Abstract
We construct a stochastic model showing the relationship between noise, gradient flows and rate-independent systems. The model consists of a one-dimensional birth–death process on a lattice, with rates derived from Kramers’ law as an approximation of a Brownian motion on a wiggly energy landscape. Taking various limits, we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via ‘L log L’ gradient flows. This is achieved via Mosco-convergence of the renormalized large-deviations rate functional of the stochastic process.
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Abeyaratne R., Chu C., James R.D.: Kinetics of materials with wiggly energies: theory and application to the evolution of twinning microstructures in a Cu–Al–Ni shape memory alloy. Philos. Mag. A 73(2), 457–497 (1996)
Adams S., Dirr N., Peletier M.A., Zimmer J.: From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage. Commun. Math. Phys. 307, 791–815 (2011)
Adams S., Dirr N., Peletier M.A., Zimmer J.: Large deviations and gradient flows. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 371(2005), 20120341 (2013)
Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems, vol. 254. Clarendon Press Oxford, Oxford (2000)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2008)
Baldi, P.: Large deviations for diffusion processes with homogenization and applications. Ann. Probab. 19(2), 509–524 (1991)
Basinski Z.S.: Thermally activated glide in face-centred cubic metals and its application to the theory of strain hardening. Philos. Mag. 4(40), 393–432 (1959). doi:10.1080/14786435908233412
Becker R.: Über die Plasticität amorpher und kristalliner fester Körper. Phys. Z. 26, 919–925 (1925)
Berglund, N.: Kramers’ Law: Validity, Derivations and Generalisations. Arxiv preprint arXiv:1106.5799 (2011)
Brezis H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)
Cagnetti F.: A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path. Math. Models Methods Appl. Sci. 18(07), 1027–1071 (2008)
Chen X.: Global asymptotic limit of solutions of the Cahn–Hilliard equation. J. Differ. Geom. 44, 262–311 (1996)
Dal Maso G., DeSimone A., Mora M.G.: Quasistatic evolution problems for linearly elastic–perfectly plastic materials. Arch. Ration. Mech. Anal. 180(2), 237–291 (2006)
Dal Maso G., DeSimone A., Mora M.G., Morini M.: A vanishing viscosity approach to quasistatic evolution in plasticity with softening. Arch. Ration. Mech. Anal. 189(3), 469–544 (2008)
Dirr N., Laschos V., Zimmer J.: Upscaling from particle models to entropic gradient flows. J. Math. Phys. 53(6), 063704 (2012)
Duong, M.H., Laschos, V., Renger, D.R.M.: Wasserstein gradient flows from large deviations of many-particle limits. ESAIM: Control Optim. Calc. Var. E-first (2013)
Duong M.H., Peletier M.A., Zimmer J.: GENERIC formalism of a Vlasov–Fokker–Planck equation and connection to large-deviation principles. Nonlinearity 26, 2951–2971 (2013)
Dupuis P., Spiliopoulos K.: Large deviations for multiscale diffusion via weak convergence methods. Stoch. Process. Appl. 122(4), 1947–1987 (2012)
Ethier S.N., Kurtz T.G.: Markov Processes: Characterization and Convergence, vol. 282. Wiley, Hoboken (2009)
Feng, J., Kurtz, T.G.: Large Deviations for Stochastic Processes, vol. 131. Citeseer (2006)
Fiaschi A.: A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy. Annales de l’Institut Henri Poincare (C) Non Linear Anal. 26(4), 1055–1080 (2009)
Freidlin M.I., Sowers R.B.: A comparison of homogenization and large deviations, with applications to wavefront propagation. Stoch. Process. Appl. 82(1), 23–52 (1999)
Freidlin M.I., Wentzell A.D.: Random Perturbations of Dynamical Systems, vol. 260. Springer, New York (2012)
Kramers H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4), 284–304 (1940)
