Energy distributions of Frenkel–Kontorova-type atomic chains: Transition from conservative to dissipative dynamics

We investigate energy distributions of Frenkel–Kontorova-type atomic chains generated from large number of independent identically distributed (iid) random initial atomic positionings under two cases. In the first case, atoms at the free-end chains without dissipation (conservative case) are only coupled to one other atom, whereas each atom inside the bulk is coupled to its 2 nearest neighbours. Here, atoms located at the chain are all at the same type. Such kind of systems can be modelled by conservative standard map. We show that, when the coupling is non-linear (which leads chaotic arrangement of the atoms) for energy distribution, the Boltzmann–Gibbs statistical mechanics is constructed, namely, exponential form emerges as Boltzmann factor $P\left(E\right)\propto e^{-\beta E}$. However, when the coupling is linear (which leads linear arrangement of the atoms) the Boltzmann–Gibbs statistical mechanics fails and the exponential distribution is replaced by a $q$-exponential form, which generalizes the Boltzmann factor as $P\left(E\right)\propto e_q^{-{\beta }_{q}E}={[1-(1-q){\beta }_{q}E]}^{1/(1-q)}$. We also show for each type of atom localization with $N$ number of atoms, $\beta$ (or ${\beta }_q$) values can be given as a function of $1/N$. In the second case, although the couplings among the atoms are exactly the same as the previous case, atoms located at the chain are now considered as being at different types. We show that, for energy distribution of such linear chains, each of the distributions corresponding to different dissipation parameters ($\gamma$) are in the $q$-exponential form. Moreover, we numerically verify that ${\beta }_q$ values can be given as a linear function of $1/{\sum }_{n=1}^N{(1-\gamma )}^{(n-2)}$. On the other hand, although energy distributions of the chaotic chains for different dissipation parameters are in exponential form, a linear scaling between $\beta$ and $\gamma$ values cannot be obtained. This scaling is possible if the energies of the chains are scaled with $1/{(1-\gamma )}^{-N}$. For both cases, clear data collapses among distributions are evident.