Copyright © 2022 V.V. Aulin, S.V. Lysenko, A.V. Hrynkiv, D.V. Holub. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Problems of Tribology, V. 27, No 2/104-2022, 55-63 Problems of Tribology Website: http://tribology.khnu.km.ua/index.php/ProbTrib E-mail: tribosenator@gmail.com DOI: https://doi.org/10.31891/2079-1372-2022-104-2-55-63 Thermodynamic substantiation of the direction of nonequilibrium processes in triadconjugations of machine parts based on the principles of maximum and minimum entropy V.V. Aulin*, S.V. Lysenko, A.V. Hrynkiv , D.V. Holub Central Ukrainian National Technical University , Ukraine * E-mail: AulinVV @ gmail.com Received: 05 April 2022: Revised: 05 May 2022: Accept: 27 May 2022 Abstract The article gives a thermodynamic substantiation of the direction of nonequilibrium processes in tribocouples of machine parts, in tribosystems, based on the principles of maximum and minimum entropy. It is clarified how nonequilibrium processes can be substantiated on the basis of the minimum and maximum function of entropy production: linear and nonlinear nonequilibrium processes and their different thermodynamics. The entropy production function is considered as a function of thermodynamic force flows and thermodynamic flows. The theory of nonequilibrium processes is based on the Liouville equation for classical tribosystems, taking into account external influences or perturbations. It is shown that in thermodynamic processes in tribosystems the principle of entropy maximization is realized as the second principle of synergetics. Key words: triadconjugation of details, nonequilibrium processes, thermodynamics, synergetics, entropy, thermodynamic flow Introduction The essence of the principle of maximum entropy production G. Ziegler is that the evolution of the nonequilibrium tribosystem develops in the direction of maximizing the production of entropy in it under given external constraints. The second law of thermodynamics in the language of entropy production is formulated as follows: entropy production 0 S  not only has a positive value, but also goes to the maximum. Given the statistical interpretation of entropy and the work of Boltzmann and Gibbs, entropy, and consequently its production, tends to increase to the maximum level assumed by the constraints imposed on tribosystems. The final equilibrium state of the tribosystem is the most probable and is described by the maximum number of microstates. Such a statistical interpretation allows us to consider the principle of maximum entropy production as a natural generalization of its Clausius-Boltzmann-Gibbs formulation, and in some cases as a consequence. Literature review The peculiarity of nonequilibrium thermodynamics in tribosystems is initially based on the equations of balance of entropy, momentum energy and matter and on the first two laws of thermodynamics [1,2] . Compared with the principle of I. Prigogine [ 4 ] , the principle of G. Ziegler [5] describes a wider range in the evolution of nonequilibrium tribosystems and is a more generalized approach in their study and study of the relationship of characteristics and properties with entropy (entropy production) [3] . G. Ziegler's principle makes it possible to constructively construct both linear and nonlinear thermodynamics. It follows that Onsager's variational principle is valid only for linear nonequilibrium thermodynamics of tribosystems [4,6] . At that time, the principle of Onsager-Diarmati, as a partial statement, is valid for stationary processes, in the presence of free forces. From it follows the principle of I. Prigogine [7-9]. If given thermodynamic forces (flows), then, based on http://creativecommons.org/licenses/by/3.0/ http://tribology.khnu.km.ua/index.php/ProbTrib https://doi.org/10.31891/2079-1372-2022-104-2-55-54 56 Problems of Tribology the principle of G. Ziegler, the tribo system will adjust its thermodynamic flows (forces) to max S  [10-12]. If s  it is a quadratic function, then the relationship between flows and forces in the tribosystem is adjusted as a result. If the system is in a stationary weakly nonequilibrium state, but part of the thermodynamic forces remains free, then the currents generated by Ziegler will begin to reduce the thermodynamic forces, and those in turn - thermodynamic flows. As a result, the production of entropy is minimized: minS [4]. It is possible to substantiate nonequilibrium processes in tribosystems by the methods of general statistical theory [13-15]. Classical kinetic theory is not suitable for relatively dense tribosystems with a strong interaction between their elements and particles. The problem is to create a nonequilibrium microscopic theory that can describe such systems. This is primarily to obtain the equations of energy transfer, momentum, mass and calculation of kinetic coefficients directly from the equations of classical and quantum mechanics. Such a statistical theory began to develop intensively from the middle of the twentieth century [16,17]. L. Onsager stated: the temporal evolution of the function of a given physical quantity in the equilibrium system occurs on average by the same laws as the change of the corresponding macroscopic variable in the nonequilibrium system [1,2,10,18]. Being in an unbalanced state, the tribosystem does not feel how it got into it - due to fluctuations or due to external influences, and therefore its next reaction must be the same. As a result of relaxation of the nonequilibrium tribosystem near the state of equilibrium and resorption of fluctuations will occur according to the same laws [19,20]. Purpose The aim of this work is a thermodynamic substantiation based on the maximum entropy of the direction of nonequilibrium processes occurring in the triad conjugations of machine parts. Results If the tribosystem, the conjugation of machine parts, is in some nonequilibrium state, then after some time (relaxation time) it will come to an equilibrium state from the set of possible states for which the entropy will be maximum. The change in entropy during this period of time will be the maximum among the possible, and therefore the maximum becomes the production of entropy. The variational principle gives possible relations of linear nonequilibrium thermodynamics:  k kiki XLJ ; kiik LL  , (1) where ik L – the matrix of kinetic coefficients independent of i J and k X . The system of equations (1) makes it possible to describe the transfer of entropy, momentum, mass. The above equations (1) are valid for relatively small thermodynamic forces, when the relationship between forces and flows is almost linear. This is L. Onsager's first deductive formulation of linear nonequilibrium thermodynamics. If the values of irreversible forces are given i X , then the true flows i J maximize the expression )],(),([ kikiS JJJX  . The variation in flow J at constant X is equal to: 0)],(),([  XkikiSJ JJJX ; (2)  ki kiikki JJRJJ ,2 1 ),( , (3) where  – the scattering potential ( 0 ); ikR – coefficient matrix, inverse matrix ikL , matrix ikR – can be considered a system tensor, which should be considered as the sum of symmetric ikS and antisymmetric ikA tensors:    ki kiikkiik JJAJJS ,2 1 . (4) Because for the antisymmetric tensor 0iiA and kiik AA  , the antisymmetric part of the tensor ikR in equation (3) does not contribute to the scattering potential  and the tensor ikR becomes symmetric Problems of Tribology 57 kiik RR  . Substituting the expression  i iiS JX into equation (2), as well as transforming (3) by replacing the variation derivative over the corresponding flows, we obtain: 0 2 1 ,             constxi ki kiikii i JJRJX J . (5) The equation for thermodynamic force after differentiation will look like:   k k kkjkkjjki JRJRRX )( 2 1 . (6) Since the flow function is nonnegative 0 , the solution of equation (6) with respect to unknown flows is equal to:    k k kjkkjkj XLXRJ 1 , (7) where jkjk LR  1 . In this case jk R – a symmetric matrix, it is 1 jk R also symmetric. This suggests that the expression )],(),([ kikiS JJJX  in equation (2) has one extreme point ki JX , , which is described by expressions (6) and (7). Because the flow function  is a homogeneous quadratic positive function, this point is the point of maximum. Note that the Onsager variation principle is formulated for thermodynamic flows in the tribosystem. For the space of forces in the tribosystem, according to I. Diarmati, if the values of thermodynamic flows are given i J , then the irreversible existing forces i X maximize the expressions ),(),( kiiiS XXYJX  , ie we have: 0)],(),([  JkiiiSX XXYJX ; (8)    ki kiikki XXLXXY ,2 1 , , (9) where   0,  ki XXY is the scattering potential in the force space of the tribosystem. Analysis of entropy production shows that its function is a symmetric bilinear form. Then according to the principle of Diarmati equation (2); (3) and (8), (9) are equivalent. The principle of minimum entropy production, formulated by I. Prigogine, against the background of the apparatus of nonequilibrium thermodynamics also describes various nonequilibrium processes supported by constant applications of irreversible forces jiX i ,1,  where nj  , n is the number of forces in the system and entropy production is minimal nji ,...,1 , disappear. Prigogine's principle is a simple consequence of the Onsager-Diarmati principle. The theory of linear nonequilibrium thermodynamics is widely used in tribosystems: – it becomes possible to solve the system of equations of mass transfer, momentum and energy, because the number of equations is equal to the number of unknowns; – using non-diagonal coefficients ik L , it becomes possible to describe cross-flows in chemical, electrical and other kinetic processes; – it is possible to obtain additional information about the values of kinetic coefficients; – the presence of entropy production values s  that have extreme values in the nonequilibrium state allows to obtain additional information about the characteristics and properties of the tribosystem. Note that linear nonequilibrium thermodynamics in tribosystems describes thermodynamic forces of small magnitude. Linear nonequilibrium thermodynamics cannot explain