Open access peer-reviewed chapter

Kinetic Equations of Active Soft Matter

Written By

Viktor Gerasimenko

Submitted: 14 May 2017 Reviewed: 22 August 2017 Published: 14 February 2018

DOI: 10.5772/intechopen.70667

From the Edited Volume

Kinetic Theory

Edited by George Z. Kyzas and Athanasios C. Mitropoulos

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Abstract

We consider a new approach to the description of the collective behavior of complex systems of mathematical biology based on the evolution equations for observables of such systems. This representation of the kinetic evolution seems, in fact, the direct mathematically fully consistent formulation modeling the collective behavior of biological systems since the traditional notion of the state in kinetic theory is more subtle and it is an implicit characteristic of the populations of living creatures.

Keywords

  • kinetic equation
  • marginal observables
  • scaling limit
  • active soft matter

1. Introduction

The rigorous derivation of kinetic equations for soft condensed matter remains an open problem so far. It should be noted wide applications of these evolution equations to the description of collective processes of various nature [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], in particular, the collective behavior of complex systems of mathematical biology [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. We emphasize that the considerable advance in solving the problem of rigorous modeling of the kinetic evolution of systems with a large number of constituents (entities) of mathematical biology, in particular, systems of large number of cells, is recently observed [20, 21, 22, 23, 24, 25, 26] (and see references cited therein).

In modern research, the main approach to the problem of the rigorous derivation of kinetic equation consists in the construction of scaling limits of a solution of evolution equations which describe the evolution of states of a many-particle system, in particular, a perturbative solution of the corresponding BBGKY hierarchy [2, 3, 4].

In this chapter, we review a new approach to the description of the collective behavior of complex systems of mathematical biology [17, 18] within the framework of the evolution of observables. This representation of the kinetic evolution seems, in fact, the direct mathematically fully consistent formulation modeling kinetic evolution of biological systems since the notion of the state is more subtle and it is an implicit characteristic of populations of living creatures.

One of the advantages of the developed approach is the opportunity to construct kinetic equations in scaling limits, involving initial correlations, in particular, that can characterize the condensed states of soft matter. We note also that such approach is also related to the problem of a rigorous derivation of the non-Markovian kinetic-type equations from underlying many-cell dynamics which make it possible to describe the memory effects of the kinetic evolution of cells.

Using suggested approach, we establish a mean field asymptotic behavior of the hierarchy of evolution equations for marginal observables of a large system of interacting stochastic processes of collisional kinetic theory [24], modeling the microscopic evolution of active soft condensed matter [14, 15]. The constructed scaling limit of a non-perturbative solution of this hierarchy is governed by the set of recurrence evolution equations, namely, by the dual Vlasov hierarchy for interacting stochastic processes.

Furthermore, we established that for initial states specified by means of a one-particle distribution function and correlation functions the evolution of additive-type marginal observables is equivalent to a solution of the Vlasov-type kinetic equation with initial correlations, and a mean field asymptotic behavior of non-additive-type marginal observables is equivalent to the sequence of explicitly defined correlation functions which describe the propagation of initial correlations of active soft condensed matter.

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2. On collisional dynamics of active soft condensed matter and the evolution of marginal observables

The many-constituent systems of active soft condensed matter [14, 15] are dynamical systems displaying a collective behavior which differs from the statistical behavior of usual gases [2, 4]. In the first place, their own distinctive features are connected with the fact that their constituents (entities or self-propelled particles) show the ability to retain various complexity features [14, 15, 16, 17, 18]. To specify such nature of entities, we consider the dynamical system suggested in papers [13, 24, 29] which is based on the Markov jump processes that must represent the intrinsic properties of living creatures.

A description of many-constituent systems is formulated in terms of two sets of objects: observables and states. The functional of the mean value of observables defines a duality between observables and states and as a consequence there exist two approaches to the description of the evolution of such systems, namely in terms of the evolution equations for observables and for states. In this section, we adduce some preliminary facts about dynamics of finitely many entities of various subpopulations described within the framework of non-equilibrium grand canonical ensemble [2].

