Energy landscape properties studied using symbolic sequences

Abstract

We investigate a classical lattice system with N particles. The potential energy V of the scalar displacements is chosen as a ϕ 4 on-site potential plus interactions. Its stationary points are solutions of a coupled set of nonlinear equations. Starting with Aubry’s anti-continuum limit it is easy to establish a one-to-one correspondence between the stationary points of V and symbolic sequences σ = ( σ 1 , … , σ N ) with σ n = + , 0 , − . We prove that this correspondence remains valid for interactions with a coupling constant ϵ below a critical value ϵ c and that it allows the use of a “thermodynamic” formalism to calculate statistical properties of the so-called “energy landscape” of V . This offers an explanation for why topological quantities of V may become singular, like in phase transitions. In particular, we find that the saddle index distribution is maximum at a saddle index n s max = 1 / 3 for all ϵ < ϵ c . Furthermore there exists an interval ( v ∗ , v max ) in which the saddle index n s as a function of the average energy v ̄ is analytical in v ̄ and it vanishes at v ∗ , above the ground state energy v gs , whereas the average saddle index n ̄ s as a function of the energy v is highly nontrivial. It can exhibit a singularity at a critical energy v c and it vanishes at v gs , only. Close to v gs , n ̄ s ( v ) exhibits power law behavior which even holds for noninteracting particles.