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6816 results in Condensed Matter Physics, Nanoscience and Mesoscopic Physics

Page 36 of 341

  • 12 - Divergences, Resummations, and Logarithms

  • J. R. Dorfman, University of Maryland, College Park, Henk van Beijeren, Universiteit Utrecht, The Netherlands, T. R. Kirkpatrick, University of Maryland, College Park
  • Book:
    Contemporary Kinetic Theory of Matter
    Published online:
    18 June 2021
    Print publication:
    24 June 2021, pp 467-506

  • Summary

    Cluster expansion methods provide a power series expansion in the density for the collision operator in the equation for the time dependence of the one particle distribution function. Successive terms depend on the dynamics of successively larger numbers of particles. All but the first few terms grow with time, for long times, due to contributions from correlated sequences of binary collisions. Useful expressions are obtained by summing the fastest growing terms in each order of the density. This “ring resummation” predicts that the density expansion for transport coefficients contains terms proportional to logarithms of the gas density. The leading logarithmic terms in the expansion have been calculated for several systems and are in good agreement with the results of computer simulations. The ring sum also provides a microscopic foundation for mode coupling theory needed for a description of the long time behavior of Green-Kubo correlation functions and other quantities.

Contemporary Kinetic Theory of Matter
  • J. R. Dorfman, Henk van Beijeren, T. R. Kirkpatrick
  • Published online:
    18 June 2021
    Print publication:
    24 June 2021
  • Kinetic theory provides a microscopic description of many observable, macroscopic processes and has a wide range of important applications in physics, astronomy, chemistry, and engineering. This powerful, theoretical framework allows a quantitative treatment of many non-equilibrium phenomena such as transport processes in classical and quantum fluids. This book describes in detail the Boltzmann equation theory, obtained in both traditional and modern ways. Applications and generalizations describing non-equilibrium processes in a variety of systems are also covered, including dilute and moderately dense gases, particles in random media, hard sphere crystals, condensed Bose-Einstein gases, and granular materials. Fluctuation phenomena in non-equilibrium fluids, and related non-analyticities in the hydrodynamic equations are also discussed in some detail. A thorough examination of many topics concerning time dependent phenomena in material systems, this book describes both current knowledge as well as future directions of the field.
A Unified Computational Fluid Dynamics Framework from Rarefied to Continuum Regimes

A Unified Computational Fluid Dynamics Framework from Rarefied to Continuum Regimes
  • Kun Xu
  • Published online:
    13 May 2021
    Print publication:
    10 June 2021
  • This Element presents a unified computational fluid dynamics framework from rarefied to continuum regimes. The framework is based on the direct modelling of flow physics in a discretized space. The mesh size and time step are used as modelling scales in the construction of discretized governing equations. With the variation-of-cell Knudsen number, continuous modelling equations in different regimes have been obtained, and the Boltzmann and Navier-Stokes equations become two limiting equations in the kinetic and hydrodynamic scales. The unified algorithms include the discrete velocity method (DVM)–based unified gas-kinetic scheme (UGKS), the particlebased unified gas-kinetic particle method (UGKP), and the wave and particle–based unified gas-kinetic wave-particle method (UGKWP). The UGKWP is a multi-scale method with the particle for non-equilibrium transport and wave for equilibrium evolution. The particle dynamics in the rarefied regime and the hydrodynamic flow solver in the continuum regime have been unified according to the cell's Knudsen number.

Page 36 of 341