280 research outputs found
The time-dependent Born-Oppenheimer approximation
No full textWe explain why the conventional argument for deriving the time-dependent Born-Oppenheimer approximation is incomplete and review recent mathematical results, which clarify the situation and at the same time provide a systematic scheme for higher order corrections. We also present a new elementary derivation of the correct second-order time-dependent Born-Oppenheimer approximation and discuss as applications the dynamics near a conical intersection of potential surfaces and reactive scattering
Effective dynamics for Bloch electrons: Peierls substitution and beyond
No full textWe consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, phi(epsilonx), and vector potential A(epsilonx), with xis an element ofR(d) and epsilonmuch less than1. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of L-2(R-d) and an effective Hamiltonian governing the evolution inside this subspace to all orders in epsilon. To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics
Stationary measures and hydrodynamics of zero range processes with several species of particles
No full textWe study general zero range processes with different types of particles on a ddimensional
lattice with periodic boundary conditions. A necessary and sufficient
condition on the jump rates for the existence of stationary product measures is
established. For translation invariant jump rates we prove the hydrodynamic limit
on the Euler scale using Yau’s relative entropy method. The limit equation is
a system of conservation laws, which are hyperbolic and have a globally convex
entropy. We analyze this system in terms of entropy variables. In addition we
obtain stationary density profiles for open boundaries
Hard rod hydrodynamics and the Lévy Chentsov field
No full textWe study the hydrodynamics of the hard rod model proposed by Boldrighini, Dobrushin and Soukhov by describing the displacement of each quasiparticle with respect to the corresponding ideal gas particle as a height difference in a related field. Starting with a family of nonhomo- geneous Poisson processes contained in the position-velocity-length space R3, we show laws of large numbers for the quasiparticle positions and the length fields, and the joint convergence of the quasiparticle fluctuations to a Lévy Chentsov field. We allow variable rod lengths, including negative lengths
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Large Scale Stochastic Dynamics
No full textIn focus are interacting stochastic systems with many components, ranging from stochastic partial differential equations to discrete systems as interacting particles on a lattice moving through random jumps.
More specifically one wants to understand the large scale behavior, large in spatial extent but also over long time spans, as entailed by the characterization of stationary measures, effective macroscopic evolution laws, transport of conserved fields, homogenization, self-similar structure and scaling, critical dynamics, aging, dynamical phase transitions, large deviations, to mention only a few key items
Recommended from our members
Large Scale Stochastic Dynamics
No full textIn focus are interacting stochastic systems with many components, ranging from stochastic partial differential equations to discrete systems as interacting particles on a lattice moving through random jumps. More specifically one wants to understand the large scale behavior, large in spatial extent but also over long time spans, as entailed by the characterization of stationary measures, effective macroscopic evolution laws, transport of conserved fields, homogenization, self-similar structure and scaling, critical dynamics, dynamical phase transitions, metastability, large deviations, to mention only a few key items
Large Scale Stochastic Dynamics
No full textEquilibrium statistical mechanics studies random fields distributed according to a Gibbs probability measure. Such random fields can be equipped with a stochastic dynamics given by a Markov process with the correspondingly high-dimensional state space. One particular case are stochastic partial differential equations suitably regularized. Another common version is to consider the evolution of random fields taking only values 0 or 1. The workshop was concerned with an understanding of qualitative properties of such high-dimensional Markov processes. Of particular interest are nonreversible dynamics for which the stationary measures are determined only through the dynamics and not given a priori (as would be the case for reversible dynamics). As a general observation, properties on a large scale do not depend on the precise details of the local updating rules. Such kind of universality was a guiding theme of our workshop
Recommended from our members
Large Scale Stochastic Dynamics
No full textEquilibrium statistical mechanics studies random fields distributed according to a Gibbs probability measure. Such random fields can be equipped with a stochastic dynamics given by a Markov process with the correspondingly high-dimensional state space. One particular case are stochastic partial differential equations suitably regularized. Another common version is to consider the evolution of random fields taking only values 0 or 1. The workshop was concerned with an understanding of qualitative properties of such high-dimensional Markov processes. Of particular interest are nonreversible dynamics for which the stationary measures are determined only through the dynamics and not given a priori (as would be the case for reversible dynamics). As a general observation, properties on a large scale do not depend on the precise details of the local updating rules. Such kind of universality was a guiding theme of our workshop
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