scholarly journals Rarefied particle motions on hillslopes – Part 1: Theory

2021 ◽  
Vol 9 (3) ◽  
pp. 539-576
Author(s):  
David Jon Furbish ◽  
Joshua J. Roering ◽  
Tyler H. Doane ◽  
Danica L. Roth ◽  
Sarah G. W. Williams ◽  
...  
Keyword(s):  
Kinetic Energy ◽  
Particle Surface ◽  
Generalized Pareto Distribution ◽  
Planck Equation ◽  
Particle Energy ◽  
Finite Variance ◽  
Isothermal Conditions ◽  
Scale Parameters ◽  
Size Sorting ◽  
Heavy Tailed

Abstract. We describe the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces. The particle energy balance involves gravitational heating with conversion of potential to kinetic energy, frictional cooling associated with particle–surface collisions, and an apparent heating associated with preferential deposition of low-energy particles. Deposition probabilistically occurs with frictional cooling in relation to the distribution of particle energy states whose spatial evolution is described by a Fokker–Planck equation. The Kirkby number Ki – defined as the ratio of gravitational heating to frictional cooling – sets the basic deposition behavior and the form of the probability distribution fr(r) of particle travel distances r, a generalized Pareto distribution. The shape and scale parameters of the distribution are well-defined mechanically. For isothermal conditions where frictional cooling matches gravitational heating plus the apparent heating due to deposition, the distribution fr(r) is exponential. With non-isothermal conditions and small Ki this distribution is bounded and represents rapid thermal collapse. With increasing Ki the distribution fr(r) becomes heavy-tailed and represents net particle heating. It may possess a finite mean and finite variance, or the mean and variance may be undefined with sufficiently large Ki. The formulation provides key elements of the entrainment forms of the particle flux and the Exner equation, and it clarifies the mechanisms of particle-size sorting on large talus and scree slopes. Namely, with conversion of translational to rotational kinetic energy, large spinning particles are less likely to be stopped by collisional friction than are small or angular particles for the same surface roughness.

scholarly journals Rarefied particle motions on hillslopes: 1. Theory

2020 ◽  
Author(s):  
David Jon Furbish ◽  
Joshua J. Roering ◽  
Tyler H. Doane ◽  
Danica L. Roth ◽  
Sarah G. W. Williams ◽  
...  
Keyword(s):  
Kinetic Energy ◽  
Particle Surface ◽  
Generalized Pareto Distribution ◽  
Planck Equation ◽  
Particle Energy ◽  
Finite Variance ◽  
Isothermal Conditions ◽  
Scale Parameters ◽  
Size Sorting ◽  
Heavy Tailed

Abstract. We describe the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces. The particle energy balance involves gravitational heating with conversion of potential to kinetic energy, frictional cooling associated with particle-surface collisions, and an apparent heating associated with preferential deposition of low energy particles. Deposition probabilistically occurs with frictional cooling in relation to the distribution of particle energy states whose spatial evolution is described by a Fokker-Planck equation. The Kirkby number Ki – defined as the ratio of gravitational heating to frictional cooling – sets the basic deposition behavior and the form of the probability distribution fr(r) of particle travel distances r, a generalized Pareto distribution. The shape and scale parameters of the distribution are well-defined mechanically. For isothermal conditions where frictional cooling matches gravitational heating plus the apparent heating due to deposition, the distribution fr(r) is exponential. With non-isothermal conditions and small Ki this distribution is bounded and represents rapid thermal collapse. With increasing Ki the distribution fr(r) becomes heavy-tailed and represents net particle heating. It may possess a finite mean and finite variance, or the mean and variance may be undefined with sufficiently large Ki. The formulation provides key elements of the entrainment forms of the particle flux and the Exner equation, and it clarifies the mechanisms of particle-size sorting on large talus and scree slopes. Namely, with conversion of translational to rotational kinetic energy, large spinning particles are less likely to be stopped by collisional friction than are small or angular particles for the same surface roughness.

