Stochastic Source Term Research Articles

Nonlinear perturbations in multiple-field inflation

We develop a nonlinear framework for describing long-wavelength perturbations in multiple-field inflation. The basic variables describing inhomogeneities are defined in a nonperturbative manner, are invariant under changes of time slicing on large scales and include both matter and metric perturbations. They are combinations of spatial gradients generalizing the gauge-invariant variables of linear theory. Dynamical equations are derived and supplemented with stochastic source terms which provide the long-wavelength initial conditions determined from short-wavelength modes. Solutions can be readily obtained via numerical simulations or analytic perturbative expansions. The latter are much simpler than the usual second-order perturbation theory. Applications are given in a companion paper.

Computing reactive flows with a field Monte Carlo formulation and multi-scale methods

Two novel techniques are merged together to compute efficiently flows involving chemical reaction or combustion. In order to predict the evolution of the reacting flow, a probability density function approach is applied to the chemical species, which is described by a set of pure Eulerian transport equations with stochastic source terms. The proposed model, which has been extended in this case to a multi-scalar situation, has the advantage that the chemistry terms are closed, requiring no modeling. The formulation is solved by a stabilized finite element method based on the variational multi-scale theory, which enables an accurate and stable treatment of the difficult and stiff source terms employing iterative techniques. The combination of these ingredients results in a robust and accurate tool for the computation of reactive flows, as will be shown in the numerical examples.

An accurate and efficient high frequency noise simulation technique for deep submicron MOSFETs

Based on an active transmission line concept and two-dimensional (2-D) device simulations, an accurate and computationally efficient simulation technique for high frequency noise performance of MOSFETs is demonstrated. Using a Langevin stochastic source term model and small-signal equivalent circuit of the MOSFET, three intrinsic noise parameters (/spl gamma/, /spl delta/, and c) for the drain noise and induced gate noise are calculated. Validity and error analysis for the simulation are discussed by comparing the simulation results with theoretical results as well as measured data.

Stochastic dynamics of correlations in quantum field theory: From the Schwinger-Dyson to Boltzmann-Langevin equation

The aim of this paper is twofold: to probe the statistical mechanical properties of interacting quantum fields, and to provide a field theoretical justification for a stochastic source term in the Boltzmann equation. We start with the formulation of quantum field theory in terms of the set of Schwinger-Dyson equations for the correlation functions, which we describe by a closed-time-path master $(n=\ensuremath{\infty}\mathrm{PI})$ effective action. When the hierarchy is simply truncated to a certain order, one obtains the usual closed system of correlation functions up to that order, and from the $n\mathrm{PI}$ effective action, a set of time-reversal invariant equations of motion. (This is the Dyson equation, the quantum field theoretical parallel of the collisionless Boltzmann equation.) But when the effect of the higher order correlation functions is included through a causal factorization condition (such as the molecular chaos assumption in Boltzmann's theory) called slaving, the dynamics of the lower order correlations shows dissipative features, as familiar in the usual (dissipative yet noiseless) Boltzmann equation, the field-theoretical version of which being the dissipative Dyson equations. We show that a fluctuation-dissipation relation should exist for such effectively open systems, and use this fact to show that a stochastic term, which explicitly introduces quantum fluctuations in the lower order correlation functions, necessarily accompanies the dissipative term. This leads to a stochastic Dyson equation, which is the quantum field theoretic parallel of the classical Boltzmann-Langevin equation, encompassing both the dissipative and stochastic dynamics of correlation functions.

Stochastic Gravity

We give a summary of the status of current research in stochastic semiclassical gravity and suggest directions for further investigations. This theory generalizes the semiclassical Einstein equation to an Einstein-Langevin equation with a stochastic source term arising from the fluctuations of the energy-momentum tensor of quantum fields. We mention recent efforts in applying this theory to the study of black hole fluctuations and backreaction problems, linear response of hot flat space, and structure formation in inflationary cosmology. To explore the physical meaning and implications of this stochastic regime in relation to both classical and quantum gravity, we find it useful to take the view that semiclassical gravity is mesoscopic physics and that general relativity is the hydrodynamic limit of certain spacetime quantum substructures. Three basic issues - stochasticity, collectivity, correlations- and three processes - dissipation, fluctuations, decoherence- underscore the transformation from quantum micro structure and interaction to the emergence of classical macro structure and dynamics. We discuss ways to probe into the high energy activity from below and make two suggestions: via effective field theory and the correlation hierarchy. We discuss how stochastic behavior at low energy in an effective theory and how correlation noise associated with coarse-grained higher correlation functions in an interacting quantum field could carry nontrivial information about the high energy sector. Finally we describe processes deemed important at the Planck scale, including tunneling and pair creation, wave scattering in random geometry, growth of fluctuations and forms, Planck scale resonance states, and spacetime foams.

Nonequilibrium dynamics of a thermal plasma in a gravitational field

We introduce functional methods to study the non-equilibrium dynamics of a quantum massless scalar field at finite temperature in a gravitational field. We calculate the close time path (CTP) effective action and, using its formal equivalence with the influence functional, derive the noise and dissipation kernels of the quantum open system in terms of quantities in thermodynamical equilibrium. Using this fact, we formally prove the existence of a fluctuation-dissipation relation (FDR) at all temperatures between the quantum fluctuations of the plasma in thermal equilibrium and the energy dissipated by the external gravitational field. What is new is the identification of a stochastic source (noise) term arising from the quantum and thermal fluctuations in the plasma field, and the derivation of a Langevin-type equation which describes the non-equilibrium dynamics of the gravitational field influenced by the plasma. The back reaction of the plasma on the gravitational field is embodied in the FDR. From the CTP effective action the contribution of the quantum scalar field to the thermal graviton polarization tensor can also be derived and it is shown to agree with other techniques, most notably, linear response theory (LRT). We show the connection between the LRT, which is applicable for near-equilibrium conditions and the functional methods used in this work which are useful for fully nonequilibrium conditions.

