## Recursos de colección

DASH is a central, open-access repository of research by members of the Harvard community

FAS Scholarly Articles

1. #### Automatically Determining Versions of Scholarly Articles

Rothchild, Daniel Hugo; Shieber, Stuart Merrill
Background: Repositories of scholarly articles should provide authoritative information about the materials they distribute and should distribute those materials in keeping with pertinent laws. To do so, it is important to have accurate information about the versions of articles in a collection. Analysis: This article presents a simple statistical model to classify articles as author manuscripts or versions of record, with parameters trained on a collection of articles that have been hand-annotated for version. The algorithm achieves about 94 percent accuracy on average (cross-validated). Conclusion and implications: The average pairwise annotator agreement among a group of experts was 94 percent, showing that...

2. #### The work of Cédric Villani

Yau, Horng-Tzer
The starting point of C´edric Villani’s work goes back to the introduction of entropy in the nineteenth century by L. Carnot and R. Clausius. At the time, entropy was a vague concept and its rigorous definition had to wait until the fundamental work of L. Boltzmann who introduced nonequilibrium statistical physics and the famous H functional. Boltzmann’s work, though a fundamental breakthrough, did not resolve the question concerning the nature of entropy and time arrow; the debate on this central question continued for a century until today. J. von Neumann, in recommending C. Shannon to use entropy for his uncertainty...

3. #### Logarithmic Sobolev inequality for lattice gases with mixing conditions

Yau, Horng-Tzer
Let μgcΛL,λμΛL,λgc denote the grand canonical Gibbs measure of a lattice gas in a cube of sizeL with the chemical potential γ and a fixed boundary condition. Let μcΛL,nμΛL,nc be the corresponding canonical measure defined by conditioning μgcΛL,λμΛL,λgc on Σx∈Ληx=nΣx∈Ληx=n . Consider the lattice gas dynamics for which each particle performs random walk with rates depending on near-by particles. The rates are chosen such that, for everyn andL fixed, μcΛL,nμΛL,nc is a reversible measure. Suppose that the Dobrushin-Shlosman mixing conditions holds for μgcL,λμL,λgc forall chemical potentials λ ∈ γ ∈ ℝ. We prove that ∫flogfdμcΛL,n≦const.L2D(f√)∫flog⁡fdμΛL,nc≦const.L2D(f) for any probability densityf with...

4. #### (logt)2/3 law of the two dimensional asymmetric simple exclusion process

Yau, Horng-Tzer
We prove that the diffusion coefficient for the two dimensional asymmetric simple exclusion process diverges as (logt)2/3 to the leading order. The method applies to nearest and non-nearest neighbor asymmetric simple exclusion processes.

5. #### Asymptotic dynamics of nonlinear Schrödinger equations: Resonance-dominated and dispersion-dominated solutions

Tsai, Tai-Peng; Yau, Horng-Tzer
We consider a linear Schrödinger equation with a nonlinear perturbation in ℝ3. Assume that the linear Hamiltonian has exactly two bound states and its eigen-values satisfy some resonance condition. We prove that if the initial data is sufficiently small and is near a nonlinear ground state, then the solution approaches to certain nonlinear ground state as the time tends to infinity. Furthermore, the difference between the wave function solving the nonlinear Schrödinger equation and its asymptotic profile can have two different types of decay: The resonance-dominated solutions decay as t−1/2 or the dispersion-dominated solutions decay at least like t−3/2. ©...

6. #### Relaxation of excited states in nonlinear Schrödinger equations

Tsai, Tai-Peng; Yau, Horng-Tzer
We consider a nonlinear Schrödinger equation with a bounded local potential in ℝ3. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is small and is near some nonlinear excited state. We give a sufficient condition on the initial data so that the solution to the Schrödinger equation approaches to certain nonlinear ground state as the time tends to infinity. Our proof is based on a notion of outgoing estimate which measures the time-direction related information of the wave functions for the nonlinear Schrödinger equations.

7. #### Stable Directions for Excited States of Nonlinear Schrödinger Equations

Tsai, Tai-Peng; Yau, Horng-Tzer
We consider nonlinear Schrödinger equations in R3. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both non-resonant and resonant cases. In the resonant case, the linearized operators around the excited states are non-self adjoint perturbations to some linear Hamiltonians with embedded eigenvalues. Although self-adjoint perturbation turns embedded eigenvalues into resonances, this class of non-self adjoint perturbations turn an embedded eigenvalue into two eigenvalues with the distance to the continuous spectrum given to the leading order by the Fermi golden...

8. #### Bulk diffusivity of lattice gases close to criticality

Spohn, Herbert; Yau, Horng-Tzer
We consider lattice gases where particles jump at random times constrained by hard-core exclusion (simple exclusion process with speed change). The conventional theory of critical slowing down predicts that close to a critical point the bulk diffusivity vanishes as the inverse compressibility. We confirm this claim by proving a strictly positive lower bound for the conductivity.

