American Institute of Mathematical Sciences

Advanced
June  2014, 9(2): 217-238. doi: 10.3934/nhm.2014.9.217

Variational evolution of one-dimensional Lennard-Jones systems

Andrea Braides 1, , Anneliese Defranceschi 2, and  Enrico Vitali 3,

1. 

Dipartimento di Matematica, Università di Roma 'Tor Vergata', via della Ricerca Scientifica, 00133 Roma
2. 

Dipartimento di Matematica, Università di Trento, via Sommarive 14, 38123 Povo, Italy
3. 

Dipartimento di Matematica 'F. Casorati', Università di Pavia, via Ferrata, 1-27100 Pavia

Received  October 2013 Revised  May 2014 Published  July 2014

We analyze Lennard-Jones systems from the standpoint of variational principles beyond the static framework. In a one-dimensional setting such systems have already been shown to be equivalent to energies of Fracture Mechanics. Here we show that this equivalence can also be given in dynamical terms using the notion of minimizing movements.
Keywords: minimizing movements, Fracture Mechanics., Lennard-Jones systems, variational evolution.
Mathematics Subject Classification: Primary: 35K90, 74A45; Secondary: 49J45, 74R1.
Citation: Andrea Braides, Anneliese Defranceschi, Enrico Vitali. Variational evolution of one-dimensional Lennard-Jones systems. Networks & Heterogeneous Media, 2014, 9 (2) : 217-238. doi: 10.3934/nhm.2014.9.217
References:
[1]

L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246.  Google Scholar

[2]

L. Ambrosio and A. Braides, Energies in $SBV$ and variational models in fracture mechanics, in Homogenization and Applications to Material Sciences (Nice, 1995) (eds. D. Cioranescu, A. Damlamian, and P. Donato), GAKUTO Internat. Ser. Math. Sci. Appl., 9, Gakkōtosho, Tokyo, 1995, 1-22.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.  Google Scholar

[4]

L. Ambrosio and N. Gigli, A user's guide to optimal transport, in Modelling and Optimisation of Flows on Networks (eds. B. Piccoli and M. Rascle), Lecture Notes in Mathematics, 2062, Springer, Berlin, 2013, 1-155. Google Scholar

[5]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH, Zürich, Birkhhäuser, Basel, 2008.  Google Scholar

[6]

A. Braides, $\Gamma$-convergence for Beginners, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[7]

A. Braides, Local Minimization, Variational Evolution and $\Gamma$-Convergence, Lecture Notes in Mathematics, 2094, Springer, Berlin, 2014. doi: 10.1007/978-3-319-01982-6.  Google Scholar

[8]

A. Braides, M. S. Gelli and M. Novaga, Motion and pinning of discrete interfaces, Arch. Ration. Mech. Anal., 95 (2010), 469-498. doi: 10.1007/s00205-009-0215-z.  Google Scholar

[9]

A. Braides, A. J. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes, Arch. Ration. Mech. Anal., 180 (2006), 151-182. doi: 10.1007/s00205-005-0399-9.  Google Scholar

[10]

A. Braides and L. Truskinovsky, Asymptotic expansions by Gamma-convergence, Cont. Mech. Therm., 20 (2008), 21-62. doi: 10.1007/s00161-008-0072-2.  Google Scholar

[11]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[12]

E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., 29, Masson, Paris, 1993, 81-98.  Google Scholar

[13]

N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calc. Var. Partial Differential Equations, 39 (2010), 101-120. doi: 10.1007/s00526-009-0303-9.  Google Scholar

[14]

M. Gobbino, Gradient flow for the one-dimensional Mumford-Shah functional, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 145-193.  Google Scholar

[15]

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

show all references

References:
[1]

L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 19 (1995), 191-246.  Google Scholar

[2]

L. Ambrosio and A. Braides, Energies in $SBV$ and variational models in fracture mechanics, in Homogenization and Applications to Material Sciences (Nice, 1995) (eds. D. Cioranescu, A. Damlamian, and P. Donato), GAKUTO Internat. Ser. Math. Sci. Appl., 9, Gakkōtosho, Tokyo, 1995, 1-22.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.  Google Scholar

[4]

L. Ambrosio and N. Gigli, A user's guide to optimal transport, in Modelling and Optimisation of Flows on Networks (eds. B. Piccoli and M. Rascle), Lecture Notes in Mathematics, 2062, Springer, Berlin, 2013, 1-155. Google Scholar

[5]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH, Zürich, Birkhhäuser, Basel, 2008.  Google Scholar

[6]

A. Braides, $\Gamma$-convergence for Beginners, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[7]

A. Braides, Local Minimization, Variational Evolution and $\Gamma$-Convergence, Lecture Notes in Mathematics, 2094, Springer, Berlin, 2014. doi: 10.1007/978-3-319-01982-6.  Google Scholar

