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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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