A Revision on Classical Solutions to the Cauchy Boltzmann Problem for Soft Potentials

This short note complements the recent paper of the authors (Alonso, Gamba in J. Stat. Phys. 137(5–6):1147–1165, 2009). We revisit the results on propagation of regularity and stability using L ^p estimates for the gain and loss collision operators which had the exponent range misstated for the loss operator. We show here the correct range of exponents. We require a Lebesgue’s exponent α>1 in the angular part of the collision kernel in order to obtain finiteness in some constants involved in the regularity and stability estimates. As a consequence the L ^p regularity associated to the Cauchy problem of the space inhomogeneous Boltzmann equation holds for a finite range of p≥1 explicitly determined.     

The next-to-leading order (NLO) gluon distribution from DGLAP equation and the logarithmic derivatives of the proton structure functionF 2(x, Q 2) at lowx

At lowx, an analytic solution of the DGLAP equation for gluon in the next-to-leading order (NLO) is obtained by applying the method of characteristics. Its compatibility with double leading logarithmic approximation (DLLA) asymptotics is discussed and comparison with the exact ones like GRV98NLO is made. The solution is then utilized to calculate the derivatives∂F ₂ (x,Q ²)/∂ lnQ ² and ∂ lnF ₂(x,Q ²)/∂ ln (1/x) and compared with the recent HERA data. Our solution is found to reproduce most of the essential features of the data on the derivatives.     

A solution of the DGLAP equation for gluon at low x

We obtain a solution of the DGLAP equation for the gluon at low x first by expanding the gluon in a Taylor series and then using the method of characteristics. We test its validity by comparing it with that of Glück, Reya and Vogt. The convergence criteria of the approximation used are also discussed. We also calculate εF ₂(x,Q)²/ε In Q ² using its approximate relations with the gluon distribution at low x. The predictions are then compared with the HERA data.     

Global Existence of Classical Solutions to the Fokker-Planck-BGK Equation

A kinetic model of the Fokker-Planck-Boltzmann equation is introduced by replacing the original Boltzmann collision operator with the Bhatnagar-Gross-Krook collision model (BGK collision model). This model equation, which we call the Fokker-Planck-BGK equation, has many physical features that the Fokker-Planck-Boltzmann equation possesses. We first establish an L ^∞ existence result for this equation, by which we construct the approximate solutions. Then, by means of the regularizing effects of the linear Fokker-Planck operator and L ^p estimates of local Maxwellians, we obtain some uniform estimates of the approximate solutions. Finally, combining those estimates and regularizing effects, we prove by a compactness argument that the equation has a global classical solution under rather general initial conditions. Supported by the Scientific Research Foundation of Huazhong University of Science and Technology (HUST-SRF).     

Pseudo-Hermitian Hamiltonians: tale of two potentials

For the non-Hermitian, PT-symmetric potentials V = m ² x ² + gx ²(ix)^ν with ν = 1 and 2, we construct the Q operator which gives both the positive-definite metric and an equivalent Hermitian Hamiltonian h. For the case ν = 1, where the theory may be defined on the real axis, h is reasonable but complicated. For the case ν = 2, where the theory must initially be defined on a contour in the complex x plane, we first introduce a real parametrization of the contour, and then calculate Q and h as an expansion in an angle θ. Theresultant h has less desirable properties. However, Q is not uniquely determined, and it may be possible to exploit this ambiguity to produce a more acceptable equivalent Hamiltonian. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

On Diffusive Equilibria in Generalized Kinetic Theory

We investigate the solvability of equations Q(f,f)+ ² f=0 in term of nonnegative integrable densities fL ¹ ₊(R ³). Here, Q(f, f) is a generalized collision operator. If Q is the Boltzmann operator, the only solution is 0. In contrast, we show that if Q is the pseudo-Maxwellian collision operator for granular flow, then there are non -trivial weak solutions of ().     

On solutions to the linear Boltzmann equation with general boundary conditions and infinite-range forces

This paper considers the linear space-inhomogeneous Boltzmann equation in a convex, bounded or unbounded bodyD with general boundary conditions. First, mildL ¹-solutions are constructed in the cutoff case using monotone sequences of iterates in an exponential form. Assuming detailed balance relations, mass conservation and uniqueness are proved, together with anH-theorem with formulas for the interior and boundary terms. Local boundedness of higher moments is proved for soft and hard collision potentials, together with global boundedness for hard potentials in the case of a nonheating boundary, including specular reflections. Next, the transport equation with forces of infinite range is considered in an integral form. Existence of weakL ¹-solutions are proved by compactness, using theH-theorem from the cutoff case. Finally, anH-theorem is given also for the infinite-range case.     

Target-mass and finite <Emphasis Type="Italic">t</Emphasis> corrections to diffractive deep-inelastic scattering

The quantum field theoretic treatment of inclusive deep-inelastic diffractive scattering given in a previous paper (Blümlein et al. in Nucl. Phys. B 755:112–136, 2006) is discussed in detail using an equivalent formulation with the aim to derive a representation suitable for data analysis. We consider the off-cone twist-2 light-cone operators to derive the target-mass and finite t corrections to diffractive deep-inelastic scattering and deep-inelastic scattering. The corrections turn out to be at most proportional to x|t|/Q ², xM ²/Q ², x=x _BJ or x _ℙ, which suggests an expansion in these parameters. Their contribution varies in size considering diffractive scattering or meson-exchange processes. Relations between different kinematic amplitudes which are determined by one and the same diffractive GPD or its moments are derived. In the limit t,M ²→0 one obtains the results of (Blümlein and Robaschik in Phys. Lett. B 517:222, 2001) and (Blümlein and Robaschik in Phys. Rev. D 65:096002, 2002).     

Precise Nuclear Quadrupole Moments of 8B and 13B

The electric quadrupole coupling constants eqQ/h of ⁸B (, T _1/2 = 769 ms) and ¹³B (, T _1/2 = 17.4 ms) in single crystal TiO₂ have been precisely measured by the β-NQR technique. The ratios of these Q moments to Q(¹²B) were determined as ∣Q(⁸B)/Q(¹²B)∣ = 4.882(32) and ∣Q(¹³B)/Q(¹²B)∣ = 2.768(24).     

Phase space quantization as a moment problem

We consider questions related to the following quantization scheme: a classical variable f: Ω → ℝ on a phase space Ω is associated with a unique semispectral measure E ^f , such that the kth moment operator of E ^f is required to coincide with the operator integral L(f ^k , E) of f ^k with respect to a certain fixed phase space semispectral measure E. Mainly, we take the phase space Ω to be a locally compact unimodular group. In the concrete case where Ω = ℝ² and E is a translation covariant semispectral measure, we determine explicitly the relevant operators L(f ^k , E) for certain variables f. In addition, we consider the question under what conditions a positive operator measure is projection valued. The text was submitted by the author in English.     

On the Cauchy Problem of the Vlasov-Poisson-BGK System: Global Existence of Weak Solutions

Global-in-time existence of weak solutions to the Cauchy problem of the three dimensional Vlasov-Poisson-BGK system is shown for initial data belonging to the space L ^p (ℝ³×ℝ³) with p>9 and having finite second order velocity moments. This result solves partially the well-posed problem for the Vlasov-Poisson-BGK system proposed by B. Perthame: “Higher moments for kinetic equations: the Vlasov-Poisson and Fokker-Planck cases,” Math. Meth. Appl. Sci. 13:441–452, 1990.     

A Counter Example to Cercignani’s Conjecture for the <Emphasis Type="Italic">d</Emphasis> Dimensional Kac Model

Kac’s d dimensional model gives a linear, many particle, binary collision model from which, under suitable conditions, the celebrated Boltzmann equation, in its spatially homogeneous form, arise as a mean field limit. The ergodicity of the evolution equation leads to questions about the relaxation rate, in hope that such a rate would pass on the Boltzmann equation as the number of particles goes to infinity. This program, starting with Kac and his one dimensional ‘Spectral Gap Conjecture’ at 1956, finally reached its conclusion in the Maxwellian case in a series of papers by authors such as Janvresse, Maslen, Carlen, Carvalho, Loss and Geronimo, but the hope to get a limiting relaxation rate for the Boltzmann equation with this linear method was already shown to be unrealistic (although the problem is still important and interesting due to its connection with the linearized Boltzmann operator). A less linear approach, via a many particle version of Cercignani’s conjecture, is the grounds for this paper. In our paper, we extend recent results by the author from the one dimensional Kac model to the d dimensional one, showing that the entropy-entropy production ratio, Γ _N , still yields a very strong dependency in the number of particles of the problem when we consider the general case.     

Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ^∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.     