We consider a system of entities of various M subpopulations introduced in paper [24] in case of non-fixed, i.e., arbitrary, but finite average number of entities. Every i th entity is characterized by: ui=jiuiJ×U, where jiJ1M is a number of its subpopulation, and uiURd is its microscopic state [24]. The stochastic dynamics of entities of various subpopulations is described by the semigroup etΛ=n=0etΛn of the Markov jump process defined on the space Cγ of sequences b = (b0, b1,  … , bn, …) of measurable bounded functions bn(u1,  … , un) that are symmetric with respect to permutations of the arguments u1 ,  …  , un and equipped with the norm:

bCγ=maxn0γnn!bnCn=maxn0γnn!maxj1,,jnmaxu1,,unbnu1un,

where γ < 1 is a parameter. The infinitesimal generator Λn of collisional dynamics (the Liouville operator of n entities) is defined on the subspace Cn of the space Cγ and it has the following structure [24]:

Λnbn)(u1unm=1Mεm1i1im=1nΛmi1imbn)(u1un=m=1Mεm1i1im=1namui1uim(J×UAmvui1uim×bnu1ui11vui1+1undvbnu1un),E1

where ε > 0 is a scaling parameter [28], the functions a[m](ui1,  … , uim) , m ≥ 1, characterize the interaction between entities, in particular, in case of m = 1 it is the interaction of entities with an external environment. These functions are measurable positive bounded functions on J×Un such that: 0amui1uimam, where am is some constant. The functions A[m](v; ui1,  … , uim) , m ≥ 1, are measurable positive integrable functions which describe the probability of the transition of the i1 entity in the microscopic state ui1 to the state v as a result of the interaction with entities in the states ui2 ,  …  , uim (in case of m = 1 it is the interaction with an external environment). The functions A[m](v; ui1,  … , uim) , m ≥ 1, satisfy the conditions: J×UAmvui1uimdv=1,m1. We refer to paper [24], where examples of the functions a[m] and A[m] are given in the context of biological systems.

In case of M = 1 generator (1) has the form i1=1nΛn1i1 and it describes the free stochastic evolution of entities, i.e., the evolution of self-propelled particles. The case of M = m ≥ 2 corresponds to a system with the m -body interaction of entities in the sense accepted in kinetic theory [30]. The m-body interaction of entities is the distinctive property of biological systems in comparison with many-particle systems, for example, gases of atoms with a pair-interaction potential.

On the space Cn the one-parameter mapping etΛn is a bounded -weak continuous semigroup of operators.

The observables of a system of a non-fixed number of entities of various subpopulations are the sequences O = (O0, O1(u1),  … , On(u1,  … , un), …) of functions On(u1,  … , un) defined on J×Un and O0 is a real number. The evolution of observables is described by the sequences O(t) = (O0, O1(t, u1),  … , On(t, u1,  … , un), …) of the functions

Ont=etΛnOn0,n1,

that is, they are the corresponding solution of the Cauchy problem of the Liouville equations (or the Kolmogorov forward equation) with corresponding initial data On0:

tOnt=ΛnOnt,Ontt=0=On0,n1,

or in case of n noninteracting entities (self-propelled particles) these equations have the form.

tOntu1un=i=1na1ui(J×UA1vuiOntu1ui1vui+1undv.
Ontu1un),n1.

The average values of observables (mean values of observables) are determined by the following positive continuous linear functional defined on the space Cγ:

Ot=ID01OtD0ID01n=01n!J×Undu1dunOntDn0,E2

where D0=1D10Dn0 is a sequence of nonnegative functions Dn0 defined on J×Un that describes the states of a system of a non-fixed number of entities of various subpopulations at initial time and ID0=n=01n!J×Undu1dunDn0 is a normalizing factor (the grand canonical partition function).

