December  2017, 10(6): 1393-1411. doi: 10.3934/dcdss.2017074

On the modeling of transport phenomena in continuum and statistical mechanics

1. 

Accademia Nazionale dei Lincei, Palazzo Corsini, Via della Lungara, 10 -00165 Roma, Italy

2. 

Department of Mathematics, University of Rome TorVergata, Via della Ricerca Scientifica, 1 -00133 Roma, Italy

Dedicated to Tomáš Roubíček on the occasion of his sixtieth birthday

Received  November 2016 Revised  February 2017 Published  June 2017

The formulation of balance laws in continuum and statistical mechanics is expounded in forms that open the way to revise and review the correspondence instituted, in a manner proposed by Irving and Kirkwood in 1950 and improved by Noll in 1955 and 2010, between the basic balance laws of Cauchy continua and those of standard Hamiltonian systems of particles.

Citation: Paolo Podio-Guidugli. On the modeling of transport phenomena in continuum and statistical mechanics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1393-1411. doi: 10.3934/dcdss.2017074
References:
[1]

N. C. Admal and E. B. Tadmor, Stress and heat flux for arbitrary multibody potentials: A unified framework, J. Chem. Phys., 134 (2011), 184106. Google Scholar

[2]

R. J. Bearman and J. G. Kirkwood, The statistical mechanical theory of transport processes. XI. Equations of Transport in Multicomponent Systems, J. Chem. Phys., 28 (1958), 138-145. doi: 10.1063/1.1744056. Google Scholar

[3]

D. Davydov and P. Steinmann, Reviewing the roots of continuum formulations in molecular systems. Part Ⅱ: Energy and angular momentum balance equation, Math. Mech. Solids, 19 (2014), 852-867. doi: 10.1177/1081286513490301. Google Scholar

[4]

D. Davydov and P. Steinmann, Reviewing the roots of continuum formulations in molecular systems. Part Ⅰ: Particle dynamics, statistical physics, mass and linear momentum balance equations, Math. Mech. Solids, 19 (2014), 411-433. doi: 10.1177/1081286512467790. Google Scholar

[5]

D. Davydov and P. Steinmann, Reviewing the roots of continuum formulations in molecular systems. Part Ⅲ: Stresses, couple stresses, heat fluxes, Math. Mech. Solids, 20 (2015), 1153-1170. doi: 10.1177/1081286513516480. Google Scholar

[6]

A. Einstein, Über die von der molekular-kinetischen Theorie der Wärme gefordete Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys. (Leipzig), 17 (1905), 549; English translation: On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat. In A. Einstein, Investigations on the Theory of the Brownian Movement Dover Pub. s, 1956.Google Scholar

[7]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5. Google Scholar

[8]

M. E. Gurtin and P. Podio-Guidugli, Configurational forces and the basic laws for crack propagation, J. Mechan. Phys. Solids, 44 (1996), 905-927. doi: 10.1016/0022-5096(96)00014-2. Google Scholar

[9]

M. E. Gurtin and P. Podio-Guidugli, On configurational inertial forces at a phase interface, J. Elasticity, 44 (1996), 255-269. doi: 10.1007/BF00042135. Google Scholar

[10]

M. E. Gurtin and P. Podio-Guidugli, Configurational forces and a constitutive theory for crack propagation that allows for kinking and curving, J. Mechan. Phys. Solids, 46 (1998), 1343-1378. doi: 10.1016/S0022-5096(98)00002-7. Google Scholar

[11]

R. J. Hardy, Formulas for determining local properties in molecular dynamics simulations: Shock waves, J. Chem. Phys., 76 (1982), 622-628. doi: 10.1063/1.442714. Google Scholar

[12]

J. H. Irving and J. G. Kirkwood, The statistical mechanical theory of transport processes. Ⅳ. The equations of hydrodynamics, J. Chem. Phys., 18 (1950), 817-829. doi: 10.1063/1.1747782. Google Scholar

[13]

J. G. Kirkwood and D. D. Fitts, Statistical mechanics of transport processes. XIV. Linear relations in multicomponent systems, J. Chem. Phys., 33 (1960), 1317-1324. doi: 10.1063/1.1731406. Google Scholar

[14]

P. Langevin, Sur la théorie du mouvement brownien, Comptes Rendues, 146 (1908), 530. Google Scholar

[15] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Fourth Ed. Dover Publications, New York, 1944. Google Scholar
[16] A. I. Murdoch, Physical Foundations of Continuum Mechanics, Cambridge University Press, 2012. doi: 10.1017/CBO9781139028318. Google Scholar
[17]

W. Noll, Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistischen Mechanik, Indiana Univ. Math. J. , 4 (1955), 627-646; English translation: Derivation of the fundamental equations of continuum thermodynamics from statistical mechanics. J. Elasticity, 100 (2010), 5-24. Google Scholar

[18]

W. Noll, Thoughts on the concept of stress, J. Elasticity, 100 (2010), 25-32. doi: 10.1007/s10659-010-9247-8. Google Scholar

[19]

P. Podio-Guidugli, Inertia and invariance, Ann. Mat. Pura Appl. (Ⅳ), 172 (1997), 103-124. doi: 10.1007/BF01782609. Google Scholar

[20]

P. Podio-Guidugli, La scelta dei termini inerziali per i continui con microstruttura, Rend. Lincei-Mat. Appl. Serie Ⅸ, XIV (2003), 319-326. Google Scholar

[21]

B. Seguin and E. Fried, Statistical foundations of liquid-crystal theory. Ⅰ: Discrete systems of rod-like molecules, Arch. Rational Mech. Anal., 206 (2012), 1039-1072. doi: 10.1007/s00205-012-0550-3. Google Scholar

[22]

B. Seguin and E. Fried, Statistical foundations of liquid-crystal theory. Ⅱ: Macroscopic balance laws, Arch. Rational Mech. Anal., 207 (2013), 1-37. doi: 10.1007/s00205-012-0551-2. Google Scholar

[23] E. B. Tadmor and R. E. Miller, Modeling Materials. Continuum, Atomistic and Multiscale Techniques, Cambridge University Press, 2011. Google Scholar
[24]

C. Truesdell and R. A. Toupin, The classical field theories, In Handbuch der Physik Ⅲ/1, Springer, (1960), 226-793. Google Scholar

show all references

References:
[1]

N. C. Admal and E. B. Tadmor, Stress and heat flux for arbitrary multibody potentials: A unified framework, J. Chem. Phys., 134 (2011), 184106. Google Scholar

[2]

R. J. Bearman and J. G. Kirkwood, The statistical mechanical theory of transport processes. XI. Equations of Transport in Multicomponent Systems, J. Chem. Phys., 28 (1958), 138-145. doi: 10.1063/1.1744056. Google Scholar

[3]

D. Davydov and P. Steinmann, Reviewing the roots of continuum formulations in molecular systems. Part Ⅱ: Energy and angular momentum balance equation, Math. Mech. Solids, 19 (2014), 852-867. doi: 10.1177/1081286513490301. Google Scholar

[4]

D. Davydov and P. Steinmann, Reviewing the roots of continuum formulations in molecular systems. Part Ⅰ: Particle dynamics, statistical physics, mass and linear momentum balance equations, Math. Mech. Solids, 19 (2014), 411-433. doi: 10.1177/1081286512467790. Google Scholar

[5]

D. Davydov and P. Steinmann, Reviewing the roots of continuum formulations in molecular systems. Part Ⅲ: Stresses, couple stresses, heat fluxes, Math. Mech. Solids, 20 (2015), 1153-1170. doi: 10.1177/1081286513516480. Google Scholar

[6]

