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Books on the topic 'Boltzmann Distributions'

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Published: 25 May 2024

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1

Alexeev, Boris V. Generalized Boltzmann physical kinetics. Amsterdam: Elsevier, 2004.

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2

Stochastic dynamics and Boltzmann hierarchy. Berlin: Walter de Gruyter, 2009.

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3

Hydrodynamic limits of the Boltzmann equation. Berlin: Springer, 2009.

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4

Yeh, Chou, and Langley Research Center, eds. On higher order dynamics in lattice-based models using Chapman-Enskog method. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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5

1973-, Villani Cédric, and Centre Émile Borel, eds. Entropy methods for the Boltzmann equation: Lectures from a special semester at the Centre Émile Borel, Institut H. Poincaré, Paris, 2001. Berlin: Springer, 2008.

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6

Lallemand, Pierre. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 2000.

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7

Bach, Alexander. Indistinguishable classical particles. Berlin: Springer, 1997.

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8

United States. National Aeronautics and Space Administration., ed. Spectroscopic diagnostics of an arc jet heated air plasma: Thesis ... [Washington, DC: National Aeronautics and Space Administration, 1996.

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9

United States. National Aeronautics and Space Administration., ed. Numerical investigations in the backflow region of a vacuum plume: Semi-annual scientific and technical reports, October 1991 - May 1992. [Washington, DC: National Aeronautics and Space Administration, 1992.

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10

Darrigol, Olivier. The Boltzmann Equation and the H Theorem (1872–1875). Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198816171.003.0004.

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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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11

Henriksen, Niels Engholm, and Flemming Yssing Hansen. From Microscopic to Macroscopic Descriptions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805014.003.0002.

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This chapter discusses bimolecular reactions from both a microscopic and macroscopic point of view. The outcome of an isolated reactive scattering event can be specified in terms of an intrinsic fundamental quantity, the reaction cross-section that can be measured in a molecular beam experiment. It depends on the quantum states of the molecules as well as the relative velocity of reactants and products. The relation between the cross-section and the macroscopic rate constant is derived. The rate constant is a weighted average of the product between the relative speed of the reactants and the reaction cross-section. The chapter concludes with the special case of thermal equilibrium, where the velocity distributions for the molecules are the Maxwell–Boltzmann distribution. The expression for the rate constant at temperature T is reduced to a one-dimensional integral over the relative speed of the reactants.
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12

Alexeev, Boris V. Generalized Boltzmann Physical Kinetics. Elsevier Science & Technology Books, 2004.

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13

Lecture Notes on the Mathematical Theory of Generalized Boltzmann Models (Series on Advances in Mathematics for Applied Sciences). World Scientific Publishing Company, 2000.

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14

Jr, Thorne Daniel T., Michael C. Sukop, and Daniel T. Thorne. Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers. Springer London, Limited, 2007.

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15

Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers. Springer, 2005.

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16

Wolf-Gladrow, Dieter A. Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction. Springer London, Limited, 2004.

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17

Henriksen, Niels Engholm, and Flemming Yssing Hansen. Introduction. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805014.003.0001.

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This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.
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18

On higher order dynamics in lattice-based models using Chapman-Enskog method. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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19

Succi, Sauro. The Hermite–Gauss Route to LBE. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780199592357.003.0015.

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This chapter describes the side-up approach to Lattice Boltzmann, namely the formal derivation from the continuum Boltzmann-(BGK) equation via Hermite projection and subsequent evaluation of the kinetic moments via Gauss–Hermite quadrature. From a slightly different angle, one may also interpret the Gauss–Hermite quadrature as an optimal sampling of velocity space, or, better still, an exact sampling of the bulk of the distribution function, the one contributing most to the lowest order kinetic moments (frequent events). Capturing higher–order moments, beyond hydrodynamics (rare events), requires an increasing number of nodes and weights.
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20

Rezakhanlou, Fraydoun, Cédric Villani, Stefano Olla, and François Golse. Entropy Methods for the Boltzmann Equation: Lectures from a Special Semester at the Centre Émile Borel, Institut H. Poincaré, Paris 2001. Springer London, Limited, 2007.

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21

Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction (Lecture Notes in Mathematics). Springer, 2000.

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22

Darrigol, Olivier. Consolidation (1887–1895). Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198816171.003.0007.

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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, Gustav Kirchhoff, and Max Planck. He corrected errors in alleged counterexamples of equipartition by William Burnside and William Thomson; and in 1887, when the Dutch theorist Hendrik Lorentz detected an error in his earlier derivation of the H theorem for polyatomic gases, he devised a highly ingenious alternative. In 1894, he offered a new, simplified derivation of the Maxwell–Boltzmann distribution based on an idea by the British mathematician George Bryan. Together with Bryan, he also provided a kinetic-molecular model for the equalization of the temperatures of two contiguous gases. He denounced what he believed to be an error in Gustav Kirchhoff’s derivation of Maxwell’s distribution, and he strengthened Max Planck’s alternative derivation based on time reversal.
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23

From the Vlasov-Maxwell-Boltzmann System to Incompressible Viscous Electro-Magneto-Hydrodynamics. American Mathematical Society, 2019.

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24

Darrigol, Olivier. The Probabilistic Turn (1876–1884). Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198816171.003.0005.

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This chapter deals with writings in which Boltzmann expressed the statistical nature of the entropy law and temporarily made the relation between entropy and combinatorial probability a basic constructive tool of his theory. In 1881, he discovered that this relation derived from what we now call the microcanonical distribution, and he approved Maxwell’s recent foundation of the equilibrium problem on the microcanonical ensemble. Boltzmann also kept working on problems he had tackled in earlier years. He proposed a new solution to the problem of specific heats, and he performed enormous calculations for the viscosity and diffusion coefficients in the hard-ball model. In a lighter genre, he conceived a new way of determining molecular sizes, and he speculated on a gas model in which the molecular forces would be entirely attractive.
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25

Deruelle, Nathalie, and Jean-Philippe Uzan. Kinetic theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0010.

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This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.
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26

Spectroscopic diagnostics of an arc jet heated air plasma: Thesis ... [Washington, DC: National Aeronautics and Space Administration, 1996.

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27

Generalized kinetic models in applied sciences: Lecture notes on mathematical problems. Singapore: World Scientific, 2003.

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28

Myrvold, Wayne. Probabilities in Statistical Mechanics. Edited by Alan Hájek and Christopher Hitchcock. Oxford University Press, 2017. http://dx.doi.org/10.1093/oxfordhb/9780199607617.013.26.

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This chapter reviews selected aspects of the terrain of discussion of the role of probabilities in statistical mechanics. Among the topics addressed are the reasons for introduction of probabilities into statistical mechanics, the status of the standard equilibrium distribution, and the question of interpretation of statistical mechanical probabilities. The chapter starts with a brief history of probabilities in physics and the evolution of statistical mechanics therefrom. The approaches of Boltzmann and Gibbs are presented, and then some approaches to justifying choice of probability measures. The chapter closes with a presentation of possible resolutions to some puzzling aspects of the use of standard probability measures.
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29

Clarke, Andrew. Energy and heat. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780199551668.003.0002.

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Energy is the capacity to do work and heat is the spontaneous flow of energy from one body or system to another through the random movement of atoms or molecules. The entropy of a system determines how much of its internal energy is unavailable for work under isothermal conditions, and the Gibbs energy is the energy available for work under isothermal conditions and constant pressure. The Second Law of Thermodynamics states that for any reaction to proceed spontaneously the total entropy (system plus surroundings) must increase, which is why metabolic processes release heat. All organisms are thermodynamically open systems, exchanging both energy and matter with their surroundings. They can decrease their entropy in growth and development by ensuring a greater increase in the entropy of the environment. For an ideal gas in thermal equilibrium the distribution of energy across the component atoms or molecules is described by the Maxwell-Boltzmann equation. This distribution is fixed by the temperature of the system.
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30

Rau, Jochen. Perfect Gas. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780199595068.003.0006.

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The perfect gas is perhaps the most prominent application of statistical mechanics and for this reason merits a chapter of its own. This chapter briefly reviews the quantum theory of many identical particles, in particular the distinction between bosons and fermions, and then develops the general theory of the perfect quantum gas. It considers a number of limits and special cases: the classical limit; the Fermi gas at low temperature; the Bose gas at low temperature which undergoes Bose–Einstein condensation; as well as black-body radiation. For the latter we derive the Stefan–Boltzmann law, the Planck distribution, and Wien’s displacement law. This chapter also discusses the effects of a possible internal dynamics of the constituent molecules on the thermodynamic properties of a gas. Finally, it extends the theory of the perfect gas to dilute solutions.
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31

Sherwood, Dennis, and Paul Dalby. Temperature and heat. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198782957.003.0003.

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Concepts of temperature, temperature scales and temperature measurement. The ideal gas law, Dalton’s law of partial pressure. Assumptions underlying the ideal gas, and distinction between ideal and real gases. Introduction to equations-of-state such as the van der Waals, Dieterici, Berthelot and virial equations, which describe real gases. Concept of heat, and distinction between heat and temperature. Experiments of Rumford and Joule, and the principle of the conservation of energy. Units of measurement for heat. Heat as a path function. Flow of heat down a temperature gradient as an irreversible and unidirectional process. ‘Zeroth’ Law of Thermodynamics. Definitions of isolated, closed and open systems, and of isothermal, adiabatic, isobaric and isothermal changes in state. Connection between work and heat, as illustrated by the steam engine. The molecular interpretation of heat, energy and temperature. The Boltzmann distribution. Meaning of negative temperatures.
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32

Henriksen, Niels Engholm, and Flemming Yssing Hansen. Static Solvent Effects, Transition-State Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805014.003.0010.

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This chapter discusses static solvent effects on the rate constant for chemical reactions in solution. It starts with a brief discussion of the thermodynamic formulation of transition-state theory. The static equilibrium structure of the solvent will modify the potential energy surface for the chemical reaction. This effect is analyzed within the framework of transition-state theory. The rate constant is expressed in terms of the potential of mean force at the activated complex. Various definitions of this potential and their relations to n-particle- and pair-distribution functions are considered. The potential of mean force may, for example, be defined such that the gradient of the potential gives the average force on an atom in the activated complex, Boltzmann averaged over all configurations of the solvent. It concludes with a discussion of a relation between the rate constants in the gas phase and in solution.
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33

Darrigol, Olivier, and Jürgen Renn. The Emergence of Statistical Mechanics. Edited by Jed Z. Buchwald and Robert Fox. Oxford University Press, 2017. http://dx.doi.org/10.1093/oxfordhb/9780199696253.013.26.

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This article traces the history of statistical mechanics, beginning with a discussion of mechanical models of thermal phenomena. In particular, it considers how several circumstances, including the establishment of thermodynamics in the mid-nineteenth century, led to a focus on the model of heat as a motion of particles. It then describes the concept of heat as fluid and the kinetic theory before turning to gas theory and how it served as a bridge between mechanics and thermodynamics. It also explores gases as particles in motion, the Maxwell–Boltzmann distribution, the problem of specific heats, challenges to the second law of thermodynamics, and the probabilistic interpretation of entropy. Finally, it examines how the results of the kinetic theory assumed a new meaning as cornerstones of a more broadly conceived statistical physics, along with Josiah Willard Gibbs and Albert Einstein’s development of statistical mechanics as a synthetic framework.
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34

Mann, Peter. Hamilton-Jacobi Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0019.

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This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.
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35

Henriksen, Niels E., and Flemming Y. Hansen. Theories of Molecular Reaction Dynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805014.001.0001.

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This book deals with a central topic at the interface of chemistry and physics—the understanding of how the transformation of matter takes place at the atomic level. Building on the laws of physics, the book focuses on the theoretical framework for predicting the outcome of chemical reactions. The style is highly systematic with attention to basic concepts and clarity of presentation. Molecular reaction dynamics is about the detailed atomic-level description of chemical reactions. Based on quantum mechanics and statistical mechanics or, as an approximation, classical mechanics, the dynamics of uni- and bimolecular elementary reactions are described. The first part of the book is on gas-phase dynamics and it features a detailed presentation of reaction cross-sections and their relation to a quasi-classical as well as a quantum mechanical description of the reaction dynamics on a potential energy surface. Direct approaches to the calculation of the rate constant that bypasses the detailed state-to-state reaction cross-sections are presented, including transition-state theory, which plays an important role in practice. The second part gives a comprehensive discussion of basic theories of reaction dynamics in condensed phases, including Kramers and Grote–Hynes theory for dynamical solvent effects. Examples and end-of-chapter problems are included in order to illustrate the theory and its connection to chemical problems. The book has ten appendices with useful details, for example, on adiabatic and non-adiabatic electron-nuclear dynamics, statistical mechanics including the Boltzmann distribution, quantum mechanics, stochastic dynamics and various coordinate transformations including normal-mode and Jacobi coordinates.
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36

Numerical investigations in the backflow region of a vacuum plume: Performance report, May 1994 - August 1995. Normal, Ala: Alabama A&M University, 1995.

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