Verification of model probability distributions in turbulent heated jets

The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.

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Asteroid and Comet Impact Speeds upon Mars: Significance for Panspermia and the Supply of Organics

International Astronomical Union Colloquium ◽

10.1017/s0252921100014718 ◽

1997 ◽

Vol 161 ◽

pp. 197-201 ◽

Cited By ~ 1

Author(s):

Duncan Steel

Keyword(s):

Probability Distributions ◽

Regions Of Interest ◽

Significant Fraction ◽

Organic Chemicals ◽

Impact Speed ◽

The Mean ◽

The Impact

AbstractWhilst lithopanspermia depends upon massive impacts occurring at a speed above some limit, the intact delivery of organic chemicals or other volatiles to a planet requires the impact speed to be below some other limit such that a significant fraction of that material escapes destruction. Thus the two opposite ends of the impact speed distributions are the regions of interest in the bioastronomical context, whereas much modelling work on impacts delivers, or makes use of, only the mean speed. Here the probability distributions of impact speeds upon Mars are calculated for (i) the orbital distribution of known asteroids; and (ii) the expected distribution of near-parabolic cometary orbits. It is found that cometary impacts are far more likely to eject rocks from Mars (over 99 percent of the cometary impacts are at speeds above 20 km/sec, but at most 5 percent of the asteroidal impacts); paradoxically, the objects impacting at speeds low enough to make organic/volatile survival possible (the asteroids) are those which are depleted in such species.

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An Accelerated Cue Combination Principle Accounts for Multi-cue Depth Perception

Journal of Perceptual Imaging ◽

10.2352/j.percept.imaging.2020.3.1.010501 ◽

2020 ◽

Vol 3 (1) ◽

pp. 10501-1-10501-9

Author(s):

Christopher W. Tyler

Keyword(s):

Belief Propagation ◽

Probability Distributions ◽

Three Dimensional ◽

Dimensional Structure ◽

Fusion Model ◽

Cue Combination ◽

Depth Estimate ◽

Appropriate Behavior ◽

Depth Cues ◽

Bayesian Averaging

Abstract For the visual world in which we operate, the core issue is to conceptualize how its three-dimensional structure is encoded through the neural computation of multiple depth cues and their integration to a unitary depth structure. One approach to this issue is the full Bayesian model of scene understanding, but this is shown to require selection from the implausibly large number of possible scenes. An alternative approach is to propagate the implied depth structure solution for the scene through the “belief propagation” algorithm on general probability distributions. However, a more efficient model of local slant propagation is developed as an alternative.The overall depth percept must be derived from the combination of all available depth cues, but a simple linear summation rule across, say, a dozen different depth cues, would massively overestimate the perceived depth in the scene in cases where each cue alone provides a close-to-veridical depth estimate. On the other hand, a Bayesian averaging or “modified weak fusion” model for depth cue combination does not provide for the observed enhancement of perceived depth from weak depth cues. Thus, the current models do not account for the empirical properties of perceived depth from multiple depth cues.The present analysis shows that these problems can be addressed by an asymptotic, or hyperbolic Minkowski, approach to cue combination. With appropriate parameters, this first-order rule gives strong summation for a few depth cues, but the effect of an increasing number of cues beyond that remains too weak to account for the available degree of perceived depth magnitude. Finally, an accelerated asymptotic rule is proposed to match the empirical strength of perceived depth as measured, with appropriate behavior for any number of depth cues.

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