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Kelvin played a big part in the development of statistical mechanics, both for equilibrium and non-equilibrium. This chapter reviews these developments, taking a particular interest in Kelvin's own contributions. Topics covered include Kelvin and thermoelectricity, gas modeled as a collection of molecules, the reversibility paradox, mathematical probability models, and Boltzmann's equations.

In this chapter a novel, rigorous approach to analyse the validity of continuum approximations for deterministic interacting particle systems is discussed. The focus is on the Boltzmann–Grad limit of ballistic annihilation, a topic which has has received considerable attention in the physics literature. In this situation, due to the deterministic nature of the evolution, it is possible that correlations build up and the mean–field approximation by the Boltzmann equation breaks down. A sharp condition on the initial distribution, which ensures the validity of the Boltzmann equation is given, together with an example demonstrating the failure of the mean-field theory if the condition is violated.

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 equilibrium solution, but he also gave a heuristic proof that it is the only possible one. To this end he introduced a quantity, which he denoted by E and was later denoted by H, defined in terms of the molecular velocity distribution. His result is usually quoted as the H-theorem and indicates that H must be proportional to minus the entropy. The equation is the first to govern the evolution in time of a probability. The proof for the Boltzmann equation can be extended to polyatomic gases. This chapter also discusses Loschmidt's paradox, Zermelo's paradox, the physical and mathematical resolution of the paradoxes, time's arrow and the expanding universe, and whether time irreversibility is objective or subjective.

Einstein's kinetic theory of Brownian motion, based upon light water molecules continuously bombarding a heavy pollen, provided an explanation of diffusion from Newtonian mechanics. Since the discovery of quantum mechanics it has been a challenge to verify the emergence of diffusion from Schrödinger's equation. The first step in this program is to verify the linear Boltzmann equation as a certain scaling limit of a Schrödinger equation with random potential. In the second step, a longer time scale that corresponds to infinitely many Boltzmann collisions is considered. The intuition is that the Boltzmann equation then converges to a diffusive equation similarly to the central limit theorem for Markov processes with sufficient mixing. In this chapter the mathematical tools to justify this intuition rigorously is presented. This new material relies on joint papers with H. -T. Yau and M. Salmhofer.

This chapter addresses the rich variety of physical effects linked to the Bose-Einstein condensation of exciton-polaritons, including stimulated scattering of exciton-polaritons, spontaneous symmetry breaking in polariton systems, polariton lasing, and superfluidity. A detailed theoretical description of these effects based on quantum optics tools and a density matrix formalism is presented. It is demonstrated theoretically that Bose condensation at room temperature can be achieved in wide-band gap microcavities, and how it can be used for practical applications in opto-electronics. The dynamical aspects of Bose-condensation of the polaritons, including the bottleneck effect, acoustic phonon, and polariton-polariton interactions are considered.

This chapter presents the basic equations of the relativistic kinetic theory; it proves local existence theorems for Einstein equations coupled with kinetic matter. It then provides thermodynamic properties linked with the Boltzmann equation, proves the H-theorem, and indicates how perturbation around a Maxwell–Jütner equilibrium distribution gives possible equations for dissipative fluids. Finally, the chapter indicates how the theory of extended thermodynamics circumvents the difficulty of generalizing to Relativity the dissipative fluids equations.

The problem of irreversibility came to the forefront in kinetic theory with Ludwig Boltzmann. In 1872, Boltzmann not only derived the equation that bears his name, but also introduced a definition of entropy in terms of the distribution function of the molecular velocities. By 1868, Boltzmann had already extended James Clerk Maxwell's distribution to the case where the molecules are in equilibrium in a force field with potential, including the case of polyatomic molecules. Boltzmann interprets Maxwell's distribution function in two different ways: the first way is based on the fraction of a sufficiently long time interval, during which the velocity of a specific molecule has values within a certain volume element in velocity space; whereas the second way is based on the fraction of molecules which, at a given instant, have a velocity in the said volume element. This chapter discusses the Boltzmann equation, irreversibility and kinetic theory, and Boltzmann's 1872 paper on the thermal equilibrium of gas molecules.

The notions of Chapter 1 are extended here to systems of many individuals (thus many degrees of freedom), now seen from a statistical viewpoint. In the phase space, the chapter shows Liouville’s theorem of volume preservation and deduce the Boltzmann equation for the probability density of molecular velocities. The chapter then introduces the statistical concepts of entropy and temperature, and prove that the entropy increases (H-Theorem). Thermodynamical quantities are deduced. Finally, dissipative systems are treated, the energy being transferred from large scales to small scales, i.e. from macroscopic to internal energy. Friction forces are introduced from Onsager’s theory of kinetic coefficients, and these concepts are applied to dissipative forces in a system of macroscopic particles. Lastly, fluctuations are considered in the Fokker–Plank formalism.

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.

The analysis of the microscopic processes by which electrons and phonons are scattered is only half the calculation of the macroscopic transport coefficients. To discuss the flow of electric and thermal currents, a further complicated calculation with its own special methods may be needed. This chapter presents a general discussion of these methods, which will be applied in later chapters to different types of solid under various physical conditions. Topics covered include the kinetic method, distribution functions, the Boltzmann equation, macroscopic transport coefficients, Kelvin-Onsager relations, the variational principle, and Matthiesson's rule.

