Fluctuations of extensive functions of quenched random couplings

An extensive quantity is a family of functions _v of random parameters, indexed by the finite regionsV (subsets of ^d) over which _v are additive up to corrections satisfying the boundary estimate stated below. It is shown that unless the randomness is nonessential, in the sense that lim _v/|V| has a unique value in the absolute (i.e., not just probabilistic) sense, the variance of such a quantity grows as the volume ofV. Of particular interest is the free energy of a system with random couplings; for such _v bounds are derived also for the generating functionE(e ^t ). In a separate application, variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment.     

Determining Liquid Structure from the Tail of the Direct Correlation Function

In important early work, Stell showed that one can determine the pair correlation function h(r) of the hard-sphere fluid for all distances r by specifying only the tail of the direct correlation function c(r) at separations greater than the hard-core diameter. We extend this idea in a very natural way to potentials with a soft repulsive core of finite extent and a weaker and longer ranged tail. We introduce a new continuous function T(r) which reduces exactly to the tail of c(r) outside the (soft) core region and show that both h(r) and c(r) depend only on the out projection of T(r): i.e., the product of the Boltzmann factor of the repulsive core potential times T(r). Standard integral equation closures can thus be reinterpreted and assessed in terms of their predictions for the tail of c(r) and simple approximations for its form suggest new closures. A new and very efficient variational method is proposed for solving the Ornstein–Zernike equation given an approximation for the tail of c. Initial applications of these ideas to the Lennard-Jones and the hard-core Yukawa fluid are discussed.     

On the Distribution of Free Path Lengths for the Periodic Lorentz Gas III

 For r(0,1), let Z _r ={xR ² |dist(x,Z ²)>r/2} and define τ _r (x,v)=inf{t>0|x+tv∂Z _r }. Let Φ _r (t) be the probability that τ _r (x,v)≥t for x and v uniformly distributed in Z _r and §¹ respectively. We prove in this paper that

Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation

We consider the Cauchy problem for the two-dimensional vorticity equation. We show that the solution behaves like a constant multiple of the Gauss kernel having the same total vorticity as time tends to infinity. No particular structure of initial data ₀=(x, 0) is assumed except the restriction that the Reynolds numberR=|₀|dx/v is small, wherev is the kinematic viscosity. Applying a time-dependent scale transformation, we show a stability of Burgers' vortex, which physically implies formation of a concentrated vortex.Partly supported by Grant-in-Aid for Scientific Research No. B60460042, the Japan Ministry of Education, Science and Culture     

Spherical harmonics and Cartesian tensor scalar product expansion in the Fokker-Planck equation

On the basis of the expansion of the distribution functionf(v, r,t) in a sum of spherical harmonics, which is equivalent to a Cartesian tensor scalar product expansion of the distribution function, i.e.,f(v, r, t)=f ₀(v,r,t)+v. f ₁(v,r,t)+vvf ₂(v,r,t)+vvvf ₃(v,r,t)+ wheref _k (k=2, 3) arek-th order irreducible tensors, the Rosenbluth potential functions and the Fokker-Planck collision term are expanded in a similar sum. Collisions termsJ _Fk (k=0, 1, 2) and the equations forf _k (k=0, 1, 2) for the case of the Coulomb interactions are also determined.Technická 2, Praha 6, Czechoslovakia.The autor wishes to express his thanks to Prof. J. Kracík, DrSc. for valuable advice and suggestion.     

On the cauchy problem for the discrete Boltzmann equation with initial values inL 1 + (ℝ)

We prove a global existence theorem for a discrete velocity model of the Boltzmann equation when the initial values _i (x) have finite entropy and, for some constant>0, (1+|x|) _i (x)L ₁ ⁺ ().     

Conservation of Energy,Entropy Identity,and Local Stability for the Spatially Homogeneous Boltzmann Equation

For nonsoft potential collision kernels with angular cutoff, we prove that under the initial condition f ₀(v)(1+|v|²+|logf ₀(v)|)L ¹(R ³), the classical formal entropy identity holds for all nonnegative solutions of the spatially homogeneous Boltzmann equation in the class L ([0, ); L ¹ ₂(R ³))C ¹([0, ); L ¹(R ³)) [where L ¹ _s (R ³)={ff(v)(1+|v|²) ^s/2L ¹(R ³)}], and in this class, the nonincrease of energy always implies the conservation of energy and therefore the solutions obtained all conserve energy. Moreover, for hard potentials and the hard-sphere model, a local stability result for conservative solutions (i.e., satisfying the conservation of mass, momentum, and energy) is obtained. As an application of the local stability, a sufficient and necessary condition on the initial data f ₀ such that the conservative solutions f belong to L ¹ _loc([0, ); L ¹ ₂₊ (R ³)) is also given.     

Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I.

Inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schrödinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential ${v(x)= \epsilon \chi(x) |x|^{-1}}Inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schr?dinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential v(x) = ec(x) |x|^-1{v(x)= \epsilon \chi(x) |x|^{-1}}, where e{\epsilon} is sufficiently small and c ? C₀^￥{\chi \in C_0^{\infty}} even, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (Part II) of this paper.     

Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation

This paper deals with the trend to equilibrium of solutions to the spacehomogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has previously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of order 2+, then the convergence to equilibrium in his metric is exponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to the Boltzmann equation, by proving that the Fourier-transformed solutions are at least as regular as the Fourier transform of the initial data. This is also used to prove that even if the initial data only possess a second moment, then _v>R f(v, t) v² dv0 asR, and this convergence is uniform in time.     

Finite Time Approach to Equilibrium in a Fractional Brownian Velocity Field

We consider the solution of the equation r(t) = W(r(t)), r(0) = r ₀ > 0 where W(⋅) is a fractional Brownian motion (f.B.m.) with the Hurst exponent α∈ (0,1). We show that for almost all realizations of W(⋅) the trajectory reaches in finite time the nearest equilibrium point (i.e. zero of the f.B.m.) either to the right or to the left of r ₀, depending on whether W(r ₀) is positive or not. After reaching the equilibrium the trajectory stays in it forever. The problem is motivated by studying the separation between two particles in a Gaussian velocity field which satisfies a local self-similarity hypothesis. In contrast to the case when the forcing term is a Brownian motion (then an analogous statement is a consequence of the Markov property of the process) we show our result using as the principal tools the properties of time reversibility of the law of the f.B.m., see Lemma 2.4 below, and the small ball estimate of Molchan, Commun. Math. Phys. 205 (1999) 97–111.     

On the Laplace transform of the Weinberg type sum rules and on the properties of pseudoscalar mesons

We reanalyse the Weinberg type sum rules in the light of the Laplace transform sum rule and including the leading non-perturbative effects. We then estimate the light quark invariant masses \(\hat m_s /\hat m_d ,\hat m_u \hat m_d \) ; the pion decay amplitudef _π and theA ₁ coupling and mass. Using other type of sum rules, we derive bounds on the pseudoscalar decay amplitudesf _p (P≡π, K, D, F) as well as on the gluon component of theU(1) _A meson mass. Our main results are summarized in Sect. 4.     

Reliability Polynomials and Their Asymptotic Limits for Families of Graphs

We present exact calculations of reliability polynomials R(G,p) for lattice strips G of fixed widths L _y 4 and arbitrarily great length L _x with various boundary conditions. We introduce the notion of a reliability per vertex, r({G},p)=lim_|V|R(G,p)^1/|V| where |V| denotes the number of vertices in G and {G} denotes the formal limit lim_|V|G. We calculate this exactly for various families of graphs. We also study the zeros of R(G,p) in the complex p plane and determine exactly the asymptotic accumulation set of these zeros , across which r({G}) is nonanalytic.     

Influence of second kind collisions on the electron distribution function in the positive column of low pressure nitrogen discharge

A numerical solution of the Boltzmann equation for the electron gas in the positive column of a DC discharge in nitrogen is presented. The Boltzmann equation was solved with the inclusion of the second kind (superelastic) collisions proceeding from the first six excited vibrational levels of molecular nitrogen. The vibrational level population is supposed to follow a Boltzmann distribution for a given vibrational temperatureT _v, with a possible deviation of the ground level, which can be overpopulated in a given ratio. Apart from the electron distribution functions, which were gained for various values ofE/p ₀,T _v and, the values of some production frequencies and kinetic coefficients are presented in form of tables and plots. It is found that the electron distribution (and also the corresponding production rates) depends above a certain energy limit onT _v and through the normalization constant only.     

