The Mechanics of the Systems of Structured Particles and Irreversibility
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THE MECHANICS OF THE SYSTEMS OF STRUCTURED PARTICLES ANDIRREVERSIBILITY
V.M. Somsikov ∗ Laboratory of Physics of the geoheliocosmic relation, Institute of Ionosphere, Almaty, Kazahstan. (Dated: November 20, 2018)Dynamics of systems of structured particles consisting of potentially interacting material pointsis considered in the framework of classical mechanics. Equations of interaction and motion ofstructured particles have been derived. The expression for friction force has been obtained. It hasbeen shown that irreversibility of dynamics of structured particles is caused by increase of theirinternal energy due to the energy of motion. Possibility of theoretical substantiation of the laws ofthermodynamics has been considered.
PACS numbers: 05.45; 02.30.H, J
I. INTRODUCTION
Collective properties of natural systems are to be re-lated to laws of dynamics and properties of their con-stituent elements. Revealing of these relations is amongthe main tasks of physics. There are obstacles of prin-cipal nature towards solving this task mainly caused byabsence of connection between the laws of classical me-chanics and thermodynamics. Within this context, theirreversibility problem formulated by Boltzmann about150 years ago is the key one [1-4]. The essence of thisproblem is in irreversible nature of dynamics in naturalsystems while in Newton equation the dynamics of ele-ments in a system is reversible.Boltzmann’s explanations of the irreversibility arebased on extremely small probability of noticeable devi-ation of thermodynamic system from equilibrium [2, 5].Entirely, this explanation uses statistical laws which arealien to classical mechanics. Attempts made to find morestrict substantiation of irreversibility employing Boltz-mann equation or H-theorems encountered, as a rule,Poincare’s theorem of reversibility. This theorem forbidsequilibrium in Hamiltonian systems [3]. Nevertheless,Boltzmann methods of the kinetic theory have appearedvery effective when studying development of irreversibleprocesses, as well as in development and substantiationof kinetic methods for description of nonequilibrium sys-tems [6-12].Aspiring to solve the irreversibility problem withinclassical mechanics, Liouville has proved that only sys-tems which by canonical transformations of independentvariables are split into one-degree-of- freedom systemsare integrable [3]. Such splitting is only possible in thesituation with no interaction between elements of the sys-tem. At the same time Poincare proved that, as a rule,dynamic systems are not integrable. Assuming the po-tentiality condition for elements’ interaction forces leadsto potentiality of forces between systems of these ele- ∗ Electronic address: [email protected] ments, it follows from Hamiltonian formalism that thetask could always be reduced to integrable systems ofone degree of freedom. So, we face a contradiction: onone hand it is proved that the class of integrable systemsis very narrow, and on the other hand a priori conditionaccepted for interaction forces should assure integrationof any natural systems. In spite of remaining impossibil-ity within Hamiltonian formalism to bind time-reversiblemicroscopic equations with time-irreversible macroscopicequations of thermodynamics, symmetry still allows us toexpand application of the formalism for thermodynamics[13].Intensive attempts to understand the essence of irre-versibility were undertaken by Planck. He has tried toprove irreversibility of radiation (black body radiation)but he failed to overcome Poincare’s reversibility theo-rem. And though the irreversibility problem remained,Planck has come to the well-known formula of a blackbody equilibrium radiation, quantum of energy conceptand to creation of the quantum theory [14].Gibbs has developed the fundament of strict statisti-cal theory for equilibrium systems within the frame ofclassical mechanics splitting equilibrium systems into so-called ensembles. He has thereby connected the proba-bilistic meaning of initial and boundary conditions withdeterminism of classical mechanics. But applicability ofhis theory is restricted by equilibrium systems since thetheory assumed zero energy exchange between the sub-systems [5, 15].After Kolmogorov, Arnold and Mazur and theirso-called theory (Kolmogorov-Arnold-azur), the non-integrability problem was conventionally considered asa starting point for further development of stochasticdynamics [3, 4, 16]. Investigations of Hamiltonian sys-tems and chaos due to exponential instability has ledto creation of a deterministic chaos theory. This theoryhas been very fruitful. Employing the chaotic mechan-ics methods, the relation between entropy