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 ${\varphi }^4$ 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 $\mathbf{\sigma }=({\sigma }_1,...,{\sigma }_N)$ with ${\sigma }_n=+,0,-$. We prove that this correspondence remains valid for interactions with a coupling constant $\epsilon$ below a critical value ${\epsilon }_c$ 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 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 $\epsilon <{\epsilon }_c$. Furthermore there exists an interval ($v^{\ast },v_{max}$) in which the saddle index $n_s$ as function of average energy $\bar{v}$ is analytical in $\bar{v}$ and it vanishes at $v^{\ast }$, 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.

arXiv comment15 pages, 2 figures

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