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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 & Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215
References:
[1]

Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008.  Google Scholar

[2]

Cambridge University Press, Cambridge, 2004.  Google Scholar

[3]

Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, 177 (1996), 187-200.  Google Scholar

[4]

SIAM J. Math. Anal., 29 (1998), 1-17.  Google Scholar

[5]

Ph. D. thesis, California Institute of Technology, Pasadena, California, USA, 2003. Google Scholar

[6]

Evolutionary Equations. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559.  Google Scholar

[7]

January 2007, Lipschitz Lecture held in Bonn: http://www.wias-berlin.de/people/mielke/papers/Lipschitz07Mielke.pdf. Google Scholar

[8]

Discrete Contin. Dyn. Syst., 25 (2009), 585-615.  Google Scholar

[9]

NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.  Google Scholar

[10]

Bull. Soc. Math. France, 93 (1965), 273-299.  Google Scholar

[11]

Arch. Math. (Basel), 38 (1982), 158-166.  Google Scholar

[12]

Ann. of Math., 125 (1987), 537-643.  Google Scholar

[13]

Ph. D. thesis, Mathematitics Institute, University of Warwick, Coventry, UK, 2009. Google Scholar

[14]

J. Mech. Phys. Solids, 57 (2009), 1058-1077.  Google Scholar

[15]

Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965.  Google Scholar

show all references

References:
[1]

Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008.  Google Scholar

[2]

Cambridge University Press, Cambridge, 2004.  Google Scholar

[3]

Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., Dekker, New York, 177 (1996), 187-200.  Google Scholar

[4]

SIAM J. Math. Anal., 29 (1998), 1-17.  Google Scholar

[5]

Ph. D. thesis, California Institute of Technology, Pasadena, California, USA, 2003. Google Scholar

[6]

Evolutionary Equations. Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, II (2005), 461-559.  Google Scholar

[7]

January 2007, Lipschitz Lecture held in Bonn: http://www.wias-berlin.de/people/mielke/papers/Lipschitz07Mielke.pdf. Google Scholar

[8]

Discrete Contin. Dyn. Syst., 25 (2009), 585-615.  Google Scholar

[9]

NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189.  Google Scholar

[10]

Bull. Soc. Math. France, 93 (1965), 273-299.  Google Scholar

[11]

Arch. Math. (Basel), 38 (1982), 158-166.  Google Scholar

[12]

Ann. of Math., 125 (1987), 537-643.  Google Scholar

[13]

Ph. D. thesis, Mathematitics Institute, University of Warwick, Coventry, UK, 2009. Google Scholar

[14]

J. Mech. Phys. Solids, 57 (2009), 1058-1077.  Google Scholar

[15]

Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965.  Google Scholar

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