and Lyapunovexponents has been revealed and there were also stud-ied the mechanisms of mixing and correlation splitting[3, 17-19]. But nevertheless the problem of irreversibil-ity has remained unsolved due to the difficulties of theso-called ”coarse grain” of the phase space explanations.Essential contribution to studies of irreversibility wasmade by I. Prigogine and his so-called Brussels school.In particular, they suggested and developed a methodfor analysis of chaotic systems based on projection op-erator in Hilbert space [3, 20]. Within this method ir-reversibility arose due to finite predictability horizon inthe dynamics of Hamilton systems. Like A. Poincare,I. Prigogine also admitted possible limitation of classicalmechanics [3, 21].Trying to expand applications of statistical physicsthere was developed a method of non-extensive ther-modynamics used for stationary nonequilibrium systems.The method allows to define the distribution function forweak nonequilibrium systems and to study relations be-tween thermodynamic and mechanical parameters [22].But like any other statistical method, this method can-not be used to explain irreversibility.Considerable contribution to studies of nonequilibriumsystems was made by Yu. Klimontovich who has devel-oped a statistical theory of open systems [23]. The core ofhis approach verified in our research is in consideration ofcontinuous medium structure at all levels of its descrip-tion. Still, he has remained within statistical descriptionof systems.Essential progress in solution of a problem of inter-relation between micro- and macro-processes has beenachieved with a so-called Fluctuation Theorem [24] whichmade it possible to describe statistically entropy produc-tion in systems of limited number of elements.Considerable attention to entropy production mecha-nism in stationary nonequilibrium systems was paid in[12]. It is interesting that based on these works the phe-nomenological equation of motion for a system with in-ternal potential and external non-potential componentsis currently used.Thus, all widely known explanations of irreversibilityare based on certain hypotheses and these hypotheses gooutside the constraints of fundamental laws and princi-ples of classical mechanics. Therefore it is very importanteither to find a deterministic explanation of irreversibil-ity or to prove impossibility of its existence. In this con-text it is interesting to consider dynamics of structuredparticles when they are defined as equilibrium subsys-tems (ES) consisting of potentially interacting materialpoints. It turns out that under certain conditions dynam-ics of such systems is irreversible [25-27]. The conditionsare formulated as follows:1). The energy of an ES must be presented as a sum ofinternal energy and the energy of ES motion as a whole.2). Each material point in the system must be con-nected with a certain ES independent of its motion inspace.3). During all the process the subsystems are consid-ered to be equilibrium.The first condition is necessary to introduce internalenergy in the description of system dynamics as a newkey parameter charactering energy variations in ES. Thesecond condition enables not to redefine ES after mixing of material points. The last condition is taken from ther-modynamics. It is equivalent to the condition of weakinteractions in the ES, which do not violate ES equi-librium. Moreover, it implies that each ES contains somany elements that it can be described using the conceptof equilibrium system.Objectives of this paper were to analyze the dynamicsof systems of structured particles when such systems aredefined as ES and to determine the difference betweentheir dynamics and dynamics of Hamiltonian systemsrepresented by potentially interacting material points.For that, let us derive an equitation which describesthe dynamics of two interacting equilibrium systems rep-resented by potentially interacting material points. Werefer to this equation hereafter as to the equation of mo-tion for structured particles. This equation explains themechanism of friction in classical mechanics. It is shownwhy the model of a group of ES enables to describe fric-tion forces. It is also shown how Lagrange, Hamiltonand Liouville equations for ES are derived from the equa-tion of motion of interacting ES. We consider below howsuch equations are different from similar ones for systemsof material points. It is shown how classical mechanicscould be linked with thermodynamics by means of theequation of ES interaction and how the concept of en-tropy arises in classical mechanics. It is shown also howbased on the hypothesis of local equilibrium, which en-ables to represent non-equilibrium systems as an ensem-ble of ES, one can generalize the obtained results for twointeracting ES.
