Stochastic Mixtures Research Articles

Publisher’s Note: “Benchmark simulations of radiative transfer in participating binary stochastic mixtures in two dimensions” [Matter Radiat. Extremes 9, 067802 (2024)

Publisher’s Note: “Benchmark simulations of radiative transfer in participating binary stochastic mixtures in two dimensions” [Matter Radiat. Extremes 9, 067802 (2024)

Benchmark simulations of radiative transfer in participating binary stochastic mixtures in two dimensions

We study radiative transfer in participating binary stochastic mixtures in two dimensions (2D) by developing an accurate and efficient simulation tool. For two different sets of physical parameters, 2D benchmark results are presented, and it is found that the influence of the stochastic mixture on radiative transfer is clearly parameter-dependent. Our results confirm that previous multidimensional results obtained in different studies are basically consistent, which is interpreted in terms of the relationship between the photon mean free path lp and the system size L. Nonlinear effects, including those due to scattering and radiation–material coupling, are also discussed. To further understand the particle size effect, we employ a dimensionless parameter lp/L, from which a critical particle size can be derived. On the basis of further 2D simulations, we find that an inhomogeneous mix is obtained for lp/L > 0.1. Furthermore, 2D material temperature distributions reveal that self-shielding and particle–particle shielding of radiation occur, and are enhanced when lp/L is increased. Our work is expected to provide benchmark results to verify proposed homogenized models and/or other codes for stochastic radiative transfer in realistic physical scenarios.

Deep Conservative Policy Iteration

Conservative Policy Iteration (CPI) is a founding algorithm of Approximate Dynamic Programming (ADP). Its core principle is to stabilize greediness through stochastic mixtures of consecutive policies. It comes with strong theoretical guarantees, and inspired approaches in deep Reinforcement Learning (RL). However, CPI itself has rarely been implemented, never with neural networks, and only experimented on toy problems. In this paper, we show how CPI can be practically combined with deep RL with discrete actions, in an off-policy manner. We also introduce adaptive mixture rates inspired by the theory. We experiment thoroughly the resulting algorithm on the simple Cartpole problem, and validate the proposed method on a representative subset of Atari games. Overall, this work suggests that revisiting classic ADP may lead to improved and more stable deep RL algorithms.

Cauchy formulas for linear transport in random media

In the context of linear transport in homogeneous non-stochastic media, the well-known Cauchy formulas express a remarkably elegant relation between the contributions to the angular flux within an arbitrary body due to particles starting from the body surface and those due to particles starting from the interior of the body. In this letter, we will extend these results to binary stochastic mixtures, a prototype class of random media that emerges in a broad spectrum of applications in physical and life sciences, ranging from hydrodynamical instabilities to tracer diffusion in biological tissues. In particular, for both quenched disorder and annealed disorder models we will establish generalized Cauchy-like formulas that appear to have a universal character and bear a close similarity to those previously derived for non-stochastic media.

The evolution of collaboration in symmetric 2×2-games with imperfect recognition of types

A recent series of papers has introduced a fresh perspective on the problem of the evolution of human cooperation by suggesting an amendment to the concept of cooperation itself: instead of thinking of cooperation as playing a particular strategy in a given game, usually C in the prisoner's dilemma, we could also think of cooperation as collaboration, i.e. as coalitional strategy choice, such as jointly switching from (D,D) to (C,C). The present paper complements previous work on collaboration by expanding on its genericity while relaxing the assumption that collaborators are able to perfectly identify their own kind. Conditions for the evolutionary viability of such collaboration under fairly undemanding assumptions about population and interaction structure are derived. Doing so, this paper shows that collaboration is an adaptive principle of strategy choice in a broad range of niches, i.e., stochastic mixtures of games.

On the decomposition of stochastic cellular automata

In this paper we present two interesting properties of stochastic cellular automata that can be helpful in analyzing the dynamical behavior of such automata. The first property allows for calculating cell-wise probability distributions over the state set of a stochastic cellular automaton, i.e. images that show the average state of each cell during the evolution of the stochastic cellular automaton. The second property shows that stochastic cellular automata are equivalent to so-called stochastic mixtures of deterministic cellular automata. Based on this property, any stochastic cellular automaton can be decomposed into a set of deterministic cellular automata, each of which contributes to the behavior of the stochastic cellular automaton.

Using a multinomial tree model for detecting mixtures in perceptual detection.

