Research Article A Derivation of the Main Relations of Nonequilibrium Thermodynamics

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1 ISRN Thermodynamic Volume 2013, Article ID , 9 page Reearch Article A Derivation of the Main Relation of Nonequilibrium Thermodynamic Vladimir N. Pokrovkii Mocow State Univerity of Economic, Statitic and Informatic, Mocow , Ruia Correpondence hould be addreed to Vladimir N. Pokrovkii; vpok@comtv.ru Received 9 July 2013; Accepted 5 September 2013 Academic Editor: C. D. Daub, P. Epeau, A. Ghoufi, and P. Tren Copyright 2013 Vladimir N. Pokrovkii. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. The principle of nonequilibrium thermodynamic are dicued, uing the concept of internal variable that decribe deviation of a thermodynamic ytem from the equilibrium tate. While conidering the firt law of thermodynamic, work of internal variable i taken into account. It i hown that the requirement that the thermodynamic ytem cannot fulfil any work via internal variable i equivalent to the conventional formulation of the econd law of thermodynamic. Thee tatement, in line with the axiom introducing internal variable can be conidered a baic principle of nonequilibrium thermodynamic. While conidering tationary nonequilibrium ituation cloe to equilibrium, it i hown that known linear paritie between thermodynamic force and fluxe and alo the production of entropy, a a um of product of thermodynamic force and fluxe, are conequence of fundamental principle of thermodynamic. 1. Introduction The modern nonequilibrium thermodynamic i formulated [1 3] a a generaliation of equilibrium thermodynamic, a adding ome concept and principle, in particular, the concept of fluxe and thermodynamic force, pecific for nonequilibrium. Depite many different approache the problem, reviewed recently by Muchik [4], the extenion of equilibrium thermodynamic to non-equilibrium thermodynamic eem to need ome jutification. We are going to follow the approach [5, 6], which exploit additional variable, ocalled internal variable, 1 todecribedeviationofatateof thermodynamic ytem from equilibrium. It can be thought, that thi approach allow one to explore the principle of nonequilibrium thermodynamic, providing ome jutification of the known linear relation and open opportunitie for nonlinear generalization. We have to note that formulation of the main principle of non-equilibrium thermodynamic in term of internal variable i diputable, there i, at leat, two explicit verion. In particular, one of the approache [6] take into conideration only ditinctive internal variable, thoe that can be called [7] complexity internal variable.the other approach, which i followed in thi paper, conider all quantitie, which decribe the deviation of the ytem from the equilibrium, to be internal variable. In thi paper, we are trying to how advantage of our decription [7]a compared with the alternative formulation [4, 6]. Section 2 begin with a decription of a et of variable needed for the depiction of a non-equilibrium tate of a thermodynamic ytem. Further, reproducing partly the previou paper [7], we pay a pecial attention to formulation of the firt and econd principle of thermodynamic in term of internal variable. Dynamic of internal variable i dicued in Section 3.Section 4 i devoted to conideration of tationary tate, while it i hown, that, in the area cloe to equilibrium, the known linear relation between thermodynamic force and fluxe and alo expreion for production of entropy, a a umofproductoffluxeandthermodynamicforce,follow the formulated general principle. Concluion contain a dicuion of the reult. 2. The Main Concept and Principle Let u conider a thermodynamic ytem, the equilibrium tate of which, on definition, i characterized by abolute temperature 2 T and contitutive variable x 1,x 2,...,x n.the pecified contitutive variable determine thermodynamic