II. SUBSTANTIATION OF THE APPROACH
Irreversibility is caused by dissipative processes.Therefore, if in the framework of classical mechanics onemanaged to explain the friction force and find its analyt-ical expression; it would be equivalent to the existence ofirreversible dynamics.In practice it is not difficult to introduce friction forcesempirically into the equation of motion, though in theframes of classical mechanics they have not got any ex-planation and a corresponding model. It is caused by thefact that classical mechanics is based on Newton laws.Newton equation of motion gives a relation between thechanges in the body momentum and potential forces act-ing on it. The friction forces are not potential forces.They transform the motion energy of the body into theenergy of chaotic motion of elements of this body.It is obvious that analytical description of frictionforces in the framework of classical mechanics is basedon the fact that all bodies have certain microstructure.In such a case friction may be defined as a process ofexcitation of chaotic motion of a body elements as a re-sult of its interaction with external bodies or field. Workof external field is spent to change internal energy of astructured particle and to move it as determined by thesystems’ CM trajectory. Part of energy converted intointernal energy of the system produces no work on mo-tion of the system. This energy only results in higherchaotic velocities of material points with respect to CM.Friction forces make similar transformation of energy. So,we would call forces which change internal energy as fric-tion forces. Then the description of friction is reduced tothe description of transfer of energy of relative motion ofbodies into the energy of chaotic motion of their elementsas a result of action of fundamental forces between theelements. It means that friction forces cause changes ininternal energy. It turns out that for some models of sys-tems one can built such description based on the law ofenergy conservation. For this purpose the energy of bod-ies must be presented as a sum of their energy of motionand internal energy.We will derive the expression for the friction force usingthe following method. As a model of the system we willtake two interacting structured particles moving with re-spect to each other. Let us assume that each structuredparticle is an ES and consists of a finite number of poten-tially interacting material points. This model will enableus to combine micro-description of the motion of materialpoints and macro-description of motion of ES.To derive an equation for transformation of the energyof motion of ES into their internal energy it is necessaryto define the energy of each ES as a sum of internal energyand energy of its motion. Differentiating the system en-ergy over time and using the condition of its conservation,we derive the equation for energy exchange between ESand based on it determine the equation of motion. Thisequation contains the friction coefficient as a measure oftransformation of the energy of system motion into itsinternal energy.
III. THE EQUATION OF EQUILIBRIUMSYSTEMS MOTION
The equation of motion for two ES can be obtainedin two stages. At the first stage, based on the conditionof energy conservation we obtain the equation of motionfor the system in the field of external forces. Then weconsider a non-equilibrium system consisting of two ESand obtain their equations of motion when the externalfield for one ES is the field of forces of the other ES.Forces acting between the ES can be obtained from theirenergy of interaction.Let us show how the equation of motion for a sys-tem of N material points with weights m = 1 can beobtained [3-5]. As it is generally accepted in classi-cal mechanics, we consider forces between two materialpoints as potential ones. The energy of the system E is equal to the sum of kinetic energies of material points T N = N P i =1 mv i /