A. Einstein, Über die von der molekular-kinetischen Theorie der Wärme gefordete Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys. (Leipzig), 17 (1905), 549; English translation: On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat. In A. Einstein, Investigations on the Theory of the Brownian Movement Dover Pub. s, 1956.Google Scholar

[7]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. doi: 10.1016/0167-2789(95)00173-5. Google Scholar

[8]

M. E. Gurtin and P. Podio-Guidugli, Configurational forces and the basic laws for crack propagation, J. Mechan. Phys. Solids, 44 (1996), 905-927. doi: 10.1016/0022-5096(96)00014-2. Google Scholar

[9]

M. E. Gurtin and P. Podio-Guidugli, On configurational inertial forces at a phase interface, J. Elasticity, 44 (1996), 255-269. doi: 10.1007/BF00042135. Google Scholar

[10]

M. E. Gurtin and P. Podio-Guidugli, Configurational forces and a constitutive theory for crack propagation that allows for kinking and curving, J. Mechan. Phys. Solids, 46 (1998), 1343-1378. doi: 10.1016/S0022-5096(98)00002-7. Google Scholar

[11]

R. J. Hardy, Formulas for determining local properties in molecular dynamics simulations: Shock waves, J. Chem. Phys., 76 (1982), 622-628. doi: 10.1063/1.442714. Google Scholar

[12]

J. H. Irving and J. G. Kirkwood, The statistical mechanical theory of transport processes. Ⅳ. The equations of hydrodynamics, J. Chem. Phys., 18 (1950), 817-829. doi: 10.1063/1.1747782. Google Scholar

[13]

J. G. Kirkwood and D. D. Fitts, Statistical mechanics of transport processes. XIV. Linear relations in multicomponent systems, J. Chem. Phys., 33 (1960), 1317-1324. doi: 10.1063/1.1731406. Google Scholar

[14]

P. Langevin, Sur la théorie du mouvement brownien, Comptes Rendues, 146 (1908), 530. Google Scholar

[15] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Fourth Ed. Dover Publications, New York, 1944. Google Scholar
[16] A. I. Murdoch, Physical Foundations of Continuum Mechanics, Cambridge University Press, 2012. doi: 10.1017/CBO9781139028318. Google Scholar
[17]

W. Noll, Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistischen Mechanik, Indiana Univ. Math. J. , 4 (1955), 627-646; English translation: Derivation of the fundamental equations of continuum thermodynamics from statistical mechanics. J. Elasticity, 100 (2010), 5-24. Google Scholar

[18]

W. Noll, Thoughts on the concept of stress, J. Elasticity, 100 (2010), 25-32. doi: 10.1007/s10659-010-9247-8. Google Scholar

[19]

P. Podio-Guidugli, Inertia and invariance, Ann. Mat. Pura Appl. (Ⅳ), 172 (1997), 103-124. doi: 10.1007/BF01782609. Google Scholar

[20]

P. Podio-Guidugli, La scelta dei termini inerziali per i continui con microstruttura, Rend. Lincei-Mat. Appl. Serie Ⅸ, XIV (2003), 319-326. Google Scholar

[21]

B. Seguin and E. Fried, Statistical foundations of liquid-crystal theory. Ⅰ: Discrete systems of rod-like molecules, Arch. Rational Mech. Anal., 206 (2012), 1039-1072. doi: 10.1007/s00205-012-0550-3. Google Scholar

[22]

B. Seguin and E. Fried, Statistical foundations of liquid-crystal theory. Ⅱ: Macroscopic balance laws, Arch. Rational Mech. Anal., 207 (2013), 1-37. doi: 10.1007/s00205-012-0551-2. Google Scholar

[23] E. B. Tadmor and R. E. Miller, Modeling Materials. Continuum, Atomistic and Multiscale Techniques, Cambridge University Press, 2011. Google Scholar
[24]

C. Truesdell and R. A. Toupin, The classical field theories, In Handbuch der Physik Ⅲ/1, Springer, (1960), 226-793. Google Scholar

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