Second-order corrections to mean field evolution of weakly interacting Bosons. II

We study the evolution of an N-body weakly interacting system of Bosons. Our work forms an extension of our previous paper Grillakis, Machedon, and Margetis (2010) [13], in which we derived a second-order correction to a mean-field evolution law for coherent states in the presence of small interaction potential. Here, we remove the assumption of smallness of the interaction potential and prove global existence of solutions to the equation for the second-order correction. This implies an improved Fock-space estimate for our approximation of the N-body state.     

On the evolution of continued fractions in a fixed quadratic field

We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the ‘normal’ statistics given by the Gauss-Kuzmin measure. As a byproduct, the growth rate of the period is analyzed and, for example, it is shown that for a fixed integer k and a quadratic irrational α, the length of the period of the continued fraction expansion of k ⁿ α equals ck ⁿ + o(k^15n/16) for some positive constant c. This improves results of Cohn, Lagarias, and Grisel, and settles a conjecture of Hickerson. The results are derived from the main theorem of the paper, which establishes an equidistribution result regarding single periodic geodesics along certain paths in the Hecke graph. The results are effective and give rates of convergence and the main tools are spectral gap (effective decay of matrix coefficients) and dynamical analysis on S-arithmetic homogeneous spaces.     

On the mean field limit of the Random Batch Method for interacting particle systems

The Random Batch Method proposed in our previous work(Jin et al.J Comput Phys,2020)is not only a numerical method for interacting particle systems and its mean-field limit,but also can be viewed as a model of the particle system in which particles interact,at discrete time,with randomly selected mini-batch of particles.In this paper,we investigate the mean-field limit of this model as the number of particles N→∞.Unlike the classical mean field limit for interacting particle systems where the law of large numbers plays the role and the chaos is propagated to later times,the mean field limit now does not rely on the law of large numbers and the chaos is imposed at every discrete time.Despite this,we will not only justify this mean-field limit(discrete in time)but will also show that the limit,as the discrete time intervalτ→0,approaches to the solution of a nonlinear Fokker-Planck equation arising as the mean-field limit of the original interacting particle system in the Wasserstein distance.     

Baryon resonances in the relativistic mean field approach

We suggest a new approach according to which baryon resonances can be viewed as collective excitations about “intrinsic” one-quark excitations in a mean field of definite symmetry. This standpoint is justified in the limit of a large number N_c of colors. Although N_c = 3 in the real world, we obtain a good agreement with the observed resonance spectrum up to 2 GeV. A possible consequence of the scheme is the existence of new exotic charmed (and bottom) baryons that may be stable against strong decays.     

On the number of directions determined by a pair of functions over a prime field

A three-dimensional analogue of the classical direction problem is proposed and an asymptotically sharp bound for the number of directions determined by a non-planar set in AG(3,p), p prime, is proved. Using the terminology of permutation polynomials the main result states that if there are more than pairs with the property that f(x)+ag(x)+bx is a permutation polynomial, then there exist elements c,d,e∈Fp with the property that f(x)=cg(x)+dx+e.     

Clusters die hard: Time-correlated excitation in the Hamiltonian mean field model

The Hamiltonian mean field (HMF) model has a low-energy phase where N particles are trapped inside a cluster. Here, we investigate some properties of the trapping/untrapping mechanism of a single particle into/outside the cluster. Since the single particle dynamics of the HMF model resembles the one of a simple pendulum, each particle can be identified as a high-energy particle (HEP) or a low-energy particle (LEP), depending on whether its energy is above or below the separatrix energy. We then define the trapping ratio as the ratio of the number of LEP to the total number of particles and the “fully-clustered” and “excited” dynamical states as having either no HEP or at least one HEP. We analytically compute the phase-space average of the trapping ratio by using the Boltzmann–Gibbs stable stationary solution of the Vlasov equation associated with the N → ∞ limit of the HMF model. The same quantity, obtained numerically as a time average, is shown to be in very good agreement with the analytical calculation. Another important feature of the dynamical behavior of the system is that the dynamical state changes transitionally: the “fully-clustered” and “excited” states appear in turn. We find that the distribution of the lifetime of the “fully-clustered” state obeys a power law. This means that clusters die hard, and that the excitation of a particle from the cluster is not a Poisson process and might be controlled by some type of collective motion with long memory. Such behavior should not be specific of the HMF model and appear also in systems where itinerancy among different “quasi-stationary” states has been observed. It is also possible that it could mimick the behavior of transient motion in molecular clusters or some observed deterministic features of chemical reactions.     

On a pair of equations occurring in swirling viscous flow with an applied magnetic field

Summary The differential equations governing the similarity solutions of the boundary layer induced by a swirling flow over an infinite disk are investigated. A weak magnetic field is applied, and this can be decisive for the existence of similarity solutions. If the external swirling flow decays liker ^–n , wherer is the distance from the axis, then the character of the solution changes atn=n ₀ 0.1217, ,n=1, and possibly atn=3. A ladder structure develops forn ₀ , with a remarkable similarity to the ladder structure investigated in [7].

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Zusammenfassung Es werden die Differentialgleichungen untersucht, die die Ähnlichkeitslösungen einer Grenzschicht an einer unendlichen Scheibe beschreiben, wenn die äussere Strömung einen Wirbel darstellt. Die Existenz der Ähnlichkeitslösungen wird wesentlich beeinflusst durch das angelegte Magnetfeld. Wenn die Wirbelgeschwindigkeit wier ^–n abfällt (r ist die Distanz von der Drehachse), so ändert der Lösungscharakter bein=n ₀ 0.1217, ,n=1, undn=3. Fürn ₀ ergibt sich eine Doppelstruktur, die überraschende Ähnlichkeit zeigt mit der Leiterstruktur, wie sie in [7] untersucht worden ist.

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Dedicated to our friend, Professor N. Rott.     

Free energy of the spherical mean field model

We compute at any temperature the free energy of the multi p-spin spherical model when only terms for p even are considered. Work partially supported by an NSF grant     

Partial mean field limits in heterogeneous networks

We investigate systems of interacting stochastic differential equations with two kinds of heterogeneity: one originating from different weights of the linkages, and one concerning their asymptotic relevance when the system becomes large. To capture these effects, we define a partial mean field system, and prove a law of large numbers with explicit bounds on the mean squared error. Furthermore, a large deviation result is established under reasonable assumptions. The theory will be illustrated by several examples: on the one hand, we recover the classical results of chaos propagation for homogeneous systems, and on the other hand, we demonstrate the validity of our assumptions for quite general heterogeneous networks including those arising from preferential attachment random graph models.     

On the stability of rods for stochastic excitations

The stability of motion of an elastic rod in a viscous medium compressed by a randomly acting force is studied. The conditions of stability of the rod acted upon by a stationary process with bilinear spectral density are obtained. The dependence of the statistical moments of the amplitude of the finite flexure of the rod under stationary-motion conditions on the parameters of the compressing force and the amplitude of the initial deformation is analysed. A number of problems concerning the stability of longitudinal flexure of viscoelastic constructions acted upon by random loads were discussed in /1–3/.