2 2 ISRN Thermodynamic ytem and define it border. In the implet cae, a volume V, which contain a certain part of ubtance in a ga or liquid phae, i conidered a a contitutive variable. To define a nonequilibrium tate of the ame ytem, one need, apart from temperature (in thi cae the concept of temperature ought to be dicued pecially [6]) and contitutive variable, in an indefinite number of other variable ξ 1,ξ 2,...Whenalimitedlitofn contitutive variable i fixed, the other variable are called internal variable; in an equilibrium ituation there are no internal variable or, otherwie, we can conider value of internal variable whatever their et i, are equal to zero, which correpond to the agreement that the internal variable decribe deviation of theytemfromtheequilibrium. 3 The internal variable characterize local heterogeneity of ytem and, conequently, can be named a tructural variable alo. The thermodynamic ytem ha no predetermined et of internal parameter; they arie, generally peaking, at any influence on the ytem and, conequently, it i poible to note alo that internal variable in the ytem repreent diappearing memory of the pat interaction with the environment. Thu, the tate pace of a ytem i defined a (T, x 1, x 2,...x n ; ξ 1,ξ 2...), where temperature T i a unique characteritic for both equilibrium and non-equilibrium, and any pace deviation from thi quantity are conidered a internal variable. It i not the unique way of determining the ytem, but it i convenient to adopt the lited variable for the beginning. Under unchanging condition, the thermodynamic ytem i trending to a unique teady macrotate, which i referred to a the equilibrium tate.thetrendofthe ytem to the equilibrium tate mean a trend of all internal variable to zero value The Firt Principle of Thermodynamic. Firt of all, let u conider the balance of all influence on the ytem. Taking into account energy coming into the ytem with fluxe of heat and matter from the environment, ΔQ and ΔN j,and work of the ytem 4 through contitutive variable (work, performed by the ytem, i conidered poitive), one can write for a change of total energy U of the ytem that ΔU = ΔQ n i=1 X i Δx i + N μ j ΔN j. (1) Thi i a conventional form of the balance of energy [4, 6]. In line with the total internal energy U, which include poible potential energy of agitated internal variable, internal thermal energy E, which doe not depend on internal variable, can be introduced [7]. In the equilibrium ituation, total energy U coincide with internal energy E, but,inthe non-equilibrium ituation, thee quantitie are different. 5 Then, the law of conervation of energy (1) canberewritten in the form ΔE = ΔQ + N n i=1 X i Δx i Ξ i Δξ i μ j ΔN j, i (2) where work of the ytem via internal variable i included. Thi form of thefirtprincipleofthermodynamicdemontrate, that the external influence (work via contitutive variable, heat energy and chemical energy of particle) contribute to a change of internal energy E and emerging of internal variable. It i important to take into account the procee of reallocation of energy due to the work of the internal variable in the explicit form The Second Principle of Thermodynamic. The work connected with the internal variable hould be conidered a an eential element of the decription. Internal variable can be agitated by external influence, but the ytem cannot perform any poitive work via internal variable, which mean Ξ i Δξ i 0. (3) i Thi tatement, formulated earlier [7], introduce impoibilityofthereverionofevolutionofthethermodynamicytem in time and can be conidered a a formulation of the econd principle of thermodynamic the formulation, which i, a it will be hown below, equivalent to the formulation of the principle in term of entropy (ee Section 2.4) The Introduction of Entropy. To define thermodynamic quantitie a function of tate, that i, function of the temperature and contitutive variable, one need in the concept of entropy with help of which a change of internal energy E, which doe not depend on internal variable on the definition, can be written a the total differential of variable of tate: de = TdS n i=1 X i dx i. (4) The relation i valid both for reverible and irreverible procee. Entropy S can be defined here, on a comparion of relation (2) and(4), a a quantity, a change of which i determined by external influence and internal change: TdS = ΔQ i Ξ i Δξ i + K μ j ΔN j. (5) Remarkably, when the external fluxe are abent, entropy, in virtue of (5), can be conidered a a function of internal variable, wherea the change of entropy (5) can be conidered a a total differential of the function. To calculate entropy S, in thi cae, one can conider proce of changing of the ytem from a nonequilibrium tate with the fixed value of variable ξ 1,ξ 2,...ξ to the equilibrium tate. The reult doe not depend on the way of integration, and the difference between value of entropy in equilibrium and nonequilibriumtatecanbewrittenymbolicallya S (T, x, ξ) S(T, x,0) = 1 T ξ 1,ξ 2,...ξ 0 j Ξ j dξ j 0. (6)

