Energy landscape properties studied by 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&#39;s anti-continuum limit it is easy to establish a one-to-one correspondence between the stationary points of $V$ and symbolic sequences $\bmσ = (σ_1,...,σ_N)$ 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 &#39;&#39;thermodynamic&#39;&#39; formalism to calculate statistical properties of the so-called ``energy landscape&#39;&#39; of $V$. This offers an explanation why topological quantities of $V$ may become singular, like in phase transitions. Particularly, we find 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 function of average energy $\bar{v}$ is analytical in $\bar{v}$ and it vanishes at $v^*$, above the ground state energy $v_{gs}$, whereas the average saddle index $\bar{n}_s$ as function of 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 $v_{gs}, \bar{n}_s(v)$ exhibits power law behavior which even holds for noninteracting particles.