Weakly Nonlocal Non-equilibrium Thermodynamics - Variational Principles and Second Law

A general, uniform, rigorous and constructive thermodynamic approach to weakly nonlocal non-equilibrium thermodynamics is reviewed. A method is given to construct and restrict the evolution equations of physical theories according to the Second Law of thermodynamics and considering weakly nonlocal constitutive state spaces. The evolution equations of internal variables, the classical irreversible thermodynamics and Korteweg fluids are treated.

[1]  S. Forest,et al.  Thermodynamical Frameworks for Higher Grade Material Theories with Internal Variables or Additional Degrees of Freedom , 2006 .

[2]  Miroslav Grmela,et al.  Weakly nonlocal heat conduction in rigid solids , 1996 .

[3]  G. Szamosi.  Variational principle in thermodynamics , 1973 .

[4]  P. Beda.  Dynamical systems, rate and gradient effects in material instability , 2000 .

[5]  Paolo Maria Mariano,et al.  Multifield theories in mechanics of solids , 2002 .

[6]  P. Mazur,et al.  Non-equilibrium thermodynamics, , 1963 .

[7]  Dick Bedeaux,et al.  The nonequilibrium van der Waals square gradient model. (III). Heat and mass transfer coefficients , 2004 .

[8]  M. Gurtin,et al.  Thermodynamics with Internal State Variables , 1967 .

[9]  J. Vendelin.  MICROSTRUCTURE DESCRIBED BY HIERARCHICAL INTERNAL VARIABLES , 2003 .

[10]  Lars Onsager,et al.  Fluctuations and Irreversible Process. II. Systems with Kinetic Energy , 1953 .

[11]  Gérard A. Maugin,et al.  The thermomechanics of nonlinear irreversible behaviors : an introduction , 1999 .

[12]  Raphael Andreas Hauser,et al.  A historical note on the entropy principle of Müller and Liu , 2002 .

[13]  Can material time derivative be objective , 2005, math-ph/0510037.

[14]  P. Ván,et al.  Hamilton formalism and variational principle construction , 1999 .

[15]  Gérard A. Maugin,et al.  THERMODYNAMICS WITH INTERNAL VARIABLES , 1999 .

[16]  József Verhás,et al.  Thermodynamics and Rheology , 1997 .

[17]  P. M. Mariano,et al.  Multifield theories: an introduction , 2001 .

[18]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[19]  Wolfgang Muschik,et al.  Thermodynamics with Internal Variables. Part II. Applications , 1994 .

[20]  Miroslav Grmela,et al.  Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism , 1997 .

[21]  P. Ván.  One- and two-component fluids: Restrictions from the second law , 2004 .

[22]  J. Kestin.  Internal Variables in the Local-Equilibrium Approximation , 1992 .

[23]  P. Ván.  Objective time derivatives in nonequilibrium thermodynamics , 2008 .

[24]  I. Oppenheim.  Beyond Equilibrium Thermodynamics , 2006 .

[25]  Wolfgang Muschik,et al.  A sketch of continuum thermodynamics , 2001 .

[26]  G. Maugin,et al.  Internal variables and the thermodynamics of macromolecule solutions , 1983 .

[27]  K. Hutter.  The foundations of thermodynamics, its basic postulates and implications. A review of modern thermodynamics , 1977 .

[28]  P. Ván.  The Ginzburg-Landau equation as a consequence of the Second Law , 2005 .

[29]  P. Ván,et al.  Weakly non-local fluid mechanics: the Schrödinger equation , 2003, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[30]  Wolfgang Muschik,et al.  Thermodynamics with Internal Variables. Part I. General Concepts , 1994 .

[31]  Vito Antonio Cimmelli,et al.  On the evolution of higher order fluxes in non-equilibrium thermodynamics , 2007, Math. Comput. Model..

[32]  W. Kosinski,et al.  Gradient generalization to internal state variable approach , 1995 .

[33]  István Gyarmati,et al.  Non-equilibrium Thermodynamics , 1970 .

[34]  István Gyarmati,et al.  Non-equilibrium thermodynamics : field theory and variational principles , 1970 .

[35]  G. Maugin.  Nonlocal theories or gradient-type theories - A matter of convenience , 1979 .

[36]  P. Hohenberg,et al.  Theory of Dynamic Critical Phenomena , 1977 .

[37]  P. Glansdorff,et al.  Thermodynamic theory of structure, stability and fluctuations , 1971 .

[38]  J. E. Dunn,et al.  On the Thermomechanics of Interstitial Working , 1985 .

[39]  V. Cimmelli.  An extension of Liu procedure in weakly nonlocal thermodynamics , 2007 .

[40]  P. Ván.  Internal Thermodynamic Variables and Failure of Microcracked Materials , 2001 .

[41]  Dick Bedeaux,et al.  The nonequilibrium van der Waals square gradient model. (II). Local equilibrium of the Gibbs surface , 2003 .

[42]  Peter Van,et al.  WEAKLY NONLOCAL IRREVERSIBLE THERMODYNAMICS , 2001 .

[43]  Arkadi Berezovski,et al.  Internal Variables and Dynamic Degrees of Freedom , 2006, cond-mat/0612491.

[44]  Peter Ireman,et al.  Using the gradients of temperature and internal parameters in Continuum Thermodynamics , 2004 .

