Graphene thermal break-down induced by anharmonic bending mode.

A novel approach is proposed allowing us to prove, self-consistently, that free-standing graphene reaching a certain temperature loses its mechanical stability resulting in abrupt breakdown, which can be interpreted as melting. Our study is based on the idea of the crucial role of the anomalously soft bending 'sound' mode in the jump transition of graphene from the state with relatively small bending fluctuations to a state with fluctuations close in amplitude to the graphene lattice constant. The acme of the developed theory is in establishing a quantitative relationship connecting the graphene elastic moduli of second, third (negative!), and fourth orders at the melting temperature T m that permits us to calculate T m. The results obtained lay a theoretical foundation for an analog of Lindemann criterion for graphene.

Ising-like criticality derived from the theory of fluids.

It is shown that the conventional theory based on the integral equations for the correlation functions of fluid allows for a reasonable scaling analysis of the critical phenomena. The calculated critical exponents are in rather good agreement with the experimental data for real media, but some of the exponents (first of all, Fisher's exponent eta ) differ from the values predicted by the three-dimensional Ising model. The possibility to obtain Ising-like criticality from the statistical theory of fluids means that the latter does not disagree with the Kadanoff-Wilson-Fisher renormalization group approach.

Statistical theory of noble-gas crystals and the phenomenon of sublimation.

We propose a consistent (without any fitting parameters) statistical theory of classical noble-gas crystals with pair interaction between atoms. Using the equation for the single-particle distribution function of the statistical system, we demonstrate the existence of an infinite number of exact sum rules for the amplitudes of the space-periodic solutions. Even the first sum rule leads to the solution which turns into the exact one at the absolute zero temperature. For the pair distribution function, we obtained the physically correct solution using the well-known exact relation for the compressibility as the self-consistent condition. As a result, we succeeded in recovering the equation of state of the crystal, and starting from the Lennard-Jones potential with the "gaseous" parameters, we calculated the temperature dependencies of the lattice constant and the isothermal compressibility of the crystal at the sublimation line. These calculations (including the form of the sublimation line itself) agree rather well with the corresponding experimental data for the argon-type media in the "classical" temperature region. The question about the bifurcation of the solutions is considered. Ways to further develop the theory are discussed.