Theory of phase transitions and critical phenomena

The work on the development of a theory of phase transitions and critical phenomena was initiated by I. R. Yukhnovskii in the mid-1980s. The method of collective variables he developed for the study of spin systems opened the route to the construction of a consistent microscopic theory of phase transitions for a number of physical systems. To name some of the achivements, magnetic and segnetoelectric systems, liquid-gas in the vicinity of the critical point, systems with layering phase transitions, and a number of other physical objects were succesfully described (I. R. Yukhnovskii, I. O. Vakarchuk, Yu. K. Rudavskii, M. P. Kozlovskii, I. M. Mryglod, I. V. Stasyuk, M. A. Korynevskii, O. V. Patsahan, I. V. Pylyuk).

In the last decade interesting results were obtained in the theory of critical phenomena with the use of the field-theoretical approach (Yu. V. Holovatch, M. A. Shpot, Z. Ye. Usatenko, M. L. Dudka, V. B. Blavatska).

Microscopic theory of phase transitions. A theory of phase transitions for a magnet with one-component order parameter was built on the microscopic level. By means of the tree-dimensional Ising model critical behaviour of similar models with a second order phase transition was studied. All thermodynamic characteristics of a three-dimensional Ising-like system were derived from the first principles with the confluent corrections taken into account. The microscopic analogue of the Landau free energy was found and a method for obtaining the equation of state in the vicinity of the critical point was proposed. The influence of the exponentially decaying interaction potential on the critical temperature, critical region size, thermodynamic functions behaviour was studied.

Yukhnovskii's theory of phase transitions was generalized for the case of an external field. The developed theoretical approach enabled description of the thermodynamic charasteristics dependance on the temperature and external field as well as on microscopic parameters of the system (lattice spacing, interaction potential parameters) near the phase transition point. For the first time an equation of state for a three-dimensional Ising-like system was proposed and the explicite form of the corresponding scaling function was obtained. It was shown that in the two limiting cases of large and small external fields this equation of state takes the well known forms.

Explicite analitical expressions were obtained for the thermodynamic functions of a classical n-component model of a magnet in three dimensions. The dependence of the critical amplitudes of the basic thermodynamic characteristics on the normalized temperature and microscopic parameters of the system in high- and low-temperature phases was studied. The cases with particular values of the spin component number were considered. The mechanism of the heat capacity peak formation near Tc was described in the classical Heisenberg model and its value dependence on microscopic parameters was found.

A microscopic approach to the phase transition description in multicomponent continuous systems was developed. The theory that was proposed based on the collective variables method with a marked reference system. The explicite form of the Ginsburg-Landau-Wilson microscopic Hamiltonian was obtained in the vicinity of the phase transition point of a two-component continuous system. The problem of the order parameter determination in a binary fluid mixture was studied in detail. Explicite expressions for the thermodynamic functions and equation of state were obtained in the case of a symmetric binary fluid mixture in the vicinity of the liquid-gas critical point.

Disorder and criticality in magnets. The phenomena that take place in the proximity of the Curie point under the influence of structural disorder were studied. A change in the critical behaviour under the influence of different kinds of disorder (substitution disorder, random anisotropy, frustrations) was predicted and quantitative characteristics of this behaviour were obtained. The spontaneous order parameter appearance was studied in the two-dimensional finite-size systems with continuous symmetry. The quasi-long-range ordering that appears in these systems in the thermodynamic limit was studied as well.

Dependence of the effective critical exponents of m-vector magnets with extended impurities on the renormalization group flow parameter connected to the distance to the critical temperature was studied. It was established that even weak dilution of an Ising magnet with impurities in the form of parallel lines leads faster to the transition into a new universality class than that with point-like impurities. Within the general thermodynamic formalism framework these phase transitions were proved to classify as second order phase transitions.

For the systems with thermodynamic restraints the existence of completely new scaling corrections, proportional to the heat capacity critical exponent of the system without restraints, was established. Besides the well known renormalization of the asymptotic critical exponents (Fisher renormalization) other corrections appear in such systems and they are dominant when leaving the critical region and define the pecularities of the behaviour of a system with restraints. The inclusion of these new corrections to the scaling allowed to eliminate the inconsistencies which were observed in the computer calculations of critical exponents in a number of systems.

Scaling of polymers with complex topology. A theoretical description of the scaling properties of multisorted polymer nets and stars was proposed. The results obtained were applied to the construction of phase diagrams for polymers with complex topology, study of the polymers interaction in a good solvent, description of the catalysis reactions near polymer macromolecules, description of the structural transitions in DNA molecules.

Complex networks. Following the complex networks theory based approach, it was shown that public transport networks are scale-free small worlds. A model was proposed to describe scaling properties of such networks and their stability with respect to random damages and directed attacts was analized.

In anisotropic and spatially confined systems with competing exchange interactions first non-trivial orders of the 1/N-expansions for correlation critical exponents in m-fold Lifshitz points were obtained. New conclusions about the qualitative behaviour of these exponents with the change of space dimensionality were made. Non-classical surface exponents of an ordinary transition in such systems were calculated. Numerical estimations for the Kasimir amplitudes in three dimensions were made. Surface critical behaviour was studied in semi-confined n-component systems in the presence of cubic anisotropy (m=1) at an ordinary phase transition.

Thermodynamic and structural characteristics of mixed segneto- and antisegnetoelectric systems were studied. On the basis of an original model in cluster approximation with non-equilibrium distribution of the structure elements that experience ordering, temperature-concentration phase diagrams were obtained. It is proved that segneto- and antisegnetoelectric phases can only exist separately. The threshold values of the component concentration at which the ordered state of each type vanishes — percolation thresholds — are found. The developed theory explains experimentally observed properties of solid mixtures of segnetoelectric systems.