In the area of memory research there have been two rival approaches for memory measurement—signal detection theory (SDT) and multinomial processing trees (MPT). Both approaches provide measures for the quality of the memory representation, and both approaches provide for corrections for response bias. In recent years there has been a strong case advanced for the MPT approach because of the finding of stochastic mixtures on both target-present and target-absent tests. In this paper a case is made that perceptual detection, like memory recognition, involves a mixture of processes that are readily represented as a MPT model. The Chechile (2004) 6P memory measurement model is modified in order to apply to the case of perceptual detection. This new MPT model is called the Perceptual Detection (PD) model. The properties of the PD model are developed, and the model is applied to some existing data of a radiologist examining CT scans. The PD model brings out novel features that were absent from a standard SDT analysis. Also the topic of optimal parameter estimation on an individual-observer basis is explored with Monte Carlo simulations. These simulations reveal that the mean of the Bayesian posterior distribution is a more accurate estimator than the corresponding maximum likelihood estimator (MLE). Monte Carlo simulations also indicate that model estimates based on only the data from an individual observer can be improved upon (in the sense of being more accurate) by an adjustment that takes into account the parameter estimate based on the data pooled across all the observers. The adjustment of the estimate for an individual is discussed as an analogous statistical effect to the improvement over the individual MLE demonstrated by the James–Stein shrinkage estimator in the case of the multiple-group normal model.

When One Plus One Does Not Equal Two: Fluorescence Anisotropy in Aggregates and Multiply Labeled Proteins

The behavior of fluorescence anisotropy and polarization in systems with multiple dyes is well known. Homo-FRET and its consequent energy migration cause the fluorescence anisotropy to decrease as the number of like fluorophores within energy transfer distance increases. This behavior is well understood when all subunits within a cluster are saturated with fluorophores. However, incomplete labeling as might occur from a mixture of endogenous and labeled monomer units, incomplete saturation of binding sites, or photobleaching produces stochastic mixtures. Models in widespread and longstanding use that describe these mixtures apply an assumption of equal fluorescence efficiency for all sites first stated by Weber and Daniel in 1966. The assumption states that fluorophores have the same brightness when free in solution as they do in close proximity to each other in a cluster. The assumption simplifies descriptions of anisotropy trends as the fractional labeling of the cluster changes. However, fluorophores in close proximity often exhibit nonadditivity due to such things as self-quenching behavior or exciplex formation. Therefore, the anisotropy of stochastic mixtures of fluorophore clusters of a particular size will depend on the behavior of those fluorophores in clusters. We present analytical expressions for fractionally labeled clusters exhibiting a range of behaviors, and experimental results from two systems: an assembled tetrameric cluster of fluorescent proteins and stochastically labeled bovine serum albumin containing up to 24 fluorophores. The experimental results indicate that clustered species do not follow the assumption of equal fluorescence efficiency in the systems studied with clustered fluorophores showing reduced fluorescence intensity. Application of the assumption of equal fluorescence efficiency will underpredict anisotropy and consequently underestimate cluster size in these two cases. The theoretical results indicate that careful selection of the fractional labeling in strongly quenched systems will enhance opportunities to determine cluster sizes, making accessible larger clusters than are currently considered possible.

A novel method for assessing rival models of recognition memory

A general comparison is made between the multinomial processing tree (MPT) approach and a strength-based approach for modeling recognition memory measurement. Strength models include the signal-detection model and the dual-process model. Existing MPT models for recognition memory and a new generic MPT model, called the Multistate (MS) model, are contrasted with the strength models. Although the ROC curves for the MS model and strength model are similar, there is a critical difference between existing strength models and MPT models that goes beyond the assessment of the ROC. This difference concerns the question of stochastic mixtures for foil test trials. The hazard function and the reverse hazard function are powerful methods for detecting the presence of a probabilistic mixture. Several new theorems establish a novel method for obtaining information about the hazard function and reverse hazard function for the latent continuous distributions that are assumed in the strength approach to recognition memory. Evidence is provided that foil test trials involve a stochastic mixture. This finding occurred for both short-term memory procedures, such as the Brown–Peterson task, and long-term list-learning procedures, such as the paired-associate task. The effect of mixtures on foil trials is problematic for existing strength models but can be readily handled by MPT models such as the MS model. Other phenomena, such as the mirror effect and the effect of target-foil similarity, are also predicted accurately by the MPT modeling framework.

Risk sensitivity for amounts of and delay to rewards: Adaptation for uncertainty or by-product of reward rate maximising?

