About Appearance of the Irreversibility

The inevitability of arising in equations of kinetics and hydrodynamics irreversibility not contained in original equations of classic mechanics is substantiated. It is established that transfer of information about the direction of system evolution from initial conditions to resulting equations is the consequence of losing information about the position of an individual particle in space, which takes place at roughening description. It is shown that the roughening with respect to impact parameters of colliding particles is responsible for appearance of the irreversibility in resulting equations. Direct equations of kinetics and hydrodynamics are the result of roughening distribution functions with respect to impact parameters of particles, which have not yet reached the domain of their interaction. The direct equations are valid for the progressive direction of timing on the time axis pointing from the past to the future. Reverse equations of kinetics and hydrodynamics are the result of roughening distribution functions with respect to impact parameters of particles, which have already left the domain of their interaction. The reverse equations are valid for the progressive direction of timing on the time axis pointing from the future to the past.


Introduction
The detailed comparison of the results of direct numerical integration of the Navier-Stokes equations with the experimental data in the problem on flow around a sphere is given in [1]. The comparison demonstrated that the solutions to the classic hydrodynamics equations interpret successfully the experiment until a first critical Reynolds number value is reached. However, after passing first critical Reynolds number value, these solutions become inapplicable to interpreting the instability [1]. In accordance with ideas given in [1]- [3] the solutions to the classic hydrodynamics equations attain successfully the boundary of the instability field. However, these solutions are unable to cross this boundary and, consequently, enter the instability field.
The problems encountered by classic hydrodynamics when interpreting observed evolution of unstable process, are proposed to be solved on the way toward an increase in the number of principle hydrodynamic values [4]. In [5]- [8], the multimoment hydrodynamics equations are used to study the phenomena of instability appearance and development in problem on flow around a solid sphere at a wide range of Reynolds number values.
The studies [5]- [8] demonstrated that when interpreting each of unstable regimes in problem on flow around a sphere the need emerges to involve the so-called reverse multimoment hydrodynamics equations [9]. The evolution of the system after losing its stability that is described by the direct hydrodynamics equations advances in direction of the system departure from the statistical equilibrium state. The evolution of the system after losing its stability that is described by the reverse hydrodynamics equations advances in direction of the system approaching the statistical equilibrium state.
Both the direct and the reverse multimoment hydrodynamics equations are irreversible. The main purpose of the present work is to elucidate the physical meaning of appearance of the irreversibility in equations of kinetics and hydrodynamics. A. Einstein believed that irreversibility is not a fundamental law of nature, as far as it is not incorporated in the basic laws of physics represented by original equations. He saw no reasons for emergence of the irreversibility in the resulting equations as a result of any transformation of original equations because the initial conditions are responsible for one or another direction of evolution. Thus, A. Einstein believed that appearance of the irreversibility in resulting equations is not sufficiently substantiated [10]. Unlike A. Einstein, I. Prigogine treated, the irreversibility, as a fundamental law of nature. I. Prigogine introduced the irreversibility into original equation, thereby, like A. Einstein, he denied the feasibility of achieving the irreversibility when starting with the original equations [11].
The present work shows that during transition from the classic mechanics equations to equations of kinetics and hydrodynamics the irreversibility is incapable not to appear, it must appear. By another words, the present work substantiates appearance of the irreversibility in resulting equations. Section 2 recounts the ideas on levels and stages that specify the degree of accuracy for medium description. Section 3 represents the only roughening operation, which is responsible for appearance of the irreversibility in kinetics and hydrodynamics equations. The necessity to introduce this roughening operation is substantiated.

Medium Description Levels and Stages
Suppose that some physical system consists of N structureless particles. Let by the time t 0 . Having once more the operation of inversion, make sure that the equations of classic mechanics returned the system to the same point of the phase space from which they took her.
Having described by classic mechanics equations the motion of each system particle, we will be able to calculate any macroscopic characteristic of system at any time moment, i.e., to describe the evolution of system as a whole. This description corresponds to the dynamic deterministic level. This level is the most complete and overly detailed. As a rule, the necessity in such a detailed description does not occur. When solving practical tasks we need information not about the velocities and coordinates of individual particles but about the macroscopic properties of a system. It is therefore advisable to switch from the equations of classic mechanics, which describes the motion of each system particle, to equations for macroscopic characteristics.
In (2.1), m is the mass of the particle, , i j Φ is the force of action of the i-particle on the j-particle. , , , , , , , is equivalent to 6N Newton (Hamilton, Lagrange) equations [12].
