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Time irreversibility and the H-theorem

CARLO CERCIGNANI

in Ludwig Boltzmann: The Man Who Trusted Atoms

Published in print:
2006
Published Online:
January 2010
ISBN:
9780198570646
eISBN:
9780191717949
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570646.003.0005
Subject:
Physics, History of Physics

Ludwig Boltzmann not only showed that his Boltzmann equation, which captures the essence of the second law of thermodynamics in mathematical form, admits James Clerk Maxwell's distribution as an ... More


The Critical Turn (1895–1899)

Olivier Darrigol

in Atoms, Mechanics, and Probability: Ludwig Boltzmann's Statistico-Mechanical Writings - An Exegesis

Published in print:
2018
Published Online:
March 2018
ISBN:
9780198816171
eISBN:
9780191853661
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198816171.003.0008
Subject:
Physics, Atomic, Laser, and Optical Physics, History of Physics

In the writings of this period, we see Boltzmann responding to criticism by British kinetic theorists and by the German mathematician Ernst Zermelo regarding the equipartition of energy and the H ... More


The problem of polyatomic molecules

CARLO CERCIGNANI

in Ludwig Boltzmann: The Man Who Trusted Atoms

Published in print:
2006
Published Online:
January 2010
ISBN:
9780198570646
eISBN:
9780191717949
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570646.003.0008
Subject:
Physics, History of Physics

Polyatomic molecules created difficulties in kinetic theory from the start, with the problem of thermal capacities or specific heats. If the problem of specific heats was initially solved with the ... More


Littlewood-Paley Theory, Lp-Multipliers and Function Spaces

Ali Taheri

in Function Spaces and Partial Differential Equations: Volume 2: Contemporary Analysis

Published in print:
2015
Published Online:
September 2015
ISBN:
9780198733157
eISBN:
9780191797729
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198733157.003.0017
Subject:
Mathematics, Analysis

In this chapter, Littlewood-Paley theory is developed using the theory of vector-valued singular integral operators. Various applications including multiplier theorem and Strichartz estimates for the ... More


Boltzmann’s relation and the statistical interpretation of entropy

CARLO CERCIGNANI

in Ludwig Boltzmann: The Man Who Trusted Atoms

Published in print:
2006
Published Online:
January 2010
ISBN:
9780198570646
eISBN:
9780191717949
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570646.003.0006
Subject:
Physics, History of Physics

Ludwig Boltzmann first used probabilistic arguments in his answer to Johann Loschmidt's objection to his Boltzmann equation and his assumptions about entropy in the H-theorem. Up to that moment, ... More


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

Olivier Darrigol

in Atoms, Mechanics, and Probability: Ludwig Boltzmann's Statistico-Mechanical Writings - An Exegesis

Published in print:
2018
Published Online:
March 2018
ISBN:
9780198816171
eISBN:
9780191853661
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198816171.003.0004
Subject:
Physics, Atomic, Laser, and Optical Physics, History of Physics

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 ... More


A Kinetic-Theory Description of Fluids

Luciano Rezzolla and Olindo Zanotti

in Relativistic Hydrodynamics

Published in print:
2013
Published Online:
January 2014
ISBN:
9780198528906
eISBN:
9780191746505
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528906.003.0002
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter illustrates how the foundations of the fluid description are rooted in statistical mechanics and in kinetic theory. This approach, which is appropriate for those systems composed of a ... More


Consolidation (1887–1895)

Olivier Darrigol

in Atoms, Mechanics, and Probability: Ludwig Boltzmann's Statistico-Mechanical Writings - An Exegesis

Published in print:
2018
Published Online:
March 2018
ISBN:
9780198816171
eISBN:
9780191853661
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198816171.003.0007
Subject:
Physics, Atomic, Laser, and Optical Physics, History of Physics

This chapter covers a period in which Boltzmann returned to the collision-based approach and consolidated it in answer to criticism and suggestions by William Thomson, Hendrik Lorentz, George Bryan, ... More