scholarly journals Rarefied particle motions on hillslopes: 3. Entropy

2020 ◽  
Author(s):  
David Jon Furbish ◽  
Sarah G. W. Williams ◽  
Tyler H. Doane
Keyword(s):  
Kinetic Energy ◽  
Maximum Entropy ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Individual Particle ◽  
Energetic Cost ◽  
Isothermal Conditions ◽  
Generalized Pareto ◽  
Heavy Tailed ◽  
Travel Distances

Abstract. Theoretical and experimental work (Furbish et al., 2020a, 2020b) indicates that the travel distances of rarefied particle motions on rough hillslope surfaces are described by a generalized Pareto distribution. The form of this distribution varies with the balance between gravitational heating, due to conversion of potential to kinetic energy, and frictional cooling, due to particle-surface collisions; it varies from a bounded form associated with rapid thermal collapse to an exponential form representing isothermal conditions to a heavy-tailed form associated with net heating of particles. The generalized Pareto distribution in this problem is a maximum entropy distribution constrained by a fixed energetic cost – the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. That is, among all possible accessible microstates – the many different ways to arrange a great number of particles into distance states where each arrangement satisfies the same fixed total energetic cost – the generalized Pareto distribution represents the most probable arrangement. Because this idea applies equally to the accessible microstates associated with net cooling, isothermal conditions and net heating, the fixed energetic cost provides a unifying interpretation for these distinctive behaviors, including the abrupt transition in the form of the generalized Pareto distribution in crossing isothermal conditions. The analysis therefore represents a novel generalization of an energy-based constraint in using the maximum entropy method to infer non-exponential distributions of particle motions. Moreover, the energetic costs of individual particle motions follow an extreme-value distribution that is heavy-tailed for net cooling and light-tailed for net heating. The relative contribution of different travel distances to the total energetic cost is reflected by the product of the travel distance distribution and the cost of individual particle motions – effectively a frequency-magnitude product.

scholarly journals Rarefied particle motions on hillslopes – Part 3: Entropy

2021 ◽  
Vol 9 (3) ◽  
pp. 615-628
Author(s):  
David Jon Furbish ◽  
Sarah G. W. Williams ◽  
Tyler H. Doane
Keyword(s):  
Kinetic Energy ◽  
Maximum Entropy ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Individual Particle ◽  
Energetic Cost ◽  
Isothermal Conditions ◽  
Generalized Pareto ◽  
Heavy Tailed ◽  
Travel Distances

Abstract. Theoretical and experimental work (Furbish et al., 2021a, b) indicates that the travel distances of rarefied particle motions on rough hillslope surfaces are described by a generalized Pareto distribution. The form of this distribution varies with the balance between gravitational heating, due to conversion of potential to kinetic energy, and frictional cooling, due to particle–surface collisions; it varies from a bounded form associated with rapid thermal collapse to an exponential form representing isothermal conditions to a heavy-tailed form associated with net heating of particles. The generalized Pareto distribution in this problem is a maximum entropy distribution constrained by a fixed energetic “cost” – the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. That is, among all possible accessible microstates – the many different ways to arrange a great number of particles into distance states where each arrangement satisfies the same fixed total energetic cost – the generalized Pareto distribution represents the most probable arrangement. Because this idea applies equally to the accessible microstates associated with net cooling, isothermal conditions and net heating, the fixed energetic cost provides a unifying interpretation for these distinctive behaviors, including the abrupt transition in the form of the generalized Pareto distribution in crossing isothermal conditions. The analysis therefore represents a novel generalization of an energy-based constraint in using the maximum entropy method to infer non-exponential distributions of particle motions. Moreover, the energetic costs of individual particle motions follow an extreme-value distribution that is heavy-tailed for net cooling and light-tailed for net heating. The relative contribution of different travel distances to the total energetic cost is reflected by the product of the travel distance distribution and the cost of individual particle motions – effectively a frequency–magnitude product.

scholarly journals Rarefied particle motions on hillslopes: 4. Philosophy

2020 ◽  
Author(s):  
David Jon Furbish ◽  
Tyler H. Doane
Keyword(s):  
Kinetic Energy ◽  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Particle Energy ◽  
Controlling Factors ◽  
Rate Of Change ◽  
Energetic Cost ◽  
Generalized Pareto

