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Mathematical and computer models of natural sciences systems

Priority areas of development: mathematics
The project has been carried out as part of the HSE Program of Fundamental Studies.

In this project we investigate certain mathematical models of natural sciences systems which are important for high-tech and intensively studying in the world.

Various quantum phenomena in nano- and low-dimensional systems, based on graphene, semiconductor heterostructures and superconducting microstructures, were studied. In particular, renormalization of the Fermi velocity in a system of correlated massless Coulomb-interacting electrons in graphene was considered. Calculations of the renormalized Fermi velocity were carried out in four different approximations and compared with experimental data. They can be used to model analog transistors based on graphene  and to calculate their characteristics.

Violation of conventional virial theorem for particles with linear dispersion was found in the case of graphene.

Spaser spectroscopy based on spaser consisting of plasmon resonator represented by a graphene flake and active medium of quantum dots was considered. Sensitivity of the spaser was estimated and turned out to be rather high and allowing to use it as a sensor of microscope. Nonlinear internal oscillations in a Bose-Einstein condensate of excitonic polaritons were studied. They were shown to occur in two qualitatively different regimes. Transition from one regime to another can be observed in experiments.

Dynamical Lamb effect in a superconducting system consisting of qubit and resonator was also studied. The method of essential amplification of this effect by means of periodically modulated coupling between the qubit and resonator was proposed. This effect can be realized experimentally and used to create quantum memory based on superconducting qubits.

The finite element method was used to implement an algorithm for calculating the processes which occur in the case of thermal emission from nanocathodes of complex geometry. The process parameters were computed for the nanocathode geometry realized as a ball placed at the cathode top. The results can be used to design and predict the properties of nanoelectron devices intended for different purposes.

A formal asymptotic solution was constructed for the problem of flow of a viscous incompressible liquid around a plane half-infinite plate with a small localized irregularity on its surface, like a small hump, step, or corner-shaped break, for large Reynolds numbers. It was shown that the boundary layer has a double-deck structure, i.e., it consists of a thin near-wall boundary layer and a ``thick’’ boundary layer. An algorithm, based on a difference scheme satisfying the maximum principle, is constructed to solve the equations describing the flow in the thin boundary layer is presented and a numerical simulation is performed. The influence of irregularity parameters on the character of the flow in the thin boundary layer is investigated. It is shown that in the case of a compressible liquid flowing around the plate, the double-deck structure of the flow is also possible, and the parameters representing this structure are the same as in the case of incompressible liquid. The results can also find application in the processes of chemical engineering (interaction between surfaces and active liquids) and in biophysics (flows in vessels).

In the model of resonance planar Penning nanotrap, the effective Hamiltonian was investigated near the points of extrema and stable extremum points were selected. The explicit analytic dependence of normal frequencies at the extremum points on the external parameters of the trap was obtained.  The results can be used in design of nanoelectronics devices and in qubit prototypes used in quantum computations.

Biaxial nematic liquid crystals are studying through the textural transformations in external electric or magnetic fields. We employ theoretical approaches which could have bearing on the minimization problem of the multi-parametric free energy. The direct free energy minimization (widely used for uniaxial nematics) requires taking into account non-linear constraints. We have worked out the method to overcome this difficulty by employing the curvature of the texture, which describes space rotations of the order parameter. We have obtained the tool to find all possible one-dimensional textures of biaxial nematics in external field. To illustrate the method we calculate the critical fields corresponding to the basic configurations of textural transitions in the biaxial nematics. The method could be useful for determining the intrinsic degree of biaxiality for liquid crystalline materials. In fact, it has turned to be effective in the computer modeling the texture conformations in external electric fields. The above results has been applied for studying the Frederics effect and finding the threshold external fields. The problems indicated above are intimately related to the compact sets formed by the molecules of the DNA packed in cells and fags. The sets have the toroidal shape, and it is usual to call them the DNA toroids. It is important that at a small scale there is the orientational ordering similar to that specific for the liquid crystals.    