[45]  Bernard D. Coleman,et al.  Existence of Caloric Equations of State in Thermodynamics , 1964 .

[47]  Wolfgang Muschik,et al.  Exploitation of the Second Law: Coleman–Noll and Liu Procedure in Comparison , 2008 .

[48]  Weakly nonlocal continuum theories of granular media: restrictions from the Second Law , 2003, cond-mat/0310520.

[49]  W. Muschik,et al.  An Amendment to the Second Law , 1996 .

[50]  A. Morro,et al.  A phase-field approach to non-isothermal transitions , 2008, Math. Comput. Model..

[51]  M. Frewer.  More clarity on the concept of material frame-indifference in classical continuum mechanics , 2008 .

[52]  T. Matolcsi.  ON THE OBJECTIVITY OF TIME DERIVATIVES , 2008 .

[53]  Dick Bedeaux,et al.  The nonequilibrium van der Waals square gradient model. (I). The model and its numerical solution , 2003 .

[54]  H. B. G. Casimir,et al.  On Onsager's Principle of Microscopic Reversibility , 1945 .

[55]  E. Madelung,et al.  Quantentheorie in hydrodynamischer Form , 1927 .

[56]  Weakly nonlocal hydrodynamics. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[57]  J. Verhás.  An application of Gyarmati's wave approach , 1989 .

[58]  Absolute time derivatives , 2006, math-ph/0608065.

[59]  Miroslav Grmela,et al.  Dynamics and thermodynamics of complex fluids. I. Development of a general formalism , 1997 .

[60]  D. Bedeaux,et al.  Non-equilibrium Thermodynamics of Heterogeneous Systems , 2008, Series on Advances in Statistical Mechanics.

[61]  I. Aranson,et al.  The world of the complex Ginzburg-Landau equation , 2001, cond-mat/0106115.

[62]  Carl Eckart,et al.  The Thermodynamics of Irreversible Processes. III. Relativistic Theory of the Simple Fluid , 1940 .

[63]  V. A. Cimmelli,et al.  Gradient generalization to internal state variables and a theory of superfluidity , 1997 .

[64]  W. Muschik,et al.  Structure of variational principles in nonequilibrium thermodynamics. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[65]  E. Aifantis,et al.  On the role of microstructure in the behavior of soils: Effects of higher order gradients and internal inertia , 1994 .

[66]  Stephen C. Cowin,et al.  A continuum theory for granular materials , 1972 .

[67]  K. C. Valanis,et al.  A gradient theory of internal variables , 1996 .

[68]  H. S. Allen.  The Quantum Theory , 1928, Nature.

[69]  ΓΛ Ja.  On the Entropy Current , 1983 .

[70]  Gérard A. Maugin,et al.  Infernal Variables and Dissipative Structures , 1990 .

[71]  M. Brokate,et al.  Hysteresis and Phase Transitions , 1996 .

[72]  K. Hutter,et al.  Elasto-plastic behaviour of a granular material with an additional scalar degree of freedom , 2002 .

[73]  M. Gurtin.  Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance , 1996 .

[74]  I-Shih Liu,et al.  Method of Lagrange multipliers for exploitation of the entropy principle , 1972 .

[75]  K. Frischmuth,et al.  Gradient generalization to the extended thermodynamic approach and diffusive-hyperbolic heat conduction , 2007 .

[76]  G. Maugin,et al.  Application of diffusive internal variables in complex fluids , 2001 .

[77]  J. D. van der Waals,et al.  Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung , 1894 .

[78]  I. Müller.  On the frame dependence of stress and heat flux , 1972 .

[79]  P. Holland.  The Quantum Theory of Motion , 1993 .

[80]  Schrutka.  Geometrie der Zahlen , 1911 .

[81]  P. Ván.  Exploiting the second law in weakly non-local continuum physics , 2005 .

[82]  Hans Christian Öttinger,et al.  General projection operator formalism for the dynamics and thermodynamics of complex fluids , 1998 .

[83]  Paul C. Fife,et al.  Thermodynamically consistent models of phase-field type for the kinetics of phase transitions , 1990 .

[84]  P. Ván,et al.  On the objectivity of time derivaties , 2008 .

[85]  Gérard A. Maugin,et al.  The method of virtual power in continuum mechanics: Application to coupled fields , 1980 .

[86]  Lars Onsager,et al.  Fluctuations and Irreversible Processes , 1953 .

[87]  Spacetime without reference frames: An application to the kinetic theory , 1996 .

[88]  I. Bialynicki-Birula,et al.  Nonlinear Wave Mechanics , 1976 .

[89]  I. Müller.  A thermodynamic theory of mixtures of fluids , 1968 .

[90]  P. V'an.  Internal energy in dissipative relativistic fluids , 2007, 0712.1437.

[91]  P V An,et al.  Weakly nonlocal irreversible thermodynamics , 2003, Annalen der Physik.

[92]  K. Valanis.  A gradient thermodynamic theory of self-organization , 1998 .

[93]  G. Maugin.  On the thermomechanics of continuous media with diffusion and/or weak nonlocality , 2006 .

[94]  V. A. Cimmelli,et al.  The effects of nonlocality on the evolution of higher order fluxes in nonequilibrium thermodynamics , 2004, cond-mat/0409254.