Generalized Kinetic Models in Applied Sciences

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 eachfluid-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 nonequilibrium 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.Keywords: Multicomponent fluid of living cells and related macromolecules; generalized kinetic theory; stochastic differential equation; probability distribution of parameters of the fluid particles; mode of probability distribution and multimodal distributions.

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“…10 20). This treatment does allow for PC and is in line with the thesis formulated on p. 430 of Ref.…”

Section: Introductionsupporting

confidence: 69%

Modelling Living Fluids With the Subdivision Into the Components in Terms of Probability Distributions

Willander

Mamontov

Chiragwandi

2004

Math. Models Methods Appl. Sci.

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“…In particular we study possible blow-ups and global existence in the space homogeneous case for variable interacting rates. Moreover we state the corresponding results for the nonlocal equation (2). The paper generalizes the previous approach for a simpler case of constant interaction rates in Ref.…”

Section: Introductionmentioning

confidence: 70%

“…We may consider a simplification of the operator Q N in Eq. (2). Denote by (β * f ) the convolution-like product in Ω × V, i.e.…”

mentioning

confidence: 99%

Blow-up and global existence for a kinetic equation of swarm formation

Lachowicz

Leszczyński

Parisot

2017

Math. Models Methods Appl. Sci.

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“…[10,2] and references therein. There is however an important difference: in the case of biological systems a basic microscopic theory, like the Newtonian Laws in the kinetic theory case, it is not available.…”

Section: Introductionmentioning

confidence: 99%

Individually-based Markov processes modeling nonlinear systems in mathematical biology

Lachowicz¹

2011

Nonlinear Analysis: Real World Applications

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Generalized Kinetic Models in Applied Sciences

Cited by 20 publications

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Modelling Living Fluids With the Subdivision Into the Components in Terms of Probability Distributions

Modelling Living Fluids With the Subdivision Into the Components in Terms of Probability Distributions

Blow-up and global existence for a kinetic equation of swarm formation

Individually-based Markov processes modeling nonlinear systems in mathematical biology

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