To reach the statistical deterministic level at some time t it is necessary to describe the set of systems (Gibbs ensemble) rather than single concrete system. The systems in ensemble differ from each other by values of coordinates and velocities of particles that form the system. The variables ( ) 3) The (  )   1 1  2  2 , , , , , , , statistical distribution function has a meaning of probability to find at some time t one of medium particles within the unit element of phase space near point 1 x , 1 ξ , another particle near point 2 x , 2 ξ ,  , and the last particle near point N x , N ξ [12]. The , , , , , , , , , , , , , , x ξ x ξ x ξ  function described specifically the K-system of ensemble constitutes the fluctuation of the N-particle distribution function. The (  )   1 1  2  2 , , , , , , , function contains statistical information about all N particles of the system. The reduction of the ordinal number of the particle distribution function accompanies by the loss of information about the position in phase space of the particle in turn. The s-particle distribution function ( ) has a meaning of the probability that at some time t one particle, say particle 1, finds itself within unit element of phase space near point 1 x , 1 ξ , another particle, say particle 2, within unit element near point 2 x , 2 ξ ,  , and particle s-near point s Analyzing the hierarchy (2.4), N. Bogolyubov [13] introduced a concept of characteristic intervals (scales) in gas medium. Three temporal intervals were distinguished in [13]: 0 τ , k τ , and h τ . Interval 0 τ is equivalent to the characteristic time of particle collisions 0 θ . The spatial scale l 0 corresponding to it is identical to the characteristic particle size d. Interval k τ is the characteristic time between collisions τ . The spatial interval l k corresponding to it is identical to the characteristic free path length λ . Temporal interval h τ and spatial interval l h corresponding to it are equivalent to the characteristic temporal scale of flow Θ and the characteristic spatial scale of flow L correspondently. The above three intervals specify three Bogolyubov accuracy stages of gas description: initial l 0 scale, kinetic l k scale, and hydrodynamic l h scale. Initial stage equations are the most detailed. The solutions to these equations describe the system at the finest initial stage as well as at the kinetic and hydrodynamic stages. Passage to less detailed kinetic description stage is implemented by neglecting the information about a sharp change of the distribution functions on the initial scale. Namely, the distribution function governed by the equations of kinetic description stage, varies slightly on l 0 scale. After transition to the most coarse hydrodynamic description stage the distribution function varies strongly on l h scale only.
The BBGKY hierarchy (2.4) describes the medium with accuracy that satisfies the initial stage. The At the kinetic and hydrodynamic stages, the gas is described by the one-particle distribution function ( )  , , , , F t x ξ x ξ responsible for interaction of particle 1 with some particle 2. There are several variants of the derivation of the Boltzmann equation directly from the first equation of the BBGKY hierarchy [12]. Each of them reaches inevitably the equation: The collision integral ( ) Equations (2.5), (2.6) are valid for a rarefied gas medium, where d λ  , x ξ x a ξ a x ξ x ρ ξ x a ξ x + a ρ ξ a (2.7) In (2.6), (2.7), vector   , , NF t x ξ product differs by the negligibly small value from: Otherwise, let the ( ) , , F t x ξ function on the l 0 scale. Therefore, when passing the kinetic and hydrodynamic description stages, information about spatial position of an individual particle is lost. The , , F t x ξ distribution function is the coordinate of particle 1. The 1 x argument of the ( ) 1 1 1 , , f t x ξ distribution function is not the coordinate of individual particle. The x 1 argument of the ( ) 1 1 1 , , f t x ξ function marks the place in space in the vicinity of which a set of particles is concentrated within init volume. The Boltzmann equation is not invariant with respect to inversion of velocities and time: i , t t → − , that is, the Boltzmann equation is irreversible. The operation responsible for appearance of the irreversibility is represented in Section 3.
The second equation of the BBGKY hierarchy (2.4), like the first one, is not closed. The integral term of the second hierarchy equation contains a three-particle distribution function responsible for the interaction of particles 1 and 2 with some third particle. The absence of closeness of the second hierarchy equation prevents us from the transition to the hydrodynamic stage from the phase space of two particles. It was, however, found that in the gas medium without the triple collisions of particles ( ) d λ  , the necessity of taking the third particle into account can be obviated. This possibility opens up prospects for the transition to the hydrodynamic description stage without invoking additional hypotheses.