Abstract. Theoretical and experimental work (Furbish et al., 2020a, 2020b, 2020c) indicates that the travel distances of rarefied particle motions on rough hillslope surfaces are described by a generalized Pareto distribution. The form of this distribution varies with the balance between gravitational heating due to conversion of potential to kinetic energy and frictional cooling by particle-surface collisions. The generalized Pareto distribution in this problem is a maximum entropy distribution constrained by a fixed energetic cost – the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. The analyses leading to these results provide an ideal case study for highlighting three key elements of a statistical mechanics framework for describing sediment particle motions and transport: the merits of probabilistic versus deterministic descriptions of sediment motions; the implications of rarefied versus continuum transport conditions; and the consequences of increasing uncertainty in descriptions of sediment motions and transport that accompany increasing length and time scales. We use the analyses of particle energy extraction, the spatial evolution of particle energy states, and the maximum entropy method applied to the generalized Pareto distribution as examples to illustrate the mechanistic yet probabilistic nature of the approach. These examples highlight the idea that the endeavor is not simply about adopting theory or methods of statistical mechanics off the shelf, but rather involves appealing to the style of thinking of statistical mechanics while tailoring the analysis to the process and scale of interest. Under rarefied conditions, descriptions of the particle flux and its divergence pertain to ensemble conditions involving a distribution of possible outcomes, each realization being compatible with the controlling factors. When these factors change over time, individual outcomes reflect a legacy of earlier conditions that depends on the rate of change in the controlling factors relative to the intermittency of particle motions. The implication is that landform configurations and associated particle fluxes reflect an inherent variability (weather) that is just as important as the expected (climate) conditions in characterizing system behavior.

scholarly journals Rarefied particle motions on hillslopes – Part 4: Philosophy

2021 ◽  
Vol 9 (3) ◽  
pp. 629-664
Author(s):  
David Jon Furbish ◽  
Tyler H. Doane
Keyword(s):  
Kinetic Energy ◽  
Statistical Mechanics ◽  
Maximum Entropy ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Particle Energy ◽  
Controlling Factors ◽  
Rate Of Change ◽  
Energetic Cost ◽  
Generalized Pareto

Abstract. Theoretical and experimental work (Furbish et al., 2021a, b, c) indicates that the travel distances of rarefied particle motions on rough hillslope surfaces are described by a generalized Pareto distribution. The form of this distribution varies with the balance between gravitational heating due to conversion of potential to kinetic energy and frictional cooling by particle–surface collisions. The generalized Pareto distribution in this problem is a maximum entropy distribution constrained by a fixed energetic “cost” – the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. The analyses leading to these results provide an ideal case study for highlighting three key elements of a statistical mechanics framework for describing sediment particle motions and transport: the merits of probabilistic versus deterministic descriptions of sediment motions, the implications of rarefied versus continuum transport conditions, and the consequences of increasing uncertainty in descriptions of sediment motions and transport that accompany increasing length scales and timescales. We use the analyses of particle energy extraction, the spatial evolution of particle energy states, and the maximum entropy method applied to the generalized Pareto distribution as examples to illustrate the mechanistic yet probabilistic nature of the approach. These examples highlight the idea that the endeavor is not simply about adopting theory or methods of statistical mechanics “off the shelf” but rather involves appealing to the style of thinking of statistical mechanics while tailoring the analysis to the process and scale of interest. Under rarefied conditions, descriptions of the particle flux and its divergence pertain to ensemble conditions involving a distribution of possible outcomes, each realization being compatible with the controlling factors. When these factors change over time, individual outcomes reflect a legacy of earlier conditions that depends on the rate of change in the controlling factors relative to the intermittency of particle motions. The implication is that landform configurations and associated particle fluxes reflect an inherent variability (“weather”) that is just as important as the expected (“climate”) conditions in characterizing system behavior.