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 very large number of free particles and extending over a length-scale much larger than the inter-particles separation, is first presented in the Newtonian framework and then extended to the relativistic regime. A number of fundamental conceptual steps are taken and treated in detail: the introduction of a distribution function that depends on the positions and on the four-momentum of the constituent particles, the definition of the energy–momentum tensor as the second moment of the distribution function, the discussion of the relativistic Maxwell–Boltzmann equation with the corresponding H-theorem and transport equations. Finally, equations of state are described for all possible cases of relativistic or non-relativistic, degenerate or non-degenerate fluids.

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 free streaming and binary collisions. Notably, the kinetic equation that is obtained turns out to be irreversible despite the use of concepts of classical reversible mechanics. The origin of the irreversibility, quantified by the H-theorem, is explained and justified. The irreversibility manifests in that after a few collisions, the gases reach local thermal equilibrium described by Maxwellian distributions. For long times, it is shown that the system evolves via hydrodynamic equations and the transport coefficients, viscosity, and thermal conductivity, are computed in terms of the collision properties. The Boltzmann equation is extended to describe dense gases and granular media. Finally, the concepts presented in the chapter are used to explain the cooling of particles in the expanding universe.

Diffusive electronic transport in degenerately doped semiconductors at low magnetic fields can be successfully described within the Drude theory of electrical conduction. This chapter introduces the basic concepts for two-dimensional electron gases based on a three-level approach: on the first level, the relevant conductivity formulae are derived based on very elementary kinetic theory elucidating the role of classical dynamics and the statistical influence of scattering. This view is refined on the second level, on which Boltzmann's equation is used in the relaxation time approximation to reproduce the same results. On the third level, ionized impurity scattering is treated as an example quantum mechanically, and the transport scattering rate is explicitly calculated. Conductivity is related to diffusion via the Einstein relation, and the relation between scattering time and scattering cross section is discussed. Beyond that, the influence of the device geometry on measured resistances, as well as four-terminal and two-terminal measurement techniques are introduced. A variety of scattering mechanisms is discussed. At the end, the conductivity of graphene is discussed in the framework of the Drude model.

The first atomic theory is credited to Democritus of Abdera, who lived in the 5th century BC. The actual development of the kinetic theory of gases accordingly took place much later, in the 19th century. With his transfer equations, James Clerk Maxwell had come very close to an evolution equation for the distribution function, but this last step must beyond any doubt be credited to Ludwig Boltzmann. The equation under consideration is usually called the Boltzmann equation, but sometimes the Maxwell-Boltzmann equation. Rudolf Clausius took kinetic theory to a mature stage with the explicit recognition that thermal energy is but the kinetic energy of the random motion of the molecules and the explanation of the first law of thermodynamics in kinetic terms. In any case there remained the important unsolved problem of deducing the second law of thermodynamics, the basis of the modern idea of irreversibility in physical processes.

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, although he mentioned probability in his papers, Boltzmann seemed to think that the distribution function was a way of utilising the techniques of mathematical analysis in order to count the actual numbers of molecules, and no hidden probabilistic assumption was contained in his arguments. If Boltzmann had already begun to hint at the important role of probability in 1871, the priority in stressing the necessity of a statistical interpretation of the second law of thermodynamics must certainly be credited to James Clerk Maxwell because of his invention of the demon now named after him. This chapter discusses the probabilistic interpretation of thermodynamics, explicit use of probability for a gas with discrete energies, energy as a continuous phemonenon, and the so-called H-curve.

All the fundamental laws of physics are symmetric with respect to the inversion of the time arrow; the cup which reunites itself from the broken pieces, recovers the coffee from the floor and jumps back to the table, does not violate any law of mechanics. The man who first gave a convincing explanation of this paradox was Austrian physicist Ludwig Boltzmann. Boltzmann, who was born in Vienna in 1844 and committed suicide in Duino in 1906, was one of the main figures in the development of the atomic theory of matter. His fame will be forever related to two basic contributions to science: the interpretation of the concept of entropy as a mathematically well-defined measure of what one can call the disorder of atoms, and the equation aptly known as the Boltzmann equation. This equation describes the statistical properties of a gas made up of molecules and is, from a historical standpoint, the first equation ever written to govern the time evolution of a probability.

The evolution of the velocity of the distribution function is governed by the Boltzmann equation. This chapter derives the Boltzmann equation for the homogeneous cooling granular gas and discusses the properties of the collision in general.

This chapter explores the evolution of an ensemble of electrons under stimulus, classically and quantum-mechanically. The classical Liouville description is derived, and then reformed to the quantum Liouville equation. The differences between the classical and the quantum-mechanical description are discussed, emphasizing the uncertainty-induced fuzziness in the quantum description. The Fokker-Planck equation is introduced to describe the evolution of ensembles and fluctuations in it that comprise the noise. The Liouville description makes it possible to write the Boltzmann transport equation with scattering. Limits of validity of the relaxation time approximation are discussed for the various scattering possibilities. From this description, conservation equations are derived, and drift and diffusion discussed as an approximation. Brownian motion arising in fast-and-slow events and response are related to the drift and diffusion and to the Langevin and Fokker-Planck equations as probabilistic evolution. This leads to a discussion of Markov processes and the Kolmogorov equation.