2, their potential energy in the field of ex-ternal forces, U N env , and potential energy of their inter-action U N ( r ij ) = N − P i =1 N P j = i +1 U ij ( r ij ), where r ij = r i − r j , r i , v i are coordinates and velocities of the i -th particle.Thus, E = E N + U env = T N + U N + U env = const .By substituting variables we represent the energy ofthe system as a sum of the motion energy of the CMand the internal energy. Differentiating this energy withrespect to time, we will obtain [27]: V N M N ˙ V N + ˙ E insN = − V N F env − Φ env (1)Here F env = N P i =1 F envi ( R N , ˜ r i ), ˙ E insN = ˙ T insN (˜ v i ) +˙ U insN (˜ r i )= N P i =1 ˜ v i ( m ˙˜ v i + F (˜ r ) i ), Φ env = N P i =1 ˜ v i F envi ( R N , ˜ r i ), r i = R N + ˜ r i , M N = mN , v i = V N + ˜ v i , F envi = ∂U env /∂ ˜ r i ), ˜ r i , ˜ v i are the coordinates and velocity of i -thparticle in the CM system, R N , V N are the coordinatesand velocity of the CM system.The equation (1) represents the balance of the energyof the system of material points in the field of externalforces.The first term in the left-hand side of the equationdetermines the change of kinetic energy of the system- ˙ T trN = V N M N ˙ V N . The second term determines thechange of internal energy of the system, ˙ E insN . This en-ergy dependent on coordinates and velocities of materialpoints relative to the center of mass of the system.The right-hand side corresponds to the work of exter-nal forces changing the energy of the system. The firstterm changes the systems motion energy. The secondterm determines the work of forces changing the internalenergy - ˙ E insN .Let us determine the condition when the work whichchanges internal energy is not equal to zero. We musttake into account that F env = F env ( R + ˜ r i ) where R is the distance from the source of force to the center ofmass of the system. Let us assume that R >> ˜ r i . Inthis case the force F env can be expanded with respectto a small parameter. Leaving in the expansion termsof zero and first order we can write: F envi = F envi | R +( ∇ F envi ) | R ˜ r i ≡ F envi + ( ∇ F envi )˜ r i . Taking into accountthat N P i =1 ˜ v i = N P i =1 ˜ r i = 0 and N P i =1 F envi = N F envi = F env ,we get from (1): V N ( M N ˙ V N ) + N X i =1 m ˜ v i ( ˙˜ v i + F (˜ r ) i ) ≈≈ − V N F env − ( ∇ F envi ) N X i =1 ˜ v i ˜ r i (2)In the right-hand side of equation (2) the force F env inthe first term depends on R . It is a potential force. Thesecond term depending on coordinates of material pointsand their velocities relative to the CM of the system de-termines changes in the internal energy of the system. Itis proportional to the divergence of the external force.Therefore, in spite of the condition R >> ˜ r i the valuesof ˜ v i may be not small, and the second term cannot beomitted. Forces corresponding to this term are not po-tential forces. So, the change in the internal energy willbe not equal to zero only if the characteristic scale ofinhomogeneities of the external field is commensurablewith the system scale.Equation (2) confirms assumption of A. Poincare thatit is necessary to take into account structures of interact-ing bodies at rather small distances between them [21].Dynamics of an individual material point as well as dy-namics of a system of material points can be derived fromequation (1). A material point does not have an internalenergy, and forces acting on it are caused by potentialforces of interaction with other material points and theexternal force. Therefore the motion of a material pointis determined by the work of potential forces transform-ing the energy of the external field and other materialpoints into its kinetic energy.Unlike material points, a system has its internal en-ergy. Therefore the work of external forces over the sys-tem causes changes in its T trN and E insN , i.e. the externalforce breaks up into two components. The first compo-nent is a potential force. It changes momentum of theCM. The second component is non-potential. Its workchanges E insN . Hence, the motion of the system is deter-mined by the work of potential and non-potential forcestransforming the external field energy into the energy ofCM motion and internal energy.Multiplying eq.(1) by V N and dividing by V N we findthe equation of system motion [27]: M N ˙ V N = − F env − α N V N (3)where α N = [ ˙ E insN + Φ env ] /V N is a coefficient deter-mined by the change of internal energy.The equation (3) is a motion equation for ES. The firstterm in the right-hand side of the equation determinesthe system acceleration, and the second term determinesthe change of its internal energy. The motion equationfor ES is reduced to the Newton equation if it is possibleto neglect variation in the internal energy.Thus, the system state in the external field is deter-mined by two parameters: the energy of motion and theinternal energy. Each type of energy has its own force.The change in the motion energy is caused by the po-tential component of the force, whereas the change inthe internal energy is caused by the non-potential com-ponent.It is known from the kinetics that the