scholarly journals Rarefied particle motions on hillslopes: 2. Analysis

2020 ◽  
Author(s):  
David Jon Furbish ◽  
Sarah G. W. Williams ◽  
Danica L. Roth ◽  
Tyler H. Doane ◽  
Joshua J. Roering
Keyword(s):  
Surface Roughness ◽  
Particle Size ◽  
Kinetic Energy ◽  
Particle Surface ◽  
Size And Shape ◽  
Granular Gas ◽  
Heavy Tailed Distribution ◽  
Heavy Tailed ◽  
Particle Size And Shape ◽  
Travel Distances

Abstract. We examine a theoretical formulation of the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces using measurements of particle travel distances obtained from laboratory and field-based experiments, supplemented with high-speed imaging and audio recordings that highlight effects of particle-surface collisions. The formulation, presented in a companion paper (Furbish et al., 2020a), is based on a description of the kinetic energy balance of a cohort of particles treated as a rarefied granular gas, and a description of particle deposition that depends on the energy state of the particles. Both laboratory and field-based measurements are consistent with a generalized Pareto distribution of travel distances and predicted variations in behavior associated with the balance between gravitational heating due to conversion of potential to kinetic energy and frictional cooling due to particle-surface collisions. For a given particle size and shape these behaviors vary from a bounded distribution representing rapid thermal collapse with small slopes or large surface roughness, to an exponential distribution representing approximately isothermal conditions, to a heavy-tailed distribution representing net heating of particles with large slopes. The transition to a heavy-tailed distribution likely involves an increasing conversion of translational to rotational kinetic energy leading to larger travel distances with decreasing effectiveness of collisional friction. This energy conversion is strongly influenced by particle shape, although the analysis points to the need for further clarity concerning how particle size and shape in concert with surface roughness influence the extraction of particle energy and the likelihood of deposition.

scholarly journals Rarefied particle motions on hillslopes – Part 2: Analysis

2021 ◽  
Vol 9 (3) ◽  
pp. 577-613
Author(s):  
David Jon Furbish ◽  
Sarah G. W. Williams ◽  
Danica L. Roth ◽  
Tyler H. Doane ◽  
Joshua J. Roering
Keyword(s):  
Surface Roughness ◽  
Particle Size ◽  
Kinetic Energy ◽  
Particle Surface ◽  
Size And Shape ◽  
Granular Gas ◽  
Heavy Tailed Distribution ◽  
Heavy Tailed ◽  
Particle Size And Shape ◽  
Travel Distances

Abstract. We examine a theoretical formulation of the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces using measurements of particle travel distances obtained from laboratory and field-based experiments, supplemented with high-speed imaging and audio recordings that highlight effects of particle–surface collisions. The formulation, presented in a companion paper (Furbish et al., 2021a), is based on a description of the kinetic energy balance of a cohort of particles treated as a rarefied granular gas, as well as a description of particle deposition that depends on the energy state of the particles. Both laboratory and field-based measurements are consistent with a generalized Pareto distribution of travel distances and predicted variations in behavior associated with the balance between gravitational heating due to conversion of potential to kinetic energy and frictional cooling due to particle–surface collisions. For a given particle size and shape these behaviors vary from a bounded distribution representing rapid thermal collapse with small slopes or large surface roughness, to an exponential distribution representing approximately isothermal conditions, to a heavy-tailed distribution representing net heating of particles with large slopes. The transition to a heavy-tailed distribution likely involves an increasing conversion of translational to rotational kinetic energy leading to larger travel distances with decreasing effectiveness of collisional friction. This energy conversion is strongly influenced by particle shape, although the analysis points to the need for further clarity concerning how particle size and shape in concert with surface roughness influence the extraction of particle energy and the likelihood of deposition.

scholarly journals Sample correlations of infinite variance time series models: an empirical and theoretical study

1998 ◽  
Vol 11 (3) ◽  
pp. 255-282 ◽  
Author(s):  
Jason Cohen ◽  
Sidney Resnick ◽  
Gennady Samorodnitsky
Keyword(s):  
Time Series ◽  
Correlation Function ◽  
Random Permutation ◽  
Infinite Variance ◽  
Finite Variance ◽  
More Care ◽  
Sample Correlation ◽  
Heavy Tailed ◽  
Random Limit ◽  
Permutation Scheme

