Eugen Mamontov

MODELLING LIVING FLUIDS WITH THE SUBDIVISION INTO THE COMPONENTS IN TERMS OF PROBABILITY DISTRIBUTIONS

As it follows from the results of C. H. Waddigton, F. E. Yates, A. S. Iberall, and other well-known bio-physicists, living fluids cannot be modelled within the frames of the fundamental assumptions of the statistical-mechanics formalism. One has to go beyond them. The present work does it by means of the generalized kinetics (GK), the theory enabling one to allow for the complex stochasticity of internal properties and parameters of the fluid particles. This is one of the key features which distinguish living fluids from the nonliving ones. It creates the disparity of the particles and hence breaks the each-fluid-component-uniformity requirement underlying statistical mechanics. The work deals with the corresponding modification of common kinetic equations which is in line with the GK theory and is the complement to the latter. This complement allows a subdivision of a fluid into the fluid components in terms of nondiscrete probability distributions. The treatment leads to one more equation that describes the above internal parameters. The resulting model is the system of these two equations. It appears to be always nonlinear in case of living fluids. In case of nonliving fluids, the model can be linear. Moreover, the living-fluid model, as a whole, cannot have the thermodynamic equilibrium, only partial equilibriums (such as the motional one) are possible. In contrast to this, in case of nonliving fluids, the thermodynamic equilibrium is, of course, possible. The number of the fluid components is treated as the number of the modes of the particle-characteristic probability density. In so doing, a fairly general extension of the notion of the mode from the one-dimensional case to the multidimensional case is proposed. The work also discusses the variety of the time-scales in a living fluid, the simplest quantum-mechanical equation relevant to living fluids, and the non-equilibrium nonlinear stochastic hydrodynamics option. The latter is simpler than, but conceptually comparable to, stochastic kinetic equations. A few directions for future research are suggested. The work notes a cohesion of mathematical physics and fluid mechanics with the living-fluid-related fields as a complex interdisciplinary problem.

Dynamic-equilibrium solutions of ordinary differential equations and their role in applied problems

The work introduces the notion of an dynamic-equilibrium (DE) solution of an ordinary differential equation (ODE) as the special (limit) version of the ODE general solution. The dynamic equilibrium is understood as independence of the initial point. The work explains the special importance of ODEs which have DE solutions. The criteria for the existence and global attraction of these solutions are developed. A few examples illustrate different aspects of the DE-solution theory and application. The work discusses the role of these solutions in applied problems (related to ODEs in both Euclidean and function Banach spaces) with the emphasis on advanced models for living systems (such as the active-particle generalized kinetic theory). This discussion also concerns a few directions for future research.

Modelling homeorhesis by ordinary differential equations

Homeorhesis is a necessary feature of any living system. If a system does not perform homeorhesis, it is nonliving. The present work develops the sufficient conditions for the ODE model to describe homeorhesis and suggests the structure of the model. The proposed homeorhesis model is fairly general. It treats homeorhesis as piecewise homeostasis. The model can be specified in different ways depending on the specific system and specific purposes of this analysis. An example of the specification is the PhasTraM model, the homeorhesis-aware nonlinear reaction-diffusion model for hyperplastic oncogeny in the previous works of the author. The qualitative agreement of the developed homeorhesis model with the living-system experimental results is noted. The work also shows that the basic mathematical models (such as the active-particle generalized kinetic theory) are substantially more important for the living-matter studies than in the case of nonliving matter. A few directions for future research are suggested as well.

Homeorhesis-based modelling and fast numerical analysis for oncogenic hyperplasia under radiotherapy

A few previous works of the authors derived and discussed the space-time mathematical description, the PhasTraM model, for oncogenic hyperplasia regarded as a genotoxically activated homeorhetic dysfunction. The model is based on the fluid-to-solid-and-back transitions and nonlinear reaction-diffusion equations relevant to a series of the key biomedical facts, and some distinguishing features of living systems. The first computer-simulation results have also been reported. The present work generalizes the PhasTraM model for the effect of radiation therapies (RTs), both external and internal. The resulting model also includes the autocrine mechanism promoting oncogenic hyperplasia and the suppression of this process by certain drugs. The autocrine signalling is implemented by the transforming-growth-factor-@a (TGF-@a) molecules released by the cells and bound to the epidermal-growth-factor receptors (EGFRs) at the cell surface. The suppression can be carried out by a drug deactivating the mentioned molecules. The work also presents and discusses examples of the computer-simulation results for four different settings of the applied RT. A few directions for future research, as well as prospective applications of the model and developed software, are also discussed.

Stochastic mechanics in the context of the properties of living systems

Many features of living systems prevent the application of fundamental statistical mechanics (FSM) to study such systems. The present work focuses on some of these features. After discussing all the basic approaches of FSM, the work formulates an extension of the kinetic theory paradigm (based on the reduced one-particle distribution function) that exhibits all of the living-system properties considered. This extension appears to be a model within the generalized kinetic theory developed by N. Bellomo and his co-authors. In connection with this model, the work also stresses some other features necessary for making the model relevant to living systems. A mathematical formulation of homeorhesis is also derived. An example discussed in the work is a generalized kinetic equation coupled with a probability-density equation representing the varying component content of a living system. The work also suggests a few directions for future research.

The nonzero minimum of the diffusion parameter and the uncertainty principle for a Brownian particle

The limits of applicability of many classical (non-quantum-mechanical) theories are not sharp. These theories axe sometimes applied to the problems which axe, in their nature, not very well suited ...

Nonsearch Paradigm for Large-Scale Parameter-Identification Problems in Dynamical Systems Related to Oncogenic Hyperplasia

In many engineering and biomedical problems there is a need to identify parameters of the systems from experimental data. A typical example is the biochemical-kinetics systems describing oncogenic hyperplasia where the dynamical model is nonlinear and the number of the parameters to be identified can reach a few hundreds. Solving these large-scale identification problems by the local- or global-search methods can not be practical because of the complexity and prohibitive computing time. These difficulties can be overcome by application of the non-search techniques which are much less computation- demanding. The present work proposes key components of the corresponding mathematical formulation of the nonsearch paradigm. This new framework for the nonlinear large-scale parameter identification specifies and further develops the ideas of the well-known approach of A. Krasovskii. The issues are illustrated with a concise analytical example. The new results and a few directions for future research are summarized in a dedicated section.