nonequilibriumsystem in approach of the local equilibrium can be pre-sented as a set of ES in motion relative to each other[15]. In this case dynamic processes in nonequilibriumsystems will be defined by local values of energy of ESrelative motion and their internal energy. Therefore inthis case the description of dynamics of system by meansof the equation of ES motion (3) can be carried out if theexternal forces in it equation to replace with the forcesbetween ES. Let us show how to obtain the equation for interac-tion between two ES. For this purpose we take the sys-tem consisting of two ES- L and K . L is the numberof elements in the L -ES and K is the number of ele-ments in K -ES, i.e. L + K = N . Let LV L + KV K = 0,where V L and V K are velocities of L and K equilibriumsubsystems relative to the CM system. Differentiatingthe energy of the system with respect to time, we ob-tain: N P i =1 v i ˙ v i + N − P i =1 N P j = i +1 v ij F ij = 0, where F ij = U ij = ∂U/∂r ij .In order to derive the equation for L -ES, in the left-hand side of the equation we leave only terms determin-ing change of kinetic and potential energy of interactionof L -ES elements among themselves. All other terms wedisplace into the right-hand side of the equation and com-bine the groups of terms in such a way that each groupcontains the terms with identical velocities. In accor-dance with Newton equation, the groups which containterms with velocities of the elements from K -ES are equalto zero. As a result the right-hand side of the equationwill contain only the terms which determine the inter-action of the elements L -ES with the elements K -ES.Thus we will have: L P i L =1 v i L ˙ v i L + L − P i L =1 L P j L = i L +1 F i L j L v i L j L = L P i L =1 K P j K =1 F i L j K v j K where double indexes are introducedto denote that a particle belongs to the correspondingsystem. If we make substitution v i L = ˜ v i L + V L , where˜ v i L is the velocity of i L particle relative to the CM of L -ES, we obtain the equation for L -ES. The equation for K -ES can be obtained in the same way. The equationsfor two interacting systems can be written as [26]: V L M L ˙ V L + ˙ E insL = − Φ L − V L Ψ (4) V K M K ˙ V K + ˙ E insK = Φ K + V K Ψ (5)Here M L = mL, M K = mK, Ψ = L P i L =1 F Ki L ; Φ L = L P i L =1 ˜ v i L F Ki L ; Φ K = K P i K =1 ˜ v i K F Li K ; F Ki L = K P j K =1 F i L j K ; F Lj K = L P i L =1 F i L j K ; ˙ E insL = L − P i L =1 L P j L = i L +1 v i L j L [ m ˙ v iLjL L ++ F i L j L ]; ˙ E insK = K − P i K =1 K P j K = i K +1 v i K j K [ m ˙ v iKjK K ++ F i K j K ].The equations (4, 5) are equations for interactionsbetween ES. They describe energy exchange betweenES. Independent variables are macro-parameters andmicro-parameters. Macro-parameters are coordinatesand velocities of the motion of CM of systems. Micro-parameters are relative coordinates and velocities of ma-terial points.Therefore the equation of system interaction binds to-gether two types of description: on the macrolevel andon the microlevel. The description on the macrolevel de-termines dynamics of an ES as a whole and descriptionon the microlevel determines dynamics of the elements ofan ES.The potential force, Ψ, determines the motion of anES as a whole. This force is the sum of potential forcesacting on the elements of one ES from the other system.The forces determined by terms Φ L and Φ K transformthe motion energy of ES into their internal energy asa result of chaotic motion of elements of one ES in thefield of forces of the other ES. As in the case of the sys-tem in the external field, these terms are not zero onlyif the characteristic scale of inhomogeneity of forces ofone system is commeasurable with the scale of the othersystem. The work of such forces causes violation of timesymmetry for ES dynamics.The equations for systems motion corresponding to theequations (4,5) can be written as: M L ˙ V L = − Ψ − α L V L (6) M K ˙ V K = Ψ + α K V K (7)where α L = ( ˙ E insL + Φ L ) /V L , α K = (Φ K − ˙ E insK ) /V K ,The equations (6,7) are motion equations for two in-teracting ES. The second terms in the right-hand side ofthe equations determine the forces changing the internalenergy of the ES. These forces are equivalent to the fric-tion forces. Their work is a sum of works of forces actingon the material points of one ES from the other ES.The coefficients ” α L ”, ” α K ” determine efficiency oftransformation of the energy of ES motion into their in-ternal energy. These coefficients are friction coefficients.Therefore equations (6, 7) enable to determine analyt-ical form of non-potential forces in the non-equilibriumsystem causing changes in the internal energy of the equi-librium system. IV. THE GENERALS OF LAGRANGE,HAMILTON AND LIOUVILLE EQUATIONS FORA SET OF EQUILIBRIUM SYSTEMS