A Field Monte Carlo Formulation for Calculating the Probability Density Function of a Single Scalar in a Turbulent Flow

The probability density function (PDF) formulation of one scalar field undergoing diffusion, turbulent convection and chemical reaction is restated in terms of stochastic fields. These fields are smooth in space as they have a length scale similar to that of the PDF. Their evolution is described by a set of stochastic partial differential equations, which are solved using a finite volume scheme with a stochastic source term. The application of this methodology to a particular flow is shown first for a linear source term, with exact analytical solution for the mean and standard deviation, and then for a nonlinear reaction.

Interaction between external and internal noise sources

Harmonic noise (also called quasimonochromatic noise) is used as an external stochastic source term in a non-Markovian oscillatory system with memory-damping. The stationary second moments are obtained by using the adjoint Fokker-Planck operator method. If the internal noise is harmonic then the occurrence of two maxima in the mean energy of the driven system seems to be obvious and depends on the selected relations of the frequencies of the three oscillators. The effects of the frequency-dependent strength of the external harmonic noise on the resulting energy and some generic cases like the interactions between internal harmonic noise and external white noise or exponentially correlated noise have been investigated.

Uncertainties in predicted radionuclide body burdens and doses from discrete stochastic source terms.

Expressions are derived in three cases for the expectation and uncertainty of body burdens and doses calculated from a linear model of environmental transport and human metabolism in terms of expectation and uncertainty in inputs of discrete, stochastic random variables. Three cases are compared to determine the relationship of the expectations and uncertainties under varying assumptions. In the constant input case, the input is selected randomly at the outset of the simulation period [O,T] from the distribution to which the population is exposed and then is held constant throughout [O,T]. In the two time-varying cases, random and autoregressive, it was assumed that N discrete stochastic exposures to the input were made uncorrelated and partially correlated, respectively. Each exposure was constant during each time interval of length T/N. The expectation values of the body burdens and doses in the constant input case were identical to those in the random case and the autoregressive cases for stationary inputs. The uncertainties of the body burdens and the doses in the constant input case were identical in the limit of rapid metabolism to those of the random case and the autoregressive cases for stationary inputs. In the limit of slow metabolism, the uncertainties of the body burdens and the doses in the constant input case were N1/2 and (3N/4)1/2, respectively, greater than those in the random case and were ((1 + alpha)/[(1 - alpha)N])1/2 and (4(1 + alpha)/[3(1 - alpha)N])1/2, respectively, greater than those in the autoregressive case for stationary inputs and autocorrelation coefficient alpha. That is, increasing the number of sampling periods decreases the uncertainty and increasing the autocorrelation increases the uncertainty. In an example application for ingestion of 137Cs at Bikini Island, a weak form of both slow and rapid metabolism limits apply and give the result that the uncertainty of the body burden in the constant input case is 18 times greater than the random case. For alpha = 0.5, the uncertainty of the body burden in the autoregressive case is 1.7 times greater than the random case. The smooth transition of the autoregressive case from the random case to the constant input case is shown as alpha increases from 0 (random) to 1 (constant).

Segmentation and Filtering into Neighborhoods as Processes of Percolation and Diffusion: Stochastic Processes (Randomness) as the Null Hypothesis

A set of models is reviewed in which orderly urban phenomena are accounted for in terms of noise (randomness, stochastic processes): segmentation as a process of runs in random processes (percolation); filtering as a process of random walk and so diffusion, and as a Markov process. In particular, ‘continuity’ of the spatial price envelope, a truism of real estate, leads to the diffusion equation. No foreordained centers, boundaries, or gravity (or entropy) is needed to obtain clustering in these models. The models are combined in a simulation, implemented in a Lotus 1-2-3 spreadsheet, as well as in a partial differential equation driven by a stochastic source term.

Harmonic noise: Effect on bistable systems

The coordinate of a white noise driven harmonic oscillator is used as a stochastic source term in bistable dynamics. This new kind of Gaussian colored noise gives rise to resonance phenomena due to a peak in the spectrum. We investigate its effect on linear and bistable systems. We derive a Markovian approximation for driven bistable oscillators and overdamped systems. In the resonance region computer simulations were carried out using an extension of Fox' algorithm procedure for colored noise. We find an increase of the transition rates in bistable systems as compared with the case of bistable systems driven by white and exponentially correlated noise.

Stochastic quantization of Einstein gravity.

We determine a one-parameter family of covariant Langevin equations for the metric tensor of general relativity corresponding to DeWitt's one-parameter family of supermetrics. The stochastic source term in these equations can be expressed in terms of a Gaussian white noise upon the introduction of a stochastic tetrad field. The only physically acceptable resolution of a mathematical ambiguity in the ansatz for the source term is the adoption of Ito's calculus. By taking the formal equilibrium limit of the stochastic metric a one-parameter family of covariant path-integral measures for general relativity is obtained. There is a unique parameter value, distinguished by any one of the following three properties: (i) the metric is harmonic with respect to the supermetric, (ii) the path-integral measure is that of DeWitt, (iii) the supermetric governs the linearized Einstein dynamics. Moreover the Feynman propagator corresponding to this parameter is causal. Finally we show that a consistent stochastic perturbation theory gives rise to a new type of diagram containing ``stochastic vertices.''

A comment on chemical Langevin equations

It is noted that the diffusion Langevin stochastic sources in chemical reaction-diffusion theories should really arise from a stochastic source term added to the deterministic form of Fick's law. This gives rise to results for correlation functions which agree with those from stochastic master equations provided parameters are appropriately chosen.