3 ISRN Thermodynamic 3 Relation (6) can be conidered a a definition of entropy of a nonequilibrium tate of thermodynamic ytem. The definition i applicable to any ytem, including open ytem in ituation far from equilibrium. Note that relation (5) i a generaliation of Prigogine relation, given by him for the ytem with chemical reaction [1, Equation 3.52] Two Part of the Entropy Variation. The total change of entropy in the ytem ds, defined by (5), can be plit, due to Prigogine [1], into two component: ds = d e S+. (7) Change of Entropy due to the Fluxe. The component d e S i connected with flow of heat and ubtance through the boundarie of the ytem: d e S= 1 N T (ΔQ + μ j ΔN j ), (8) where N j i a number of molecule of ubtance j in the ytem, and μ j i a chemical potential. The quantity d e S can be both poitive (the flux of heat and/or ubtance into the ytem)andnegative(thefluxofheatand/orubtanceoutof the ytem). In an iolated ytem, when there are neither heat fluxe nor matter fluxe between the ytem and environment, d e S=0. The fluxe of heat and ubtance into the ytem or from the ytem appear at abence of balance between the ytem and environment. A characteritic quantitie, real tream of heat and/or ubtance through border of the ytem are fixed. The actual fluxe into thermodynamic ytem are pecified by a problem under conideration, but, in any cae, it i poible to define empirically a et of fluxe J 1,J 2,...,J r which, it i uppoed, are given. It i convenient to write expreion (8)for a change,due to external fluxe into the ytem, of entropy in a tandardized form: d e S dt = 1 T r K j J j, (9) where quantitie K j arecharacteriticoftheytem,thati, function of temperature, contitutive variable, and alo internal variable. Production of Entropy. The internal part of the change of entropy iconnectedwithomeproceewithinthe ytem: = 1 T Ξ j Δξ j, 0. (10) j Thatmean,thatomeinternalvariable,whichcannotbe identified in advance and the number of which i not known, appear to be agitated during tranition from one macrocopic tate to another and then relax according to their internal law. Under reverible procee, when the ituation change in uch a low way (characteritic time of proce t characteritic time of relaxation τ i ), that all internal variable actually have their equilibrium value, internal production of entropy i equal to zero, =0.Byvirtueof(3), the quantity can be only nonnegative, and thi i a conventional expreion of the econd law of thermodynamic. From (10), expreion for the production of entropy i a follow: dt = 1 T dξ j Ξ j j dt, dt 0. (11) Entropy increae, when relaxation of internal variable in the ytem occur, o that the production of entropy i a manifetation of preence of internal variable, that i a manifetation of an internal complexity of the ytem. 3. Dynamic of Internal Variable To move further, our peculation hould be completed by equation, which allow to determine evolution of internal variable. Let u conider a ytem in a non-equilibrium tate (T, x, ξ) and under external influence, which can be a change of contitutive variable and/or a change of temperature and concentration of ubtance in the environment. In the cae of reverible procee, the external influence are conidered to be weak, o that the change of T and x follow external influence. In a general cae, to decribe the actual ituation, one ha to conider emerging and evolution of the internal variable, which are not influenced by the external force directly. A an example, we can point to a ituation, when temperature of the environment goe up and heat tart to get into the thermodynamic ytem. In thi ituation, gradient of temperature, that i, in our interpretation, internal variable are emerging and internal procee appear; the ytem trend to a new equilibrium, but the tate of the ytem i determined by a play between external influence and internal procee. It i convenient to conider thee contribution eparately Preferable Value of Internal Variable. Under external influence, the ytem i found to be, generally peaking, in a non-equilibrium tate, and for the decription of the ituation, one need on introduction of ome ditinctive internal variable ξ, a wa explained, for example, by Ketin [8] and Muchik [9]. A change of entropy of the ytem S(T, x, ξ) occur due to changing of argument of the function and, apartfromit,duetoanincomingfluxofentropy,thati,a contribution defined by expreion (9). Similar to the cae of iolated ytem, when a maximum value of entropy define the equilibrium tate, extreme value of entropy under the preence of external influence define pecial point in the tatepace.theepointaredefinedbyarelationobtainedby equating total variation of entropy to zero: S dt T dt + S dx x dt + S dξ ξ dt (12) + 1 T r K j (T, x, ξ) J j =0.