Let us show qualitative difference of Lagrange, Hamil-ton and Liouville equations for the systems of materialpoints from similar equations for a system which consistsfrom a set of interacting ES.Using Newton equation one can derive Hamilton prin-ciple for material points from differential D’Alambertprinciple [28, 29]. For this purpose the time integral ofvirtual work done by effective forces is equated to zero.Integration over time is carried out provided that exter-nal forces possess a power function. It means that thecanonical principle of Hamilton is valid only for caseswhen P F i δR i = − δU , where i is a particle number, and F i is a force acting on this particle. But for interactingES the condition of conservation of forces is not fulfilledbecause of the presence of a non-potential component. Therefore Hamiltonian principle for a set of ES as wellas Lagrange, Hamilton and Liouville equations must bederived using motion equation for ES.The equation for distribution function for a set of ESis written as [25]: df /dt = − R X L =1 ∂F L /∂V L (8)Here f is a distribution function for a set of ES, F L isa non-potential part of collective forces acting on the ES, V L is the velocity of L -ES.We call the equation (8) as Liouville equation for dis-tribution function of ES. Like the well-known Liouvilleequation for potential interaction of material points, theequation (8) is derived from D’Alambert equation withthe equation of motion for equilibrium subsystems substi-tuted into D’Alambert equation instead of Newton equa-tion for material points [28]. So, Liouville equation forES distribution function gets a nonzero right-hand sidedue to changes in internal energy of equilibrium subsys-tems. Thus, equation (8) actually defines the distributionfunction for particles with internal degrees of freedom.The right-hand side of the equation is determined bythe efficiency of transformation of the ES motion energyinto their internal energy. For non-equilibrium systemsthe right-hand side is not equal to zero because of non-potentiality of forces changing the internal energy.The state of the system as a set of ES can be definedin the phase space which consists of 6 R − R is the number of ES.Location of each ES is given by three coordinates andtheir moments. Let us call this space an S -space for ESin order to distinguish it from the usual phase space formaterial points. Unlike the usual phase space the S -space is not conserved. It is caused by transformationof the energy of relative motion in ES into their internalenergy. The internal energy cannot be transformed intothe energy of motion as ES momentum cannot changedue to the motion of its material points [8]. Therefore S -space is compressible. V. THE EQUATIONS OF INTERACTION OFSYSTEMS AND THERMODYNAMICS
Equations (1-8) give relationship between mechanicsand thermodynamics [15,26]. According to the basicequation of thermodynamics the work of external forcesacting on the system splits into two parts. The first partcorresponds to reversible work. In our case it correspondsto the change of the motion energy of the system as awhole. The second part of energy goes on heating. Itcorresponds to the internal energy of the system.Let us take a motionless non-equilibrium system con-sisting of ” R ” ES. Each ES consists of a great number ofelements N L >>
1, where L = 1 , , ...R, N = R P L =1 N L .In this case it is possible to define the temperature, T forES. It is average kinetic energy of a material points ofES. Let dE be work done over the system. In thermody-namics energy E is called internal energy (in our case itis equal to the sum of all energies of ES). It is known fromthermodynamics that dE = dQ − P dY . Here, accordingto generally accepted terminology, E is the energy of thesystem; Q is the thermal energy; P is the pressure; Y is the volume. The equation of interaction between twosystems is also a differential of two types of energy. Itmeans that dE in the ES is redistributed in such a waythat some part of it changes energy of relative motionof the ES and the other part changes the internal en-ergy. Thus, it follows that entropy may be introducedinto classical mechanics if it is considered as a quantitycharacterizing increase in the internal energy of an ES atthe expense of energy of their motion. Then the increasein entropy can be written as [26, 27]:∆ S = R X L =1 { N L N L X k =1 Z [ X s F Lks v k /E L ] dt } (9)Here E L is the kinetic energy of L -ES; N L is the num-ber of elements in L -ES; L = 1 , , ...R ; R is the numberof ES; s is the number of external elements which in-teract with k element belonging to the L -ES; F Lks is theforce acting on the k -element; v k is the velocity of the k -element.Based on the generally accepted definition of entropywe can derive expression for its production and definenecessary conditions for stationarity of a non-equilibriumsystem [27]. VI. CONCLUSION
