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About some effects of linear transformation of fluxes and forces in extended irreversible thermodynamics
Volume 115, number 9
PHYSICS LETTERS A
12 May 1986
ABOUT S O M E EFFECTS OF LINEAR T R A N S F O R M A T I O N OF FLUXES AND F O R C E S IN EXTENDED IRREVERSIBLE T H E R M O D Y N A M I C S J.R. RAMOS-BARRADO and P. G A L A N - M O N T E N E G R O Departamento de Flsica, Facultad de Ciencias, Universidad de M&laga, 29071 Malaga, Spain Received 28 January 1986; accepted for publication 17 March 1986
The effects of linear transformations of fluxes and forces in the framework of EIT are examined.
1. Introduction. A phenomenological approach to the behaviour of a system in a non-equilibrium state is the EIT theory [ 1 - 3 ] . In this irreversible process theory, the entropy is a function of classical variables and independent dissipative fluxes; a suitable choice of dissipative fluxes has to be assumed to describe the behaviour of the system [3-5]. In general, we choose those fluxes from energetic equations which satisfy a bilinear form on the entropy production. Nevertheless, the generalized forces and fluxes can be selected in various ways; a given set of forces and fluxes is chosen on the basis of its convenience for carrying out a given analysis. For this reason, the transformations of fluxes and forces, and their effects in the symmetry relations among the phenomenological coefficients, are a key feature in the non-equilibrium thermodynamic theory [6]. The aim, in this letter, is to provide a phenomenological analysis of some effects of linear transformations of fluxes and forces, when the system is not far from equilibrium [7], and where the nonvanishing relaxation times of the dissipative fluxes are taken into account. 2. Transformation o f the second order state equation. When u is the specific internal energy, c k the mass fraction of the kth component, and Ji the ith dissipative independent flux, we obtain:
(aS/au)ck,jl = 0-1 (u, Ck, ,1i),
i, k = 1,2, 3 ..... (2)
(OS/OCk )u, Jl = --0 -1 rl(u, c k , Ji), i , k = 1,2, 3 , . . . ,
(3)
where 0 is the generalized temperature and 7/the generalized chemical potentials of non-equilibrium. Developed in series up to second order in fluxes, we have the following expressions: o - l ( u , Ck, Ji)= T - l ( u , c k )
1 ~ (a20_l/3Ji3Jk)JiJk,
+ 2 i,k
(4)
0-1r/(u, c k, Ji) = T-1 ~ ~i(u, c i) i
1 ~
+ -2 i, k
(a20_lrl/Oji~Jk)JiJk,
(5)
with T and/a the temperature and the chemical potential of equilibrium. On the other hand, the state equations of second order are (aS/aJi)u, clc,ji = T -1 v ( ~
7ikJk) ,
(6)
in matrix form:
m
dS = (aS/Ou)ck,ji du + ~ (OS/aCk)u,,i i dc k k=l
S(J k ) = T -1 v ? J ,
n
+
with the first order state equations:
(1)
(7)
7i1 being a function of temperature, phenomenological
i=1
426
0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 115, number 9
PHYSICS LETTERS A
coefficient and relaxation time [5] : Ti/ =
-TrikRik ,
(8)
with
12 May 1986
k being the Boltzmann constant. The ot-type variables in this study are the deviation of the dissipative fluxes, and g is the matrix: g=T -I v¥.
R = L -1
,
(9)
L being the phenomenological coefficient matrix. For a change in fluxes and forces such as
(19)
Under a linear transformation such as (10), the distribution function f(ot) dot and g:ot~t remain invariant, therefore we have
J* = A J,
(10)
g* = (At) -1 gA -1
x* = (At)-aX,
(11)
and the expression (15) is obtained again in a different way. We have chosen the method in the above section because it is within the phenomenological framework of EIT. The expression for the relaxation times transformations follows with the help of (8), (12), (17):
then (12)
L* = ALA t .
From this transformation, the entropy can be written as a function of the classical variables and the new dissipative fluxes J~:
S=S(u,ck,J~. ),
i=1,2,3
.....
(13)
so the new second order state equations can be written as a function of the original dissipative fluxes by means of their jacobian:
( a~ )
= ~(J/*ck, S)/a(J~ ) u,ek, J~
c3(j~)/a(jk)
(14)
(lS)
which allow us to obtain the new second order state equations as a function of the original dissipative fluxes.
3. Relaxation time transformation. In the new framework of variables, the second order state equations are: S(J~) = T -1 v y ' J *
(16)
and with (7) and (15) 7" = (At) -1YA -1 •
(17)
The above results follow also from the expression (7) and the one time correlation of a-type variables [8], which satisfy (otot) = kg -1 ,
-1
~'j,k(Aij) r/kRjkAmk "qm = t -1 -1 ~j,k(Aij) R/kArnk
(21)
and, from the A matrix expression A/] 1 = (Atij.,"W1 ,
(22)
we have ,
and since (10), (11) lead to linear transformations of fluxes [6] we have S(J*) = ( A t ) - I s ( j ) ,
t -1
(20)
(18)
rik = r~i .
(23)
Hence we can formulate:
Theorem 1. The linear transformations of fluxes and forces, which satisfy the Meixner theorems, retain the validity of the Onsager reciprocal relations and the equality between the relaxation times of crossed fluxes. When transformation matrix A satisfies: Aij=O ,
Vi=/=], A i i = O ,
(24)
then the thermodynamic force Xi is invariant in the transformation, and we can now formulate:
Theorem 2. The relaxation times of the direct fluxes are invariant in the transformation when this transformation keeps the direct associated force unimpaired: 7ii = T~ ~ X i = X? . 4. Example: Heat reduced flux. One of the most 427
Volume 115, number 9
PHYSICS LETTERS A
useful flux transformations leads to a different definition of heat flux. In fact, in diffusing mixtures, the concept of heat flux can be defined in different ways, and the heat reduced flux is a usual alternative heat flux definition [9]. When we have a system with a diffusion flux only, the transformation matrix A will be: (
1
0)
A =
,
-h
(25)
1
h being the specific enthalpy, so the conjugated force for the heat flux is not modified. In this case, the transformations of the second order state equations are
(aS/aJ;)u,el,j; = (t)S/aJ1)U,Cl,Jq + hk(aS/aJq)u,cl,al
'
(as/aJ~)u,el,j ~ = (as/a,lq)u,c,,u I .
(26) (27)
From (17), the transformations of "Yii are * * = r11Rll + h(3'21 + T12 + h3'22) , TllRI1
(28)
* * T12R12 = TI2R12 +hR22r22,
(29)
* * r21R21 = r21R21 + hR22r22,
(30)
428
r22R22 = r22R22,
12 May 1986 (31)
where (12), (9) lead to r~2 = r ~ l ,
(32)
r~2 = r 2 2 .
(33)
Only if the relaxation time of heat flux r22 is zero, the relaxation times r12 or r21 , of conjugated dissipative fluxes, are invariant in the heat flux transformation given by (25).
References [1 ] J. Keizet, J. Chem. Phys. 69 (1978) 2609. [2] B. Eu, J. Chem. Phys. 73 (1980) 225. [3] M. Lopez de Haro, R.F. Rodriguez and L.S. Gaxeia-Colin, Physica A 128 (1984) 535. [4] D. Jou, J.M. Rubi and J. Casas-Vazquez,J. Phys. A 12 (!979) 2515. [5 ] J.E. Llebot, Ph.D. thesis Universitat Autonoma Barcelona (1981). [6] J. Meixn¢~, Adv. Mol. Relax. Phenom. 5 (1973) 319. [7] D. Jou and J. Casas-Vazquez,Physica A 101 (1980) 588. [8] S.R. de (;toot and P. Mazut, Non-equilibrium thermodynamics (North-Holland, Amsterdam, 1969). [9] R. l-Iaas¢,Thermodynamics of irreversible processes (Addison-W~ley, Reading, 1969).