Observations that humans and other species are sensitive to variability in the outcome of their choices has led to the widespread assumption that this sensitivity reflects adaptations to cope with risk (stochasticity of action consequences). We question this assumption in experiments with starlings. We show that choices between outcomes that are risky in both amount and delay to food are predictable from preferences in the absence of risk. We find that the overarching best predictor of an option's value is the average of the ratios of amount to delay across its (frequency weighted) outcomes, an expression known as "Expectation of the Ratios", or EoR. Most tests of risk sensitivity focus on the predicted impact of energetic state on preference for risk. We show instead that under controlled state conditions subjects are variance- and risk-neutral with respect to EoR, and this implies variance neutrality for amounts and variance-proneness for delays. The weak risk aversion for amounts often reported requires a small modification of EoR. EoR is consistent with associative learning: acquisition of value for initially neutral stimuli is roughly proportional to the magnitude of their consequences and inversely proportional to the interval between the stimulus and its consequence's onset. If, as is likely, the effect of amount on acquisition is sublinear, the result is a deviation from EoR towards risk aversion for amount. In 3 experiments, we first establish individual birds' preferences between pairs of fixed options that differ in both amount and delay (small-sooner vs. large-later), and then examine choices between stochastic mixtures that include these options. Experiment 1 uses a titration to establish certainty equivalents, while experiments 2 and 3 measure degree of preference between options with static parameters. The mixtures differ in the coefficient of variation of amount, delay, or both, but EoR is sufficient to predict all results, with no additional explanatory role for riskiness.

A benchmark comparison of Monte Carlo particle transport algorithms for binary stochastic mixtures

We numerically investigate the accuracy of two Monte Carlo algorithms originally proposed by Zimmerman [1] and Zimmerman and Adams [2] for particle transport through binary stochastic mixtures. We assess the accuracy of these algorithms using a standard suite of planar geometry incident angular flux benchmark problems and a new suite of interior source benchmark problems. In addition to comparisons of the ensemble-averaged leakage values, we compare the ensemble-averaged material scalar flux distributions. Both Monte Carlo transport algorithms robustly produce physically realistic scalar flux distributions for the benchmark transport problems examined. The base Monte Carlo algorithm reproduces the standard Levermore–Pomraning model [3,4] results. The improved Monte Carlo algorithm generally produces significantly more accurate leakage values and also significantly more accurate material scalar flux distributions. We also present deterministic atomic mix solutions of the benchmark problems for comparison with the benchmark and the Monte Carlo solutions. Both Monte Carlo algorithms are generally significantly more accurate than the atomic mix approximation for the benchmark suites examined.

On the application and extension of system signatures in engineering reliability

AbstractFollowing a review of the basic ideas in structural reliability, including signature‐based representation and preservation theorems for systems whose components have independent and identically distributed (i.i.d.) lifetimes, extensions that apply to the comparison of coherent systems of different sizes, and stochastic mixtures of them, are obtained. It is then shown that these results may be extended to vectors of exchangeable random lifetimes. In particular, for arbitrary systems of sizes m < n with exchangeable component lifetimes, it is shown that the distribution of an m‐component system's lifetime can be written as a mixture of the distributions of k‐out‐of‐n systems. When the system has n components, the vector of coefficients in this mixture representation is precisely the signature of the system defined in Samaniego, IEEE Trans Reliabil R–34 (1985) 69–72. These mixture representations are then used to obtain new stochastic ordering properties for coherent or mixed systems of different sizes. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008

The probability distribution associated with the dichotomic Markov process

We have calculated the probability distribution function for the dichotomic Markov process using the work of Pomraning [Linear kinetic theory and particle transport in stochastic mixtures. Singapore: World Scientific; 1991] and have studied the rate of convergence to this exact form as the number of terms in a series approximation is increased. Each term in the series involves an additional stochastic moment in the hierarchy of moments. It is observed that convergence is fast near the source but, as the distance from the source increases, more and more moments are required to obtain a specified accuracy.

A “two-grid” acceleration of binary stochastic mixture deterministic transport iterations in slab geometry

Particle transport in stochastic mixtures has been an area of active research during the last decade. The primary focus has been on the development of simple, efficient models that approximate the behavior of transport solutions which have been ensemble-averaged over a large number of realizations of the mixing statistics. These approximate models are often written as coupled systems of transport equations. The stochastic nature of this problem can be naturally incorporated into Monte Carlo transport codes, but there have also been efforts to solve these coupled equations deterministically. In this paper, we develop an efficient iterative technique for the deterministic solution of the coupled transport equations of the Levermore–Pomraning mix model. The basis for its development is the observation that these equations are similar in structure to traditional multigroup transport equations with upscattering. We provide the results of a Fourier analysis of this technique, and compare these with observed convergence rates from the implemented algorithm.