4 4 ISRN Thermodynamic The incoming flux of entropy i different, generally peaking, from the flux of entropy, coming out of the environment. The relation (12) at given temperature, contitutive variable, and external fluxe can be conidered a an equation for unknown internal variable, which, in thi cae, determine preferable value of internal variable ξ. The rate of change of internal variable in a point of quaiequilibrium can be replaced by rate of change of preferable variable (ee (16)): dξ dt ξ=ξ = dξ dt. (13) Thi allow, uing alo definition of thermodynamic force, to write the equation for preferable value a Ξ (ξ ) dξ dt =T( S dt T + r dt + S x K j (T, x, ξ )J j. dx dt ) ξ=ξ (14) The preferable value of internal variable, correponding to the actual value of other variable and external influence, are determined, apparently, by hitory of the application of external influence. In general cae, (14) cannot unambiguouly define preferable value. However, the ituation i not o hopele, when conidering pecial cae. If temperature and contitutive variable of the ytem do not change and external influence are given by contant fluxe of heat and ubtance into the ytem J 1,J 2,...,J,itipoibletomakeomeaumption. We conider, that each flux correpond to the only one of the internal variable, o that the number of ditinctive preferable variable i equal to the number of fluxe, that i, ξ l =0 at l=1,2,...,and ξ l =0at l=+1,+2,...takingalothe arbitrarine in the fluxe into account, (14) canbereducedto a et of equation: dξ i dt =B ij i, i=1,2,...,. (15) Value of factor B i are ettled by a choice of fluxe in repect to a choice of internal variable. Quantitie B i have only numerical value, when an appropriate choice of fluxe in the form of J j ξ j /τ j,whereτ j i a time of relaxation, ha been made. By a choice of the internal variable and the fluxe, it i poible to reduce value B i to unit, which i accepted further Evolution of Internal Variable. The actual value of the internal variable ξ 1,ξ 2,...differ from the preferable value. The deviation of internal variable from their current preferable value ξ i ξ i, (i = 1, 2,...) determine the trending of the internal variable to a preferable value. The change of internal variable ξ 1,ξ 2,...i determined by internal law of movement of particle of the thermodynamic ytem, o that an equation for a change of the internal variable can be written in the general form: d(ξ i ξ i ) dt = R ij (T, x 1,x 2,...,x n ;ξ 1,ξ 2,...;ξ 1,ξ 2,...) (ξ j ξ j ), i=1,2,... (16) Thi equation decribe relaxation of internal variable to preferable value ξ, which are defined by (14), in general form, or, for the imple ituation, by (15). The right-hand ide of (16) decribe change of the variable, according to internal law of the ytem, the external influence are preented via the preferable value. The ign minu i choen for convenience: matrix R ij in the ituation cloe to equilibrium (at ξ i =0, i=1,2,...) i poitive definite in many imple cae. 7 Let u notice, that the internal variable are depending generally on the pace coordinate ξ i =ξ i (t,x,y,z),othat the equation for change of internal variable can contain alo term (omitted here) for the decription of procee of diffuion of internal variable. Moreover, the equation of evolution of internal variable hould include random force. Each internal variable can be preented a a um of the regular (averaged in ome way) and random component. Here the regular component of variable are conidered only. Dicuion of random component, alo a well a fluctuation of thermodynamic variable, i omitted. For expanion of the decription, it i poible to ue the mathematical apparatu of tochatic nonlinear nonequilibrium thermodynamic decribed by Stratonovich [10]. 4. The Stationary Nonequilibrium State The ituation i being implified in a teady-tate cae, when aetofcontantfluxej 1,J 2,...,J i fixed. It i aumed that the number of fluxe correpond to the number of complexity variable, which have ditinctive contant value ξ 1,ξ 2,...,ξ. Apart of it, an indefinite number of ome other internal variable ξ +1,ξ +2,... can appear, and thermodynamic characteritic of the ytem are function of the all internal variable. The non-equilibrium tationary tate of thermodynamic ytem repreent pecial interet, and it i remarkable that thee tate, alo a equilibrium tate of the thermodynamic ytem, can be conidered in a general way. 4.1.DynamicofInternalVariable. To formulate the dynamic equation for internal variable for a tationary tate, we take (16) with definition of the derivative of preferable variable (15), in which the quantitie B j i ettled to be equal unitie. Thiallowonetowritetheequationfordynamicofinternal variable near a teady-tate point: dξ i dt =J i R il (ξ l ξ l ) R ik ξ l (ξ k ξ k )(ξ l ξ l )+, i=1,2,...,,+1,+2,..., (17)

