February  2017, 10(1): 1-35. doi: 10.3934/dcdss.2017001

On microscopic origins of generalized gradient structures

1. 

Weierstra$\beta $-Institut für Angewandte Analysis und Stochastik, Mohrenstra$\beta $e 39, 10117 Berlin, Germany

2. 

Department of Mathematics and Computer Science and Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands

* Corresponding author:Alexander Mielke

Received  July 2015 Revised  February 2016 Published  December 2016

Fund Project: The research was partially supported by Einstein Stiftung Berlin, ERC AdG 267802, and DFG via SFB 1114

Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures.

A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials.

A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary $Γ$-limit. As examples we treat (ⅰ) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ⅱ) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

Citation: Matthias Liero, Alexander Mielke, Mark A. Peletier, D. R. Michiel Renger. On microscopic origins of generalized gradient structures. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 1-35. doi: 10.3934/dcdss.2017001
References:
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R. AbeyaratneC.-H. Chu and R. James, Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape memory alloy, Phil. Mag. A, 73 (1996), 457-497. doi: 10.1080/01418619608244394.

[2]

S. Adams, N. Dirr, M. Peletier and J. Zimmer, Large deviations and gradient flows Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013), 20120341, 17pp. doi: 10.1098/rsta.2012.0341.

[3]

S. AdamsN. DirrM. A. Peletier and J. Zimmer, From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage, Comm. Math. Phys., 307 (2011), 791-815. doi: 10.1007/s00220-011-1328-4.

[4]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.

[5]

S. ArnrichA. MielkeM. A. PeletierG. Savaré and M. Veneroni, Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction, Calc. Var. Part. Diff. Eqns., 44 (2012), 419-454. doi: 10.1007/s00526-011-0440-9.

[6]

M. A. Biot, Variational principles in irreversible thermodynamics with applications to viscoelasticity, Phys. Review, 97 (1955), 1463-1469. doi: 10.1103/PhysRev.97.1463.

[7]

G. A. Bonaschi and M. A. Peletier, Quadratic and rate-independent limits for a large-deviations functional, Contin. Mech. Thermodyn., 28 (2016), 1191-1219. doi: 10.1007/s00161-015-0470-1.

[8]

E. De GiorgiA. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68 (1980), 180-187.

[9]

K. Disser and M. Liero, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, Networks Heterg. Media, 10 (2015), 233-253. doi: 10.3934/nhm.2015.10.233.

[10]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems North Holland, 1976.

[11]

M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Rational Mech. Anal., 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z.

[12]

M. Feinberg, On chemical kinetics of a certain class, Arch. Rational Mech. Anal., 46 (1972), 1-41.

[13]

W. Fenchel, On conjugate convex functions, Canadian J. Math., 1 (1949), 73-77. doi: 10.4153/CJM-1949-007-x.

[14]

J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes vol. 131 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/131.

[15]

N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Analysis, 45 (2013), 879-899. doi: 10.1137/120886315.

[16]

A. Glitzky and A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. angew. Math. Phys. (ZAMP), 64 (2013), 29-52. doi: 10.1007/s00033-012-0207-y.

[17]

A. N. GorbanI. V. KarlinV. B. Zmievskii and S. V. Dymova, Reduced description in the reaction kinetics, Physica A, 275 (2000), 361-379. doi: 10.1016/S0378-4371(99)00402-1.

[18]

M. Grmela, Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering, Adv. Chem. Eng., 39 (2010), 75-129. doi: 10.1016/S0065-2377(10)39002-8.

[19]

K. Hackl and F. D. Fischer, On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials, Proc. R. Soc. A, 464 (2008), 117-132. doi: 10.1098/rspa.2007.0086.

[20]

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, J. Mécanique, 14 (1975), 39-63.

[21]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[22]

M. Liero, Variational Methods for Evolution PhD thesis, Institut für Mathematik, Humboldt-Universität zu Berlin, 2012.

[23]

M. Liero, Passing from bulk to bulk-surface evolution in the Allen-Cahn equation, Nonl. Diff. Eqns. Appl. (NoDEA), 20 (2013), 919-942. doi: 10.1007/s00030-012-0189-7.

[24]

J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009.

