The imprint of GWs on the CMB

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.

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

10.1093/oso/9780198816171.003.0004 ◽

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.

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Coda

10.1093/oso/9780199592357.003.0036 ◽

2018 ◽

Author(s):

Sauro Succi

Keyword(s):

Fluid Dynamics ◽

Boltzmann Equation ◽

Broad Spectrum ◽

Lattice Boltzmann ◽

Lattice Boltzmann Equation ◽

Real Difference ◽

Complex Flow ◽

Flow Problems ◽

Subjective View

This section of the book revisits a question from the book The Lattice Boltzmann Equation (for fluid dynamics and beyond). This question is: What did we learn through lattice Boltzmann? Did LB make a real difference to our understanding of the physics of fluids and flowing matter in general? Here, the text aims to offer a subjective view, without the presumption of being right. Besides being routinely used for a broad spectrum of complex flow problems, there are, in the opinion expressed in this part of the book, a few precious instances in which LB has made a palpable difference.

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Approach to Equilibrium, the H-Theorem and Irreversibility

10.1093/oso/9780199592357.003.0003 ◽

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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Boltzmann’s Kinetic Theory

10.1093/oso/9780199592357.003.0002 ◽

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.

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