American Institute of Mathematical Sciences

Advanced
March  2006, 1(1): 85-107. doi: 10.3934/nhm.2006.1.85

Phase and anti-phase boundaries in binary discrete systems: a variational viewpoint

Roberto Alicandro 1, , Andrea Braides 2, and  Marco Cicalese 3,

1. 

DAEIMI, Università di Cassino, via Di Biasio, 03043 Cassino (FR)
2. 

Dipartimento di Matematica, Università di Roma 'Tor Vergata', via della Ricerca Scientifica, 00133 Roma
3. 

Dipartimento di Matematica e Applicazioni "R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli, Italy

Received  September 2005 Revised  October 2005 Published  January 2006

We provide a variational description of nearest-neighbours and next-to-nearest neighbours binary lattice systems. By studying the $\Gamma$-limit of proper scaling of the energies of the systems, we highlight phase and anti-phase boundary phenomena and show how they depend on the geometry of the lattice.
Keywords: surface energies, binary alloys., anti-phase boundaries, $\Gamma$-convergence, spin systems, Lattice systems.
Mathematics Subject Classification: Primary: 49S05, 49J45, 82B20, 82B24; Secondary: 49M25, 49Q20, 49Q2.
Citation: Roberto Alicandro, Andrea Braides, Marco Cicalese. Phase and anti-phase boundaries in binary discrete systems: a variational viewpoint. Networks and Heterogeneous Media, 2006, 1 (1) : 85-107. doi: 10.3934/nhm.2006.1.85
[1]

Ken Shirakawa. Stability analysis for phase field systems associated with crystalline-type energies. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 483-504. doi: 10.3934/dcdss.2011.4.483
[2]

I-Liang Chern, Chun-Hsiung Hsia. Dynamic phase transition for binary systems in cylindrical geometry. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 173-188. doi: 10.3934/dcdsb.2011.16.173
[3]

Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741
[4]

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
[5]

Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic and Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1
[6]

Marco Cicalese, Matthias Ruf. Discrete spin systems on random lattices at the bulk scaling. Discrete and Continuous Dynamical Systems - S, 2017, 10 (1) : 101-117. doi: 10.3934/dcdss.2017006
[7]

Lei Li, Jianping Wang, Mingxin Wang. The dynamics of nonlocal diffusion systems with different free boundaries. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3651-3672. doi: 10.3934/cpaa.2020161
[8]

Yihong Du, Mingxin Wang, Meng Zhao. Two species nonlocal diffusion systems with free boundaries. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1127-1162. doi: 10.3934/dcds.2021149
[9]

Andrea Braides, Margherita Solci, Enrico Vitali. A derivation of linear elastic energies from pair-interaction atomistic systems. Networks and Heterogeneous Media, 2007, 2 (3) : 551-567. doi: 10.3934/nhm.2007.2.551
[10]

Alessia Berti, Claudio Giorgi, Elena Vuk. Free energies and pseudo-elastic transitions for shape memory alloys. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 293-316. doi: 10.3934/dcdss.2013.6.293
[11]

Lucia Scardia, Anja Schlömerkemper, Chiara Zanini. Towards uniformly $\Gamma$-equivalent theories for nonconvex discrete systems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 661-686. doi: 10.3934/dcdsb.2012.17.661
[12]

Peng Cheng, Feng Pan, Yanyan Yin, Song Wang. Probabilistic robust anti-disturbance control of uncertain systems. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2441-2450. doi: 10.3934/jimo.2020076
[13]

Silviu-Iulian Niculescu, Peter S. Kim, Keqin Gu, Peter P. Lee, Doron Levy. Stability crossing boundaries of delay systems modeling immune dynamics in leukemia. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 129-156. doi: 10.3934/dcdsb.2010.13.129
[14]

Nuno Costa Dias, Andrea Posilicano, João Nuno Prata. Self-adjoint, globally defined Hamiltonian operators for systems with boundaries. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1687-1706. doi: 10.3934/cpaa.2011.10.1687
[15]

Xiao-Qiang Zhao, Shengfan Zhou. Kernel sections for processes and nonautonomous lattice systems. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 763-785. doi: 10.3934/dcdsb.2008.9.763
[16]

Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure and Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803
[17]

Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226
[18]

Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 445-467. doi: 10.3934/dcds.2011.31.445
[19]

Zonglin Jia, Youde Wang. Global weak solutions to Landau-Lifshtiz systems with spin-polarized transport. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1903-1935. doi: 10.3934/dcds.2020099
[20]

Kathryn Lindsey, Rodrigo Treviño. Infinite type flat surface models of ergodic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5509-5553. doi: 10.3934/dcds.2016043

2021 Impact Factor: 1.41

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (35)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy