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The Consequences of Extensivity

Robert H. Swendsen

in An Introduction to Statistical Mechanics and Thermodynamics

Published in print:
2012
Published Online:
December 2013
ISBN:
9780199646944
eISBN:
9780191775123
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199646944.003.0013
Subject:
Physics, Condensed Matter Physics / Materials

Although extensivity is not an essential property of thermodynamic systems, it is a very useful concept for analysing the properties of materials when we can ignore surface and boundary effects. This ... More


ROTATIONAL DIFFUSION

Robert M. Mazo

in Brownian Motion: Fluctuations, Dynamics, and Applications

Published in print:
2008
Published Online:
January 2010
ISBN:
9780199556441
eISBN:
9780191705625
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199556441.003.0015
Subject:
Physics, Condensed Matter Physics / Materials

This chapter treats rotational diffusion in the Smoluchowski limit, but not limited to uniaxial diffusion. The rotational Langevin equation for a spherical particle with a homogeneous external field ... More


Mathematical Tools

Hans-Peter Eckle

in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

Published in print:
2019
Published Online:
September 2019
ISBN:
9780199678839
eISBN:
9780191878589
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780199678839.003.0019
Subject:
Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

Chapter 19 introduces the mathematical techniques required to extract analytic infor- mation from the Bethe ansatz equations for a Heisenberg quantum spin chain of finite length. It discusses how the ... More


Fluid Mechanics: A Geometrical Point of View

S. G. Rajeev

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.001.0001
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

Starting with a review of vector fields and their integral curves, the book presents the basic equations of the subject: Euler and Navier–Stokes. Some solutions are studied next: ideal flows using ... More


Geometric Integrators

S. G. Rajeev

in Fluid Mechanics: A Geometrical Point of View

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.003.0015
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

Generic methods for solving ordinary differential equations (ODEs, e.g., Runge-Kutta) can break the symmetries that a particular equation might have. Lie theory can be used to get Geometric ... More


Finite Heisenberg Quantum Spin Chain

Hans-Peter Eckle

in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

Published in print:
2019
Published Online:
September 2019
ISBN:
9780199678839
eISBN:
9780191878589
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780199678839.003.0020
Subject:
Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of ... More


The Consequences of Extensivity

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/9780198853237.003.0013
Subject:
Physics, Condensed Matter Physics / Materials, Theoretical, Computational, and Statistical Physics

While not all thermodynamic systems are extensive, those that are homogeneous satisfy the useful postulate of extensivity. In this chapter we return to the thermodynamic postulates and consider the ... More


Nonlinear Bending of Straight Beams

J. N. Reddy

in An Introduction to Nonlinear Finite Element Analysis, 2nd Edn: with applications to heat transfer, fluid mechanics, and solid mechanics

Published in print:
2014
Published Online:
June 2015
ISBN:
9780199641758
eISBN:
9780191789557
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199641758.003.0005
Subject:
Physics, Condensed Matter Physics / Materials

This chapter discusses two different theories to model the kinematic behaviour of beams – the Euler–Bernoulli beam theory and the Timoshenko beam theory – and their finite element models are ... More


Nonlinear Elasticity, Plasticity, and Viscoelasticity

J. N. Reddy

in An Introduction to Nonlinear Finite Element Analysis, 2nd Edn: with applications to heat transfer, fluid mechanics, and solid mechanics

Published in print:
2014
Published Online:
June 2015
ISBN:
9780199641758
eISBN:
9780191789557
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199641758.003.0012
Subject:
Physics, Condensed Matter Physics / Materials

This chapter discusses finite element models of materially nonlinear elastic and plastic models of one-dimensional problems. It presents efficient and accurate locking-free linear viscoelastic beam ... More


Bernoulli’s equation

Marcel Escudier

in Introduction to Engineering Fluid Mechanics

Published in print:
2017
Published Online:
January 2018
ISBN:
9780198719878
eISBN:
9780191840180
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198719878.003.0007
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

In this chapter Newton’s second law of motion is used to derive Euler’s equation for the flow of an inviscid fluid along a streamline. For a fluid of constant density ρ‎ Euler’s equation can be ... More


Euler’s Equations

S. G. Rajeev

in Fluid Mechanics: A Geometrical Point of View

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.003.0002
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

Euler derived the fundamental equations of an ideal fluid, that is, in the absence of friction (viscosity). They describe the conservation of momentum. We can derive from it the equation for the ... More


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