Generalization of correlated electron-ion dynamics from nonequilibrium Green's functions

We present a new formulation of the correlated electron-ion dynamics (CEID) by using equations of motion for nonequilibrium Green's functions, which generalizes CEID to a general nonequilibrium statistical ensemble that allows for a variable total number of electrons. We make a rigorous connection between CEID and diagrammatic perturbation theory, which furthermore allows the key approximations in CEID to be quantified in diagrammatic terms, and, in principle, improved. We compare analytically the limiting behavior of CEID and the self-consistent Born approximation (SCBA) for a general dynamical nonequilibrium state. This comparison shows that CEID and SCBA coincide in the weak electron-phonon coupling limit, while they differ in the large ionic mass limit where we can readily quantify their difference. In particular, we illustrate the relation between CEID and SCBA by perturbation theory at the fourth-order in the coupling strength.Comment: 21 pages, 2 figure

Harnack Inequalities for Stochastic Equations Driven by L\'evy Noise

By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a L\'evy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by L\'evy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for L\'evy processes or linear equations driven by L\'evy noise. The main results are also extended to semi-linear stochastic equations in Hilbert spaces.Comment: 15 page

Diffuse PeV neutrinos from gamma-ray bursts

The IceCube collaboration recently reported the potential detection of two cascade neutrino events in the energy range 1-10 PeV. We study the possibility that these PeV neutrinos are produced by gamma-ray bursts (GRBs), paying special attention to the contribution by untriggered GRBs that elude detection due to their low photon flux. Based on the luminosity function, rate distribution with redshift and spectral properties of GRBs, we generate, using Monte-Carlo simulation, a GRB sample that reproduce the observed fluence distribution of Fermi/GBM GRBs and an accompanying sample of untriggered GRBs simultaneously. The neutrino flux of every individual GRBs is calculated in the standard internal shock scenario, so that the accumulative flux of the whole samples can be obtained. We find that the neutrino flux in PeV energies produced by untriggered GRBs is about 2 times higher than that produced by the triggered ones. Considering the existing IceCube limit on the neutrino flux of triggered GRBs, we find that the total flux of triggered and untriggered GRBs can reach at most a level of ~10^-9 GeV cm^-2 s^-1 sr^-1, which is insufficient to account for the reported two PeV neutrinos. Possible contributions to diffuse neutrinos by low-luminosity GRBs and the earliest population of GRBs are also discussed.Comment: Accepted by ApJ, one more figure added to show the contribution to the diffuse neutrino flux by untriggered GRBs with different luminosity, results and conclusions unchange

Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift

Consider the stochastic evolution equation in a separable Hilbert space with a nice multiplicative noise and a locally Dini continuous drift. We prove that for any initial data the equation has a unique (possibly explosive) mild solution. Under a reasonable condition ensuring the non-explosion of the solution, the strong Feller property of the associated Markov semigroup is proved. Gradient estimates and log-Harnack inequalities are derived for the associated semigroup under certain global conditions, which are new even in finite-dimensions.Comment: 36 page

Log-Sobolev inequalities: Different roles of Ric and Hess

Let $P_t$ be the diffusion semigroup generated by $L:=\Delta +\nabla V$ on a complete connected Riemannian manifold with $\operatorname {Ric}\ge-(\sigma ^2\rho_o^2+c)$ for some constants $\sigma, c>0$ and $\rho_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided $-\operatorname {Hess}_V\ge\delta$ holds outside of a compact set for some constant $\delta >(1+\sqrt{2})\sigma \sqrt{d-1}.$ This indicates, at least in finite dimensions, that $\operatorname {Ric}$ and $-\operatorname {Hess}_V$ play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.Comment: Published in at http://dx.doi.org/10.1214/08-AOP444 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org