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Onsager reciprocity in the nonlinear regime
Physica 132A (1985) 143-163 North-Holland, Amsterdam
ONSAGER
RRCR’ROCITY
IN THE NONLINEAR
REGIME
R.E. NEmTON Department of Physics, University of the Witwatersrand, Johannesburg, South Africa
Received 28 September 1984
A system is assumed, in the framework of extended non-equilibrium thermodynamics, to be described by state variables ai which are ensemble averages of dynamical functions, Ai, and their time-derivatives ui = o;i. A Fokker-Planck type equation is derived by the Zwanzig projection operator technique for g(a. o), the amplitude of the probability that A = a and A = a. The first moments of this equation give the phenomenological equations which can be cast in OnsagerCasimir canonical form, with forces and coefficients exhibiting reciprocity and non-linear in both (Iand n-variables. It is necessary to assume that g is sharply peaked in equilibrium and that a Markovian approximation holds for the particular sets of variables and time-scale chosen. The arguments are illustrated by applying them to heat conduction in a liquid or solid.
1. Introduction
In the traditional approach’) to non-equilibrium thermodynamics, we specify a non-equilibrium state by a finite set {(Ye}of time-dependent variables, n in number, and introduce an entropy S({cu,}).If the rates, bi, are cast in the form
then Lii = Lji in zero magnetic field provided the [Y~are all symmetric or all anti-symmetric under time-reversal, and the equations are linear in the set {(Y~ - aiO}, where the {ai,,} are the values assumed by the variables in equilibrium. Under these circumstances Lij = Lt({aio}). The conclusion that reciprocity relations are not limited to the linearized form of eq. (1) is reached in a derivation of the phenomenological equations from the classical Liouville equation given by Zwanzig2”) and more recently extended by Grabert4). In this approach, the ‘Ye are averages (Ai), in a time-dependent ensemble, of dynamical functions Ai which depend on the phase x of the system. By operating with a projection operator, one obtains from the Liouville equation a Fokker-Planck-type equation obeyed by the 037%4371/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
144
R.E. NETnETON
function values
g({a,})
which
is the
amplitude
a,. Eqs. (1) are the first moments
to hold
with
L,
and
W&X,
for the
probability
of the equation
non-linear
functions
that
A, assume
for g. They are shown
of {oi - (Y;,,} and
L,,= L,
provided: 1) The equations
memory
kernel
decays
fast enough
can be used in Markovian
2) Contributions
o(A’)
so that
the phenomenological
approximation.
to eqs. (1) are neglected,
where
A, = iLA,,
and L is
the self-adjoint Liouville operator. An extension of the Zwanzig derivation has been made5) for the case where the set {Ai}is assumed to be even under time-reversal and is augmented by the set {A,}. Under these circumstances, to the set of state variables {a,} we now add the set {vi = bi}. This extension was made5) to serve as a basis for what is “extended non-equilibrium thermodynamics”, in which the now called phenomenological equations have inertial terms, 7ji. It was possible to demonstrate Onsager-Casimir reciprocity in this extension provided assumptions (1) and (2) were retained, and we make the additional restrictions: 3) Only terms linear in the v-variables are kept. 4) g is sharply peaked and given by the Einstein function, g - exp ~*S/K, where 6*S represents quadratic terms in the expansion of S(a, u) about S(cy, 7). S(a, v) is the entropy when A = a,A = u. The possible non-linearity in {(Y~- (Y,} is retained. Assumption equivalent to taking dissipative forces linear in velocity. This restricted the phenomenological
equations
derive an equivalent Lagrangian of-motion of continuous media.