The key idea which enables for us to submit the expla-nation of irreversibility without usage of the hypothesisof random fluctuations is the idea that all surroundingobjects have structure. In other words all bodies con-sist of the structured particles. The energy of the struc-tured particle consists of two principally different types.It is the energy of particle motion in the external fieldand its internal energy. When particles interact witheach other, both types of their energies change. How-ever, these changes are different. Thus, kinetic energy ofmotion of a particle changes as a result of work of poten-tial component of the external force. The internal energychanges as a result of work of non-potential componentof the force of interacting particles transforming the en-ergy of their motion into internal energy. Therefore ir-reversibility is related to the structureness of interacting bodies.We describe transformation of both types of energiesin the framework of the model of the system consistingof ES of potentially interacting material points. Suchmodel enables to connect micro-processes and macropro-cesses of energy transfer during interaction of ES. Indeed,the equation for motion of ES includes microparameters-coordinates and velocities of material points of which ESare composed, whereas macroparameters include coordi-nates and velocities of ES.The evolution of closed non-equilibrium system rep-resented by a set of ES is determined by potential andnon-potential forces acting between ES. Potential forceschange kinetic energy of motion of ES. The work of non-potential forces transforms energy of motion of an ES intoits internal energy. The phase space determined by coor-dinates and velocities of such system we called S -space.It is compressible. Compression of S -space is determinedby Liouville equation for non-equilibrium system from aset of ES. The system acquires an equilibrium state whenall energy of ES motion transforms into its internal en-ergy.The obtained equations for structured particles giverelationship between classical mechanics and thermody-namics. Therefore according to the motion equation forES the first law of thermodynamics follows from the factthat the work of external forces changes both the energyof particle motion and their internal energy. The secondlaw of thermodynamics follows from irreversible transfor-mation of energy of relative motion of system’s particlesinto their internal energy.The motion equation for ES also states impossibility ofexistence of structureless particles in classical mechanics,which is equivalent to infinite divisibility of matter.As an example of importance of the equation of motionfor structured particles we should note the following. Instrong interactions the internal degrees of freedom of in-teracting particles are exited. According to the equationof motion for structured particles it causes changes intheir internal energy and corresponding violation of timesymmetry. It is impossible to take into account thesefacts only using canonic Hamiltonian formalism withoutany empirical corrections.The offered explanation of irreversibility for the struc-tured particles is applicable for nonequilibrium systemsonly at possibility of their representation by model inthe form of set of ES. The clusters and other structurespresence in continuous environments tell us about gen-erality of such model [30]. Nevertheless, it is necessaryto investigate in future the question about how this con-dition limits applicability of the offered mechanism ofirreversibility. [1] Cohen E.G., Boltzmann and statistical mechanics, Dy-namics: Models and Kinetic Methods for Nonequilibrium Many Body systems. 1998, NATO Sci. Series E: AppliedSci., 371, p. 223[1] Cohen E.G., Boltzmann and statistical mechanics, Dy-namics: Models and Kinetic Methods for Nonequilibrium Many Body systems. 1998, NATO Sci. Series E: AppliedSci., 371, p. 223