On the Linear Spectrum Mixing Model Over Discrete Random Media

The linear spectrum-mixing model for hyperspectral data analysis is evaluated in the context of the stochastic radiative transfer equations. The stochastic radiative transfer equations that treat media as binary stochastic mixtures are reformed and then solved within a discrete random layer over a rough interface. The layer constituents are randomly distributed within ellipsoidal spatial cells of various sizes. The solution of the stochastic radiative transfer equations is used to investigate the validity domain of the linear spectrum-mixing model. The solution is also used to estimate the deviations in the predictions of the linear spectrum-mixing model for the layer brightness temperature.

Monte-Carlo simulation of the chord length distribution function across convex bodies, non-convex bodies and random media

The study of chord length distributions across various kinds of geometrical shapes, including stochastic mixtures, is a topic of great interest in many research fields ranging from ecology to neutronics. We have tried here to draw links between theoretical results and actual simulations for simple objects like disks, circular rings, spheres, hollow spheres, as well as for random media consisting of stochastic mono- or polydisperse spheres packing (three different packing algorithms were tested). The Monte Carlo simulations which were performed for simple objects fit perfectly theoretical formulas. For stochastic binary mixtures the simulations were still in rather good agreement with known analytical results.

Transport in discrete stochastic mixtures

We consider steady-state particle transport through a stochastic mixture of two (or more) immiscible materials. Certain exact results, as well as approximate models, are described for ⟨ψ⟩, the ensemble average of the particle angular flux. In the simplest case of transport in a no-scattering medium, an exact description for ⟨ψ⟩ and higher stochastic moments is easily constructed. If the material mixing is taken as Markovian, the Liouville master equation leads to two (for a binary mixture) coupled deterministic differential transport equations describing ⟨ψ⟩, with similar results for higher stochastic moments, such as the variance. For non-Markovian mixing, renewal analysis leads to four coupled deterministic integral equations describing ⟨ψ⟩ and higher moments. In the more general problem involving scattering, no exact results (except in very special cases) are available, but reasonably accurate models can be developed. Certain asymptotic limits of these models lead to a single, renormalized, transport equation for ⟨ψ⟩ of the standard form, with effective values of the cross sections and source which account for the stochasticity of the problem. Diffusive approximations also result from asymptotic analyses.

Mixtures of Kr and Xe physisorbed on Pt(111): a prototype of a stochastic two-dimensional alloy

Krypton and xenon co-adsorbed on the Pt(111) surface at 40 K are found to form homogenous stochastic mixtures for all composition ratios and in the entire coverage regime below one monolayer. These quasi two-dimensional Kr 1− x Xe x alloys (0 ≤ χ ≤ 1) are characterized by narrow diffraction peaks indicating a high degree of positional as well as orientational order. The position of the diffraction peaks of the mixture varies linearly as a function of the molar fraction χ between those of the pure Kr and Xe components (Vegard's law). The half-widths of the diffraction peaks are larger than for the pure Kr system but even at χ = 0.5 quasi crystalline order still extends over a length scale of about 100 Å. The diffraction line-shapes are well described within a simple model of lattice disorder.

Mixtures of Kr and Xe physisorbed on Pt(111): a prototype of a stochastic two-dimensional alloy

Krypton and xenon co-adsorbed on the Pt(111) surface at 40 K are found to form homogenous stochastic mixtures for all composition ratios and in the entire coverage regime below one monolayer. These quasi two-dimensional Kr 1− χ − Xe χ alloys (0⩽ χ⩽ 1) are characterized by narrow diffraction peaks indicating a high degree of positional as well as orientational order. The position of the diffraction peaks of the mixture varies linearly as a function of the molar fraction χ between those of the pure Kr and Xe components (Vegard's law). The half-widths of the diffraction peaks are larger than for the pure Kr system but even at χ = 0.5 quasi crystalline order still extends over a length scale of about 100 Å. The diffraction line-shapes are well described within a simple model of lattice disorder.

Diffusive transport in binary anisotropic stochastic mixtures

It has recently been shown that a linear kinetic theory model for particle transport in a binary Markovian stochastic mixture possesses several different diffusive limits. This work was restricted to isotropic mixing statistics and a common scale length for the chord lengths of the two immiscible components of the mixture. Here we allow anisotropic statistics as well as independent length scales for the two components. The formalism used is that of asymptotic expansions, and we find an additional richness of asymptotic diffusive limits by relaxing the prior limitations on the mixing. A short discussion of the derivation of asymptotically consistent initial and boundary conditions for the diffusion equations is also given.