The (N -1)-dimensional Maxwell fish-eye is an optical system with an SO(N-1) manifest symmetry and a SO(N) hidden symmetry, but also an SO(N,1) potential group and the SO(N, 2) group of the (N-1)-dimensional Kepler and point... more
ABSTRACT The (N -1)-dimensional Maxwell fish-eye is an optical system with an SO(N-1) manifest symmetry and a SO(N) hidden symmetry, but also an SO(N,1) potential group and the SO(N, 2) group of the (N-1)-dimensional Kepler and point rotor systems. The optical Hamiltonian is proportional to the Casimir invariant. We use a stereographic map extended to a canonical transformation between the two phase spaces of the rotor and the fish-eye. The groups permit a succint [(q)\vec]1\vec q_1 and [(q)\vec]2\vec q_2 are conjugate when they are antiparallel and their magnitudes relate as g1g2 = p. They are the stereographic images of a pair of antipodal points on the sphere of radius p.
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