Analytical solution of the Kolmogorov-Feller equations that describe the ultrametric diffusion in a centrally symmetric potential field with a global minimum is constructed. The solution of such equations is of interest to ultrametric modeling of dynamics on multidimensional rugged energy landscapes, which are exploited in many modern applications of the "complex systems" paradigm, in particular, in mathematical modeling of molecular machines. The solutions for ultrametric diffusion on self-similar hierarchical energy landscapes typical for biological molecular machines were obtained and studied. It is shown that in this case the relaxation to the global minimum on intermedium time scales is described by a power function.

Random networks of different topology (the prototype of molecular machines) are studied. We have considered an equilibrium ensemble of large Erdhos-Renyi topological random networks with two types of vertices, "black" and "white", and fixed vertex degree. The system energy is a sum of all unicolor triples (either all black or all white), weighted with a chemical potential. Minimizing the system energy, we see at any positive value of chemical potential the formation of two unicolor clusters, linked by a "string".  The system exhibits a critical behavior manifested in emergence of a wide plateau in the dependence of the number of black-white links on the chemical potential.  The results permit discovering which of the topological characteristics (distribution over the connectivity degrees, distribution of the local clusterization coefficient) are responsible for the formation of the cluster structure in the network.

Supercomputing of the exascale era is inevitably limited by power efficiency.  Nowadays, different CPU architectures are considered as possible choices for these purposes. Recently, the development of ARM processors has come to the point when their floating point performance can be seriously considered for  a range of scientific applications. In this work we present an analysis of the floating point performance of the latest ARM cores and their efficiency for the algorithms of classical molecular dynamics. The results of these studies show that the use of ARM processors is very perspective for high-performance computations. The perspectives of supercomputer architecture development are considered.

The investigations performed in this project are based on original ideas.  The research in all subsections of this project was performed at the highest scientific level comparable with the first-rate investigations in the world. The obtained results are mainly published in authoritative international reviewed journals and were delivered at representative conferences. The investigations performed in the framework of this project can be further extended and developed which can lead to new important theoretical achievements and find new actual applications in high technologies.


Berman O. L., Kezerashvili R. Y., Yurii E. Lozovik. Graphene-Based Photonics and Plasmonics, in: Nanoscale Materials and Devices for Electronics, Photonics and Solar Energy. Springer, 2015. P. 93-126.
Голо В. Л., Блинов В. Н., Крупянский Ю. Ф. ТОРИЧЕСКИЕ КОНФОРМАЦИИ МОЛЕКУЛ ДНК // Наноструктуры. Математическая физика и моделирование. 2015. Т. 12. № 2. С. 5-20.
Voronova N., Elistratov A., Yurii Lozovik. Detuning-controlled internal oscillations in an exciton-polariton condensate // Physical Review Letters. 2015. Vol. 115. No. 18. P. 186402. doi
Shapiro D., Zhukov A., Pogosov W., Lozovik Y. Dynamical Lamb effect in a tunable superconducting qubit-cavity system // Physical Review A: Atomic, Molecular, and Optical physics. 2015. Vol. 91. No. 6. P. 063814. doi
Valba O. V., Nechaev S., Sterken M. G., Snoek L. B., Kammenga J. E., Vasieva O. O. On predicting regulatory genes by analysis of functional networks in C. elegans // BioData Mining. 2015. Vol. 8. No. 33. P. 1-16. doi
Danilov V., Gaydukov R. Vortices in the Prandtl boundary layer induced by irregularities on a plate // Russian Journal of Mathematical Physics. 2015. Vol. 22. No. 2. P. 161-173. doi
M.V.Karasev, E.M.Novikova. Planar Penning trap with combined resonance and top dynamics on quadratic algebra // Russian Journal of Mathematical Physics. 2015. Vol. 22. No. 4. P. 463-468.
Nikolskiy V., Stegailov V. Floating-point performance of ARM cores and their efficiency in classical molecular dynamics // Journal of Physics: Conference Series. 2016. Vol. 681. No. 012049. P. 1-6. doi
Yu. E. Lozovik, A. A. Sokolik, Zabolotskiy A. D. Quantum capacitance and compressibility of graphene: The role of Coulomb interactions // Physical Review B: Condensed Matter and Materials Physics. 2015. Vol. 91. No. 075416. P. 1-8. doi