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.   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), 1.   Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford University Press, (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), (2062), 1.   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, (2008).   Google Scholar

[6]

A. Braides, $\Gamma$-convergence for Beginners,, Oxford University Press, (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, (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.  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.  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.  doi: 10.1007/s00161-008-0072-2.  Google Scholar

[11]

G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäuser, (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, (1993), 81.   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.  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.   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.  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.   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), 1.   Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford University Press, (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), (2062), 1.   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, (2008).   Google Scholar

[6]

A. Braides, $\Gamma$-convergence for Beginners,, Oxford University Press, (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, (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.  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.  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.  doi: 10.1007/s00161-008-0072-2.  Google Scholar

[11]

G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäuser, (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, (1993), 81.   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.  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.   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.  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 - A, 1997, 3 (2) : 153-174. doi: 10.3934/dcds.1997.3.153
[5]

Antonio Tribuzio. Perturbations of minimizing movements and curves of maximal slope. Networks & Heterogeneous Media, 2018, 13 (3) : 423-448. doi: 10.3934/nhm.2018019
[6]

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
[7]

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
[8]

Mariano Giaquinta, Paolo Maria Mariano, Giuseppe Modica. A variational problem in the mechanics of complex materials. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 519-537. doi: 10.3934/dcds.2010.28.519
[9]

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
[10]

Eliot Fried. New insights into the classical mechanics of particle systems. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1469-1504. doi: 10.3934/dcds.2010.28.1469
[11]

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
[12]

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
[13]

Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023
[14]

Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317
[15]

Oscar E. Fernandez, Anthony M. Bloch, P. J. Olver. Variational Integrators for Hamiltonizable Nonholonomic Systems. Journal of Geometric Mechanics, 2012, 4 (2) : 137-163. doi: 10.3934/jgm.2012.4.137
[16]

Paulo Cesar Carrião, Olimpio Hiroshi Miyagaki. On a class of variational systems in unbounded domains. Conference Publications, 2001, 2001 (Special) : 74-79. doi: 10.3934/proc.2001.2001.74
[17]

Fei Xu, Ross Cressman, Vlastimil Křivan. Evolution of mobility in predator-prey systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3397-3432. doi: 10.3934/dcdsb.2014.19.3397
[18]

Gianni Dal Maso, Flaviana Iurlano. Fracture models as $\Gamma$-limits of damage models. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1657-1686. doi: 10.3934/cpaa.2013.12.1657
[19]

François Gay-Balma, Darryl D. Holm, Tudor S. Ratiu. Variational principles for spin systems and the Kirchhoff rod. Journal of Geometric Mechanics, 2009, 1 (4) : 417-444. doi: 10.3934/jgm.2009.1.417
[20]

Sergio Grillo, Marcela Zuccalli. Variational reduction of Lagrangian systems with general constraints. Journal of Geometric Mechanics, 2012, 4 (1) : 49-88. doi: 10.3934/jgm.2012.4.49

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences