Similarity conditions of the processes in systems of interacting particles

The hypotheses and conclusions in this paper result from investigating the feasibility of parametrization in mathematical models which describe the interaction of particles in the liquid media of living systems. The method of parametrization proposed is to consider similarities of such interactions in identical systems which differ only by the intensity of micromotion of particles. Parametrization in the mathematical models of physical processes is necessary for at least two reasons, viz. to simplify the problem of parametric identification of a model [1] and adapt a model for describing a concrete object, which can be called the 'personalization of a moder. Asachenkov and others [1] discuss the idea of parametrization in mathematical models of immuno-physiological processes and try to theoretically justify the parametrization proposed. The application of their approach to the modelling of the processes that take place in an individual organism and in the population as a whole [1,4,6] has shown that the hypotheses that underlie the approach do not contradict the observational data. Therefore it has become necessary to study the processes in greater detail and mathematically rigorously justify the approach to parametrization in mathematical models of immuno-physiological processes. The results of this study are given in the present paper. 1. A SYSTEM OF INTERACTING PARTICLES We consider a system in which a liquid containing particles of m kinds is circulating at volume speed V. The particles of each kind are continuously generated by their own source and arrive at the reactor due to the fluid motion, where because of stirring they collide and interact with each other to form complexes which are absorbed by the environment. The stirring of particles in the reactor is due to its inhomogeneous structure. Thus, the particles which travel due to the ordered fluid motion acquire the properties of the Brownian motion as a result of their multiple collisions with the stationary parts of the reactor. Therefore we consider that the motion of a particle in the reactor is described by the stochastic differential equation (see [2]) where q(t) e R is the generalized coordinates of a particle, S is the number of its degrees of freedom, Ψ(ί) is the drift vector, D=BB is the diffusion matrix, W(t) is the Wiener process. The first term in the right-hand side of this equation describes * Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 117334, Russia 42 G.I. Marchuk, I.B. Pogozhev, and S. M. Zuev the ordered motion of a particle, the second term describes the diffusion motion due to the inhomogeneous structure of the reactor. Both the first and second terms are due to the fluid circulation. 2. MICROMOTION SIMILARITY We consider two systems, viz. the system studied and the base system. Let q(t), q(t) e R, be the generalized coordinates of a particle for the system studied and the base system, respectively. In what follows we underline the designations referring to the base system. We consider the systems for which the relation holds^: d9(0 = «d2(0 (2.1) which is called the similitude relationship. Thus, we have defined the similarity ratio of the system studied with respect to the base system: T/2 where V is the speed of the fluid circulation. Then, using (2.1), we can write for the system studied: and <P(0 = Vtf ! Ρ(0, D=HD. (2.3) This implies that an increase in the velocity of the fluid motion leads not only to an increase in the velocity the ordered motion of particles but also to their more chaotic motion in the reactor, which in turn increases the probability of their collisions and interactions. Let q(t) satisfy the equation «-no. Then it is convenient to write the motion of a particle inside the flow in the new coordinates U(t) = q(f) q(t). According to (1.1) dU(t)=BaW(t) (2.4) and hence we have the lemma. Lemma 2.1. For the system studied and the base system the increments in the generalized coordinates satisfy the equalities U(Ai) = U(AtH), U(At) = Vtf £7(210. (2.5) 'For random processes the equality implies stochastic equivalence. It holds up to the distributions in the space of trajectories, which induce these processes. Similarity conditions of processes 43 Proof. The continued equalities follow from the properties of the Wiener process: Ευ(Δί)υ(Δί) = D At = HD At = Ε^υ(Αή^υ(Αί) Ευ(Αί)υ(Αί) = DAt = HD At = D(HAt) = EU(HAt)U(HAt) which are the proof of the lemma. Here E is the operator of mathematical expectation. Note that the parameter H is thus analogous to absolute temperature at which the Brownian motion of particles in a fluid takes place, to the effective temperature which is introduced by Myasnikov for multicomponent media [3], and to the fluctuation temperature proposed by Regirer [5]. The particle collisions which lead to their interactions result from their stirring, i.e. their diffusion motion. The intensity of particle interactions A is defined as the average number of their interactions in a unit volume in a unit of time: A lim (2.6) Jf-0 At ^ * where Z(At) is a random number of the particle interactions in At. Using equalities (2.5) we can prove that the following lemma is valid. Lemma 2.2. The intensities of interactions in the system studied and the base system satisfy the relation λ=Ηλ. (2.7) Proof. By definition ΕΖ(Αί) = Φ(υτ), 0<τ<Αί i.e. it is a functional of the particle path. Equalities (2.5) allow us to change the time scale: EZ(At) = Φ(υ(θ}) = Φ(υ(ΗΘ)) = Φ(ϋ(τ))=ΕΖ(ΗΑή Hence, taking into account (2.6), we have the equalities a H ̂ t-*Q At At-Q HAt which proves the statement. Let xt be the concentration of free particles of some kind in the reactor at time f, and Xt be their number. By definition where β is the effective volume of interactions, i.e. the measure of the domain Ω which is formed in the reactor by moving particles. The model (1.1) and similitude relationships (2.3) lead to the lemma. 44 G.I. Marchuky LB. Pogozhev, and S.M. Zuev Lemma 2.3. The effective volumes of interactions , Q in the system studied and in the base system satisfy the similarity condition (2.8) Proof. Let p(t,x) =p(t,x,x,x'*) be the distribution of particles in the reactor, i.e. Ω From (2.3) it follows that p(t,x)=e(t9x/jH). (2.9) Indeed according to the law of motion (1.1) the functions p(t,x) and p(t,x) satisfy the equations (see [2]) These equations yield (2.9). Therefore the number of particles contained in an arbitrary domain Ω* is defined as X? ' = JJJ p(t,u) au = JJJ p(tjU/^H)au = /// JJJ p(t,u) du = Ω* Ω* Ω' where Ω* is the domain that corresponds to Ω* and depends on H. It is reasonable to compare the effective volumes in the systems considered only at identical concentrations, i.e. if x( =x(. If this equality holds, then xp' xp' Q* Q* 0* Hence, This proves that relation (2.9) is valid. Lemma 2.4. The stationary concentrations jc^, x^ and the number of particles of some kind X^, X^ in the system studied and in the base system are related by *oo=#~*co (2-lOa)