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February  2013, 6(1): 215-233. doi: 10.3934/dcdss.2013.6.215

Thermalization of rate-independent processes by entropic regularization

T. J. Sullivan 1, , M. Koslowski 2, , F. Theil 3, and  Michael Ortiz 4,

1. 

Applied & Computational Mathematics and Graduate Aerospace Laboratories, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125-9400, United States
2. 

School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088, United States
3. 

Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
4. 

Graduate Aerospace Laboratories, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125, United States

Received  May 2011 Revised  July 2011 Published  October 2012

We consider the effective behaviour of a rate-independent process when it is placed in contact with a heat bath. The method used to ``thermalize'' the process is an interior-point entropic regularization of the Moreau--Yosida incremental formulation of the unperturbed process. It is shown that the heat bath destroys the rate independence in a controlled and deterministic way, and that the effective dynamics are those of a non-linear gradient descent in the original energetic potential with respect to a different and non-trivial effective dissipation potential.
Keywords: non-linear evolution equations, Gradient descent, thermodynamics..
Mathematics Subject Classification: Primary: 47J35, 82C3.
Citation: T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215
References:
[1]

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, second ed., 2008.

[2]

S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004.

[3]

R. Jordan and D. Kinderlehrer, An extended variational principle, Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, 177 (1996), 187-200.

[4]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.

[5]

M. Koslowski, "A Phase-Field Model of Dislocations in Ductile Single Crystals," Ph. D. thesis, California Institute of Technology, Pasadena, California, USA, 2003.

[6]

A. Mielke, Evolution of rate-independent systems, Evolutionary Equations. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559.

[7]

\bysame, Modeling and analysis of rate-independent processes, January 2007, Lipschitz Lecture held in Bonn: http://www.wias-berlin.de/people/mielke/papers/Lipschitz07Mielke.pdf.

[8]

A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615.

[9]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.

[10]

J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299.

[11]

J. P. Penot and M. Théra, Semicontinuous mappings in general topology, Arch. Math. (Basel), 38 (1982), 158-166.

[12]

D. Preiss, Geometry of measures in $\mathbb{R}^{N}$distribution, rectifiability, and densities: , Ann. of Math., 125 (1987), 537-643.

[13]

T. J. Sullivan, "Analysis of Gradient Descents in Random Energies and Heat Baths," Ph. D. thesis, Mathematitics Institute, University of Warwick, Coventry, UK, 2009.

[14]

T. J. Sullivan, M. Koslowski, F. Theil and M. Ortiz, On the behavior of dissipative systems in contact with a heat bath: application to Andrade creep, J. Mech. Phys. Solids, 57 (2009), 1058-1077.

[15]

K. Yosida, "Functional Analysis," Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965.

show all references

References:
[1]

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, second ed., 2008.

[2]

S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004.

[3]

R. Jordan and D. Kinderlehrer, An extended variational principle, Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, 177 (1996), 187-200.

[4]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.

[5]

M. Koslowski, "A Phase-Field Model of Dislocations in Ductile Single Crystals," Ph. D. thesis, California Institute of Technology, Pasadena, California, USA, 2003.

[6]

A. Mielke, Evolution of rate-independent systems, Evolutionary Equations. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559.

[7]

\bysame, Modeling and analysis of rate-independent processes, January 2007, Lipschitz Lecture held in Bonn: http://www.wias-berlin.de/people/mielke/papers/Lipschitz07Mielke.pdf.

[8]

A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst., 25 (2009), 585-615.

[9]

A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.

[10]

J. J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299.

[11]

J. P. Penot and M. Théra, Semicontinuous mappings in general topology, Arch. Math. (Basel), 38 (1982), 158-166.

[12]

D. Preiss, Geometry of measures in $\mathbb{R}^{N}$distribution, rectifiability, and densities: , Ann. of Math., 125 (1987), 537-643.

[13]

T. J. Sullivan, "Analysis of Gradient Descents in Random Energies and Heat Baths," Ph. D. thesis, Mathematitics Institute, University of Warwick, Coventry, UK, 2009.

[14]

T. J. Sullivan, M. Koslowski, F. Theil and M. Ortiz, On the behavior of dissipative systems in contact with a heat bath: application to Andrade creep, J. Mech. Phys. Solids, 57 (2009), 1058-1077.

[15]

K. Yosida, "Functional Analysis," Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965.

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