Let Lα1=n=0αnLn1 be the space of sequences f = (f0, f1,  … , fn, …) of the integrable functions fn(u1,  … , un) defined on J×Un, that are symmetric with respect to permutations of the arguments u1 ,  …  , un, and equipped with the norm:

fLα1=n=0αnfnLn1=n=0αnj1JjnJUndu1dunfnu1un,

where α > 1 is a parameter. Then for D(0) ∈ L1 and O(t) ∈ Cγ average value functional (2) exists and it determines a duality between observables and states.

As a consequence of the validity for functional (2) of the following equality:

ID01OtD0=ID01etΛO0D0=IetΛD01O0etΛD0IDt1O0Dt,

where etΛ=n=0etΛn is the adjoint semigroup of operators with respect to the semigroup etΛ=n=0etΛn, it is possible to describe the evolution within the framework of the evolution of states. Indeed, the evolution of all possible states, i.e. the sequence D(t) = (1, D1(t, u1),  … , Dn(t, u1,  … , un), …) ∈ L1 of the distribution functions Dn(t) , n ≥ 1, is determined by the formula:

Dnt=etΛnDn0,n1,

where the generator Λn is the adjoint operator to operator (1) and on Ln1 it is defined as follows:

Λnfnu1unm=1Mεm1i1im=1n(J×UAmui1vui2uimamvui2uimfnu1ui11vui1+1undvamui1uimfnu1un),E3

where the functions A[m] and a[m] are defined as above in (1).

The function Dn(t) is a solution of the Cauchy problem of the dual Liouville equation (or the Kolmogorov backward equation).

On the space Ln1 the one-parameter mapping etΛn is a bounded strong continuous semigroup of operators [26].

For the description of microscopic behavior of many-entity systems we also introduce the hierarchies of evolution equations for marginal observables and marginal distribution functions known as the dual BBGKY hierarchy and the BBGKY hierarchy, respectively [26]. These hierarchies are constructed as the evolution equations for one more method of the description of observables and states of finitely many entities.

An equivalent approach to the description of observables and states of many-entity systems is given in terms of marginal observables B(t) = (B0, B1(t, u1),  … , Bs(t, u1,  … , us), …) and marginal distribution functions F0=1F10,εu1Fs0,εu1usLα1.

Considering formula (2), marginal observables and marginal distribution functions are introduced according to the equality:

Ot=ID01OtD0=BtF0,

where (I, D(0)) is a normalizing factor defined as above. If F0Lα1 and B(0) ∈ Cγ, then at t ∈ ℝ the functional (B(t), F(0)) exists under the condition that: γ > α−1.

Thus, the relationship of marginal distribution functions F0=1F10,εFs0,ε and the distribution functions D0=1D10Dn0 is determined by the formula:

Fs0,εu1usID01n=01n!J×Undus+1dus+nDs+n0u1us+n,s1,

and, respectively, the marginal observables are determined in terms of observables as follows:

Bstu1usn=0s1nn!j1jn=1sOsntu1us\uj1ujn,s1.E4

Two equivalent approaches to the description of the evolution of many interacting entities are the consequence of the validity of the following equality for the functional of mean values of marginal observables:

BtF0=B0Ft,

where B0=1B10,εBs0,ε is a sequence of marginal observables at initial moment.

We remark that the evolution of many-entity systems is usually described within the framework of the evolution of states by the sequence F(t) = (1, F1(t, u1),  … , Fs(t, u1,  … , us), …) of marginal distribution functions Fs(t, u1,  … , us) governed by the BBGKY hierarchy for interacting entities [13, 24].

