Convergence to equilibrium for many particle systems

The goal of this paper is to give a short review of recent results of the authors concerning classical Hamiltonian many particle systems. We hope that these results support the new possible formulation of Boltzmann's ergodicity hypothesis which sounds as follows. For almost all potentials, the minimal contact with external world, through only one particle of $N$, is sufficient for ergodicity. But only if this contact has no memory. Also new results for quantum case are presented

Cell mechanics in flow: algorithms and applications

The computer simulations are pervasively used to improve the knowledge about biophysical phenomena and to quantify effects which are difficult to study experimentally. Generally, the numerical methods and models are desired to be as accurate as possible on the chosen length and time scales, but, at the same time, affordable in terms of computations. Until recently, the cell mechanics and blood flow phenomena on the sub-micron resolution could not be rigorously studied using computer simulations. However, within the last decade, advances in methods and hardware catalyzed the development of models for cells mechanics and blood flow modeling which, previously, were considered to be not feasible. In this context, a model should accurately describe a phenomenon, be computationally affordable, and be flexible to be applied to study different biophysical changes. This thesis focuses on the development of the new methods, models, and high-performance software implementation that expand the class of problems which can be studied numerically using particle-based methods. Microvascular networks have complex geometry, often without any symmetry, and to study them we need to tackle computational domains with several inlets and outlets. However, an absence of appropriate boundary conditions for particle- based methods hampers study of the blood flow in these domains. Another obstacle to model complex blood flow problems is the absence the highperformance software. This problem restricts the applicability of the of particlebased cell flow models to relatively small systems. Although there are several validated red blood cell models, to date, there are no models for suspended eukaryotic cells. The present thesis addresses these issues. We introduce new open boundary conditions for particle-based systems and apply them to study blood flow in a part of a microvascular network. We develop a software demonstrating outstanding performance on the largest supercomputers and used it to study blood flow in microfluidic devices. Finally, we present a new eukaryotic cell model which helps in quantifying the effect of sub-cellular components on the cell mechanics during deformations in microfluidic devices

The problem with peaking in the atmospheric magnetohydrodynamics. limiting cases

For the mathematical modelling of highly nonequilibrium and nonlinear processes in the atmosphere based on the equations of momentum and charge transfer, a thermodynamic approach is used with the model function of sources and sinks, which is characteristic for problems with peaking, where the maximum of the velocity distribution and charge distribution in space can increase without bound for a limited time. It allows to consider the general case, taking into account the interaction between the components of the velocity vector and the electromagnetic field in the presence of sources and sinks of momentum in a flat layer. As a limiting case, we consider the transfer of momentum when its source in a nonlinear medium leads to the regime with peaking, and the development of the regime generated by a nonlinear medium itself leads to self-organization. The competition between the process increment and the propagation of momentum and charge can result in appearance of new medium characteristics, such as the spatial diameter of tornado (lightning core), in which these processes balance each other. Another limiting case is the process of charge transfer in an atmosphere considered. As a result, a more general problem may be formulated, and a joint system of equations, which not only describes the behaviour of the velocity vector for an incompressible medium in the form of parabolic equation of momentum, but also takes into account the influence of electromagnetic field, may be derived