Approach to Equilibrium, the H-Theorem and Irreversibility

Sauro Succi

in The Lattice Boltzmann Equation: For Complex States of Flowing Matter

Published in print:
2018
Published Online:
June 2018
ISBN:
9780199592357
eISBN:
9780191847967
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780199592357.003.0003
Subject:
Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

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 ... More


Entropic Lattice Boltzmann

Sauro Succi

in The Lattice Boltzmann Equation: For Complex States of Flowing Matter

Published in print:
2018
Published Online:
June 2018
ISBN:
9780199592357
eISBN:
9780191847967
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780199592357.003.0021
Subject:
Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

The LB with enhanced collisions opened the way to the top-down design of LB schemes, with major gains in flexibility and computational efficiency. However, compliance with the second principle was ... More


The Boltzmann equation for dilute gases

Rodrigo Soto

in Kinetic Theory and Transport Phenomena

Published in print:
2016
Published Online:
June 2016
ISBN:
9780198716051
eISBN:
9780191824401
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198716051.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The Boltzmann equation, which is the first and best known kinetic equation, describes the dynamics of classical dilute gases. For its derivation, the motion of the atoms and molecules is separated in ... More


Irreversibility

Robert H. Swendsen

in An Introduction to Statistical Mechanics and Thermodynamics

Published in print:
2019
Published Online:
February 2020
ISBN:
9780198853237
eISBN:
9780191887703
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198853237.003.0022
Subject:
Physics, Condensed Matter Physics / Materials, Theoretical, Computational, and Statistical Physics

The phenomenon of irreversibility is explained on the basis of an analysis by H. L. Frisch. The history of the debate over irreversibility is briefly discussed, including Boltzmann’s H-theorem, ... More


BBGKY Hierarchy, Kinetic Theories and the Boltzmann Equation

A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo

in Physics of Long-Range Interacting Systems

Published in print:
2014
Published Online:
October 2014
ISBN:
9780199581931
eISBN:
9780191787140
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199581931.003.0007
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces to the study of the dynamics of many-body systems. The reduced distribution functions and their dynamics, determined by the equations of the BBGKY hierarchy, are presented. It ... More


Nonequilibrium Quantum Hydrodynamics

Klaus Morawetz

in Interacting Systems far from Equilibrium: Quantum Kinetic Theory

Published in print:
2017
Published Online:
February 2018
ISBN:
9780198797241
eISBN:
9780191838743
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198797241.003.0015
Subject:
Physics, Condensed Matter Physics / Materials

The balance equations resulting from the nonlocal kinetic equation are derived. They show besides the Landau-like quasiparticle contributions explicit two-particle correlated parts which can be ... More


Nonextensive Entropies and Sensitivity to Initial Conditions of Complex Systems

Marcelo L. Lyra

in Nonextensive Entropy: Interdisciplinary Applications

Published in print:
2004
Published Online:
November 2020
ISBN:
9780195159769
eISBN:
9780197562024
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195159769.003.0009
Subject:
Earth Sciences and Geography, Atmospheric Sciences

Tsallis generalized statistics has been successfully applied to describe some relevant features of several natural systems exhibiting a nonextensive character. It is based on an extended form for ... More


Statistical Fluid Dynamics

Rick Salmon

in Lectures on Geophysical Fluid Dynamics

Published in print:
1998
Published Online:
November 2020
ISBN:
9780195108088
eISBN:
9780197561034
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195108088.003.0008
Subject:
Earth Sciences and Geography, Geophysics: Earth Sciences

In the previous chapter, we considered fluid turbulence from a heuristic viewpoint, combining ideas based on fundamental conservation laws with more qualitative ideas about irreversible behavior. ... More


Boltzmann: the probabilistic interpretation of entropy

Robert T. Hanlon

in Block by Block: The Historical and Theoretical Foundations of Thermodynamics

Published in print:
2020
Published Online:
April 2020
ISBN:
9780198851547
eISBN:
9780191886133
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198851547.003.0042
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Boltzmann’s collective work was a mathemetical tour de force. Building on Clausius and Maxwell, he demonstrated that the distribution of gas phase atoms and molecules follows from probability theory. ... More


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