PHYSICS LETTERS A
12 May 1986
ABOUT S O M E EFFECTS OF LINEAR T R A N S F O R M A T I O N OF FLUXES AND F O R C E S IN EXTENDED IRREVERSIBLE T H E R M O D Y N A M I C S J.R. RAMOS-BARRADO and P. G A L A N - M O N T E N E G R O Departamento de Flsica, Facultad de Ciencias, Universidad de M&laga, 29071 Malaga, Spain Received 28 January 1986; accepted for publication 17 March 1986
The effects of linear transformations of fluxes and forces in the framework of EIT are examined.
1. Introduction. A phenomenological approach to the behaviour of a system in a non-equilibrium state is the EIT theory [ 1 - 3 ] . In this irreversible process theory, the entropy is a function of classical variables and independent dissipative fluxes; a suitable choice of dissipative fluxes has to be assumed to describe the behaviour of the system [3-5]. In general, we choose those fluxes from energetic equations which satisfy a bilinear form on the entropy production. Nevertheless, the generalized forces and fluxes can be selected in various ways; a given set of forces and fluxes is chosen on the basis of its convenience for carrying out a given analysis. For this reason, the transformations of fluxes and forces, and their effects in the symmetry relations among the phenomenological coefficients, are a key feature in the non-equilibrium thermodynamic theory [6]. The aim, in this letter, is to provide a phenomenological analysis of some effects of linear transformations of fluxes and forces, when the system is not far from equilibrium [7], and where the nonvanishing relaxation times of the dissipative fluxes are taken into account. 2. Transformation o f the second order state equation. When u is the specific internal energy, c k the mass fraction of the kth component, and Ji the ith dissipative independent flux, we obtain:
(aS/au)ck,jl = 0-1 (u, Ck, ,1i),
i, k = 1,2, 3 ..... (2)
(OS/OCk )u, Jl = --0 -1 rl(u, c k , Ji), i , k = 1,2, 3 , . . . ,
(3)
where 0 is the generalized temperature and 7/the generalized chemical potentials of non-equilibrium. Developed in series up to second order in fluxes, we have the following expressions: o - l ( u , Ck, Ji)= T - l ( u , c k )
1 ~ (a20_l/3Ji3Jk)JiJk,
+ 2 i,k
(4)
0-1r/(u, c k, Ji) = T-1 ~ ~i(u, c i) i
1 ~
+ -2 i, k
(a20_lrl/Oji~Jk)JiJk,
(5)
with T and/a the temperature and the chemical potential of equilibrium. On the other hand, the state equations of second order are (aS/aJi)u, clc,ji = T -1 v ( ~
7ikJk) ,
(6)
in matrix form:
m
dS = (aS/Ou)ck,ji du + ~ (OS/aCk)u,,i i dc k k=l
S(J k ) = T -1 v ? J ,
n
+
with the first order state equations:
(1)
(7)
7i1 being a function of temperature, phenomenological
i=1
426
0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 115, number 9
PHYSICS LETTERS A
coefficient and relaxation time [5] : Ti/ =
-TrikRik ,
(8)
with
12 May 1986
k being the Boltzmann constant. The ot-type variables in this study are the deviation of the dissipative fluxes, and g is the matrix: g=T -I v¥.
R = L -1
,
(9)
L being the phenomenological coefficient matrix. For a change in fluxes and forces such as
(19)
Under a linear transformation such as (10), the distribution function f(ot) dot and g:ot~t remain invariant, therefore we have
J* = A J,
(10)
g* = (At) -1 gA -1
x* = (At)-aX,
(11)
and the expression (15) is obtained again in a different way. We have chosen the method in the above section because it is within the phenomenological framework of EIT. The expression for the relaxation times transformations follows with the help of (8), (12), (17):
then (12)
L* = ALA t .
From this transformation, the entropy can be written as a function of the classical variables and the new dissipative fluxes J~:
S=S(u,ck,J~. ),
i=1,2,3
.....
