Research / Main fundamental results / Methods of statistical physics and computer simulations

# Methods of statistical physics and computer simulations

**Method of collective variables.** Lviv school of statistical physics is well known in physics community mainly due to the method of collective variables elaborated for description of collective effects in classical systems of interacting particles. An extremely fruitful idea was the separation of a basic system by means of an average of the Jacobian of transition to collective variables over the short-range interactions. The suggested approach was applied to description of ion-dipole systems, that was an initial step in construction of a microscopic theory of electrolyte solutions, that is based on equal accounting for all the possible interactions between ions of electrolyte and molecules of solvent.

A modification of the method of collective variables for its application to description of quantum systems of interacting particles is known in the literature as a method of displacements and collective variables. The main idea of the method is in a separation of a part from the statistical operator, that characterizes interaction of quantum wave packets of particles. The suggested approach appeared extremely fruitful for description of systems of interacting Bose and Fermi particles.

**Theoreical description of phase transitions.** Based on the method of collective variables there has been suggested a method for theoretical description of phase transitions and connected with them critical phenomena for a broad range of problems of statistical physics. This method is based on a suggested by I.R.Yukhnovskii original scheme for calculation of partition function of the three-dimensional Ising model, that makes use of idea of existence of specific collective variables in the phase space, average values of which are connected with order parameter. Namely this has opened a way to construction of a consistent theory of phase transitions on microscopic level, in particular for magnetic and ferroelectric systems, region of liquid-gas critical point, demixing phenomena etc. A generalization of grand canonical distribution was constructed.

A method of calculation of free energy in vicinity of a point of the phase transition of second order with applied external field. An effect of the external field on behaviour of physical quantities of the system (susceptibility, specific heat, order parameter etc) close to the temperature of phase transition was studied and their dependence on microscopic parameters was established. There was formulated a method of calculation of critical indices of “ordinary ” and “special” phase transitions in massive field theory for three-dimensional semi-confined systems. It was shown, that the surface disorder does not effect the special transition, while bulk disorder changes critical indices of the ordinary transition. Based on an original approach there were calculated critical indices of the Lifshitz point with arbitrary number of anisotropy axes and shown that the anisotropy index is non-classical.

**Multi-density formalism.** Within a concept of association in statistical theory of liquids was developed a multi-density formalism, that permitted to extend the methods of simple fluids onto complex liquids. In this scheme the potentials of intermolecular interaction are separated into three parts: short-range repulsion that defines the structure of simple fluids; long-range interactions that have electrostatic origin and define energetic properties of liquids; short-range strongly attractive interactions that are responsible for formation of various associative complexes and clusters. The theory is based on a diagram technique and combination of expansions in activity and density fro correlation functions that are used for description of associate and non-associate interactions, respectively.

**Analytical approach in the theory of dynamic mean field** in the theory of strongly correlated electron systems – materials and compounds for which the single-electron approach cannot be applied – were developed **original tight binding approaches that make use the technique of Hubbard operators and diagram expansions for Green functions**. Within a theory of dynamic mean field was developed a new analytic scheme that is based on both the formalism of auxiliary Fermi-field and approach of generating functional. On this basis for the models of the class of Hubbard model was constructed a sequence of analytical approximations (like generalization of the Hubbard-III approximation), in framework of which were studied the thermodynamics and features of electron spectrum in the region of metal-nonmetal transitions as well as mixed valence transitions.

There was developed a general approach to construction of **spectral relations for many-time correlation functions** with a special focus on treatment of non-ergodic contributions. A representation of many-time Green functions was obtained via spectral densities and an inverse problem was solved. By making use of these spectral relations were obtained general relations that connect cross-section of inelastic scattering of electromagnetic waves with many-time Green functions with all the contributions – non-resonant as well as resonant and mixed ones – taken into account.

**Method of generalized collective modes.** An approach of generalized collective modes within the statistical hydrodynamics of simple and many-component liquids was suggested for investigation of collective behaviour and generalized transport coefficients. Within this approach the time correlation functions are represented as a separable sum of contributions with different weight coefficients from dynamic eigenmodes of the system, that describe relaxing and propagating processes in liquids. A **theory of non-hydrodynamic collective excitations** in simple and many-component liquids was constructed, that permitted for the first time to describe consistently such collective processes in liquids as structural relaxation, heat waves and excitations of optic phonon type.

Based on N.Bogolyubov’s ideas on reduced description of non-equilibrium processes and the method of non-equilibrium statistical operator was developed a methodology of consistent description of kinetic and hydrodynamic processes for dense gases, plasma and liquids. There was suggested a formulation of non-equilibrium thermofield dynamics for quantum-field systems. A combination of the methods of Zubarev’s non-equilibrium statistical operator and Green functions was formulated and developed for quantum spatially non-uniform electron systems and shown its connection to time-dependent density and current-density functional theories.

A method of calculation of electron structure of transition and rare-earth metals was suggested on the basis of a **formalism of completely orthogonalized plane waves (COPW)**. Within such an approach the ab initio pseudopotentials are represented via unitary transformation of initial atomic potentials on a complete and orthogonal set of basis functions. Due to absence of overcompleteness of the basis set, as it was for the OPW set, it was possible to obtain ab initio COPW pseudopotentials free of drawbacks that contain majority of ab initio pseudopotentials.

**New numerical methods.** In the field of numerical schemes for integration of equations of motion for classical and quantum simulations were proposed a class of high-precision algorithms. Within the approach of factorization of evolution operators there was performed the complete classification and consistent derivation of all explicit decomposition algorithms with a number of exponential operators up to 11. Hence in addition to the 8 known earlier algorithms were obtained 37 new schemes with improved precision up to 6 orders in time step. It was shown, that some new algorithms can be more than two orders effective in comparison with well known integrators as Verlet, Forest-Ruth, Suzuki, Lee etc.

Algorithms for integration of equations of motion were developed for polymer liquid crystal systems in the method of molecular dynamics. In particular, were generalized the methods with a thermostat and barostat within the ensemble with constant pressure and temperature for a system of spherical and anisometric particles. This permitted to describe effectively the internal structure and intramolecular dynamics of a number of complex polymer liquid crystals, in particular brush-like and dendritic systems. Application of the proposed algorithms was successful for description of the main mechanisms of experimentally observed phenomenon of photo-induced deformations in azobenzene films that was caused by the process of photoisomerization.