If the ( ) , , , , 0 The idea that leads to the concept of a pair of particles is given in [14] [15]. In [15], Equation (2.9) describing the evolution of a pair of particles is derived directly from the main statistical mechanics postulates. The heuristic derivation of Equation (2.9) was given in [3]. The collects all the positions of particles 1 and 2 with respect to each other in which they turn out to be after a collision with each other. The integration of 2 collects all the positions of particles 1 and 2 with respect to each other in which they turn out to be before a collision with each other, see Figure 3 from [2]. The integration of ( ) div app p F (2.11) with respect to a within region W having the characteristic linear size d l λ   removes strong spatial dependence of the functions on the initial scale having the particle size d. The multiplication of ( ) At the hydrodynamic stage, the description of gas medium may be carried out not in terms of distribution function but in terms of hydrodynamic values that are the moments of distribution functions. In order to derive the hydrodynamics equations from the equations for the distribution functions it is necessary to refuse the study of behavior of the distribution function on the kinetic scale, k l , k τ kinetic scale. Under the hydrodynamic description, the Knudsen layers, the initial layers and the shock waves, where the scale of the change of the distribution function is k l , k τ , are excluded from consideration.

Inevitability of Appearance of the Irreversibility
To understand the physical meaning of appearance of the irreversibility in equations of kinetics and hydrodynamics, two concepts discussed above should be compared. First, since the time of L. Bolzmann, the responsibility for direction of evolution of the system rests with the initial conditions, namely, the set of initial values of coordinates and velocities of all the particles [16]. At a certain mutual arrangement of the particles, the system evolves in the direction that we see everywhere and every second. However, there exist such arrangements of particles that send the system along an extremely unlike, rarely realized direction. Refer to the example. Let after removal of partition at the time relaxes from non-equilibrium state (Figure 1(a)) to the state of statistical equilibrium (Figure 1(b)) that is achieved by the time 1 t t′ = . The mutual arrangement of particles at the time 0 t t′ = sends the system along the usually observed direction that corresponds to approaching the state of statistical equilibrium. However, mutual arrangement of the particles at the time 1 t t′ = after inversion in velocity space (Figure 1(c)) sends the system along extremely unlike direction that corresponds to departure from the state of statistical equilibrium. Even a negligible change in mutual arrangement of the particles (on distance much smaller than particle size d) can change the direction of evolution [17].
Secondly, after transition from the initial to the kinetic and hydrodynamic description stages the distribution functions lose the information about spatial position of individual particle and hence, about mutual arrangement of particles. The variables x i , 1, , i s =  both in the s-particle distribution function ( ) contain the information about direction of evolution. The x 1 variable in distribution functions and initial conditions of the kinetic and hydrodynamic stages specifies the location in space. It means that distribution functions and initial conditions of kinetic and hydrodynamic stages lose the information about the direction of evolution. If the roughening operations, transforming the equations and the initial conditions of the initial stage into equations and the initial conditions of the kinetic and hydrodynamic stages, are performed correctly, the information about direction of evolution of the system must somehow be preserved. And this information preserved, it disappeared from the coarse distribution functions and the initial conditions, but appeared in the coarse equations of the kinetic and hydrodynamic stages. The equations of the kinetic and hydrodynamic stages acquired a new characteristic property that is not contained in the classic mechanics equations as well as in the BBGKY hierarchy equations. The irreversibility identified with the direction of the system evolution appeared there.
To clear up the mechanism of appearance of the irreversibility in equations of kinetics and hydrodynamics let us carry out the roughening the reversible Equation (2.9) that describes the motion and interaction of two particles of pair. Suppose that v is the velocity of relative motion of particles which after an elastic collision with the impact parameter b and azimuth angle ε acquire velocity v. Trajectories of relative motion of In order to leave the interaction domain with the same velocity 1 v , and parameter 2 1 b b ≠ , particles should en- Figure 2). If the particles have velocity 1 ′ v and at the point of entry, they will emerge from the interaction domain with parameters n b and , , , ,π , , , ,π , , ,π , , , , , ,π , , , , ,π , d . Here, Velocities of relative motion of particles appearing in Equation (3.1) are displayed in , , , , ,π , , , ,π , , ,π , , , , ,π ! Here, Reversibility of the equations for pair functions will be examined in the seven- The seven-dimensional space is a combination of the C Gv six-dimensional space of velocities defined by vectors G and v and the t C one-dimensional time space. The positive direction of the time axis runs from the past to the future.
Suppose that at some instant of time, we have a chance to reverse the directions of the velocities of all the particles, without changing the directions of the reference axes in space C Gv .