5 ISRN Thermodynamic 5 where a matrix R il and it derivative are fixed in a conidered preferable point. The matrix R ij depend on temperature T and contitutive variable x 1,x 2,...,x n andaumedtobe poitive definite, that i to have poitive eigenvalue. 8 Let u remind, that the preferable tate i defined in uch a way, that ξ l =0at l=1,2,...,and ξ l =0at l = + 1, + 2,.... In tationary tate, value of preferable variable are contant;howeveritireachedbybalanceofproceeofrelaxation and permanent excitation of internal variable by external fluxe. Conidering the procee of relaxation and excitation in a tationary ituation independent, intead of (17), one can write two equation, in linear approximation: dξ i dt = R ijξ j, J i = R ij ξ j, i=1,2,...,. (18) The rate of change of internal variable ξ j depend on deviation of value of internal variable from the equilibrium value, but preferable value ξ j,which,ontheaumption,are cloe to ξ j, are determined by fluxe. An analyi of an empirical ituation pecifie the econd et of equation from (18), thu, etablihing the et of internal complexity variable Entropy Near a Stationary State. In ituation cloe to equilibrium, an expanion of entropy into erie with repect to internal variable contain no term of the firt order, in the implet approach: S (ξ) =S(0) 1 2 i,j S ij = ( 2 S ξ i ξ j ) S ij ξ i ξ j +, T,x,ξ =0. (19) By virtue of (6),valueofentropyoftheyteminanonequilibrium tate i le than value of entropy of the ame ytem in the equilibrium tate, o that one ha to conider matrix S to be nonnegative determined. Component of the matrix S are function of temperature and contitutive variable. An expanion of entropy of the thermodynamic ytem near a preferable (tationary) point begin with linear term: S (ξ) =S(ξ ) S j (ξ j ξ j ) 1 2 i,j S ij (ξ i ξ i )(ξ j ξ j )+, S j = ( S ξ j ) T,x,ξ, S ij = ( 2 S ξ i ξ j ) T,x,ξ,,2,...,, i,,2,... (20) Ititakenintoaccountthat,apartfromofthecomplexity internal variable ξ 1,ξ 2,...,ξ, there i a et of internal variable withnumber +1,+2,..., that decribe deviation of the ytem from the tationary tate. Matrixe S j and S lk are calculated not in equilibrium point, a in expreion (19), but in the tationary point and depend, apart from temperature and contitutive variable, on the complexity internal variable. A a function of the internal complexity variable, entropy ha no feature, which would allow to characterize the tationary tate. Conidering the variable of complexity to be fixed, expanion of entropy, a a function of all other poible internal variable, i reduced to a form: S (ξ) =S(ξ ) 1 2 S ij = ( 2 S ξ i ξ j ) T,x,ξ i,j=+1 S ij ξ i ξ j +, i,j=+1,+2,... (21) It i aumed that a tationary tate of thermodynamic ytem near the equilibrium tate i teady, o that matrix S lk (ξ 1,ξ 2,...,ξ ) in expreion (21) i poitive definite; entropy, a a function of internal variable, in the fixed tationary tate ha a maximum. The propertie of matrix S lk in a tationary point, which i far from equilibrium, remain not certain. Relation (21) allow u to write an expreion for thermodynamicforceinapointnearthetationarytate: Ξ j (ξ) = T S ξ j =Ξ j (ξ ) +TS jl (ξ l ξ l )+,,2,...,. (22) The firt term of the expanion of the thermodynamic forcearecontant.intheituationcloetotheequilibrium tate, in the implet approach, thermodynamic force are connected linearly with internal variable: Ξ j = T S ξ j =TS jk ξ k +. (23) Let u remember that, in the tate of equilibrium, value of internal variable are conidered to be zero, and thermodynamic force Ξ j diappear Production of Entropy. In a tationary nonequilibrium tate, value of temperature and all contitutive variable of the thermodynamic ytem are contant. Value of thermodynamic function of ytem, including entropy, alo are contant; however, there i production of entropy inide the ytem and correponding decreae in entropy of the ytem, due to fluxe of heat and/or ubtance, o that one can write dt = d es dt. (24)

