Stochastic inflation with interacting noises

This paper describes a unified method to quantify and compare amplitude and phase noise induced by white and colored noise sources in electrical oscillators. The proposed approach relies on the Floquet theory of linear time-varying systems and on the application of averaging techniques to simplify the stochastic equations. Closed-form expressions for the phase and amplitude power spectrum are provided. These results are validated via comparisons with SpectreRF simulations and measurements.

We propose a new approach for calculating the curvature perturbations produced during inflation in the stochastic formalism. In our formalism, the fluctuations of the e-foldings are directly calculated without perturbatively expanding the inflaton field and they are connected to the curvature perturbations by the δN formalism. The result automatically includes the contributions of the higher order perturbations because we solve the equation of motion non-perturbatively. In this paper, we analytically prove that our result (the power spectrum and the nonlinearity parameter) is consistent with the standard result in single field slow-roll inflation. We also describe the algorithm for numerical calculations of the curvature perturbations in more general inflation models.

Magnetically confined plasmas are far from equilibrium and pose considerable challenges in statistical analysis. We discuss a non-perturbative statistical method, namely a time-dependent probability density function (PDF) approach that is potentially useful for analysing time-varying, large, or non-Gaussian fluctuations and bursty events associated with instabilities in the low-to-high confinement transition and the H-mode. Specifically, we present a stochastic Langevin model of edge-localized modes (ELMs) by including stochastic noise terms in a previous ODE ELM model. We calculate exact time-dependent PDFs by numerically solving the Fokker-Planck equation and characterize time-varying statistical properties of ELMs for different energy fluxes and noise amplitudes. The stochastic noise is shown to introduce phase-mixing and plays a significant role in mitigating extreme bursts of large ELMs. Furthermore, based on time-dependent PDFs, we provide a path-dependent information geometric theory of the ELM dynamics and demonstrate its utility in capturing self-regulatory relaxation oscillations, bursts and a sudden change in the system. This article is part of a discussion meeting issue 'H-mode transition and pedestal studies in fusion plasmas'.

A key circuit in the response of cells to damage is the p53-mdm2 feedback loop. This circuit shows sustained, noisy oscillations in individual human cells following DNA breaks. Here, we apply an engineering approach known as systems identification to quantify the in vivo interactions in the circuit on the basis of accurate measurements of its power spectrum. We obtained oscillation time courses of p53 and Mdm2 protein levels from several hundred cells and analyzed their Fourier spectra. We find characteristic spectra with distinct low-frequency components that are well-described by a third-order linear model with white noise. The model identifies the sign and strength of the known interactions, including a negative feedback loop between p53 and its upstream regulator. It also implies that noise can trigger and maintain the oscillations. The model also captures the power spectra of p53 dynamics without DNA damage. Parameters such as noise amplitudes and protein lifetimes are estimated. This approach employs natural biological noise as a diagnostic that stimulates the system at many frequencies at once. It seems to be a useful way to find the in vivo design of circuits and may be applied to other systems by monitoring their power spectrum in individual cells.

The amplitude and envelope phase noise of a modelocked Ti:sapphire laser are predicted based on the power spectral density of the pump laser and the noise transfer process. Pump laser noise is found to transfer directly to the modelocked laser's amplitude and phase noise power spectra through the noise transfer function (NTF) which is independently measured. We find good agreement between the shapes and absolute values of the predicted and measured Ti:sapphire AM and PM noise spectra except in regions where additional environmental influences affect the Ti:sapphire laser. The experiments were conducted with both a single-mode and a multi-mode diode-pumped solid-state pump laser.

We extend the formalism of stochastic inflation to the setup of non-attractor inflation with a sound speed $c_s$. We obtain the Langevin equations for the superhorizon perturbations and calculate the stochastic corrections to curvature perturbation power spectrum. It is shown that the fractional stochastic corrections in mean number of e-folds and power spectrum are at the order of power spectrum. We also calculate the boundary crossing and the first hitting probabilities in a hypothetical dS space with two boundaries in field space. Furthermore, the stochastic corrections in power spectrum in a setup akin to eternal inflation with large diffusion term are calculated.