When the elements of a stationary ergodic time series have finite variance the sample correlation function converges (with probability 1) to the theoretical correlation function. What happens in the case where the variance is infinite? In certain cases, the sample correlation function converges in probability to a constant, but not always. If within a class of heavy tailed time series the sample correlation functions do not converge to a constant, then more care must be taken in making inferences and in model selection on the basis of sample autocorrelations. We experimented with simulating various heavy tailed stationary sequences in an attempt to understand what causes the sample correlation function to converge or not to converge to a constant. In two new cases, namely the sum of two independent moving averages and a random permutation scheme, we are able to provide theoretical explanations for a random limit of the sample autocorrelation function as the sample grows.

Penggunaan Taburan Pareto Umum dalam Menganalisis Nilai Ekstrim Banjir Menggunakan Siri Aliran Puncak Melebihi Paras

2012 ◽  
Author(s):  
Ani Shabri
Keyword(s):  
Poisson Process ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Flood Frequency ◽  
Flood Frequency Analysis ◽  
Annual Maximum ◽  
Peaks Over Threshold ◽  
Heavy Tailed Distributions ◽  
Probability Weighted Moment ◽  
Heavy Tailed

Siri banjir tahunan maksimum (Annual Maximum, AM) merupakan pendekatan yang begitu terkenal dalam analisis frekuensi banjir. Siri puncak melebihi paras (peaks over threshold, POT) telah digunakan sebagai alternatif kepada siri banjir tahunan maksimum. Masalah utama dalam pendekatan POT adalah berkaitan pemilihan paras yang sesuai. Dalam kajian ini, kesan perubahaan paras bagi siri POT ke atas nilai anggaran dikaji. Model POT dengan andaian bahawa bilangan puncak melebihi paras bertabur secara Poisson dan magnitud puncak melebihi paras tertabur secara Pareto Umum (General Pareto Distribution, GPD) dibincangkan. Parameter taburan GPD dianggar menggunakan kaedah kebarangkalian pemberat momen (Probability Weighted Moment, PWM) untuk paras yang diketahui. Perbandingan kesesuaian model POT dan model AM dalam menganggarkan nilai hujung atas taburan dibuat. Hasil kajian menunjukkan bahawa apabila paras siri POT boleh disuaikan oleh taburan Pareto dengan proses Poisson, model POT didapati dapat menghasilkan anggaran nilai hujung atas taburan lebih baik berbanding model aliran maksimum. Kata kunci: Siri puncak melebihi paras, proses poisson, taburan pareto umum, GEV, hujung atas taburan Annual maximum flood series remains the most popular approach to flood frequency analysis. Peaks over threshold series have been used as an alternative to annual maximum series. One specific difficulty of the POT approach is the selection of the threshold level. In this study the effect of raising the threshold of the POT series on heavy-tailed distributions estimation is investigated. The POT model described by the generalized Pareto distribution for peak magnitudes with the Poisson process for the occurrence of peaks is discussed. Estimation of the GPD parameters by the method of probability weighted moment (PWM) is formulated for known thresholds. A comparison of the efficiencies of the POT and AM models in heavy-tailed distributions is made. The result showed that when the threshold of POT series can be fitted by GPD with the Poisson process, the POT model is more efficient than the annual maximum (AM) model in estimating the highest extreme value. Key words: Peaks over threshold, poisson process, pareto distribution, GEV, heavy tailed distributions

State-dependent importance sampling for regularly varying random walks

2008 ◽  
Vol 40 (04) ◽  
pp. 1104-1128 ◽  
Author(s):  
Jose H. Blanchet ◽  
Jingchen Liu
Keyword(s):  
Importance Sampling ◽  
Large Deviation ◽  
Rare Event ◽  
Parametric Family ◽  
Finite Variance ◽  
Or Efficiency ◽  
State Dependent ◽  
Regularly Varying ◽  
Heavy Tailed ◽  
Path Dependent

Consider a sequence (X k : k ≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (S n : n ≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail of S n in a large deviation regime as n ↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

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