6 6 ISRN Thermodynamic Equation (17) and(22) allow u to write an expanion of function of production of entropy (11) near a teady-tate point: dt = 1 Ξ T i J i + 1 T + + j,l=1 j,k,l=1 j,k,l=+1 ( TS jl J j +Ξ j R jl )(ξ l ξ l ) (S jk R jl +Ξ j R jk ξ l )(ξ k ξ k )(ξ l ξ l ) S jk R jl ξ k ξ l +. (25) Value of all matrixe are determined in the conidered teady-tate point ξ 1,ξ 2,...,ξ. In the implet approximation, expreion for production of entropy can be written a dt = 1 T Ξ j J j. (26) Thi expreion repreent production of entropy in the conventional form a the um of product of fluxe and thermodynamic force. Equation (26) i conidered a one of the baic tatement of nonequilibrium thermodynamic [1 3]. The emerging of the ign minu in expreion (26)iconnected with the fact that ign of the fluxe are taken oppoite to ign of the force (the internal variable). One can note, that repreentation (26) i valid only for teady-tate ituation and for mall deviation from equilibrium tate. Comparion of (9) and(26) for a tationary tate,when relation (24) i valid, determine that quantity K j in (9)coincide with thermodynamic force, K i =Ξ i, i=1,2,..., On the Criterion of Stability of Stationary State. Expreion (25) how that, in a teady-tate point, production of entropy, a a function of the internal variable of complexity, ha no peculiar point. However, if the teady-tate point i fixed (ξ l =ξ l, l=1,2,...), expreion (25)taketheform dt = 1 Ξ T i J i + j,k,l=+1 S jk R jl ξ k ξ l +. (27) Thebehaviourofproductionofentropyinvicinityofa teady-tate point i determined by term of the econd order with repect to all internal variable, excepting the complexity variable. The term of the econd order comprie a quare form with the matrix that i a product of two matrixe R jk and S lk,whicharecalculatedintheteady-tatepointanddepend, apart from temperature and contitutive variable, on the complexity internal variable. In the equilibrium point and, aumingly, in the teady-tate point near to the equilibrium tate the matrixe are poitive definite, o that production of entropy ha a minimum, which confirm the validity of theprigogine principleofaminimumofproductionof entropy [1 3]. In the point, that are far fromthe equilibrium point, the matrixe R jk and S lk are not necearily poitive definite, o that tability of the ytem can be connected with a maximum of production of entropy, a it i tated by ome invetigator [11, 12] The Relation between Fluxe and Thermodynamic Force. Now, with help of (18)and(23),wecanwriteexpreion(26) for production of entropy in other form a 1 T j,k=1 Ξ j R jk ξ k = j,k=1 S jk ξ k J j. (28) The written equation, by virtue of the uppoed arbitrarine and independence of internal variable, i followed by a relation between fluxe and thermodynamic force: 1 Ξ T j R jk = Thi relation can be rewritten a J i = L ik Ξ k, S jk J j. (29) L ik = 1 T R ils 1 lk. (30) Conidering linear approach, the component of matrixe R jk and S lk are contant, but in more general cae it i neceary to conider them a function of internal variable. By virtue of definition, the matrix L i poitive definite. The relation between fluxe and thermodynamic force in linear approximation are fundamental relation of the nonequilibrium thermodynamic [1 3]. We have hown that thee paritie are conequence of the main principle of thermodynamic and valid under two condition: firt, deviation from an equilibrium tate are mall, and econd, the tate i tationary. Let u note in addition that there i a tatement [1 3]for the matrixl to be ymmetric or antiymmetric: L ij =±L ji. (31) Fortheproofofthitatement,towhichuuallyreferato Onager principle, one ha to addre the other principle and ome aumption conidered in the following ection Symmetry of Kinetic Coefficient. Theproofoftheymmetry of kinetic coefficient i baed on the property of invariance of correlation of fluctuation of variou quantitie with repect to reverion of time, which i fair for equilibrium ituation [1 3, 13]. Being intereted in tationary tate, it i poible to conider fluctuation of internal variable near their tationary value and to aume that time correlation of random deviation of internal variable alo are invariant with repect to reverion of time. In other word, for correlation of variou quantitie, it i poible to write a relation (ξ i ξ i ) t (ξ k ξ k ) 0 (32) =± (ξ k ξ k ) t (ξ i ξ i ) 0.

7 ISRN Thermodynamic 7 The minu ignarieinthecae,whentheinternalvariable itelf change the ign at the reverion of time. Further, one can take advantage of the equation of evolution (16), which, with ue of expreion for thermodynamic force (22) and definition of a matrix of kinetic coefficient (30),canbepreentedintheform d(ξ i ξ i ) = L dt il (Ξ l (ξ) Ξ l (ξ )). (33) Thi equation allow one, after differentiation of relation (32), to write down L il (Ξ l (ξ) Ξ l (ξ )) (ξ k ξ k ) 0 (34) =±L kl (Ξ l (ξ) Ξ l (ξ )) (ξ i ξ i ) 0. By virtue of uppoed independence of deviation of variou variable from tationary value, the required relation (31) follow the equation written previouly. Expanionoftheprincipleofinvarianceofcorrelation with repect to reverion of time for tationary ituation allow proving the principle of ymmetry of kinetic coefficient A Simple Example. Let u conider a portion of matter in a mall volume ΔxΔyΔz a a thermodynamic ytem. In addition to the cae, when a tationary tate i maintained