9. #### Superdiffusivity of Two Dimensional Lattice Gas Models

Landim, Claudio; Ramírez, José A.; Yau, Horng-Tzer
It was proved [Navier–Stokes Equations for Stochastic Particle System on the Lattice. Comm. Math. Phys. (1996) 182, 395; Lattice gases, large deviations, and the incompressible Navier–Stokes equations. Ann. Math. (1998) 148, 51] that stochastic lattice gas dynamics converge to the Navier–Stokes equations in dimension d=3 in the incompressible limits. In particular, the viscosity is finite. We proved that, on the other hand, the viscosity for a two dimensional lattice gas model diverges faster than (log t)1/2. Our argument indicates that the correct divergence rate is (log t)1/2. This problem is closely related to the logarithmic correction of the time decay...

10. #### Bulk universality for Wigner matrices

Erdos, Laszlo; Péché, Sandrine; Ramírez, José A.; Schlein, Benjamin; Yau, Horng-Tzer
We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C6( \input amssym \$\Bbb R\$) with at most polynomially growing derivatives and ν(x) ≥ Ce−C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc.

11. #### Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams

Erdos, Laszlo; Salmhofer, Manfred; Yau, Horng-Tzer
We consider random Schrödinger equations on {mathbb{R}d} for d≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψ t the solution with initial data ψ0. The space and time variables scale as {x˜ λ^{-2 -kappa/2}, t ˜ λ^{-2 -kappa}} with 0 < κ < κ0( d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ t converges weakly to the solution of a heat equation in the space variable x for arbitrary L 2 initial data. The proof is based on a rigorous analysis of...

12. #### On the Quantum Boltzmann Equation

Erdos, Laszlo; Salmhofer, Manfred; Yau, Horng-Tzer
We give a nonrigorous derivation of the nonlinear Boltzmann equation from the Schrödinger evolution of interacting fermions. The argument is based mainly on the assumption that a quasifree initial state satisfies a property called restricted quasifreenessin the weak coupling limit at any later time. By definition, a state is called restricted quasifree if the four-point and the eight-point functions of the state factorize in the same manner as in a quasifree state.

13. #### Wegner Estimate and Level Repulsion for Wigner Random Matrices

Erdos, L; Schlein, B.; Yau, Horng-Tzer
We consider N × N Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order 1/ N. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales η ≫ N−1. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result [6]. We then show a Wegner estimate, i.e., that the averaged...

14. #### Universality of random matrices and local relaxation flow

Erdos, Laszlo; Schlein, Benjamin; Yau, Horng-Tzer
Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we prove that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N−ζ for some ζ>0. The proof is based on an estimate of the entropy flow of the Dyson Brownian motion w.r.t. a “pseudo equilibrium measure”. As an application of this estimate, we prove that the eigenvalue spacing statistics in the bulk of the spectrum for N×N symmetric Wigner ensemble is the same as that...

15. #### Local Semicircle Law and Complete Delocalization for Wigner Random Matrices

Erdos, Laszlo; Schlein, Benjamin; Yau, Horng-Tzer
We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales η≫N−1(logN)8η≫N−1(log⁡N)8 . Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the ℓ∞-norm of the corresponding eigenvectors...

16. #### Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

Erdos, Laszlo; Schlein, Benjamin; Yau, Horng-Tzer
Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N²V (N(xi − xj)), where x = (x1,..., xN) denotes the positions of the particles. Let HN denote the Hamiltonian of the system and let ψN,t be the solution to the Schrödinger equation. Suppose that the initial data ψN,0 satisfies the energy condition 〈ψN,0, H k NψN,0 〉 ≤ C k N k for k = 1, 2,.... We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N → ∞. We prove that the k-particle density...

17. #### Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems

Erdos, Laszlo; Schlein, Benjamin; Yau, Horng-Tzer
We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schrödinger equation in a suitable scaling limit. The result is extended to k-particle density matrices for all positive integer k.

18. #### Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

Erdos, Laszlo; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun
We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN≫N2/3pN≫N2/3 , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest...

19. #### Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation

Erdos, Laszlo; Yau, Horng-Tzer
We study the time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation globally in time. The Boltzmann collision kernel is given by the Born approximation of the quantum differential scattering cross section.

20. #### Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II

Chen, Chiun-Chuan; Strain, Robert M.; Tsai, Tai-Peng; Yau, Horng-Tzer
Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ℝ3 with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ε ≤ 1, |v (x, t)| ≤ C ∗ r −1+ε |t|−ε/2 for − T 0 ≤ t < 0 and 0 < C ∗ < ∞ allowed to be large. We prove that v is regular at time zero.

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