(13)
so the new second order state equations can be written as a function of the original dissipative fluxes by means of their jacobian:
( a~ )
= ~(J/*ck, S)/a(J~ ) u,ek, J~
c3(j~)/a(jk)
(14)
(lS)
which allow us to obtain the new second order state equations as a function of the original dissipative fluxes.
3. Relaxation time transformation. In the new framework of variables, the second order state equations are: S(J~) = T -1 v y ' J *
(16)
and with (7) and (15) 7" = (At) -1YA -1 •
(17)
The above results follow also from the expression (7) and the one time correlation of a-type variables [8], which satisfy (otot) = kg -1 ,
-1
~'j,k(Aij) r/kRjkAmk "qm = t -1 -1 ~j,k(Aij) R/kArnk
(21)
and, from the A matrix expression A/] 1 = (Atij.,"W1 ,
(22)
we have ,
and since (10), (11) lead to linear transformations of fluxes [6] we have S(J*) = ( A t ) - I s ( j ) ,
t -1
(20)
(18)
rik = r~i .
(23)
Hence we can formulate:
Theorem 1. The linear transformations of fluxes and forces, which satisfy the Meixner theorems, retain the validity of the Onsager reciprocal relations and the equality between the relaxation times of crossed fluxes. When transformation matrix A satisfies: Aij=O ,
Vi=/=], A i i = O ,
(24)
then the thermodynamic force Xi is invariant in the transformation, and we can now formulate:
Theorem 2. The relaxation times of the direct fluxes are invariant in the transformation when this transformation keeps the direct associated force unimpaired: 7ii = T~ ~ X i = X? . 4. Example: Heat reduced flux. One of the most 427
Volume 115, number 9
PHYSICS LETTERS A
useful flux transformations leads to a different definition of heat flux. In fact, in diffusing mixtures, the concept of heat flux can be defined in different ways, and the heat reduced flux is a usual alternative heat flux definition [9]. When we have a system with a diffusion flux only, the transformation matrix A will be: (
1
0)
A =
,
-h
(25)
1
h being the specific enthalpy, so the conjugated force for the heat flux is not modified. In this case, the transformations of the second order state equations are
(aS/aJ;)u,el,j; = (t)S/aJ1)U,Cl,Jq + hk(aS/aJq)u,cl,al
'
(as/aJ~)u,el,j ~ = (as/a,lq)u,c,,u I .
(26) (27)
From (17), the transformations of "Yii are * * = r11Rll + h(3'21 + T12 + h3'22) , TllRI1
(28)
* * T12R12 = TI2R12 +hR22r22,
(29)
* * r21R21 = r21R21 + hR22r22,
(30)
428
r22R22 = r22R22,
12 May 1986 (31)
where (12), (9) lead to r~2 = r ~ l ,
(32)
r~2 = r 2 2 .
(33)
Only if the relaxation time of heat flux r22 is zero, the relaxation times r12 or r21 , of conjugated dissipative fluxes, are invariant in the heat flux transformation given by (25).
References [1 ] J. Keizet, J. Chem. Phys. 69 (1978) 2609. [2] B. Eu, J. Chem. Phys. 73 (1980) 225. [3] M. Lopez de Haro, R.F. Rodriguez and L.S. Gaxeia-Colin, Physica A 128 (1984) 535. [4] D. Jou, J.M. Rubi and J. Casas-Vazquez,J. Phys. A 12 (!979) 2515. [5 ] J.E. Llebot, Ph.D. thesis Universitat Autonoma Barcelona (1981). [6] J. Meixn¢~, Adv. Mol. Relax. Phenom. 5 (1973) 319. [7] D. Jou and J. Casas-Vazquez,Physica A 101 (1980) 588. [8] S.R. de (;toot and P. Mazut, Non-equilibrium thermodynamics (North-Holland, Amsterdam, 1969). [9] R. l-Iaas¢,Thermodynamics of irreversible processes (Addison-W~ley, Reading, 1969).