byfluxeofheatandparticleeparately[7], let u conider the imultaneou action of thee factor. We can aume that tream of heat and a ubtance are moving along the x-axi, o that 1 ΔQ J T = ΔyΔz Δt, J 1 Δm C = ΔyΔz Δt. (35) The fluxe of heat and ubtance are contributing independently in the entropy of the volume. The flux of heat, according to (8), determine the change of denity of entropy in the volume: d heat S = 1 dt Δx (J T T J T T+ΔT ) J T T 2 xt. (36) In a imilar way, one can define the change of denity of entropy due to the flux of the ubtance: d ub S dt = J C μ (c+δc) μ(c) J C μ T Δx T c xc. (37) The ytem i aumed to be characteried by chemical potential μ(c), which depend on concentration c. It i convenient to introduce pecial ymbol for the gradient of temperature and concentration, which can be conidered to be internal complexity variable: ξ T = x T= ΔT Δx, ξ C = x c= Δc Δx. (38) Thu, the change of denity of entropy, due to the external fluxe of heat and ubtance, can be written a follow: d e S dt = J T T 2 ξ T + J C T μ c ξ C. (39) The fluxe and gradient are connected with each other locally. It i well known [1 3], a an empirical fact, that, at imultaneou preence of the thermal and diffuion gradient, ξ T and ξ C cro-effect are oberved; a gradient of temperature induce a flux of the ubtance and vice vera, a gradient of concentration induce a flux of heat, o that according to the lat equation from (18), we record a linear parity connecting the fluxe and gradient for the conidered ytem: J T = R TT ξ T R TC ξ C, J C = R CT ξ T R CC ξ C. (40) Thee equation allow one to write the internal production of entropy, according to (26), in the form dt = 1 T [(R TTξ T +R TC ξ C )Ξ T +(R CT ξ T +R CC ξ C )Ξ C ]. (41) In a teady-tate ituation, the thermodynamic tate of the ytem doe not change, o that (24) ivalid,andtaking relation (39) into account, one ha another expreion for production of entropy: dt = J T T 2 ξ T J C μ T c ξ C. (42) Comparing (41)and(42), one find the relation between fluxe and thermodynamic force for a tationary cae: J T = T(R TT Ξ T +R CT Ξ C ), J C = ( μ c ) 1 (R TC Ξ T +R CC Ξ C ). (43) The condition of ymmetry of kinetic coefficient (31) i followedbyanequation: 1 T R TC = μ c R CT. (44) Entropy of the ytem i defined by formula (6), which, in conidered cae, take the form S S 0 = 1 T x T x c (Ξ T dξ T +Ξ C dξ C ). (45) 0 0 The thermodynamic force Ξ T and Ξ C,afunctionofcomplexity variable, can be found from (40) and(43). After the calculation of the integral, one get a imple formula for entropy a a function of internal variable: S S 0 = 1 2T ( 1 T ( xt) 2 + μ c ( xc) 2 ). (46) It i remarkable that the condition of integrability and exitence of entropy of the ytem appear to be identical to Onager relation, that i, equality of the nondiagonal component of a matrix in (43).

8 8 ISRN Thermodynamic 5. Concluion The uage of internal variable, playing a ditinctive role in the decription of nonequilibrium tate of the ytem, allow u to develop a generalized view on the principle of nonequilibrium thermodynamic. In thi paper, in line with the paper [7], the author apired to how that the thermodynamic of nonequilibrium tate could be formulated o conitently, a the thermodynamic of equilibrium tate. At leat, conidering the ituation cloe to equilibrium and uing the implet approach, we have obtained the fundamental relation of non-equilibrium thermodynamic. One can note the mot eential feature of our decription. (i) The reference to local equilibrium i not ued. On the contrary, it i uppoed, that local nonequilibrium exit. A et of internal variable ξ 1,ξ 2,... are introduced to decribe deviation of the ytem from equilibrium. (ii) While formulating the law of conervation of energy, the work connected with internal variable i conidered, thu, one ha to ditinguih between the total internal energy U(T,x, ξ) and internal thermal energy E(T, x) of the ytem. It allow to introduce entropy S for non-equilibrium tate of thermodynamic ytem by (4), which keep the meaning of entropy unchanged. A a conequence of thi, temperature in any ituation i defined through thermodynamic function of the ytem a T = ( E/ S) x or T = ( U/ S) x,ξ. (iii) The econd principle of thermodynamic may be formulated a impoibility of fulfilment of poitive work by internal variable (Section 2.3). Thi formulation i equivalent to conventional formulation but exclude a flavour of mytique from concept of entropy. (iv) Relation (5), that connect a change of entropy of athermodynamicytemwithfluxeofheatand ubtance and alo with change of internal variable, i valid for the cae of open ytem and irreverible procee. It i a generaliation of Prigogine relation, given