Convergence in Higher Mean of a Random Schrödinger to a Linear Boltzmann Evolution

We study the macroscopic scaling and weak coupling limit for a random Schrödinger equation on $\mathbb{Z}^3We study the macroscopic scaling and weak coupling limit for a random Schr?dinger equation on . We prove that the Wigner transforms of a large class of “macroscopic” solutions converge in r ^th mean to solutions of a linear Boltzmann equation, for any 1 ≤ r < ∞. This extends previous results where convergence in expectation was established.     

Finite-size scaling of correlation functions in finite systems

We propose the finite-size scaling of correlation functions in finite systems near their critical points.At a distance r in a ddimensional finite system of size L,the correlation function can be written as the product of|r|~(-(d-2+η))and a finite-size scaling function of the variables r/L and tL~(1/ν),where t=(T-T_c)/T_c,ηis the critical exponent of correlation function,andνis the critical exponent of correlation length.The correlation function only has a sigificant directional dependence when|r|is compariable to L.We then confirm this finite-size scaling by calculating the correlation functions of the two-dimensional Ising model and the bond percolation in two-dimensional lattices using Monte Carlo simulations.We can use the finite-size scaling of the correlation function to determine the critical point and the critical exponentη.     

Entropy dissipation and moment production for the Boltzmann equation

LetH(f/M)=flog(f/M)dv be the relative entropy off and the Maxwellian with the same mass, momentum, and energy, and denote the corresponding entropy dissipation term in the Boltzmann equation byD(f)=Q(f,f) logf dv. An example is presented which shows that |D(f)/H(f/M)| can be arbitrarily small. This example is a sequence of isotropic functions, and the estimates are very explicitly given by a simple formula forD which holds for such functions. The paper also gives a simplified proof of the so-called Povzner inequality, which is a geometric inequality for the magnitudes of the velocities before and after an elastic collision. That inequality is then used to prove that f(v) |v|^s dt<C(t), wheref is the solution of the spatially homogeneous Boltzmann equation. HereC(t) is an explicitly given function dependings and the mass, energy, and entropy of the initial data.     

The ideal polymer chain near planar hard wall beyond the Dirichlet boundary conditions

We present a new ab initio approach to describe the statistical behavior of long ideal polymer chains near a plane hard wall. Forbidding the solid half-space to the polymer explicitly (by the use of Mayer functions) without any other requirement, we derive and solve an exact integral equation for the partition function G _D(r,r′, N) of the ideal chain consisting of N bonds with the ends fixed at the points r and r′ . The expression for G(r,r′, s) is found to be the sum of the commonly accepted Dirichlet result G _D(r,r′, N) = G ₀(r,r′, N) - G ₀(r,r”, N) , where r” is the mirror image of r′ , and a correction. Even though the correction is small for long chains, it provides a non-zero value of the monomer density at the very wall for finite chains, which is consistent with the pressure balance through the depletion layer (so-called wall or contact theorem). A significant correction to the density profile (of magnitude 1/is obtained away from the wall within one coil radius. Implications of the presented approach for other polymer-colloid problems are discussed.     

On rational solutions of Yang-Baxter equations. Maximal orders in loop algebra

In 1982 Belavin and Drinfeld listed all elliptic and trigonometric solutionsX(u, v) of the classical Yang-Baxter equation (CYBE), whereX takes values in a simple complex Lie algebrag, and left the classification problem of the rational one open. In 1984 Drinfeld conjectured that if a rational solution is equivalent to a solution of the formX(u,v)=C ₂/(u–v)+r(u,v), whereC ₂ is the quadratic Casimir element andr is a polynomial inu,v, then deg _u r=deg _v r1. In another paper I proved this conjecture forg=sl (n) and reduced the problem of listing nontrivial (i.e. nonequivalent toC ₂/(u–v)) solutions of CYBE to classification of quasi-Frobenius subalgebras of g. They, in turn, are related with the so-called maximal orders in the loop algebra of g corresponding to the vertices of the extended Dynkin diagramD ^e (g). In this paper I give an algorithm which enables one to list all solutions and illustrate it with solutions corresponding to vertices ofD ^e (g) with coefficient 2 or 3. In particular I will find all solutions forg=o=(5) and some solutions forg=o(7),o(10),o(14) andg ₂.     

A comment on an exact solution with a stiff equation of state

Recently, an analytical stellar model with a stiff equation of state and density distribution= _c (1-r ²/r _o ² ) was presented. We show that such a solution cannot exist.