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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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

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

[26]

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

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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