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The paper describes the development results on one-dimensional (1D) asymptotic model of the formation kinetics for the objects (clusters) of subnuclear (quark) and subatomic (nuclear) matters. A concept of the objects distribution density wave φ(a, t) in space of sizes a lies in the basis for analytical description of the processes under consideration. The proposed formalism makes it possible to describe in an adequate way the final outcomes of the well-known catastrophic phenomena in the world of elementary particles. Mass characteristics of different processes of approach to equilibrium in nuclear reactions are calculated.

Different analytical and numerical methods are used to study the mechanism of reaching the equilibrium in nuclear processes (e.g., see [1-4]). Such investigations are based on the model concepts of the nature of nuclear matter and the dynamics of the processes under consideration. Work [

The aims of this work are as follows: 1) verification of the proposed cluster model using the example of hard collision processes of elementary particles and formation of stable hadron jets [

A concept of the objects distribution density wave φ(a, t) in space of sizes a lies in the basis of analytical method for describing the processes under consideration [5,6]. This 1D approach makes it possible to neglect deviations of the object geometric shape from the spherical one. An evolution of the wave φ(a, t) during the stochastic process of the objects aggregation can be described in the diffusion approximation with the help of Fokker-Planck kinetic equation written for the space of clusters sizes a (where t is time):

Here, and are the average rate of kinematic transfer of φ and the diffusion coefficient in the space a, respectively; m is the cluster mass; and is the reduced Planck constant. The carried out study of the asymptotic properties of function φ(a, t) showed that in the end of the irreversible aggregation of objects as a result of interaction of large clusters with small seeds in a stochastic conservative system the distribution density of large clusters with a ? a_{0} (a_{0} is a size of a seed) is inversely proportional to their masses: j µ m^{–1} (an “inverse-mass” law) [

We define two mechanisms of growth of nuclear matter clusters: 1) the small flux of seeds, when each of them has time to occupy its place on the surface of the cluster before starting interaction with the next seed; 2) the high flux of seeds, when they affect the cluster almost simultaneously (equivalent to collision of clusters). Linearization of Equation (1) makes it possible to produce analytical expressions for increase in the mean size of large clusters (a ? a_{0}) with time [

Here, t_{1} is the unit of time in the processes with small flux of seeds, t_{0} is a typical time scale of objects interaction, m_{0} is the seed mass. The second mechanism has a typical time scale equal to, where c_{0} = 5 × 10^{7} m·s^{–1} is the “sound” velocity or average nucleon thermal velocity determined by the average thermal energy of the degenerate Fermi gas (22 MeV per nucleon [8,9]). Then, the law of increasing the average size can be written as

Here, t_{2} is a unit of time in the processes with a large flow of seeds.

The size of a spherical nucleus is related with mass number A in the following way [8,9]:

Here, r_{0} = 1.3 fm is a typical space scale of strong interaction. The total time of the process is determined by the Heisenberg rule basing on the level of shell energy: [

The following formula can be written for the most probable mass numbers of cluster-nuclides [

Here, A_{0} is the mass number of the seed; λ > 1 is an arbitrary real number; and small parameter is defined by seed mass m_{0} and size a_{0}, and by the characteristic time scale t_{0} of the objects interaction. A reasonable choice for this scale is the period of high-frequency nucleon oscillations in the nucleus. This parameter can be defined as t_{0} = 2r_{0}/c_{0} » 5 × 10^{–23} s. The approximate quantity λ is found by matching the solutions for the first maximum of j, which corresponds to the seeds, and the maximum of j for small clusters [

Verification of the suggested formalism can be performed using the example of catastrophic (deeply inelastic) processes with hadrons [7,11]. One can consider these processes as the following chain of events: head-on collisions of “lepton-nucleon” or “nucleon-nucleon” ® the quark is split from the gluon cloud ® random interactions in the continuous intrahadron medium ® formation of new particles (hadron jets). New particles are considered as clusters containing the elements of a continuous intrahadron medium (partons): quarks, gluons, quark-antiquark pairs, and so on. Then the law of evolution of the mean cluster size (3) gives the following relation between the units of length a_{unit}, time t_{unit} and mass m_{unit} at deep inelastic interaction of fundamental particles:

Here, c is maximum velocity of interaction propagation in hadron medium, namely, the velocity of light in vacuum. On the basis of this formula and other known data [7,11,12] for “conventional” quark masses (uand dquarks) deep inside hadrons and for typical space scales, it is possible to get the following evaluation of a time unit for current quarks (cq) in the state of asymptotic freedom (af):, , Þ. Transition time t_{trans} for quarks from asymptotic freedom to constrained state (confinement) inside the hadron is estimated by formula (3) as

. The produced value corresponds to the time scale of strong interaction.

In the processes considered in [

.

The produced value is much higher than the life time of unstable hadrons (resonances):. This reflects the formation of stable hadron jets in the processes of catastrophic collisions being described. If we take, in compliance with the results of [_{s} = 0.3c and 0.57c, then multipliers 1.8 and 1.3 will appear in the evaluations given above for time units, the time of quark transition into the bound state, and the time of hadron formation. These corrections will not change the estimation of time scale for strong interaction and will make stronger inequality that reflects the formation of stable hadron jets.