Let us go beyond the scope of formal analysis and examine Equation (3.la). Inversion of time and velocities, Inversion of time and velocities fails to alter the number of pairs of diverging particles, which at time t τ + start approaching each other, The sets (3.1) and (3.2) are written for the progressive direction of timing along the time axis pointing from the past to the future. The sets (3.4) and (3.6) are written for the regressive direction of timing along the same time axis.
To comply with the physical scenario by which the system attains the kinetic and hydrodynamic stages and the peculiarities of this stage, let us dispose of the variables defining the mutual arrangement of particles. To this end, integrate Equation (3.la) with respect to b and ε : Integration "collects" all the trajectories of types 1 and 2 ( Here, integration is over the trajectories of types 1 and n (  (  )   app  app  2   1  , , , , , , , , , , , , π Equation (3.7) assumes therewith the following form: Dashed arrows between B and C are the trajectories of types 1 and n (Figure 2). It is these trajectories that were "collected" by integrating (3.8) to switch from the exact value of app By analogy with operation (3.9) we replace the initial function by its average, ,π , , ,π , , , ,π , In Equation (

Equation (3.13) is reversible. This inference stems from an informal analysis similar to analysis of Equation (3.la).
Following the procedure proposed in [15], we average Equation (3.10) and (3.13) over x within domain W of linear size d l λ   , Equation (2.11), multiply two-particle distribution functions by the number of ways in which a pair of particles can be chosen among N panicles, and switch from the trajectory form of the equations to their differential form, x G x G x G x (3.14) Equation (3.14) were previously derived heuristically in [14] and immediately from the Liouville equation in [15]. The functions div and app p f were defined by Equations (2.10), (2.11), and (2.12). Now we switch from space C Gv to space C * Gv , C C * → Gv Gv . All the axes of the six-dimensional frame of reference C * Gv are in opposition to the axes of C Gv . Inversion of the reference axes of the velocity space C Gv has no effect on the physical pattern: particles keep moving. In going to C * Gv , the superscript of pair distribution functions remains invariant. However, in frame of reference C * Gv , the arguments of distribution functions G and v reverse their signs: Figure 6, The time axis in space t C * points towards the past. In space t C * , t t t , , , ,π , , , ,π , , ,π Let us carry out the replacing, ,π , , ,π , , , ,π , In Equation (3.19), average function div p F σ * models pairs of particles which have collided by time t * .
As a result, Equation (2.5) with collision integral (2.6) is invariant with respect to inversion: t t t , that is, Equation (2.5) with collision integral (2.6) is reversible. The reversibility of Equation (2.5) is true for any analytical representation of two-particle distribution functions in the collision integral (2.6). Hence, the Bogolyubov boundary condition for correlation failure [13] − v x ξ x ρ ξ in terms of the two-particle distribution function at the entrance to interaction domain.

ξ x a ρ ξ a x x
Let us recast collision integral (3.39) in terms of two-particle distribution functions 2 F   , written in 1 x , 1 ξ , Equation ( Let us invoke replacing (3.41) to obtain Equation (2.5) with collision integral (3.40) in the form: In terms of two-particle distribution functions 2 F , written in 1 x , 1 ξ , 2 x , 2 ξ variables, collision integral   (3.44) is also valid with progressive timing along the same time axis. Following to Boltzmann, let us factorize two particle distribution functions in the ( ) The factorization of two particle distribution functions, that is, their representation in the form of product of two one-particle functions, closes Equation (3.43) with collision integral (3.44). The obtained classic kinetic equation for the ( ) , , f t x ξ one-particle function is called the Boltzmann equation: x ξ x ρ ξ x ξ x ξ x ξ x ρ ξ x ξ x ξ The stated ideas about the transition of information from initial conditions to resulting equations allow submitting additional argument in favor of strong spatial dependence of the one-particle distribution function ( ) , , NF t x ξ function can be identified with number of 1 ξ -particles in unit volume element near point x 1 . Then, it will be impossible to distinguish one direction of evolution from another by means of initial conditions for the ( ) , , F t x ξ one-particle distribution function obeys the first reversible equation of BBGKY hierarchy (2.4). Thus, having disappeared from the initial conditions, direction of evolution identified with irreversibility does not appear in the equation. That is why, the assumption made will not allow for a description of both directions of evolution in the terms of the ( ) , , F t x ξ function disproves the assumption was made.

Discussion
Evolution of function called entropy is the indicator of reversibility of equation. Every function presented above, Equations (2.3), (2.8), (2.10), allows to form own entropy. At the dynamic description level the system is characterized by the dynamic entropy, pair entropy (4.4), (4.5) have quite definite physical sense; they meet the volume which system occupies in the G-space [6] [19]. Thus, the system evolution is accompanied by change of its volume in the G-space.