Background noise is ubiquitous and can impair acoustic communication and influence signaling behavior in animals. Despite evidence demonstrating myriad effects of anthropogenic noise on animal communication, precisely how natural background noise influences communication and behavior remains unclear. Yet, natural sources of background noise, such as rushing rivers or crashing ocean surf, share similar power spectra to sources of anthropogenic noise and can occur at high amplitudes, potentially masking acoustic signals. To investigate the effects of water-generated noise on songbird behavior, we experimentally broadcast landscape-level playbacks of ocean surf and river noise in coastal California, USA, and riparian habitat in Idaho, USA, respectively. In Chapter 1, we conducted a conspecific playback experiment examining how territorial defense behaviors of lazuli buntings (Passerina amoena) and spotted towhees (Pipilo maculatus) vary in response to broadcast water noise. We also incorporated cicada noise from a serendipitous Okanagana spp. emergence as a biotic source influencing lazuli bunting behavior. Both species produce songs that share substantial spectral overlap with low-frequency, water-generated noise, and lazuli bunting song shares an additional high-frequency overlap with cicada calls. Thus, there is potential for background acoustic conditions to mask conspecific signals. We found that detection and discrimination of conspecific playback occurred more slowly for both species as sound level increased. Lazuli buntings also exhibited divergent flight behaviors in response to high- and low-frequency acoustic sources, both dependent and independent of sound level. In Chapter 2, we investigated how amplitude and frequency of water-generated noise influences spectral and temporal song characteristics in six songbird species. We recorded individuals defending territories across 37 sites, with each site representing one of four acoustic environments: naturally quiet ‘controls’, naturally noisy ‘positive controls’ adjacent to the ocean or a whitewater river, ‘phantom’ playback sites with continuous broadcast of low-frequency water noise, and ‘shifted’ playback sites with continuous broadcast of high-frequency water noise. We predicted that all individuals exposed to ‘positive control’, ‘phantom’, or ‘shifted’ noise would adjust song structure, but the magnitude of signal modification would be larger in noisier locations and the type of modification would depend on the spectral profile of the acoustic environment. No two species altered songs in precisely the same way. However, song structure of all six species varied with amplitude and/or frequency of background noise. Together our results demonstrate that natural noise can impair agonistic behaviors and influence vocal structure. These findings suggest that the natural acoustic environment shapes acoustic communication,

This chapter mainly aims at describing dynamical effects like diffusion or relaxation, in the framework of Markovian stochastic processes. A simple description of the formalism is provided, together with a discussion of random walk processes, as well as Langevin and Fokker-Planck equations. Anomalous diffusion processes are also briefly described, as well as some generic properties of the relaxation of stochastic processes to equilibrium.

In the present study, the effect of ‘red’, ‘white’ and ‘blue’ environmental noise on population dynamics in a simple three‐species food chain system was analyzed. The ‘colored’ noise was superimposed on the three‐species food chain model first put forth by Rosenzweig in different ways and the resulting power spectra were investigated. We showed that the amplitude of environmental noise, the trophic level at which a population is positioned and whether a population is directly affected by environmental noise, are all important with respect to the way in which a population responds to noise with different colors. For the deterministic case, all population dynamics are ‘red’ irrespective of the system dynamics. When all species are sensitive to environmental noise, the top predator's dynamics always remain ‘red’ regardless of the color of the noise and its amplitude, whereas the dynamics of the intermediate species turn ‘blue’ under disturbances of any color with high amplitude, and those of the basal species may become ‘blue’ only under ‘blue’ noise. If only one species is sensitive to environmental noise, the dynamics of the insensitive species are always ‘red’, irrespective of the color of the noise and its amplitude. Unlike previous results obtained by studying single‐species models, our results have almost nothing to do with deterministic system dynamics. In other words, changing the deterministic system dynamics from stable via periodic to chaotic does not qualitatively change the outcome. Our results are of importance in determining how we interpret the ubiquitous ‘red’ power spectrum of natural ecological time series.

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order Ω(-3∕2) for reaction systems which do not obey detailed balance and at least accurate to order Ω(-2) for systems obeying detailed balance, where Ω is the characteristic size of the system. Hence, the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order Ω(-1∕2) and variance estimates accurate to order Ω(-3∕2). This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.