has been formalism
used
by Landau
used7,*) in obtaining
and
(3) is form of
Lifshitz6)
to
the equations-
Because many mechanical models have dissipative forces linear in velocity, the references cited above3X5) establish reciprocity in a variety of nonlinear problems of interest. It has been shown’) that if the equations for the Crj are inferred from an intuitive or phenomenological model, they can be cast uniquely in Onsager-Casimir canonical form, so long as they are linear in the n-variables,
but not necessarily
in the {(Y;- Q},
and so reciprocity
can be of
practical use in such problems. For example, some coefficients not calculable from a simple molecular model can be expressed in terms of others more accessible to model calculations. Nevertheless, more general cases may be encountered in a theory of states far from equilibrium, and so we should like to remove as many as possible of the restrictions (l)-(4) listed above. Various generalisations of Onsager reciprocal relations to the nonlinear regime have been proposed and shown to fai11@‘2,‘3 ), and therefore we desire to ascertain whether a valid generalisation does indeed exist. The answer to this problem lies in the equation derived5) for the fluctuation distribution function g(a, v), defined by
145
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
g(U,v> = f(x) dX n S(Ai I
(2)
- ai)S(Ai - pi) )
with 6 a Dirac delta and f(x) the solution of the classical Liouville equation i aflat = Lf.
(3)
g IIi da,dv, is the probability that A, has value between ai and Ui+ dUi and Ai between ui and ui + dv, (all i). Use of eq. (3) implies that a closed system is assumed or, alternatively, we postulate that the variables we are discussing are “fast” and relax over a time short compared to the time required for appreciable amounts of energy or particles to diffuse into or out of the system. This means that the x-integration in eq. (2) is taken over an energy shell. As we shall point out in section 5, the phenomenological equations may have additional terms arising from coupling to the surroundings, and the OnsagerCasimir symmetry of this coupling can be inferred in general only by phenomenological reasoning. To obtain an equation for g, we define a projection operator, P, such that for arbitrary B(x) PB(x) = [W(A)]-’
1 B(x’)6(A(x)
W(A) = j S(A(x)
- A(x’)) dx' ,
S(A(x)-
A(x’)) = fl S(A,(x)
- A(x’)) dx' ,
W)
- A,(x’))S(&x)-
Ai(
.
(4c)
With this definition, we have
g(4 v) = WG Mfl,4E&&” *
(44
The equation derived for g by operating with P on the Liouville equation is [ref. 5, eq. (S)], with the definition A> = (iL)‘Aj:
(a’at)g(u9t, = - i: {(alauj)[ujg(U,t)] + (a/auj)[qjg(U,
t)])
j=l
+ ?+ i j=l k=l
j ds J du’ (a/auj)[ 0
W(u)Kjk(U,
U’, S)](a/aV;)[g(U’~
t - S) W(U’)-‘1 (54
)
146
R.E. NETILETON
K,,(a, a’, s) = (Aj{exp[ - is(1 - P)L]}(l (G; u)=PGI,=, qj=(A;;
, a).
- P)AiG(A - a’); a),
(5b) PC) (54
In these equations we have written g(u) for g({u,}, {2ri})and da for Iii da,du,. Also “A = a” means both Ai = a, and Ai = ui (all i). We shall use this abbreviated notation’extensively in what follows. The phenomenological equations obeyed by the variables LY~ and ni = cUiare obtained from the first moments of eq. @a). In the following sections, we want to show that these moment equations can be cast in the form ki =
c LFF; ,
(64
where F, = aSIaa;.and FT = aS/avj for S the entropy. This is to be done in such a way that the Lf’ are, in general, nonlinear in the n-variables and in the set {ai - a,}, while Lz’ = LF’ and Li,” = -L$’ in zero magnetic field. We can do this without approximation (3) and with a weakened version of approximation (4). To show uniqueness of LI;“, we shall neglect some contributions to the constants which are of higher order in A. This is analogous to a weakened form of approximation (2). In subsection 2.1 we propose an ansatz for g consistent with the requirement that it reduce to the Einstein function in equilibrium. In subsection 2.2, we derive the equations for the first moments, {ai} and {vi}, of g by taking moments of eq. (5a). The rates Criand 7ji depend on g(O),a generalized Einstein function, and on an additive correction g”‘G assumed in the ansatz for g. Section 3 regroups the contributions from g(O)to the moment equations so that they assume the form of eqs. (6a, b), with 3.1 demonstrating the symmetry of the principal contributions to LI;” and 3.2 the antisymmetry relation. Section 4 takes up the contributions of the additive correction g”‘G. These additional contributions are cast in Onsager-symmetric form, with neglect of terms involving fourth-order fluctuations and third and higher-order derivatives of the entropy and also terms smaller by a power of p, the second derivative of S, or of A than other terms of the same order in F*. Such terms should be small if g is sharply-peaked. The calculations are exemplified in section 5 for the special case where the A-functions are the components of the heat flux, and Onsager symmetry is demonstrated for that case. A discussion is given in section 6, which mentions extensions to include cases of coupling to external
147
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
resevoirs and the inclusion of “slow” variables such as temperature energy in an open system.