Krausz A.S., Eyring H.: Deformation Kinetics. Wiley, New York (1975)
Mainik A., Mielke A.: Existence results for energetic models for rate-independent systems. Cal. Var. Partial Differ. Equ. 22(1), 73–99 (2005)
Mielke, A.: Handbook of Differential Equations: Evolutionary Differential Equations, chap. Evolution in Rate-Independent Systems, pp. 461–559. North-Holland, Amsterdam (2005)
Mielke A.: Emergence of rate-independent dissipation from viscous systems with wiggly energies. Contin. Mech. Thermodyn. 24(4-6), 591–606 (2012)
Mielke, A.: On Evolutionary Gamma-Convergence for Gradient Systems. Tech. Rep. 1915, WIAS, Berlin (2014)
Mielke, A., Peletier, M.A., Renger, D.R.M.: On the Relation Between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion. arXiv preprint arXiv:1312.7591 (2013)
Mielke, A., Rossi, R., Savaré, G.: Modeling solutions with jumps for rate-independent systems on metric spaces. Discrete Contin. Dyn. Syst. A 25(2) (2009)
Mielke A., Rossi R., Savaré G.: BV solutions and viscosity approximations of rate-independent systems. ESAIM Control Optim. Calc. Var. 18(01), 36–80 (2012)
Mielke A., Rossi R., Savaré G.: Variational convergence of gradient flows and rate-independent evolutions in metric spaces. Milan J. Math. 80(2), 381–410 (2012)
Mielke, A., Rossi, R., Savaré, G.: Balanced-Viscosity (bv) Solutions to Infinite Dimensional Rate-Independent Systems. arXiv preprint arXiv:1309.6291 (2013)
Mielke, A., Theil, F.: A mathematical model for rate-independent phase transformations with hysteresis. In: Proceedings of the Workshop on “Models of Continuum Mechanics in Analysis and Engineering, pp. 117–129 (1999)
Mielke A., Theil F.: On rate-independent hysteresis models. Nonlinear Differ. Equ. Appl. 11(2), 151–189 (2004)
Mielke A., Theil F., Levitas V.I.: A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162(2), 137–177 (2002)
Mielke A., Truskinovsky L.: From discrete visco-elasticity to continuum rate-independent plasticity: rigorous results. Arch. Ration. Mech. Anal. 203(2), 577–619 (2012)
Orowan E.: Problems of plastic gliding. Proc. Phys. Soc. 52, 8–22 (1940)
Puglisi, G., Truskinovsky, L.: Thermodynamics of rate-independent plasticity. J. Mech. Phys. Solids 53(3), 655–679 (2005). doi:10.1016/j.jmps.2004.08.004. http://www.sciencedirect.com/science/article/pii/S0022509604001425
Renger, D.R.M.: Microscopic Interpretation of Wasserstein Gradient Flows. Ph.D. thesis, Technische Universiteit Eindhoven (2013). http://alexandria.tue.nl/extra2/749143.pdf
Shwartz A., Weiss A.: Large Deviations for Performance Analysis: Queues, Communications, and Computing. Chapman & Hall/CRC, London (1995)
Sullivan, T.J.: Analysis of Gradient Descents in Random Energies and Heat Baths. Ph.D. thesis, University of Warwick (2009)
Sullivan T.J., Koslowski M., Theil F., Ortiz M.: On the behavior of dissipative systems in contact with a heat bath: application to andrade creep. J. Mech. Phys. Solids 57(7), 1058–1077 (2009)
Trotter H.F.: Approximation of semi-groups of operators. Pac. J. Math. 8(4), 887–919 (1958)
Varadhan S.R.S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math. 19(3), 261–286 (1966)
Wentzell A.D.: Rough limit theorems on large deviations for Markov stochastic processes I. Theory Probab. Appl. 21(2), 227–242 (1977)
Wentzell A.D.: Limit Theorems on Large Deviations for Markov Stochastic Processes, vol. 38. Springer, New York (1990)
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Communicated by Andreas Öchsner.
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Bonaschi, G.A., Peletier, M.A. Quadratic and rate-independent limits for a large-deviations functional. Continuum Mech. Thermodyn. 28, 1191–1219 (2016). https://doi.org/10.1007/s00161-015-0470-1
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DOI: https://doi.org/10.1007/s00161-015-0470-1