by him for the ytem with chemical reaction [1, Equation 3.52]. (v) The known linear relation of nonequilibrium thermodynamic (in particular, the parity between force and the fluxe) follow from the general principle for the cae of teady-tate procee running in the region of mall deviation from equilibrium. Onecaneethatallknownreultoflinearnon-equilibrium thermodynamic follow the preented approach; however, the theory i open for nonlinear generaliation. Endnote 1. Perhap, the firt who ued internal variable for the conecutive formulation of thermodynamic wa Leontovich [5]. The firt Ruian edition of Introduction to Thermodynamic, baed on hi prewar- and war-time lecture,haappeared,aaeparatebook,in Temperature i uually meaured in degree, but, according to relationhip with other phyical quantitie, it i energy. To omit the factor of tranition from degree to energy, it i convenient to agree that the abolute temperature T i meaured in energy unit a it wa propoed by Landau and Lifhitz [13] and ued here through the paper. 3. We do not include here into conideration the fluctuation in thermodynamic ytem (local deviation of denity, temperature, concentration of ubtance, and otherquantitiefromequilibriumvalue)whichcanbe treated a huge number of uncontrollable internal variable. 4. In nonequilibrium ituation, work of the ytem i not equal to the work of the environment on the ytem [14]. To balance the inner and outer force, one need, a it wa hown by Gujrati [14], in ome internal variable. 5. The alternative theory [4, 6] of non-equilibriumin term of internal variable doe not ditinguih between the total and internal energie, thu, miinterpreting the role of internal variable. 6. The definition of entropy in the alternative theory [4, 6] of non-equilibrium in term of internal variable doe not include fluxe of ubtance (the lat term in (5)), thu, failing to reproduce the Prigogine reult. Moreover, one need in the omitted term to conider the procee of diffuion correctly, a wa dicued earlier [7]. 7. There are many important excluion from thi rule. A decription of thermodynamic ytem with ocillating chemical reaction, for example, require matrix R ij not to be poitive definite. 8. It i known, that uch combination (tranformation) of variable can be choen that the equation of dynamic (16) for thee variable get the form of the equation of a relaxation: dξ i dt = 1 τ i ξ j, i=1,2,...,, (47) where τ i =τ i (T, x 1,x 2,...,x n ) i a time of relaxation of correponding variable. Changing under internal law of the movement, the agitated internal variable trend to their unique equilibrium value, o that the ytem i moving to it equilibrium tate. Acknowledgment The author i grateful to the anonymou reviewer of the paper for their helpful and contructive comment. Reference [1] I. Prigogine, Introduction To Thermodynamic of Irreverible Procee,EditedbyC.C.Thoma,Sprinfild,Geneeo,Ill,USA, 1955.

9 ISRN Thermodynamic 9 [2] S.R.deGrootandP.Mazur,Non-Equilibrium Thermodynamic, NorthHolland,Amterdam,TheNetherland,1962. [3] D. Kondepudi and I. Prigogine, Modern Thermodynamic. From Heat Machine to Diipative Structure, JohnWiley&Son, Chicheter, UK, [4]W.Muchik, Whyomany chool ofthermodynamic? Forchung im Ingenieurween,vol.71,no.3-4,pp ,2007. [5] M.A.Leontovich,Introduction to Thermodynamic. Statitical Phyic, Nauka, Glavnaja redaktzija fiiko-matematichekoi literatury, Mocow, Ruia, [6] G. A. Maugin and W. Muchik, Thermodynamic with internal variable Part I. general concept, Non-Equilibrium Thermodynamic,vol.19,no.3,pp ,1994. [7] V.N.Pokrovki, Extendedthermodynamicinadicrete-ytem approach, European Phyic, vol. 26, no. 5, pp , [8] J. Ketin, the local-equilibrium approximation, Non- Equilibrium,vol.18,no.4,pp ,1993. [9]W.Muchik, CommenttoJ.Ketin:internalvariableinthe local-equilibrium approximation, Non-Equilibrium Thermodynamic, vol. 18, pp , [10] R. L. Stratonovich, Nonlinear Nonequilibrium Thermodynamic, Springer, Heidelberg, Germany, [11] A. Kleidon, Nonequilibrium thermodynamic and maximum entropy production in the Earth ytem: application and implication, Naturwienchaften,vol.96,no.6,pp ,2009. [12] K. Hackl and F. D. Ficher, On the relation between the principle of maximum diipation and inelatic evolution given by diipation potential, Proceeding of the Royal Society A, vol. 464,no.2089,pp ,2008. [13] L. D. Landau and E. M. Lifhitz, Statitical Phyic, vol. 1, Pergamon Pre, Oxford, UK, 3rd edition, [14] P. D. Gujrati, Nonequilibrium thermodynamic. Symmetric and unique formulation of the firt law, tatitical definition of heat and work, adiabati theorem and the fate of the Clauiu inequality: a microcopi view,

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