The evolution of a non-fixed number of interacting entities of various subpopulations within the framework of marginal observables (4) is described by the Cauchy problem of the dual BBGKY hierarchy [25]:

ddtBt=Λ+n=11n![[Λ,a+],,a+ntimes]Bt,E5
Btt=0=B0,E6

where on Cγ the operator a+ (an analog of the creation operator) is defined as follows

a+bsu1usj=1sbs1u1uj1uj+1us,

the operator Λ=n=0Λn is defined by (1), and the symbol [·, ·] denotes the commutator of operators.

In the componentwise form, the abstract hierarchy (5) has the form:

tBstu1us=ΛsBstu1us+n=1s1n!k=n+1s1kn!××j1jk=1sεk1Λkj1jki1inj1jkBsntu1us\ui1uin,E7
Bstu1ust=0=Bs0,εu1us,s1,E8

where the operators Λs and Λ[k] are defined by formulas (1) and the functions Bs0,ε,s1, are scaled initial data.

A solution B(t) = (B0 , B1(t, u1) ,  …  , Bs(t , u1 ,  … , us) ,  … ) of the Cauchy problem of recurrence evolution Eqs (7), (8) is given by the following expansions [26]:

Bstu1us=n=0s1n!j1jn=1sA1+ntY\ZZBsn0,εu1uj11uj1+1ujn1ujn+1us,s1,E9

where the (1 + n)th-order cumulant of the semigroups {etΛk}t ∈ ℝ , k ≥ 1, is determined by the formula [25]:

A1+ntY\ZZP:YZZ=iZi1P1P1!ZiPetΛθZi,E10

the sets of indexes are denoted by Y ≡ (1,  … , s), Z ≡ (j1,  … , jn) ⊂ Y, the set {Y \ Z} consists from one element Y \ Z = (1,  … , j1 − 1, j1 + 1,  … , jn − 1, jn + 1,  … , s) and the mapping θ(·) is the declusterization operator defined as follows: θ({Y \ Z}, Z) = Y.

The simplest examples of expansions for marginal observables (9) have the following form:

B1tu1=A1t1B1ε,0u1,
B2tu1u2=A1t12B2ε,0u1u2+A2t12B1ε,0u1+B1ε,0u2,

and, respectively:

A1t12=etΛ212,
A2t12=etΛ212etΛ11etΛ12.

For initial data B0=B0B10,εBs0,εCγ the sequence B(t) of marginal observables given by expansions (9) is a classical solution of the Cauchy problem of the dual BBGKY hierarchy for interacting entities (7), (8).

We note that a one-component sequence of marginal observables corresponds to observables of certain structure, namely the marginal observable B10=0b1εu10 corresponds to the additive-type observable, and a one-component sequence of marginal observables Bk0=00bkεu1uk0 corresponds to the k-ary-type observable [25]. If in capacity of initial data (8) we consider the additive-type marginal observables, then the structure of solution expansion (9) is simplified and attains the form

Bs1tu1us=Ast1sj=1sb1εuj,s1.E11

In the case of k-ary-type marginal observables solution expansion (9) has the form

Bsktu1us=1sk!j1jsk=1sA1+sk(t,1s\j1jsk,j1jskbkεu1uj11uj1+1ujsk1ujsk+1us,sk,E12

and, if 1 ≤ s < k, we have Bskt=0.

We remark also that expansion (9) can be also represented in the form of the perturbation (iteration) series [25] as a result of applying of analogs of the Duhamel equation to cumulants of semigroups of operators (10).

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3. A mean field asymptotic behavior of the marginal observables and the kinetic evolution of states

To consider mesoscopic properties of a large system of interacting entities we develop an approach to the description of the kinetic evolution within the framework of the evolution equations for marginal observables. For this purpose we construct the mean field asymptotics [9] of a solution of the Cauchy problem of the dual BBGKY hierarchy for interacting entities, modeling of many-constituent systems of active soft condensed matter [26, 27].

We restrict ourself by the case of M = 2 subpopulations to simplify the cumbersome formulas and consider the mean field scaling limit of non-perturbative solution (9) of the Cauchy problem of the dual BBGKY hierarchy for interacting entities (7), (8).

