On the asymptotic theory of the Boltzmann and Enskog equations a rigorous H-theorem for the Enskog equation

1991 ◽  
pp. 15-30 ◽  
Author(s):  
N. Bellomo ◽  
M. Lachowicz
Keyword(s):  
Asymptotic Theory ◽  
Enskog Equation ◽  
H Theorem

H-theorem for the (modified) nonlinear Enskog equation

1978 ◽  
Vol 19 (6) ◽  
pp. 593-609 ◽  
Author(s):  
P. Resibois
Keyword(s):  
Enskog Equation ◽  
H Theorem

An H-theorem for the Enskog equation of a binary mixture of dissimilar hard spheres

1997 ◽  
Vol 106 (1) ◽  
pp. 236-246 ◽  
Author(s):  
Patricia Goldstein ◽  
L. S. Garcı́a-Colı́n
Keyword(s):  
Binary Mixture ◽  
Hard Spheres ◽  
Enskog Equation ◽  
H Theorem

Lectures on the Asymptotic Theory of Ideals

1988 ◽  
Author(s):  
D. Rees
Keyword(s):  
Asymptotic Theory

The Asymptotic Theory of the Kakwani Class of Poverty Measures

2007 ◽  
Author(s):  
Gane Samb Lo ◽  
Serigne Touba Sall ◽  
Cheikh Tidiane Seck
Keyword(s):  
Asymptotic Theory ◽  
Poverty Measures

Asymptotic Theory for Renewal Based High-Frequency Volatility Estimation

2018 ◽  
Author(s):  
Yifan Li ◽  
Ingmar Nolte ◽  
Sandra Nolte (Lechner)
Keyword(s):  
High Frequency ◽  
Asymptotic Theory ◽  
Volatility Estimation

Standard Asymptotic Theory

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Asymptotic Theory ◽  
Asymptotic Variance ◽  
Statistical Significance ◽  
Regularity Conditions ◽  
Parameter Estimates ◽  
Asymptotic Results ◽  
Ml Estimation ◽  
True Parameter ◽  
Log Likelihood ◽  
Ml Estimator

This book relies on maximum likelihood (ML) estimation of parameters. Asymptotic theory assumes regularity conditions hold when the ML estimator is consistent. Typically an additional third derivative condition is assumed to ensure that the ML estimator is also asymptotically normally distributed. Standard asymptotic results that then hold are summarized in this chapter; for example, the asymptotic variance of the ML estimator is then given by the Fisher information formula, and the log-likelihood ratio, the Wald and the score statistics for testing the statistical significance of parameter estimates are all asymptotically equivalent. Also, the useful profile log-likelihood then behaves exactly as a standard log-likelihood only in a parameter space of just one dimension. Further, the model can be reparametrized to make it locally orthogonal in the neighbourhood of the true parameter value. The large exponential family of models is briefly reviewed where a unified set of regular conditions can be obtained.

The Analogical Turn (1884–1887)

2018 ◽  
Author(s):  
Olivier Darrigol
Keyword(s):  
Boltzmann Equation ◽  
Statistical Mechanics ◽  
Mechanical System ◽  
Thermal Equilibrium ◽  
Special Kind ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Equipartition Theorem ◽  
Royal Road ◽  
H Theorem

This chapter recounts how Boltzmann reacted to Hermann Helmholtz’s analogy between thermodynamic systems and a special kind of mechanical system (the “monocyclic systems”) by grouping all attempts to relate thermodynamics to mechanics, including the kinetic-molecular analogy, into a family of partial analogies all derivable from what we would now call a microcanonical ensemble. At that time, Boltzmann regarded ensemble-based statistical mechanics as the royal road to the laws of thermal equilibrium (as we now do). In the same period, he returned to the Boltzmann equation and the H theorem in reply to Peter Guthrie Tait’s attack on the equipartition theorem. He also made a non-technical survey of the second law of thermodynamics seen as a law of probability increase.

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.

Approach to Equilibrium, the H-Theorem and Irreversibility

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Boltzmann Equation ◽  
Detailed Knowledge ◽  
Approach To Equilibrium ◽  
The Boltzmann Equation ◽  
H Theorem ◽  
Irreversible Evolution

Like most of the greatest equations in science, the Boltzmann equation is not only beautiful but also generous. Indeed, it delivers a great deal of information without imposing a detailed knowledge of its solutions. In fact, Boltzmann himself derived most if not all of his main results without ever showing that his equation did admit rigorous solutions. This Chapter illustrates one of the most profound contributions of Boltzmann, namely the famous H-theorem, providing the first quantitative bridge between the irreversible evolution of the macroscopic world and the reversible laws of the underlying microdynamics.

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