Abstract
We consider a one-dimensional classical many-body system with interaction potential of Lennard–Jones type in the thermodynamic limit at low temperature 1/beta in (0,infty ). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of Nexp (- beta e_mathrm {surf}/2) with e_mathrm {surf}>0 a surface energy. For the proof, the system is mapped to an effective model, which is a low-density lattice gas of defects. The results require conditions on the interactions between defects. We succeed in verifying these conditions for next-nearest neighbor interactions, applying recently derived uniform estimates of correlations.
Highlights
A fundamental problem in statistical and solid mechanics is to gain insight into the structure of matter and to derive material properties from basic atomistic interaction models
Assuming that particles interact via a classical pair interaction potential such as the Lennard–Jones potential, their crystallization in ground states at zero temperature has been well understood since the pioneering contributions [26,36,37,47]
Neighboring particles and next-to-nearest neighbors interact via a pair potential v : [0, ∞) → R ∪ {∞} which is repulsive for short distances and attractive for spacings larger than a unique energy minimizing bond length
Summary
A fundamental problem in statistical and solid mechanics is to gain insight into the structure of matter and to derive material properties from basic atomistic interaction models. Thermal equilibrium is investigated within the framework of classical equilibrium statistical mechanics [34,38] This means that we study families of probability measures indexed by the number N of atoms, the length L of the chain, and a positive parameter β > 0 called inverse temperature. The number of cracks is not bounded but instead proportional to the number of atoms in the chain; each crack is of microscopic length even though the length is exponentially large in β This behavior is similar to one-dimensional Ising chains with nearest neighbor interaction at low temperature [43] or with Kac interactions and small Kac parameter [10,12].
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