Let for initial data Bs0,εCs there exists the limit function bs0Cs

wlimε0εsBs0,εbs0=0,s1,

then for arbitrary finite time interval there exists a mean field limit of solution (9) of the Cauchy problem of the dual BBGKY hierarchy for interacting entities (7), (8) in the sense of the -weak convergence of the space Cs

wlimε0εsBstbst=0,s1,

where the limit sequence b(t) = (b0, b1(t),  … , bs(t), …) of marginal observables is determined by the following expansions:

bstu1us=n=0s10tdt10tn1dtnett1Σk1=1sΛ1k1i1j1=1sΛ2i1j1et1t2Σl1=1,l1j1sΛ1l1etn1tnΣkn=1,knj1jn1)sΛ1kninjn=1,in,jnj1jn1sΛ2injnetnΣln=1,lnj1jn)sΛ1lnbsn0u1usuj1ujn,s1.E13

In particular, the limit marginal observable bs1t of the additive-type marginal observable (11) is determined as a special case of expansions (13):

bs1tu1us=0tdt10ts2dts1ett1Σk1=1sΛ1k1i1j1=1sΛ2i1j1et1t2Σl1=1,l1j1sΛ1l1ets2ts1Σks1=1,knj1jn1)sΛ1knis1js1=1,is1,js1j1js2sΛ2is1js1ets1Σls1=1,ls1j1js1)sΛ1ls1×b10u1usuj1ujs1,s1,

for example,

b11tu1=etΛ11b10u1,
b21tu1u2=0tdt1i=12ett1Λ1iΛ212j=12et1Λ1jb10uj.

The proof of this statement is based on the corresponding formulas for cumulants of asymptotically perturbed semigroups of operators (10).

If b0 ∈ Cγ, then the sequence b(t) = (b0, b1(t),  … , bs(t), …) of limit marginal observables (13) is generalized global in time solution of the Cauchy problem of the dual Vlasov hierarchy:

tbst=j=1sΛ1jbst+j1j2=1sΛ2j1j2bs1tu1uj21uj2+1us,E14
bstu1ust=0=bs0u1us,s1,E15

where in recurrence evolution Eq. (14) the operators Λ[1](j) and Λ[2](j1, j2) are determined by Formula (1).

Further we consider initial states specified by a one-particle marginal distribution function in the presence of correlations, namely

fc1f10u1g2u1u2i=12f10uigsu1usi=1sf10ui,E16

where the bounded functions gs ≡ gs(u1,  … , us) , s ≥ 2, are specified initial correlations. Such assumption about initial states is intrinsic for the kinetic description of complex systems in condensed states.

If b(t) ∈ Cγ and f10L1J×U, then under the condition that f10L1J×U<γ, there exists a mean field scaling limit of the mean value functional of marginal observables and it is determined by the following series expansion:

btfc=s=01s!J×Usdu1dusbstu1usgsu1usi=1sf10ui.

Then for the mean-value functionals of the limit initial additive-type marginal observables, i.e. of the sequences b10=0b10u10 [25], the following representation is true:

b1tfc=s=01s!J×Usdu1dusbs1tu1usgsu1usi=1sf10ui=J×Udu1b10u1f1tu1.E17

In equality (17) the function bs1t is given by a special case of expansion (13), namely

bs1tu1us=0tdt10ts2dts1ett1Σk1=1sΛ1k1i1j1=1sΛ2i1j1et1t2Σl1=1,l1j1sΛ1l1ets2ts1Σks1=1,ks1j1js2sΛ1ks1is1js1=1,is1,js1j1js2sΛ2is1js1×ets1Σls1=1,ls1j1js1sΛ1ls1b10u1usuj1ujs1,s1,

and the limit one-particle distribution function f1(t) is represented by the series expansion:

f1tu1=n=00tdt10tn1dtnJ×Undu2dun+1ett1Λ11××Λ212j1=12et1t2Λ1j1jn1=1netn1tnΛ1jn1××in=1nΛ2inn+1jn=1n+1etnΛ1jng1+nu1un+1i=1n+1f10ui,E18

where the operators Λ∗[i], i = 1, 2, are adjoint operators (3) to the operators Λ[i], i = 1, 2 defined by formula (1), and on the space Ln1 defined as follows:

Λ1ifnu1unJ×UA1uiva1v××fnu1ui1vui+1undva1uifnu1un,
Λ2ijfnu1unJ×UA2uivuja2vuj××fnu1ui1vui+1undva2uiujfnu1un,

where the functions A[m] , a[m] , m = 1 , 2, are defined above in formula (1).

For initial data f10L1J×U limit marginal distribution function (18) is the Vlasov-type kinetic equation with initial correlations:

tf1tu1=Λ11f1tu1+J×Udu2Λ212i1=12etΛ1i1g2u1u2i2=12etΛ1i2f1tu1f1tu2,E19
f1tu1t=0=f10u1,E20

where the function g2(u1, u2) is initial correlation function specified initial state (16).

For mean value functionals of the limit initial k-ary marginal observables, i.e. of the sequences bk0=00bk0u1uk0, the following representation is true:

bktfc=s=01s!J×Usdu1dusbsktu1usgsu1usi=1sf10ui==1k!J×Ukdu1dukbk0u1uk×i1=1ketΛ1i1gku1uki2=1ketΛ1i2i=1kf1tui,k2,E21

where the limit one-particle marginal distribution function f1(t, ui) is determined by series expansion (18) and the functions gk(u1,  … , uk) , k ≥ 2, are initial correlation functions specified initial state (16).

Hence in case of k-ary marginal observables the evolution governed by the dual Vlasov hierarchy (14) is equivalent to a property of the propagation of initial correlations (21) for the k-particle marginal distribution function or in other words mean field scaling dynamics does not create correlations.

In case of initial states of statistically independent entities specified by a one-particle marginal distribution function, namely fc1f10u1i=1sf10ui, the kinetic evolution of k-ary marginal observables governed by the dual Vlasov hierarchy means the property of the propagation of initial chaos for the k-particle marginal distribution function within the framework of the evolution of states [4], i.e. a sequence of the limit distribution functions has the form ft1f1tu1i=1sf1tui, where the one-particle distribution function f1(t) is governed by the Vlasov kinetic Eq. [26]

tf1tu1=Λ11f1tu1+J×Udu2Λ212f1tu1f1tu2.

We note that, according to equality (21), in the mean field limit the marginal correlation functions defined as cluster expansions of marginal distribution functions [30, 33, 34] namely,

fs(t,,,u1,,,,,,us)=P:(u1,,,,us)= iUiUiPgUi(t,Ui),s1,

has the following explicit form [27]:

g1tu1=f1tu1,E22
gstu1us=i1=1setΛ1i1gsu1usi2=1setΛ1i2j=1sf1tuj,s2,

where for initial correlation functions (16) it is used the following notations:

gsu1us=P:u1us=iUiUiPgUiUi,

the symbol ∑P‍ means the sum over possible partitions P of the set of arguments (u1,  … , us) on ∣P∣ non-empty subsets Ui, and the one-particle marginal distribution function f1(t) is a solution of the Cauchy problem of the Vlasov-type kinetic equation with initial correlations (19), (20).

Thus, an equivalent approach to the description of the kinetic evolution of large number of interacting constituents in terms of the Vlasov-type kinetic equation with correlations (19) is given by the dual Vlasov hierarchy (14) for the additive-type marginal observables.

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4. The non-Markovian generalized kinetic equation with initial correlations

Furthermore, the relationships between the evolution of observables of a large number of interacting constituents of active soft condensed matter and the kinetic evolution of its states described in terms of a one-particle marginal distribution function are discussed.