The temporal conservation of entropy for system that evolves to the state of statistical equilibrium at its detailed description (at high accuracy level) is noted by J. Gibbs on the classical example with dye [20]. According to Gibbs, the temporal change of entropy is possible only when a coarse description.
Among other things, the entropy specifies the degree of smearing system macroscopic state upon acceptable microscopic states. At the dynamic description level the microscopic state that corresponds to given macroscopic state is quite concrete. At the initial stage of the statistical level, the absence of entropy outflow through the surface confining the system ensures the conservation of the degree of smearing system macroscopic state upon acceptable microscopic states. At the kinetic and hydrodynamic stages, the system evolution is accompanied by change of the degree of smearing due to non-zero entropy production.
The transition from the classic mechanics equations to the hydrodynamics equations is a successive multistep process of loss of excess information about the system. During this transition a set of roughening operations is performed but only one of these operations transforms the reversible equations into the irreversible ones. At the first step of roughening the description of an individual system is replaced by description of the ensemble of systems. The  4) is reversible, that is, the second step of roughening also does not lead to appearance of the irreversibility. The irreversibility appears at the next roughening step after transition from the initial description stage to the kinetic and hydrodynamic stages. Equation (3.7) is invariant with respect to inversion of time and velocities. Replacing the exact function on the left hand side of Equation (3.7) by its average over impact parameters (3.9) eliminates the reversibility of Equation (3.7), that is, Equation (3.10) becomes irreversible. Replacing (3.9) closes one of two directions for time reckoning. It means that both Equation (3.10) and the direct set (3.14) are valid for the progressive direction of timing on the time axis pointing from the past to the future.
Equation (3.16) is invariant with respect to inversion of time and velocities. Replacing the exact function on the right hand side of Equation (3.16) by its average over impact parameters (3.17) eliminates the reversibility of Equation (3.16), that is, Equation (3.19) becomes irreversible. Replacing (3.17) closes one of two directions for time reckoning. It means that both Equation (3.19) and the reverse set (3.23) are valid for the progressive direction of timing on the time axis pointing from the future to the past. Note that solutions to the set (3.23) are also suitable for modeling observed system evolution with regressive timing along the time axis pointing from the past to the future.
The local pair entropy corresponding to the direct equations for pair distribution functions (3.14) and the multimoment hydrodynamics equations they yield can only be produced in the system due to binary collisions at any space point x and at any instant t, time Suppose that each system in ensemble consists of N particles. Let the evolution of ensemble systems be described by the direct set (3.14). At some time t, let us reverse the sign of velocities of all N particles of all the systems in ensemble, + i i i → = − ξ ξ ξ , 1, , i N =  , and reckon time in the direction of its decrease, t t t + → = − . As a result of this inversion, in accordance with the laws of classic mechanics, all the particles of all the systems in ensemble will move in the opposite direction. The reverse Equation (3.23) should describe the reverse movement of particles. Indeed, reverse Equation (3.23) can be obtained from direct Equation (3.14) also by the inversion of velocities and time. However, to implement the reverse movement in system, it is not necessary to change the direction of timing. It is enough to change the direction of velocities of all the particles of system, then, the system will move back with progressive direction of timing.
Processes occurring in nature are objective events, while the choice of the direction of timing on the time axis is a subjective process. Time is reckoned by an observer, while processes occurring in nature are absolutely insensitive to the direction in which the observer counts the time. Therefore, if the reverse Equation (3.23) are capable to describe the reverse motion of systems in ensemble for progressive direction of timing on the time axis pointing from the future to the past, then the same equations must describe the reverse movement of systems in ensemble for progressive direction of timing on the time axis pointing from the past to the future.
This means the followings. Let us reckon the time in progressive direction both along the time axis pointing from the future to the past and along the time axis pointing from the past to the future. Let us begin to observe some phenomenon, agreeing upon the origin for two directions of timing. On finding solution to the reverse Equations (3.23), we obtain the distribution of hydrodynamic values, their spatial and temporal derivatives. Let the calculated values agree well with the values observed in the direction of increasing time on the time axis pointing from the future to the past. Then, we also find the agreement between the calculated reverse distributions of hydrodynamic values, their spatial and temporal derivatives and the values observed in the direction of increasing time on the time axis pointing from the past to the future. However, there exist no direct equations that would satisfy the distributions of reverse hydrodynamic values, their spatial and temporal derivatives.