Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

Rossberg, Axel G.

doi:10.1239/jap/1214950365

Cited by 14 publications

(5 citation statements)

References 27 publications

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“…Prato et al 2006). Following Rossberg (2008), we rewrite the Laplace transform (Equation (5)) via the convolution integral, which will allow efficient numerical computations of nV F via ξ(T ) and vice versa. Using the change of variables s = exp(y) and t = exp(−x), let us rewrite Equation (5) in the following form: (4) can be similarly brought into the form of Equation (5) using the variable change…”

Section: Reconstruction Of Electron Distribution Function From Dem(t)mentioning

confidence: 99%

Electron Distribution Functions in Solar Flares From Combined X-Ray and Extreme-Ultraviolet Observations

Battaglia¹,

Kontar²

2013

ApJ

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Simultaneous solar flare observations with SDO and RHESSI provide spatially resolved information about hot plasma and energetic particles in flares. RHESSI allows the properties of both hot ( 8 MK) thermal plasma and nonthermal electron distributions to be inferred, while SDO/AIA is more sensitive to lower temperatures. We present and implement a new method to reconstruct electron distribution functions from SDO/AIA data. The combined analysis of RHESSI and AIA data allows the electron distribution function to be inferred over the broad energy range from 0.1 keV up to a few tens of keV. The analysis of two well observed flares suggests that the distributions in general agree to within a factor of three when the RHESSI values are extrapolated into the intermediate range 1-3 keV, with AIA systematically predicting lower electron distributions. Possible instrumental and numerical effects, as well as potential physical origins for this discrepancy are discussed. The inferred electron distribution functions in general show one or two nearly Maxwellian components at energies below ∼ 15 keV and a non-thermal tail above.

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Section: Reconstruction Of Electron Distribution Function From Dem(t)mentioning

confidence: 99%

Electron Distribution Functions in Solar Flares From Combined X-Ray and Extreme-Ultraviolet Observations

Battaglia¹,

Kontar²

2013

ApJ

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“…N (φ) for gamma, sectional and lognormal NPDFs is shown in Table 1. For the Lognormal distribution N (φ) is approximated using N (φ) ∝ 1/φ 0 n(ξ ) dξ (Rossberg, 2008) which is accurate to within 5 % for σ ϕ > 3, when compared against the direct numerical solution of Eq. (3) (not shown).…”

Section: General Theorymentioning

confidence: 99%

“…2.1, is the gamma function, and f a,k represents the fraction of the aerosol population with cumulative ice nucleation probability below 1 − exp(−ξ kφ ). The lognormal N (φ) is approximated using N (φ) ∝ 1/φ 0 n(ξ ) dξ (Rossberg, 2008).…”

Section: The Nature Of N(ξ ) In Deposition Ice Nucleationmentioning

confidence: 99%

On the ice nucleation spectrum

Barahona

2012

Atmos. Chem. Phys.

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Abstract. This work presents a novel formulation of the ice nucleation spectrum, i.e. the function relating the ice crystal concentration to cloud formation conditions and aerosol properties. The new formulation is physically-based and explicitly accounts for the dependency of the ice crystal concentration on temperature, supersaturation, cooling rate, and particle size, surface area and composition. This is achieved by introducing the concepts of ice nucleation coefficient (the number of ice germs present in a particle) and nucleation probability dispersion function (the distribution of ice nucleation coefficients within the aerosol population). The new formulation is used to generate ice nucleation parameterizations for the homogeneous freezing of cloud droplets and the heterogeneous deposition ice nucleation on dust and soot ice nuclei. For homogeneous freezing, it was found that by increasing the dispersion in the droplet volume distribution the fraction of supercooled droplets in the population increases. For heterogeneous ice nucleation the new formulation consistently describes singular and stochastic behavior within a single framework. Using a fundamentally stochastic approach, both cooling rate independence and constancy of the ice nucleation fraction over time, features typically associated with singular behavior, were reproduced. Analysis of the temporal dependency of the ice nucleation spectrum suggested that experimental methods that measure the ice nucleation fraction over few seconds would tend to underestimate the ice nuclei concentration. It is shown that inferring the aerosol heterogeneous ice nucleation properties from measurements of the onset supersaturation and temperature may carry significant error as the variability in ice nucleation properties within the aerosol population is not accounted for. This work provides a simple and rigorous ice nucleation framework where theoretical predictions, laboratory measurements and field campaign data can be reconciled, and that is suitable for application in atmospheric modeling studies.

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“…Efficient computation of the Laplace transform of other ubiquitous probability distributions of interevent statistics of renewal processes, such as the Weibull or Pareto distributions, is an open area of research, for example, see Ref. [33].…”

Section: Generalized Montroll-weiss Equation: Beyond Markovmentioning

confidence: 99%

Population density equations for stochastic processes with memory kernels

Lai

Kamps

2017

Phys. Rev. E

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We present a method for solving population density equations (PDEs)-a mean-field technique describing homogeneous populations of uncoupled neurons-where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation-a recent result from random network theory-describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky-and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process.

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Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

Cited by 14 publications

References 27 publications

Electron Distribution Functions in Solar Flares From Combined X-Ray and Extreme-Ultraviolet Observations

Electron Distribution Functions in Solar Flares From Combined X-Ray and Extreme-Ultraviolet Observations

On the ice nucleation spectrum

Population density equations for stochastic processes with memory kernels

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