Since many-particle systems in condensed states are characterized by correlations we consider initial states specified by a one-particle marginal distribution function and correlation functions, namely

Fc=1F10,εu1g2εu1u2i=12F10,εuigsεu1usi=1sF10,εui.E23

If the initial state is completely specified by a one-particle distribution function and a sequence of correlation functions (23), then, using a non-perturbative solution of the dual BBGKY hierarchy (9), in [31, 32] it was proved that all possible states at the arbitrary moment of time can be described within the framework of a one-particle distribution function governed by the non-Markovian generalized kinetic equation with initial correlations, i.e. without any approximations like in scaling limits as above.

Indeed, for initial states (23) for mean value functional (4) the equality holds

BtFc=B0FtF1t,E24

where F(t| F1(t)) = (1, F1(t), F2(t| F1(t)),  … , Fs(t| F1(t)), …) is a sequence of marginal functionals of the state with respect to a one-particle marginal distribution function

F1tu1=n=01n!J×Undu2dun+1A1+nt1n+1gn+1εu1un+1i=1n+1F10,εui.E25

The generating operator A1+nt of series (25) is the (1 + n)-order cumulant of the semigroups of operators etΛnt0,n1.

The marginal functionals of the state is defined by the series expansions:

Fstu1usF1tn=01n!J×Undus+1dus+nV1+ntYX\Yi=1s+nF1tui,E26

where the following notations used: Y ≡ (1,  … , s), X \ Y ≡ (s + 1,  … , s + n) and the generating operators V1+nt,n0, are defined by the expansions [31]:

V1+ntYX\Yk=0n1km1=1nmk=1nm1mk1n!nm1mk!×Â1+nm1mktYs+1s+nm1mkj=1kk2j=0mjknm1mj+sj=0knm1mj+s1jij=1s+nm1mj1knm1mj+s+1ijjknm1mj+s+2ijj!×Â1+knm1mj+s+1ijjknm1mj+s+2ijj(t,ij,s+nm1mj+1+ks+nm1mj+2ijj,,s+nm1mj+ks+nm1mj+1ijj),E27

where k1jmj, knm1mj+s+1j0 and the evolution operators Ânt,n1, are cumulants of the semigroups of scattering operators etΛkgkεi=1ketΛ1it0,k1. We adduce some examples of evolution operators (27):

V1tY=Â1tYetΛsgsεi=1setΛ1i,
V2tYs+1=Â2tYs+1Â1tYi1=1sÂ2ti1s+1.

If F1tL1J×U<e3s+2, then for arbitrary t ∈ ℝ series expansion (26) converges in the norm of the space Ls1 [30].

The proof of equality (24) is based on the application of cluster expansions to generating operators (10) of expansions (9) which are dual to the kinetic cluster expansions introduced in paper [35]. Then the adjoint series expansion can be expressed in terms of one-particle distribution function (25) in the form of the functional from the right-hand side of equality (24).

We emphasize that marginal functionals of the state (26) characterize the processes of the creation of correlations generated by dynamics of many-constituent systems of active soft condensed matter and the propagation of initial correlations.

For small initial data F10,εL1J×U [31], series expansion (25) is a global in time solution of the Cauchy problem of the generalized kinetic equation with initial correlations:

tF1tu1=Λ11F1tu1+k=1M1εkk!J×Ukdu2duk+1j1jk+11k+1Λk+1j1jk+1Fk+1tu1uk+1F1t,E28
F1tu1t=0=F10,εu1.E29

For initial data F10,εL1J×U it is a strong (classical) solution and for an arbitrary initial data it is a weak (generalized) solution.

In particular case M = 2 of two subpopulations kinetic Eq. (28) has the following explicit form:

tF1tu1=J×UA1u1va1vF1tvdva1u1F1tu1+J×Udu2J×UA2u1vu2a2vu2F2tvu2F1tdva2u1u2F2tu1u2F1t,

where the functions A[k] and a[k] are defined above.

We note that for initial states (23) specified by a one-particle (marginal) distribution function, the evolution of states described within the framework of a one-particle (marginal) distribution function governed by the generalized kinetic equation with initial correlations (28) is dual to the dual BBGKY hierarchy for additive-type marginal observables with respect to bilinear form (2), and it is completely equivalent to the description of states in terms of marginal distribution functions governed by the BBGKY hierarchy of interacting entities.

Thus, the evolution of many-constituent systems of active soft condensed matter described in terms of marginal observables in case of initial states (23) can be also described within the framework of a one-particle (marginal) distribution function governed by the non-Markovian generalized kinetic equation with initial correlations (28).

We remark, considering that a mean field limit of initial state (23) is described by sequence (16), a mean field asymptotics of a solution of the non-Markovian generalized kinetic equation with initial correlations (28) is governed by the Vlasov-type kinetic equation with initial correlations (19) derived above from the dual Vlasov hierarchy (14) for limit marginal observables of interacting entities [27]. Moreover, a mean field asymptotic behavior of marginal functionals of the state (26) describes the propagation in time of initial correlations like established property (22).

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5. Conclusion

We considered an approach to the description of kinetic evolution of large number of interacting constituents (entities) of active soft condensed matter within the framework of the evolution of marginal observables of these systems. Such representation of the kinetic evolution seems, in fact, the direct mathematically fully consistent formulation modeling the collective behavior of biological systems since the notion of state is more subtle and implicit characteristic of living creatures.

A mean field scaling asymptotics of non-perturbative solution (9) of the dual BBGKY hierarchy (7) for marginal observables was constructed. The constructed scaling limit of a non-perturbative solution (9) is governed by the set of recurrence evolution equations (14), namely, by the dual Vlasov hierarchy for interacting stochastic processes modeling large particle systems of active soft condensed matter.

We established that the limit additive-type marginal observables governed by the dual Vlasov hierarchy (14) gives an equivalent approach to the description of the kinetic evolution of many entities in terms of a one-particle distribution function governed by the Vlasov kinetic equation with initial correlations (19). Moreover, the kinetic evolution of non-additive-type marginal observables governed by the dual Vlasov hierarchy means the property of the propagation of initial correlations (22) within the framework of the evolution of states.

One of the advantages of suggested approach in comparison with the conventional approach of the kinetic theory [2, 3, 4] is the possibility to construct kinetic equations in various scaling limits in the presence of initial correlations which can characterize the analogs of condensed states of many-particle systems of statistical mechanics for interacting entities of complex biological systems.

We note that the developed approach is also related to the problem of a rigorous derivation of the non-Markovian kinetic-type equations from underlying many-entity dynamics which make it possible to describe the memory effects of collective dynamics of complex systems modeling active soft condensed matter.

In case of initial states completely specified by a one-particle distribution function and correlations (23), using a non-perturbative solution of the dual BBGKY hierarchy (9), it was proved that all possible states at the arbitrary moment of time can be described within the framework of a one-particle distribution function governed by the non-Markovian generalized kinetic equation with initial correlations (28), i.e. without any approximations. A mean field asymptotics of a solution of kinetic equation with initial correlations (28) is governed by the Vlasov-type kinetic equation with initial correlations (19) derived above from the dual Vlasov hierarchy (14) for limit marginal observables.

Moreover, in the case under consideration the processes of the creation of correlations generated by dynamics of large particle systems of active soft condensed matter and the propagation of initial correlations are described by the constructed marginal functionals of the state (26) governed by the non-Markovian generalized kinetic equation with initial correlations (28).

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Written By

Viktor Gerasimenko

Submitted: 14 May 2017 Reviewed: 22 August 2017 Published: 14 February 2018