or internal
2. First moment of the fluctuation distribution We proceed here to multiply eq. (5a) by uj and vi and integrate over the space of a- and v-variables. The right-hand side depends on g, and so it falls into two parts, stemming respectively from the additive terms g(O)and g”‘G in the ansatz of subsection 2.1. The explicit form of these contributions, which will be maneuvered into canonical form in sections 3 and 4, will be written down in subsection 2.2. 2.1. Ansatz for g In an equilibrium energy-shell ensemble, g(a) is proportional to W(a) = where we use the single argument “a” to represent both the aand v-variables. This proportionality holds since W(a) is the number of configurations consistent with A = a and A = v. This implies g(u) is given by the Einstein function in equilibrium, since w(a) = exp(S(u)/K). If we expand S about the values c+, and vi0 = 0 which maximise it in equilibrium, we have a sharp maximum, with W(A)(A,o,A,V
s = So(ao)+ szs
(7)
)
where the first-order terms vanish for expansion about a maximum. Because of the sharpness of the maximum, the higher terms are neglected in first approximation. In a non-equilibrium state characterised by (Y# (Ye, n f 0, a function having the desirable property of being sharply-peaked and reducing to the Einstein function in equilibrium is obtained if we replace a - (Ye-+a - (Y and v + v - n in S’S and set g - exp(S’S/K) following this replacement. This assumption has been used’4T’5)an d sh own’“) to be the first approximation to the solution of eq. (5a) in the case where g describes fluctuations in the heat flow. Another generalisation of the Einstein distribution, which reduces to it in equilibrium and agrees with exp(S’S/K) to second order, is g”‘(a) = C(a,
7) exp{S’(a) - S’((Y,7) -
where S’ =
S/K
2 [(Ui - ai)(3S'laai) + (Vi - 77i)(aS’la77i)l~
and the single argument
9
(8)
“a” denotes both a and v. C(a, 7)
148
R.E. NETTLETON
normalises expansion
g”’ to unity. Eq. (8) merely ads to S’S the higher-order terms in the of S(a, v) about S((Y, v). A suitable ansatz for g is then
g(a) = s’“‘(4P+ G(a)] , where,
I
since g(O)is normalized
(9 to unity,
g"'G da = 0,
with, as before,
da = llj du,dv,.
The function G(u) is to be determined to satisfy eq. (5a) subject to eq. (10). G must vanish in equilibrium, and so is a sum of powers and products of the forces {E;;} and {FT}, with coefficients which are polynomials in a - LYand v - 7. An inconsistency arises if we assume G = 0. To see this, note that
I
(v-n)gdu=O.
(11)
Since S(u) is a sum of even powers of v, its expansion will have terms of the type ~(v - r])3 which means that g(O)will not satisfy eq. (11). The contribution of G to the first moments of eq. (5a) involves moments of g(O)which are small if the latter is sharply-peaked. Therefore, we shall discuss the contributions to the Lr’ coefficients in section 4 as a correction believed
the main contributions
derived
in section
G-dependent to what are
3.
2.2. Moment equations Multiply
d; =
eq. (5a) by a, and integrate
I
over a and v. We find immediately
v,g da = nj.
that
(12)
This result shows that bj depends on variables odd with respect to time reversal, so that it can be expanded in forces from the set {as/an,}. We shall show in subsection 3.2 that the contribution from g”’ to the integral in eq. (12) yield a matrix Ly (cf. eq. (6a)) which satisfies the Casimir antireciprocity relation. If we multiply
eq. (5a) by v, and integrate,
the last term on the right yields
149
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
-C]dudu’j
ds W(u)Kjk(u, a’, s)(a/au;)[gco’/ W(u')][l
k
+
G(u’)]
0
=x
j ds
jdudu’ k
W(a)Kjk[g'"'/W(a')]as'/arlk
0
+F jduduj
ds w(a)G(a’)(aiaU;)Kjk(a,
d, &fV
w(d)]
(13)
0
to v; and have integrated by parts the term proportional From eq. (5b) for Kjk we find that
to G.
