Application of stochastic theory to dilute gas reactions

1976 ◽  
Vol 14 (3) ◽  
pp. 271-289 ◽  
Author(s):  
Paul D. Fleming ◽  
Julian H. Gibbs
Keyword(s):  
Stochastic Theory ◽  
Dilute Gas ◽  
Gas Reactions

In situ REM imaging of surface processes on ceramic bulk crystals from 300 to 1670 K in a conventional TEM

1991 ◽  
Vol 49 ◽  
pp. 660-661
Author(s):  
Z. L. Wang ◽  
J. Bentley
Keyword(s):  
High Temperatures ◽  
Image Resolution ◽  
Atomic Processes ◽  
Crystal Surfaces ◽  
Beam Radiation ◽  
Surface Areas ◽  
Transmission Electron ◽  
Reflection Electron Microscopy ◽  
Gas Reactions

Studying the behavior of surfaces at high temperatures is of great importance for understanding the properties of ceramics and associated surface-gas reactions. Atomic processes occurring on bulk crystal surfaces at high temperatures can be recorded by reflection electron microscopy (REM) in a conventional transmission electron microscope (TEM) with relatively high resolution, because REM is especially sensitive to atomic-height steps.Improved REM image resolution with a FEG: Cleaved surfaces of a-alumina (012) exhibit atomic flatness with steps of height about 5 Å, determined by reference to a screw (or near screw) dislocation with a presumed Burgers vector of b = (1/3)<012> (see Fig. 1). Steps of heights less than about 0.8 Å can be clearly resolved only with a field emission gun (FEG) (Fig. 2). The small steps are formed by the surface oscillating between the closely packed O and Al stacking layers. The bands of dark contrast (Fig. 2b) are the result of beam radiation damage to surface areas initially terminated with O ions.

Residual Gas Reactions in the Electron Microscope: I. Qualitative Observations on the Water Gas Reaction

1968 ◽  
Vol 26 ◽  
pp. 292-293
Author(s):  
Richard E. Hartman ◽  
Roberta S. Hartman ◽  
Peter L. Ramos
Keyword(s):  
Electron Beam ◽  
Electron Microscope ◽  
Gas Phase ◽  
Absolute Temperature ◽  
Residual Gas ◽  
Gas Reactions ◽  
Image Position ◽  
The Right ◽  
Water Gas ◽  
Thinning Rate

The action of water and the electron beam on organic specimens in the electron microscope results in the removal of oxidizable material (primarily hydrogen and carbon) by reactions similar to the water gas reaction .which has the form:The energy required to force the reaction to the right is supplied by the interaction of the electron beam with the specimen.The mass of water striking the specimen is given by:where u = gH2O/cm2 sec, PH2O = partial pressure of water in Torr, & T = absolute temperature of the gas phase. If it is assumed that mass is removed from the specimen by a reaction approximated by (1) and that the specimen is uniformly thinned by the reaction, then the thinning rate in A/ min iswhere x = thickness of the specimen in A, t = time in minutes, & E = efficiency (the fraction of the water striking the specimen which reacts with it).

The Boltzmann Equation and the H Theorem (1872–1875)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Transport Phenomena ◽  
Dilute Gas ◽  
The Boltzmann Equation ◽  
H Theorem ◽  
Irreversible Evolution

This chapter covers Boltzmann’s writings about the Boltzmann equation and the H theorem in the period 1872–1875, through which he succeeded in deriving the irreversible evolution of the distribution of molecular velocities in a dilute gas toward Maxwell’s distribution. Boltzmann also used his equation to improve on Maxwell’s theory of transport phenomena (viscosity, diffusion, and heat conduction). The bulky memoir of 1872 and the eponymous equation probably are Boltzmann’s most famous achievements. Despite the now often obsolete ways of demonstration, despite the lengthiness of the arguments, and despite hidden difficulties in the foundations, Boltzmann there displayed his constructive skills at their best.

