Energy landscape properties studied using symbolic sequences

We investigate a classical lattice system with N particles. The potential energy V of the scalar displacements is chosen as a ϕ4 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=(σ1,…,σN) with σn=+,0,−. We prove that this correspondence remains valid for interactions with a coupling constant ϵ below a critical value ϵc 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 nsmax=1/3 for all ϵ<ϵc. Furthermore there exists an interval (v∗,vmax) in which the saddle index ns as a function of the average energy v̄ is analytical in v̄ and it vanishes at v∗, above the ground state energy vgs, whereas the average saddle index n̄s as a function of the energy v is highly nontrivial. It can exhibit a singularity at a critical energy vc and it vanishes at vgs, only. Close to vgs,n̄s(v) exhibits power law behavior which even holds for noninteracting particles.