[25]

J. Maas and A. Mielke, On gradient structures for chemical reactions with detailed balance: Ⅰ. modeling and large-volume limit, In preparation.

[26]

J. Maas and A. Mielke, On gradient structures for chemical reactions with detailed balance: Ⅱ. dissiaption distances and geodesic convexity, In preparation.

[27]

A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn., 15 (2003), 351-382. doi: 10.1007/s00161-003-0120-x.

[28]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016.

[29]

A. Mielke, Emergence of rate-independent dissipation from viscous systems with wiggly energies, Contin. Mech. Thermodyn., 24 (2012), 591-606. doi: 10.1007/s00161-011-0216-7.

[30]

A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Part. Diff. Eqns., 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8.

[31]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discr. Cont. Dynam. Systems Ser. S, 6 (2013), 479-499. doi: 10.3934/dcdss.2013.6.479.

[32]

A. Mielke, Variational approaches and methods for dissipative material models with multiple scales, in Analysis and Computation of Microstructure in Finite Plasticity (eds. K. Hackl and S. Conti), vol. 78 of Lect. Notes Appl. Comp. Mechanics, Springer, 2015, chapter 5, 125-155. doi: 10.1007/978-3-319-18242-1_5.

[33]

A. Mielke, On evolutionary $Γ$-convergence for gradient systems (ch. 3), in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity (eds. A. Muntean, J. Rademacher and A. Zagaris), 187-249, Lect. Notes Appl. Math. Mech. , 3, Springer, [Cham], 2016. doi: 10.1007/978-3-319-26883-5_3.

[34]

A. Mielke, R. I. A. Patterson, M. A. Peletier and D. R. M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, J. Chem. Physics Submitted WIAS preprint 2165.

[35]

A. MielkeM. A. Peletier and D. R. M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014), 1293-1327. doi: 10.1007/s11118-014-9418-5.

[36]

A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application Applied Mathematical Sciences, Vol. 193, Springer-Verlag New York, 2015. doi: 10.1007/978-1-4939-2706-7.

[37]

A. Mielke and L. Truskinovsky, From discrete visco-elasticity to continuum rate-independent plasticity: Rigorous results, Arch. Rational Mech. Anal., 203 (2012), 577-619. doi: 10.1007/s00205-011-0460-9.

[38]

L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev., 91 (1953), 1505-1512. doi: 10.1103/PhysRev.91.1505.

[39]

L. Onsager, Reciprocal relations in irreversible processes, Ⅰ+Ⅱ, Physical Review, 37 (1931), 405-426, (part Ⅱ, 38 (1931), 2265-2279). doi: 10.1103/PhysRev.38.2265.

[40]

M. A. PeletierG. Savaré and M. Veneroni, From diffusion to reaction via $Γ$-convergence, SIAM J. Math. Analysis, 42 (2010), 1805-1825. doi: 10.1137/090781474.

[41]

M. A. PeletierG. Savaré and M. Veneroni, Chemical reactions as $Γ$-limit of diffusion revised reprint of 40, SIAM Rev., 54 (2012), 327-352. doi: 10.1137/110858781.

[42]

L. Rayleigh and Hon. J. W. Strutt, Some general theorems relating to vibrations, Proc. London Math. Soc., s1-4 (1871), 357-368. doi: 10.1112/plms/s1-4.1.357.

[43]

D. R. M. Renger, Microscopic Interpretation of Wasserstein Gradient Flows PhD thesis, Technische Universiteit Eindhoven, 2013.

[44]

F. RotersD. Raabe and G. Gottstein, Work hardening in heterogeneous alloys -a microstructural approach based on three internal state variables, Acta Materialia, 48 (2000), 4181-4189. doi: 10.1016/S1359-6454(00)00289-5.

[45]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.

[46]

S. Serfaty, Gamma-convergence of gradient flows on Hilbert spaces and metric spaces and applications, Discr. Cont. Dynam. Systems Ser. A, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427.

[47]

N. ZaafaraniD. RaabeR. N. SinghF. Roters and S. Zaefferer, Three-dimensional investigation of the texture and microstructure below a nanoindent in a Cu single crystal using 3D EBSD and crystal plasticity finite element simulations, Acta Materialia, 54 (2006), 1863-1876. doi: 10.1016/j.actamat.2005.12.014.

show all references

References:
[1]

R. AbeyaratneC.-H. Chu and R. James, Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in a Cu-Al-Ni shape memory alloy, Phil. Mag. A, 73 (1996), 457-497. doi: 10.1080/01418619608244394.

[2]

S. Adams, N. Dirr, M. Peletier and J. Zimmer, Large deviations and gradient flows Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013), 20120341, 17pp. doi: 10.1098/rsta.2012.0341.

[3]

S. AdamsN. DirrM. A. Peletier and J. Zimmer, From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage, Comm. Math. Phys., 307 (2011), 791-815. doi: 10.1007/s00220-011-1328-4.

[4]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.

[5]

S. ArnrichA. MielkeM. A. PeletierG. Savaré and M. Veneroni, Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction, Calc. Var. Part. Diff. Eqns., 44 (2012), 419-454. doi: 10.1007/s00526-011-0440-9.

[6]

M. A. Biot, Variational principles in irreversible thermodynamics with applications to viscoelasticity, Phys. Review, 97 (1955), 1463-1469. doi: 10.1103/PhysRev.97.1463.

[7]

G. A. Bonaschi and M. A. Peletier, Quadratic and rate-independent limits for a large-deviations functional, Contin. Mech. Thermodyn., 28 (2016), 1191-1219. doi: 10.1007/s00161-015-0470-1.

[8]

E. De GiorgiA. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68 (1980), 180-187.

[9]

K. Disser and M. Liero, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, Networks Heterg. Media, 10 (2015), 233-253. doi: 10.3934/nhm.2015.10.233.

[10]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems North Holland, 1976.

[11]

M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Rational Mech. Anal., 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z.

[12]

M. Feinberg, On chemical kinetics of a certain class, Arch. Rational Mech. Anal., 46 (1972), 1-41.

[13]

W. Fenchel, On conjugate convex functions, Canadian J. Math., 1 (1949), 73-77. doi: 10.4153/CJM-1949-007-x.

[14]

J. Feng and T. G. Kurtz, Large Deviations for Stochastic Processes vol. 131 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/131.

[15]

N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Analysis, 45 (2013), 879-899. doi: 10.1137/120886315.

[16]

A. Glitzky and A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. angew. Math. Phys. (ZAMP), 64 (2013), 29-52. doi: 10.1007/s00033-012-0207-y.

[17]

A. N. GorbanI. V. KarlinV. B. Zmievskii and S. V. Dymova, Reduced description in the reaction kinetics, Physica A, 275 (2000), 361-379. doi: 10.1016/S0378-4371(99)00402-1.

[18]

M. Grmela, Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering, Adv. Chem. Eng., 39 (2010), 75-129. doi: 10.1016/S0065-2377(10)39002-8.

[19]

K. Hackl and F. D. Fischer, On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials, Proc. R. Soc. A, 464 (2008), 117-132. doi: 10.1098/rspa.2007.0086.

[20]

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, J. Mécanique, 14 (1975), 39-63.

[21]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[22]

M. Liero, Variational Methods for Evolution PhD thesis, Institut für Mathematik, Humboldt-Universität zu Berlin, 2012.

[23]

M. Liero, Passing from bulk to bulk-surface evolution in the Allen-Cahn equation, Nonl. Diff. Eqns. Appl. (NoDEA), 20 (2013), 919-942. doi: 10.1007/s00030-012-0189-7.

[24]

J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009.

[25]

J. Maas and A. Mielke, On gradient structures for chemical reactions with detailed balance: Ⅰ. modeling and large-volume limit, In preparation.

[26]

J. Maas and A. Mielke, On gradient structures for chemical reactions with detailed balance: Ⅱ. dissiaption distances and geodesic convexity, In preparation.

[27]

A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances, Contin. Mech. Thermodyn., 15 (2003), 351-382. doi: 10.1007/s00161-003-0120-x.

[28]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016.

[29]

A. Mielke, Emergence of rate-independent dissipation from viscous systems with wiggly energies, Contin. Mech. Thermodyn., 24 (2012), 591-606. doi: 10.1007/s00161-011-0216-7.

