Models of Passive and Reactive Tracer Motion: an Application of Ito Calculus

By means of Ito calculus it is possible to find, in a straight-forward way, the analytical solution to some equations related to the passive tracer transport problem in a velocity field that obeys the multidimensional Burgers equation and to a simple model of reactive tracer motion.Comment: revised version 7 pages, Latex, to appear as a letter to J. of Physics

Equation-Free Dynamic Renormalization: Self-Similarity in Multidimensional Particle System Dynamics

We present an equation-free dynamic renormalization approach to the computational study of coarse-grained, self-similar dynamic behavior in multidimensional particle systems. The approach is aimed at problems for which evolution equations for coarse-scale observables (e.g. particle density) are not explicitly available. Our illustrative example involves Brownian particles in a 2D Couette flow; marginal and conditional Inverse Cumulative Distribution Functions (ICDFs) constitute the macroscopic observables of the evolving particle distributions.Comment: 7 pages, 5 figure

Preheating after N-flation

We study preheating in N-flation, assuming the Mar\v{c}enko-Pastur mass distribution, equal energy initial conditions at the beginning of inflation and equal axion-matter couplings, where matter is taken to be a single, massless bosonic field. By numerical analysis we find that preheating via parametric resonance is suppressed, indicating that the old theory of perturbative preheating is applicable. While the tensor-to-scalar ratio, the non-Gaussianity parameters and the scalar spectral index computed for N-flation are similar to those in single field inflation (at least within an observationally viable parameter region), our results suggest that the physics of preheating can differ significantly from the single field case.Comment: 14 pages, 14 figures, references added, fixed typo

The quantum theory of measurement within dynamical reduction models

We analyze in mathematical detail, within the framework of the QMUPL model of spontaneous wave function collapse, the von Neumann measurement scheme for the measurement of a 1/2 spin particle. We prove that, according to the equation of the model: i) throughout the whole measurement process, the pointer of the measuring device is always perfectly well localized in space; ii) the probabilities for the possible outcomes are distributed in agreement with the Born probability rule; iii) at the end of the measurement the state of the microscopic system has collapsed to the eigenstate corresponding to the measured eigenvalue. This analysis shows rigorously how dynamical reduction models provide a consistent solution to the measurement problem of quantum mechanics.Comment: 24 pages, RevTeX. Minor changes mad

Cosmological Inflation and the Quantum Measurement Problem

According to cosmological inflation, the inhomogeneities in our universe are of quantum mechanical origin. This scenario is phenomenologically very appealing as it solves the puzzles of the standard hot big bang model and naturally explains why the spectrum of cosmological perturbations is almost scale invariant. It is also an ideal playground to discuss deep questions among which is the quantum measurement problem in a cosmological context. Although the large squeezing of the quantum state of the perturbations and the phenomenon of decoherence explain many aspects of the quantum to classical transition, it remains to understand how a specific outcome can be produced in the early universe, in the absence of any observer. The Continuous Spontaneous Localization (CSL) approach to quantum mechanics attempts to solve the quantum measurement question in a general context. In this framework, the wavefunction collapse is caused by adding new non linear and stochastic terms to the Schroedinger equation. In this paper, we apply this theory to inflation, which amounts to solving the CSL parametric oscillator case. We choose the wavefunction collapse to occur on an eigenstate of the Mukhanov-Sasaki variable and discuss the corresponding modified Schroedinger equation. Then, we compute the power spectrum of the perturbations and show that it acquires a universal shape with two branches, one which remains scale invariant and one with nS=4, a spectral index in obvious contradiction with the Cosmic Microwave Background (CMB) anisotropy observations. The requirement that the non-scale invariant part be outside the observational window puts stringent constraints on the parameter controlling the deviations from ordinary quantum mechanics... (Abridged).Comment: References added, minor corrections, conclusions unchange

Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging

We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it Multiscale}: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by slow variables (ii) {\it Versatile}: the method is based on averaging the flows of the given dynamical system (which may have hidden slow and fast processes) instead of averaging the instantaneous drift of assumed separated slow and fast processes. This bypasses the need for identifying explicitly (or numerically) the slow or fast variables (iii) {\it Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale (iv) {\it Convergent over two scales}: strongly over slow processes and in the sense of measures over fast ones. We introduce the related notion of two-scale flow convergence and analyze the convergence of these integrators under the induced topology (v) {\it Structure preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be made to be symplectic, time-reversible, and symmetry preserving (symmetries are group actions that leave the system invariant) in all variables. They are explicit and applicable to arbitrary stiff potentials (that need not be quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy and stability over four orders of magnitude of time scales. For stiff Langevin equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs reversible, quasi-symplectic on all variables and conformally symplectic with isotropic friction.Comment: 69 pages, 21 figure

Differentiable potentials and metallic states in disordered one-dimensional systems

We provide evidence that as a general rule Anderson localization effects become weaker as the degree of differentiability of the disordered potential increases. In one dimension a band of metallic states exists provided that the disordered potential is sufficiently correlated and has some minimum degree of differentiability. Several examples are studied in detail. In agreement with the one parameter scaling theory the motion in the metallic region is ballistic if the spectral density is smooth. Finally, we study the most promising settings to observe these results in the context of cold atoms.Comment: 5 pages, 3 figures, journal versio

STOCHASTIC DYNAMICS OF LARGE-SCALE INFLATION IN DE~SITTER SPACE

In this paper we derive exact quantum Langevin equations for stochastic dynamics of large-scale inflation in de~Sitter space. These quantum Langevin equations are the equivalent of the Wigner equation and are described by a system of stochastic differential equations. We present a formula for the calculation of the expectation value of a quantum operator whose Weyl symbol is a function of the large-scale inflation scalar field and its time derivative. The unique solution is obtained for the Cauchy problem for the Wigner equation for large-scale inflation. The stationary solution for the Wigner equation is found for an arbitrary potential. It is shown that the large-scale inflation scalar field in de Sitter space behaves as a quantum one-dimensional dissipative system, which supports the earlier results. But the analogy with a one-dimensional model of the quantum linearly damped anharmonic oscillator is not complete: the difference arises from the new time dependent commutation relation for the large-scale field and its time derivative. It is found that, for the large-scale inflation scalar field the large time asymptotics is equal to the `classical limit'. For the large time limit the quantum Langevin equations are just the classical stochastic Langevin equations (only the stationary state is defined by the quantum field theory).Comment: 21 pages RevTex preprint styl

Stochastic Gravity

Gravity is treated as a stochastic phenomenon based on fluctuations of the metric tensor of general relativity. By using a (3+1) slicing of spacetime, a Langevin equation for the dynamical conjugate momentum and a Fokker-Planck equation for its probability distribution are derived. The Raychaudhuri equation for a congruence of timelike or null geodesics leads to a stochastic differential equation for the expansion parameter $\theta$ in terms of the proper time $s$. For sufficiently strong metric fluctuations, it is shown that caustic singularities in spacetime can be avoided for converging geodesics. The formalism is applied to the gravitational collapse of a star and the Friedmann-Robertson-Walker cosmological model. It is found that owing to the stochastic behavior of the geometry, the singularity in gravitational collapse and the big-bang have a zero probability of occurring. Moreover, as a star collapses the probability of a distant observer seeing an infinite red shift at the Schwarzschild radius of the star is zero. Therefore, there is a vanishing probability of a Schwarzschild black hole event horizon forming during gravitational collapse.Comment: Revised version. Eq. (108) has been modified. Additional comments have been added to text. Revtex 39 page