Moment inequalities for the boltzmann equation and applications to spatially homogeneous problems

Some inequalities for the Boltzmann collision integral are proved. These inequalities can be considered as a generalization of the well-known Povzner inequality. The inequalities are used to obtain estimates of moments of the solution to the spatially homogeneous Boltzmann equation for a wide class of intermolecular forces. We obtain simple necessary and sufficient conditions (on the potential) for the uniform boundedness of all moments. For potentials with compact support the following statement is proved: if all moments of the initial distribution function are bounded by the corresponding moments of the MaxwellianA exp(−Bv ²), then all moments of the solution are bounded by the corresponding moments of the other MaxwellianA ₁ exp[−B ₁(t)v ²] for anyt > 0; moreoverB(t) = const for hard spheres. An estimate for a collision frequency is also obtained.     

Dynamical Localization of Quantum Walks in Random Environments

We study shock statistics in the scalar conservation law ∂ _t u+∂ _x f(u)=0, x∈ℝ, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈ℝ. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

On entanglement distillation and quantum error correction for unknown states and channels

We consider the problem of invariance of distillable entanglement D and quantum capacities Q under erasure of information about single copy of quantum state or channel respectively. We argue that any 2 ⊗N two-way distillable state is still two-way distillable after erasure of single copy information. For some known distillation protocols the obtained two-way distillation rate is the same as if Alice and Bob knew the state from the very beginning. The isomorphism between quantum states and quantum channels is also investigated. In particular it is pointed out that any transmission rate down the channel is equal to distillation rate with formal LOCC-like superoperator that uses in general nonphysical Alice actions. This allows to we prove that if given channel Λ has nonzero capacity (Q _→ or Q _⟺) then the corresponding quantum state ϱ(Λ) has nonzero distillable entanglement (D _→ or D _⟺). Follwoing the latter arguments are provided that any channel mapping single qubit into N level system allows for reliable two-way transmission after erasure of information about single copy. Some open problems are discussed.     

Q value related mass determinations using a Penning trap

We report here about measurements of reaction and decay Q values by precise determination of pairs of atomic masses. These were performed with the Penning trap mass spectrometer SMILETRAP. Measurements with Penning traps give reliable and accurate masses, in particular Q values, due to the fact that certain systematic errors to a great deal cancel in the mass difference between the two atoms defining the Q value. Some Q values that are of fundamental interest will be discussed here, for example, a new Q value for the ⁶Li (n,γ) ⁷Li reaction, for the β-decay of tritium, related to properties of the electron neutrino mass, and for the neutrino-less double β-decay of ⁷⁶Ge, related to the question of whether the neutrino is a Majorana particle or not. In case of the latter two we report the most accurate Q values, namely 18,589.8(12) eV for the tritium decay and 2,038.997(46) keV for the neutrino-less double β-decay of ⁷⁶Ge.     

Magnetooptical properties and FMR in granular nanocomposites (Co84Nb14Ta2) x (SiO2)100? x

For nanodimensional magnetically inhomogeneous amorphous granular films of the system (Co₈₄Nb₁₄Ta₂) _x (SiO₂)_100−x , 30≤x≤60 at.% the concentration dependences of the magnetooptical Kerr effect (MOKE) spectra and FMR have been investigated. The observed changes in the MOKE and FMR spectra are associated with transformations of microstructure and topology of the nanocomposites. For the compositions within the percolation region the transversal Kerr effect increases by an order of magnitude.     

Some Ideas About Quantitative Convergence of Collision Models to Their Mean Field Limit

We consider a stochastic N-particle model for the spatially homogeneous Boltzmann evolution and prove its convergence to the associated Boltzmann equation when N⟶∞, with non-asymptotic estimates: for any time T>0, we bound the distance between the empirical measure of the particle system and the measure given by the Boltzmann evolution in a relevant Hilbert space. The control got is Gaussian, i.e. we prove that the distance is bigger than xN ^−1/2 with a probability of type O(e^-x2)O(e^{-x^{2}}). The two main ingredients are a control of fluctuations due to the discrete nature of collisions and a kind of Lipschitz continuity for the Boltzmann collision kernel. We study more extensively the case where our Hilbert space is the homogeneous negative Sobolev space [(H)\dot]^-s\smash {\dot {H}}^{-s}. Then we are only able to give bounds for Maxwellian models; however, numerical computations tend to show that our results are useful in practice.     

A priori estimates for the Hill and Dirac operators

The Hill operator Ty = −y″ + q′(t)y is considered in L ²(ℝ), where q ∈ L ²(0, 1) is a periodic real potential. The spectrum of T is absolutely continuous and consists of bands separated by gaps. We obtain a priori estimates of gap lengths, effective masses, and action variables for the KDV equation. In the proof of these results, the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator is used. Similar estimates for the Dirac operator are obtained.