[8]

A. Braides, M. S. Gelli and M. Novaga, Motion and pinning of discrete interfaces, Arch. Ration. Mech. Anal., 95 (2010), 469-498. doi: 10.1007/s00205-009-0215-z.  Google Scholar

[9]

A. Braides, A. J. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes, Arch. Ration. Mech. Anal., 180 (2006), 151-182. doi: 10.1007/s00205-005-0399-9.  Google Scholar

[10]

A. Braides and L. Truskinovsky, Asymptotic expansions by Gamma-convergence, Cont. Mech. Therm., 20 (2008), 21-62. doi: 10.1007/s00161-008-0072-2.  Google Scholar

[11]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[12]

E. De Giorgi, New problems on minimizing movements, in Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., 29, Masson, Paris, 1993, 81-98.  Google Scholar

[13]

N. Gigli, On the heat flow on metric measure spaces: Existence, uniqueness and stability, Calc. Var. Partial Differential Equations, 39 (2010), 101-120. doi: 10.1007/s00526-009-0303-9.  Google Scholar

[14]

M. Gobbino, Gradient flow for the one-dimensional Mumford-Shah functional, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 145-193.  Google Scholar

[15]

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

[1]

Mathias Schäffner, Anja Schlömerkemper. On Lennard-Jones systems with finite range interactions and their asymptotic analysis. Networks & Heterogeneous Media, 2018, 13 (1) : 95-118. doi: 10.3934/nhm.2018005
[2]

Irina Berezovik, Wieslaw Krawcewicz, Qingwen Hu. Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1879-1903. doi: 10.3934/dcdss.2019124
[3]

Thomas Hudson. Gamma-expansion for a 1D confined Lennard-Jones model with point defect. Networks & Heterogeneous Media, 2013, 8 (2) : 501-527. doi: 10.3934/nhm.2013.8.501
[4]

Antonin Chambolle, Francesco Doveri. Minimizing movements of the Mumford and Shah energy. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 153-174. doi: 10.3934/dcds.1997.3.153
[5]

Andrea Braides, Antonio Tribuzio. Perturbed minimizing movements of families of functionals. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 373-393. doi: 10.3934/dcdss.2020324
[6]

Antonio Tribuzio. Perturbations of minimizing movements and curves of maximal slope. Networks & Heterogeneous Media, 2018, 13 (3) : 423-448. doi: 10.3934/nhm.2018019
[7]

Andrew D. Lewis. Nonholonomic and constrained variational mechanics. Journal of Geometric Mechanics, 2020, 12 (2) : 165-308. doi: 10.3934/jgm.2020013
[8]

Christopher J. Larsen. Local minimality and crack prediction in quasi-static Griffith fracture evolution. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 121-129. doi: 10.3934/dcdss.2013.6.121
[9]

Juan Carlos Marrero, D. Martín de Diego, Diana Sosa. Variational constrained mechanics on Lie affgebroids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 105-128. doi: 10.3934/dcdss.2010.3.105
[10]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033
[11]

Mariano Giaquinta, Paolo Maria Mariano, Giuseppe Modica. A variational problem in the mechanics of complex materials. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 519-537. doi: 10.3934/dcds.2010.28.519
[12]

Pedro D. Prieto-Martínez, Narciso Román-Roy. Higher-order mechanics: Variational principles and other topics. Journal of Geometric Mechanics, 2013, 5 (4) : 493-510. doi: 10.3934/jgm.2013.5.493
[13]

Eliot Fried. New insights into the classical mechanics of particle systems. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469
[14]

Kaizhi Wang. Action minimizing stochastic invariant measures for a class of Lagrangian systems. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1211-1223. doi: 10.3934/cpaa.2008.7.1211
[15]

Zhenhai Liu, Van Thien Nguyen, Jen-Chih Yao, Shengda Zeng. History-dependent differential variational-hemivariational inequalities with applications to contact mechanics. Evolution Equations & Control Theory, 2020, 9 (4) : 1073-1087. doi: 10.3934/eect.2020044
[16]

Brian Straughan. Shocks and acceleration waves in modern continuum mechanics and in social systems. Evolution Equations & Control Theory, 2014, 3 (3) : 541-555. doi: 10.3934/eect.2014.3.541
[17]

Thomas Hagen, Andreas Johann, Hans-Peter Kruse, Florian Rupp, Sebastian Walcher. Dynamical systems and geometric mechanics: A special issue in Honor of Jürgen Scheurle. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : i-iii. doi: 10.3934/dcdss.20204i
[18]

Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 677-694. doi: 10.3934/dcdss.2020360
[19]

Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023
[20]

Vyacheslav Maksimov. The method of extremal shift in control problems for evolution variational inequalities under disturbances. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021048

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences