TY - JOUR

T1 - Ground states and critical points for Aubry-Mather theory in statistical mechanics

AU - De La Llave, Rafael

AU - Valdinoci, Enrico

PY - 2010/4/1

Y1 - 2010/4/1

N2 - We consider several models of networks of interacting particles and prove the existence of quasi-periodic equilibrium solutions. We assume (1) that the network and the interaction among particles are invariant under a group that satisfies some mild assumptions; (2) that the state of each particle is given by a real number; (3) that the interaction is invariant by adding an integer to the state of all the particles at the same time; (4) that the interaction is ferromagnetic and coercive (it favors local alignment and penalizes large local oscillations); and (5) some technical assumptions on the regularity speed of decay of the interaction with the distance. Note that the assumptions are mainly qualitative, so that they cover many of the models proposed in the literature. We conclude (A) that there are minimizing (ground states) quasi-periodic solutions of every frequency and that they satisfy several geometric properties; (B) if the minimizing solutions do not cover all possible values at a point, there is another equilibrium point which is not a ground state. These results general-ize basic results of Aubry-Mather theory (take the network and the group to be Z). In particular, we provide with a generalization of the celebrated criterion of existence of invariant circles if and only iff the Peierls-Nabarro barrier vanishes.

AB - We consider several models of networks of interacting particles and prove the existence of quasi-periodic equilibrium solutions. We assume (1) that the network and the interaction among particles are invariant under a group that satisfies some mild assumptions; (2) that the state of each particle is given by a real number; (3) that the interaction is invariant by adding an integer to the state of all the particles at the same time; (4) that the interaction is ferromagnetic and coercive (it favors local alignment and penalizes large local oscillations); and (5) some technical assumptions on the regularity speed of decay of the interaction with the distance. Note that the assumptions are mainly qualitative, so that they cover many of the models proposed in the literature. We conclude (A) that there are minimizing (ground states) quasi-periodic solutions of every frequency and that they satisfy several geometric properties; (B) if the minimizing solutions do not cover all possible values at a point, there is another equilibrium point which is not a ground state. These results general-ize basic results of Aubry-Mather theory (take the network and the group to be Z). In particular, we provide with a generalization of the celebrated criterion of existence of invariant circles if and only iff the Peierls-Nabarro barrier vanishes.

KW - Ground states

KW - Long-range interaction

KW - Peierls-Nabarro barrier

KW - Quasiperiodic equilibria

UR - http://www.scopus.com/inward/record.url?scp=77953326205&partnerID=8YFLogxK

U2 - 10.1007/s00332-009-9055-0

DO - 10.1007/s00332-009-9055-0

M3 - Article

AN - SCOPUS:77953326205

VL - 20

SP - 153

EP - 218

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 2

ER -