Transmission coefficient and heat conduction of a harmonic chain with random masses: Asymptotic estimates on products of random matrices

We find upper and lower bounds for the transmission coefficient of a chain of random masses. Using these bounds we show that the heat conduction in such a chain does not obey Fourier's law: For different temperatures at the ends of a chain containingN particles the energy flux falls off likeN ^–1/2 rather thanN ^–1.Research supported by a ZWO fellowship and in part by U.S.A.F.O.S.R. grant no. 78-3522     

Convergence of spherical harmonic expansions for the evaluation of hard-sphere cluster integrals

ForN particles (N>2), by means of a spherical harmonic expansion of Silverstone and Moats, a 3N-dimensional cluster may be reduced to 2N+1 trivial integrals andN–1 interesting integrals. For hard spheres, theN–1 interesting integrals are products of polynomials integrated between binomial bounds. With simple clusters, closed forms are obtained; for more complex clusters, infinite series inl (ofY _lm ) appear. It is here shown for representative cases that these series converge exponentially rapidly, the leading pair of terms accounting for all but a few tenths of a percent of the total cluster integral.     

Mean-Field Dynamics: Singular Potentials and Rate of Convergence

We consider the time evolution of a system of N identical bosons whose interaction potential is rescaled by N ⁻¹. We choose the initial wave function to describe a condensate in which all particles are in the same one-particle state. It is well known that in the mean-field limit N → ∞ the quantum N-body dynamics is governed by the nonlinear Hartree equation. Using a nonperturbative method, we extend previous results on the mean-field limit in two directions. First, we allow a large class of singular interaction potentials as well as strong, possibly time-dependent external potentials. Second, we derive bounds on the rate of convergence of the quantum N-body dynamics to the Hartree dynamics.     

Rigorous bounds for critical temperatures of O(N) spin models by transformations of block spin type

We obtain new upper bounds of critical temperatures of N-vector (Heisenberg) models. We apply a transformation of block spin type to random walk representations of the spin models, which was developed by Fröhlich et al. more than a decade ago. Though the transformation is applied just one time, the upper bounds are considerably improved.     

Upon the utility of spherical harmonic expansions in the evaluation of cluster integrals

A novel procedure for the analytic evaluation of cluster integrals is given. By means of a result of Silverstone and Moats which transforms the spherical harmonic expansion of a function around a given point into a new spherical harmonic expansion around a displaced point, a 3N-dimensional cluster integral forN point particles (N > 2) may be reduced to 2N+1 trivial integrals andN– 1 interesting integrals, an improvement over the usual reduction to six trivial integrals and3N–6 nontrivial integrals. For hard spheres, theN–1 integrals involve only a series of simple polynomials taken between linear algebraic bounds.This work was supported in part by the National Science Foundation under Grant No. CHE79-20389.     

Close to Close Packing

For various lattice gas models with nearest neighbor exclusion (and, in one case, second-nearest neighbor exclusion as well), we obtain lower bounds on m, the average number of particles on the nonexcluded lattice sites closest to a given particle. They are all of the form m/m _cp 1–const·(N _cp /N–1), where N is the number of occupied sites, m _cp is the value of m at close packing, and N _cp is the value of N at close packing. An analogous result exists for hard disks in the plane.     

Mean field bound and GHS inequality

A new proof of the mean field bounds for magnetizations is presented. It applies to any single-component spin system which allows GHS inequality, and to anN-vector model forN 3, and to anN-solid sphere model for all values ofN, provided that the interactions are ferromagnetic and translation invariant.     

Large-N-limit of mixed valence: Diagrammatic expansion and rigorous results

For an intermediate valence model, containing one simply and oneN-fold degeneratef-configuration at a specific lattice site, the large-N-limit is studied within a concise diagrammatic expansion and also rigorously. At zero temperatureT rigorous upper and lower bounds for the ground state energy are found, while at finiteT the large-N-expansion is shown to be equivalent to the principle of classifying the diagrams according the maximum number of spin flips, whereupon various physical quantities can be calculated.On leave of absence from Department of Physics, Beijing University, Beijing, People's Republic of China with support from DAAD     

Rate of Convergence Towards the Hartree–von Neumann Limit in the Mean-Field Regime

In the mean-field regime we prove convergence, with explicit bounds, of N-particle density matrices satisfying the time-dependent von Neumann equation with factorized initial data to a product of one particle density matrices satisfying the Hartree–von Neumann equation. To prove explicit bounds we generalize techniques developed by Pickl (in A simple derivation of mean field limits for quantum systems. ArXiv:0907.4464, 2009) and Knowles–Pickl (in Commun. Math. Phys. 298(1):101–138, 2010).     

Periodic and Quasi-Periodic Orbits¶for the Standard Map

We consider both periodic and quasi-periodic solutions for the standard map, and we study the corresponding conjugating functions, i.e. the functions conjugating the motions to trivial rotations. We compare the invariant curves with rotation numbers ω satisfying the Bryuno condition and the sequences of periodic orbits with rotation numbers given by their convergents ω _N = p _N /q _N . We prove the following results for N→ ∞: (1) for rotation numbers ω _N N we study the radius of convergence of the conjugating functions and we find lower bounds on them, which tend to a limit which is a lower bound on the corresponding quantity for ω; (2) the periodic orbits consist of points which are more and more close to the invariant curve with rotation number ω; (3) such orbits lie on analytical curves which tend uniformly to the invariant curve. Received: 14 December 2001 / Accepted: 16 March 2002?Published online: 2 October 2002     

Bounds on the Minimal Energy of Translation Invariant N-Polaron Systems

For systems of N charged fermions (e.g. electrons) interacting with longitudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fröhlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants U > 0 and α > 0 measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition ${\sqrt{2}\alpha < U}For systems of N charged fermions (e.g. electrons) interacting with longitudinal optical quantized lattice vibrations of a polar crystal we derive upper and lower bounds on the minimal energy within the model of H. Fr?hlich. The only parameters of this model, after removing the ultraviolet cutoff, are the constants U > 0 and α > 0 measuring the electron-electron and the electron-phonon coupling strengths. They are constrained by the condition ?2a < U{\sqrt{2}\alpha < U}, which follows from the dependence of U and α on electrical properties of the crystal. We show that the large N asymptotic behavior of the minimal energy E _N changes at ?2a = U{\sqrt{2}\alpha=U} and that ?2a ￡ U{\sqrt{2}\alpha\leq U} is necessary for thermodynamic stability: for ${\sqrt{2}\alpha > U}${\sqrt{2}\alpha > U} the phonon-mediated electron-electron attraction overcomes the Coulomb repulsion and E _N behaves like −N ^7/3.     

Scaling of Linking and Writhing Numbers for Spherically Confined and Topologically Equilibrated Flexible Polymers

Scaling laws for Gauss linking number Ca and writhing number Wr for spherically confined flexible polymers with thermally fluctuating topology are analyzed. For ideal (phantom) polymers each of N segments of length unity confined to a spherical pore of radius R there are two scaling regimes: for sufficiently weak confinement (R≫N ^1/3) each chain has |Wr|≈N ^1/2, and each pair of chains has average |Ca|≈N/R ^3/2; alternately for sufficiently tight confinement (N ^1/3≫R), |Wr|≈|Ca|≈N/R ^3/2. Adding segment-segment avoidance modifies this result: for n chains with excluded volume interactions |Ca|≈(N/n)^1/2 f(φ) where f is a scaling function that depends approximately linearly on the segment concentration φ=nN/R ³. Scaling results for writhe are used to estimate the maximum writhe of a polymer; this is demonstrated to be realizable through a writhing instability that occurs for a polymer which is able to change knotting topology and which is subject to an applied torque. Finally, scaling results for linking are used to estimate bounds on the entanglement complexity of long chromosomal DNA molecules inside cells, and to show how “lengthwise” chromosome condensation can suppress DNA entanglement.     

Proof of chiral symmetry breaking in lattice gauge theory

Using the method of infrared bounds and partial-integration formulas, we prove that there is a chiral phase transition in four-dimensional strongly coupled lattice gauge theory with gauge group U(N) and staggered fermions for all N5.     

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.     

Gaussian bounds for correlations in lattice spin systems

In quite generalN-component ferromagnetic spin systems, it is proved that an arbitrary correlation function is bounded by the corresponding correlation function of a Gaussian model. The bound is useful for the analysis of high-temperature behavior of the system. Similar bounds for truncated correlation functions are also obtained for a class of single-component spin systems.     

Phase transitions and positive definite couplings

The method of infrared bounds is extended to a large class of nearest neighbour interactions in classical spin systems. Temperature controlled bounds on fluctuations follow whenever the coupling function is a positive definite kernel. Existence of phase transitions is demonstrated for the RP ^Nmodel for d3.     

Simulation of CPMAS Signals at High Spinning Speeds

The spin dynamics of anS( )I_Nsystem during the CP mixing time of continuous wave and variable amplitude cross-polarization magic angle spinning (CWCPMAS and VACPMAS) experiments is discussed. The signal enhancement of a low abundantSspin, coupled to a set ofN= 6 coupled spins withI= , is evaluated as a function of the length of the mixing time. For CWCPMAS this signal is first evaluated in the frequency domain and then transformed to the time domain. These calculations provide some additional insight into the CP spin dynamics and enable a practical approach toward the evaluation of CP signals of large spin systems. In addition the adiabatic character of the ramped VACPMAS experiments is discussed andS-spin signals of a spin system withN= 6 are simulated. Estimates of the upper bounds of the CP signals as a function of the number ofIspins in anS( )I_Nsystem are given and compared with the calculated values.     

Exponential bounds and semi-finiteness of point spectrum forN-body Schrödinger operators

For a large class ofN-body Schrödinger operatorsH, we prove that eigenvalues ofH cannot accumulate from above at any threshold ofH. Our proof relies onL ² exponential upper bounds for eigenfunctions ofH with explicit constants obtained by modifying methods of Froese and Herbst.Bantrell Research Fellow in Mathematical Physics     

Asymptotic neutrality of diatomic molecules

We use the no-binding theorem of Thomas-Fermi theory to prove that a large diatomic molecule is almost neutral. That is to say, that if the total nuclear charge isZ then the numberN of electrons required for the diatomic molecule to be stable satisfies . In contrast to the atomic case the emphasis here is on the lower bound onN. Our analysis will imply a new bound on the size of the molecule. These results are proved in the Born-Oppenheimer approximation. We also give bounds onN which hold for allZ by a very elementary method, not assuming the Born-Oppenheimer approximation.Work supported by a Danish Research Academy fellowship and U.S. National Science Foundation grant PHY-85-15288-A03Adress from September 1989: Department of Mathematics, The University of Michigan, Ann Arbor, Michigan 48109 USA     

Some consequences of neutral current systematics

Two kinds of general consequences of the ΔS=0 weak hadron neutral current independent of a gauge model are presented. Firstly are results which depend on the quark parton model. These involve bounds among neutrino inclusive cross-section and a bound onQ(Z, N) in terms of these inclusive cross-sections. Secondly are results which are independent of the quark-parton model and depend only on the SU(3) structure of the most general ΔS=0 neutral current. These tests of isopin and speciallyU-spin properties of the current are given forv+N→v+hadron+anything,v+N→v+baryon+meson ande ⁺ e ⁻→baryon+anti-baryon. In addition some conjectures are made with regard to the semi-inclusive neutrino-reactions using the quark parton model.