and describe the fundamental problems of self-organization, oscillatory processes, etc. Onsager's linear thermodynamics in thermodynamic theory is generalized to the nonlinear case on the basis of the maximum entropy production (G. Ziegler's principle). In the flow space   k J we have:  k kikiS JJXJ )()( . (10) 58 Problems of Tribology To find the functional dependence )( kk JX , G. Ziegler proposed the principle of maximum entropy production S  : if an irreversible thermodynamic force i X is given, then the true flow i J that satisfies the equation  i iiiS JXJ )( contributes to the maximum entropy production. S  [5] . This principle can be widely used in the theory of plasticity in the form of the principle of maximum rate of dissipation of mechanical energy (Mises principle): the rate of dissipation of mechanical energy per unit volume during plastic deformation has maximum value for the actual stress state among all stress states. plasticity. The strain rate is considered fixed. This principle of the theory of plasticity is in fact generalized to all nonequilibrium thermodynamics. For nonequilibrium processes described by linear nonequilibrium thermodynamics, in the tribosystem at a given complex of forces there is always a maximization of the max)( is J entropy production function, ie from the Ziegler principle we can obtain the Onsager principle. G. Ziegler's principle is realized in the system of equations:                                 i ii ki kiik constX ki i ii ki kiikkiik i JXJJR JXJJRJJR j , , , , 0   . (11) Where the space of thermodynamic forces is determined by the expression:      k kiki JRX   12 . (12) Substituting the last expression for the thermodynamic force in the second equation of system (11), we obtain: 1 1 2           ie μ = 2. Given this, we have: 02 , ,                      constX ki i ii ki kiikkiikJ JXJJRJJR . (13) Taking into account equation (10) and making some transformations, we obtain: 0 2 1 ,          constXki kiikSJ JJR , or      0,,  constXkikiSJ JJJX . (14) The latter indicates that the principle of G. Ziegler follows the variational principle of Onsager. The research shows that the function of entropy production, as a function of flows, is convex, and G. Ziegler's principle proves a mutually unambiguous correspondence between flows and forces and triadconjugations of details. This is confirmed by the geometric interpretation of the function )( iS J : it )( iS J tries to go to zero when 0 i J , and the whole surface )( iS J is sign-defined. When 0)(  iS J , for arbitrary values of forces, the line of intersection of the surface )( iS J and the plane  i ii JX will lie in the negative region and max S  corresponds to this line. Based on the principle of G. Ziegler, it can be argued that there can never be physically realized states of tribosystems with negative entropy production, ie always 0 s  . In the variational construction of nonequilibrium thermodynamics, a particular species s is postulated, iJ and iX there is some freedom in expression. Problems of Tribology 59 If in the tribosystem there are two thermodynamic forces 1 X and 2 X , which are known functions of flows 1 J and 2 J . The entropy production in this case is equal to:       1212121121 ,,, JJJXJJJXJJS  . (15) The orthogonality condition is: nk J X k S k ,1; *      ; (16) 1 2 1               k k k S S J J   . (17) Converting (16) and (17), taking into account (15), we obtain: 1 1 * 1 J XX  ,, 2 2 * 2 J XX  _ where Δ is the deviation from the orthogonality condition, which is finally in these conditions equal to:                     1 1 12 2 2 21 2 2 22 1 1 11 21 J X JX J X JX J X JX J X JX JJ S   . (18) Assuming that Δ → 0, we have:         2 21 211 1 21 212 , , , , J JJ JJX J JJ JJX SS       . (19) Equation (19) defines a class of functions s for which thermodynamic forces are determined. It is valid for the quadratic function S  (15) if the Onsager reciprocity relations are valid. If we assume that the fluctuation of quantities ia near the equilibrium state occurs according to a linear law (proportional to Xi) and that they are ergodic, we can obtain reciprocity and give kinetic coefficients ijL through time correlation functions to quickly change i a the corresponding values: ,)()( 0 dtoataL jiij     (20) where )()( oata ji  is the averaging over the equilibrium ensemble of functions )(ta i  with the distribution function )(aP :           Bk aS aP )( exp)(  , (21) where k B – the Boltzmann constant; )(aS – change of entropy at fluctuation SSaS eq  )( ; eqS – entropy system in equilibrium; )......( jie aaaa  – a set of values that characterize the system. The physical meaning of expression (20) is as follows: the longer the fluctuation, ie, the slower the attenuation of the correlation function, the greater the canonical coefficient. Let the tribosystem at the moment 0t be in an unbalanced state with entropy 0S . Until the next time t, when the difference 0tt  is significantly longer than the duration of one interaction, but less than the relaxation time, the system can go to one of the states with entropy S1…SN (S1<…