[30]

A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Part. Diff. Eqns., 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8.

[31]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discr. Cont. Dynam. Systems Ser. S, 6 (2013), 479-499. doi: 10.3934/dcdss.2013.6.479.

[32]

A. Mielke, Variational approaches and methods for dissipative material models with multiple scales, in Analysis and Computation of Microstructure in Finite Plasticity (eds. K. Hackl and S. Conti), vol. 78 of Lect. Notes Appl. Comp. Mechanics, Springer, 2015, chapter 5, 125-155. doi: 10.1007/978-3-319-18242-1_5.

[33]

A. Mielke, On evolutionary $Γ$-convergence for gradient systems (ch. 3), in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity (eds. A. Muntean, J. Rademacher and A. Zagaris), 187-249, Lect. Notes Appl. Math. Mech. , 3, Springer, [Cham], 2016. doi: 10.1007/978-3-319-26883-5_3.

[34]

A. Mielke, R. I. A. Patterson, M. A. Peletier and D. R. M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, J. Chem. Physics Submitted WIAS preprint 2165.

[35]

A. MielkeM. A. Peletier and D. R. M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014), 1293-1327. doi: 10.1007/s11118-014-9418-5.

[36]

A. Mielke and T. Roubíček, Rate-Independent Systems: Theory and Application Applied Mathematical Sciences, Vol. 193, Springer-Verlag New York, 2015. doi: 10.1007/978-1-4939-2706-7.

[37]

A. Mielke and L. Truskinovsky, From discrete visco-elasticity to continuum rate-independent plasticity: Rigorous results, Arch. Rational Mech. Anal., 203 (2012), 577-619. doi: 10.1007/s00205-011-0460-9.

[38]

L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev., 91 (1953), 1505-1512. doi: 10.1103/PhysRev.91.1505.

[39]

L. Onsager, Reciprocal relations in irreversible processes, Ⅰ+Ⅱ, Physical Review, 37 (1931), 405-426, (part Ⅱ, 38 (1931), 2265-2279). doi: 10.1103/PhysRev.38.2265.

[40]

M. A. PeletierG. Savaré and M. Veneroni, From diffusion to reaction via $Γ$-convergence, SIAM J. Math. Analysis, 42 (2010), 1805-1825. doi: 10.1137/090781474.

[41]

M. A. PeletierG. Savaré and M. Veneroni, Chemical reactions as $Γ$-limit of diffusion revised reprint of 40, SIAM Rev., 54 (2012), 327-352. doi: 10.1137/110858781.

[42]

L. Rayleigh and Hon. J. W. Strutt, Some general theorems relating to vibrations, Proc. London Math. Soc., s1-4 (1871), 357-368. doi: 10.1112/plms/s1-4.1.357.

[43]

D. R. M. Renger, Microscopic Interpretation of Wasserstein Gradient Flows PhD thesis, Technische Universiteit Eindhoven, 2013.

[44]

F. RotersD. Raabe and G. Gottstein, Work hardening in heterogeneous alloys -a microstructural approach based on three internal state variables, Acta Materialia, 48 (2000), 4181-4189. doi: 10.1016/S1359-6454(00)00289-5.

[45]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.

[46]

S. Serfaty, Gamma-convergence of gradient flows on Hilbert spaces and metric spaces and applications, Discr. Cont. Dynam. Systems Ser. A, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427.

[47]

N. ZaafaraniD. RaabeR. N. SinghF. Roters and S. Zaefferer, Three-dimensional investigation of the texture and microstructure below a nanoindent in a Cu single crystal using 3D EBSD and crystal plasticity finite element simulations, Acta Materialia, 54 (2006), 1863-1876. doi: 10.1016/j.actamat.2005.12.014.

Figure 1.1.  For reversible, time-continuous Markov processes the large-deviation principle (LPD) of Section 2.4 provides a (generalized) gradient structure. This mapping commutes with taking the limit $\varepsilon\to 0$ and EDP-convergence, respectively.
Figure 3.1.  Left: Three-state Markov process with high rate of leaving state 2. Right: The limit for $\varepsilon \to 0$ gives a two-state Markov process.
Figure 1.1.  The potential $V$ along the reaction path $\Upsilon =[0,7]$.
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