Boltzmann’s Kinetic Theory

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Boltzmann Equation ◽  
Kinetic Theory ◽  
Statistical Physics ◽  
Neutron Transport ◽  
Condensed Matter ◽  
Relativistic Fluids ◽  
Classical Statistical Mechanics ◽  
Dilute Gas ◽  
Equilibrium Processes ◽  
The Boltzmann Equation

Kinetic theory is the branch of statistical physics dealing with the dynamics of non-equilibrium processes and their relaxation to thermodynamic equilibrium. Established by Ludwig Boltzmann (1844–1906) in 1872, his eponymous equation stands as its mathematical cornerstone. Originally developed in the framework of dilute gas systems, the Boltzmann equation has spread its wings across many areas of modern statistical physics, including electron transport in semiconductors, neutron transport, quantum-relativistic fluids in condensed matter and even subnuclear plasmas. In this Chapter, a basic introduction to the Boltzmann equation in the context of classical statistical mechanics shall be provided.

Innovative Closed-Cell Reactor Permits In Situ Heating and Gas Reactions with Atomic Resolution at Atmospheric Pressure

2012 ◽  
Vol 18 (S2) ◽  
pp. 1118-1119 ◽  
Author(s):  
L. Allard ◽  
S.H. Overbury ◽  
M.B. Katz ◽  
W.C. Bigelow ◽  
D. Nackashi ◽  
...  
Keyword(s):  
Atmospheric Pressure ◽  
Atomic Resolution ◽  
Closed Cell ◽  
Cell Reactor ◽  
In Situ Heating ◽  
Gas Reactions

Extended abstract of a paper presented at Microscopy and Microanalysis 2012 in Phoenix, Arizona, USA, July 29 – August 2, 2012.

scholarly journals Analysis of Stochastic Generation and Shifts of Phantom Attractors in a Climate–Vegetation Dynamical Model

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1329
Author(s):  
Lev Ryashko ◽  
Dmitri V. Alexandrov ◽  
Irina Bashkirtseva
Keyword(s):  
Multiplicative Noise ◽  
Nonlinear Dynamical Systems ◽  
Stochastic Theory ◽  
Slow Variable ◽  
Theoretical Description ◽  
Mathematical Study ◽  
Stochastic Generation ◽  
Nonlinear Dynamical ◽  
Additive And Multiplicative Noise ◽  
And Shifts

A problem of the noise-induced generation and shifts of phantom attractors in nonlinear dynamical systems is considered. On the basis of the model describing interaction of the climate and vegetation we study the probabilistic mechanisms of noise-induced systematic shifts in global temperature both upward (“warming”) and downward (“freezing”). These shifts are associated with changes in the area of Earth covered by vegetation. The mathematical study of these noise-induced phenomena is performed within the framework of the stochastic theory of phantom attractors in slow-fast systems. We give a theoretical description of stochastic generation and shifts of phantom attractors based on the method of freezing a slow variable and averaging a fast one. The probabilistic mechanisms of oppositely directed shifts caused by additive and multiplicative noise are discussed.

scholarly journals A Framework for Modelling Economic Regional Location Processes Under Uncertainty

2021 ◽  
Author(s):  
Roy Cerqueti ◽  
Eleonora Cutrini
Keyword(s):  
Theoretical Model ◽  
Theoretical Analysis ◽  
Site Selection ◽  
Decision Problem ◽  
Stochastic Theory ◽  
Spatial Location ◽  
Probabilistic Framework ◽  
Location Patterns ◽  
Regional Location ◽  
Quantitative Tool

AbstractThis paper deals with the theoretical analysis of the spatial concentration and localization of firms and employees over a set of regions. In particular, it provides a simple site-selection theoretical model to describe the probabilistic framework of the location patterns. The adopted quantitative tool is the stochastic theory of urns. The model moves from the empirical evidence of the deviation of the spatial location of companies from the uniform distribution and of employees from the distribution of firms. Factors leading to such deviations are taken into consideration. Specifically, we formalize a decision problem grounded on the economic attributes of the regions and also on the distribution of the existing firms and employees in the territory. To our purpose, the site-selection model is presented as a stepwise process.

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