Three models for scalar diffusion in turbulent flow (eddy diffusivity, random displacement, and on the Langevin equation) are briefly described. These models random velocity increment based Fokker-Planck equation is introduced as are then examined in more detail in the reverse order. The Fokker-Planck equation is the Eulerian equivalent of the Lagrangian Langevin equation, and the derivation of e outlined. The procedure for obtaining the deterministic and stochastic components of the Langevin equation from Kolmogorov`s 1941 inertial range theory and the Fokker-Planck equation is described. it is noted that a unique form of the Langevin equation can be determined for diffusion in one dimension but not in two or three. The Langevin equation for vertical diffusion in the non-Gaussian convective boundary layer is presented and successively simplified for Gaussian inhomogeneous turbulence and Gaussian homogeneous turbulence in turn. The Langevin equation for Gaussian inhomogeneous turbulence is mathematically transformed into the random displacement model. It is shown how the Fokker-Planck equation for the random displacement model is identical in form to the partial differential equation for the eddy diffusivity model. It is noted that the Langevin model is applicable in two cases in which the other two are not valid: (1) very close in time and distance to the point of scalar release and (2) the non-Gaussian convective boundary layer. The two- and three-dimensional cases are considered in Part III.

We compute the power spectrum at one-loop order in standard perturbation theory for the matter density field to which a standard Lagrangian baryonic acoustic oscillation (BAO) reconstruction technique is applied. The BAO reconstruction method corrects the bulk motion associated with the gravitational evolution using the inverse Zel’dovich approximation (ZA) for the smoothed density field. We find that the overall amplitude of one-loop contributions in the matter power spectrum substantially decreases after reconstruction. The reconstructed power spectrum thereby approaches the initial linear spectrum when the smoothed density field is close enough to linear, i.e., the smoothing scale R s ≳ 10 h − 1 Mpc . On smaller R s , however, the deviation from the linear spectrum becomes significant on large scales ( k ≲ R − 1 s ) due to the nonlinearity in the smoothed density field, and the reconstruction is inaccurate. Compared with N-body simulations, we show that the reconstructed power spectrum at one-loop order agrees with simulations better than the unreconstructed power spectrum. We also calculate the tree-level bispectrum in standard perturbation theory to investigate non-Gaussianity in the reconstructed matter density field. We show that the amplitude of the bispectrum significantly decreases for small k after reconstruction and that the tree-level bispectrum agrees well with N-body results in the weakly nonlinear regime.

We study the non-linear evolution of baryon acoustic oscillations in the matter power spectrum and correlation function from the improved perturbation theory (PT). Based on the framework of renormalized PT, we apply the {\it closure approximation} that truncates the infinite series of loop contributions at one-loop order, and obtain a closed set of integral equations for power spectrum and non-linear propagator. The resultant integral expressions keep important non-perturbative properties which can dramatically improve the prediction of non-linear power spectrum. Employing the Born approximation, we then derive the analytic expressions for non-linear power spectrum and the predictions are made for non-linear evolution of baryon acoustic oscillations in power spectrum and correlation function. A detailed comparison between improved PT results and N-body simulations shows that a percent-level agreement is achieved in a certain range in power spectrum and in a rather wider range in correlation function. Combining a model of non-linear redshift-space distortion, we also evaluate the power spectrum and correlation function in correlation function. In contrast to the results in real space, the agreement between N-body simulations and improved PT predictions tends to be worse, and a more elaborate modeling for redshift-space distortion needs to be developed. Nevertheless, with currently existing model, we find that the prediction of correlation function has a sufficient accuracy compared with the cosmic-variance errors for future galaxy surveys with volume of a few (Gpc/h)^3 at z>=0.5.

In the paper, various noise sources in delay-line optoelectronic oscillators (OEOs) with all-optical gain are analyzed. It is shown that mutual conversion of amplitude and phase noises occurs in OEO due to its non-linear time-variant dynamics. The correlation between amplitude and phase fluctuations influences the noise power spectrum density of the OEO output signal. Methods for reducing the amplitude and phase noise mutual conversion in OEOs and consequential phase noise performance improving are proposed.

From a measurement of the noise power spectral density of the pump laser, we predict and verify the amplitude and phase noise of a modelocked laser using the noise transfer function method.