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 and 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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

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

[18]

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

[19]

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

[20]

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

[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.

[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.

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

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

show all references

Dedicated to Tomáš Roubíček on the occasion of his sixtieth birthday

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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

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

[18]

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

[19]

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

[20]

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

[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.

[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.

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

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

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