Asymptotic distribution j µ m^{–1} means that: 1) the probability of nucleon fragmentation under deep inelastic scattering is higher than the chance of its preservation; 2) the number of pions in hadron jet is 1.5 times higher that the number of nucleons. These conclusions correspond to the notions on the nature of hard processes given in [^{0}p^{+}p^{+}nn. One can see that there are 3 pions and 2 neutrons in the formed hadron jet.

Thus, the proposed formalism provides adequate determination of asymptotic states reached at well-known catastrophic phenomena in the world of subnuclear objects, namely, fundamental and elementary particles. This gives us a reason for making an attempt to apply this formalism for considering the intranuclear processes mentioned in the Section 1.

The values A calculated in [_{end} » 470 calculated in [

One can rewrite expressions (2) and (3) in the following way more convenient for calculations:

Here, r is the density of nuclear matter taken as equal to r = 2.5 × 10^{17} kg·m^{–3} [_{0} from the formulas it was taken that the mass of the seed is equal to.

One can try to describe spontaneous nuclear fission as a result of excitation of the first rotational level with energy G_{rot} = 100 keV using formulas (4) and (8). The total time of the intranuclear process that corresponds to the specified value of G_{rot} is equal to t_{rot} = 6.287 × 10^{–21} s, and the average mass number of light fragments is evaluated to be. Average mass number of heavy fragments is determined by the mass conservation law, A_{mat} is the mass number of parent nucleus. In case of light actinides (Th, U) we get. The diagrams of mass distribution for the fragments of spontaneous fission of heavy nuclei (A = 235, 238) given in [

If we presume that high-speed nucleosynthesis in stars takes placed as a result of transition of the system of nucleons from the high-frequency vibration level with the period of t_{0} = 2r_{0}/c_{0} » 5 × 10^{−23} s to the first rotational level with the energy of 100 keV during the life time t_{rot} = 6.287 × 10^{−}^{21} s of this level, then formulas (4) and (7) give the evaluation of the average mass number of superheavy elements as. As for the forced fission competed with the nucleosynthesis, in this case it is possible to take the travel time t = a/c_{0} of the sound wave in the nucleus formed as a result of nucleosynthesis as a time scale t_{0}. With regard to formula (4) when we get t_{0} = 3.6 × 10^{−22} s. Then from the formulae (4) and (7) we get that, when the lifetime of the first rotational level is 6.287 × 10^{−21} s, the average mass number of light fragments of superheavy elements is about. In compliance with the mass conservation law the average mass number of heavy fragments is equal to. The produced values approximately agree with the first and the last peaks of the final element abundance in the Galaxy [

The proposed model corresponds to the problem of motion of 1D wave packet j(a, t) in the space a [

.

This circumstance reflects the expansion of wave packet and the slower-down of its propagation. At the final stage of irreversible aggregation of clusters in the closed system this wave packet with non-Gaussian shape has an attenuating discontinuity in the wave “front” related to the maximum possible size of:

The presented results and notions allow us to make a conclusion that the developed in [

In Section 3 it was found that the least value of time unit in quark-gluon medium is 10^{−29} s, and the unit of time for current quarks in the state of asymptotic freedom is 10^{−26} s. Using relation (6) one can get the following evaluation of the unit of time in the interaction processes with constituent quarks in confinement state described in [11,12]:, Þ

. The transfer time from asymptotic freedom to confinement is evaluated in Section 3 as that agrees with the time scale of strong interaction, and the formation time of stable hadrons is estimated as. Relation (6) shows that the processes in the world of fundamental and elementary particles are characterized with the spectrum of time units. Close to the upper boundary this spectrum overlaps with the “lower” area of times t = 10^{−23} - 10^{−22} s which are typical for direct nuclear reactions. Thus, formula (6), which shows that “regular” space-and-time relations are valid up to the distances of about 10^{−18} - 10^{−15} m and time of 10^{−29} - 10^{−26} s, complies with the generally accepted notions about space-and-time scales in microphysics [

We can try to determine the value of phenomenological “fundamental mass” of, if we take that the least space unit (a fundamental length) is the value of a_{fund} ~ 10^{−18} m and that this value is connected with the fundamental time scale t_{fund} by the reasonable relation. Then, from relation (6) we get the following expression for the fundamental mass:. Then we get that [^{0} should be:. Besides, this value is closed to the “critical” mass of 180 - 200 GeV, above which H^{0}-boson can decay into the pairs of Wand Z-bosons [^{0} with the zero spin. It is also important to mark approximate correspondence of obtained evaluation for to the value of the upper limit of the Higgs mass identified in [

It could be interesting to carry out a qualitative analysis of various processes with fundamental and elementary particles with the help of relation (6). For example, we could try to estimate the state of the heaviest of all quarks — t-quark (tq), which has a conventional mass of 176 GeV [^{−15} m. The produced evaluation complies with the life time of unstable resonances, therefore, t-quark does not form the stable hadron. This result complies with the common notion, according to which a t-quark is the “only quark that is ‘born and dies free’” [

In the area of nuclear scales the suggested method allows to evaluate the time of approach to equilibrium basing on the typical mass numbers and respective nuclide sizes. This problem was solved in [^{−20} s. The produced value is much less than the lifetimes of 10^{−16} - 10^{−14} s which are typical for the intermediate compound nucleus. This makes it possible to speak about comparatively quick (explosion) character of the process of deep inelastic heavy-ions interaction.

The developed cluster model makes it possible to produce adequate evaluations of space-and-time and mass characteristics for the processes in the subnuclear (quark) matter and for the intranuclear processes of approach to equilibrium.