dx 6(A - a)Aj{exp[ - is(1 - P)LJ}(l - P) 2 A:(Wv;)S(A
=
- a’)
k
=
I
dx 6(A - u)Aj{exp[-is(l-
x C [A:(alav;)
P)L]}(l-
P)
- U)
+ A,(alaa;)]s(A
k
=I
dx S(A - a)A){exp[-is(1
- P)L]}(l
= (a/as) 1 dx 6(A - u)A:{exp[-is(lHere x
we have
added
a term
S(A - a’) = 0, because (1 - P)S(A
- P)iLG(A - a’)
P)L]}S(A
(1 - P)A,(a/a+S(A - a’) = 0.
- a’). - a’) = v,(a/au;)(l
(14) -P)
Introducing eq. (14) into the second term on the right in eq. (13) we find that the latter assumes the form - 1 da I‘ dx [g(O)/W(u)]G(u)A;S(A x
{exp[-it(1
- P)L]}S(A
- a) + 1 dada’ j dx 6(A - u)A;
- a’)G(u’)[g@)(u’)/
W(u’)] .
The first term in eq. (15) cancels the contribution
(15)
from G to the integral
150
R.E. NETI’LETON
J q,g da arising the moment
from the second
term on the right in eq. @a). We are left with
equation
rjj = 1 qjg”‘(u) da + T j dada’i
ds W(a)Kj,[g”),W(a’),aS’,arlk
dx 6(A - a)A:{exp[-it(1 ’
+jdadn’l
x G(a’)[g”)(a’)/W(a’)]
- P)L]}G(A
- a’)
.
In subsection 3.2 we consider to the matrix in eq. (6a) plus (16) give rise to LF), with the and the G-dependent term since most of its contributions in the forces stemming from
(16)
the qj-term in eq. (16) and show that it gives rise contributions to Ly. The remaining terms in eq. first contribution being discussed in subsection 3.1 in section 4. The latter is considered separately, are small compared to terms of the same order g(O).
3. Reciprocity relations with G neglected We proceed
in this section
to regroup
contributions
to the first two terms on
the right in eq. (16) in order to show that these terms can be cast uniquely in a form exhibiting Onsager-Casimir reciprocity, with one application of approximation (4). L1;4’stems from both the terms we are considering in eq. (16) whereas LF comes only from the q,-term. To investigate the latter, we write, with the aid of a partial x-integration [cf. ref. 5, eq. (23)]:
I
qjgco’du =
=
dx (g’“‘/W(u))A;&4
I
da
+ J
A partial
integration
I
dx c
- u)
A,F&(~‘~‘/ W)(a/au,)S(A - a)
k
/ da Ji dx C A,A:(g(O)/W)(aiav,)s(~ k in a- and v-space
causes
- U) .
(17)
the first term on the right to be
proportional to the forces ~S’/&X, and the second to &S’/a~,. We start with the latter and determine its contribution to Ly. There is an additional contribution to L1;4’from the first term on the right in eq. (17) and we shall take that up in subsection 3.2.
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
151
3.1. Symmetriccase Define @(a, q, a) so that g”‘(a)/ W(a) = C exp[-S’(cu, v)] exp( - [ @ + 2 t;(aS~/~ll,)]}
.
(18)
j
Putting this into the second term on the right in eq. (17) and carrying out a partial integration in u-space, we have C
da
I
I
dx exp[-S’-
x c (-r(p!)-‘(
@] C A~(aS’/a~,)uj r
c u,,aS’k~q,)~ S(A - a)
p=l
n
=Ccfdaf x c
dx exp[-S’-
~#S’la~,)
I
@]Vjn,(aS’/aTm)
c (-)p(p!)-@
u,,aS’/avn)p-l~(A - a). n
PL1
The term with p = 0 does not occur because J A,S(A - a) dx is an even function of u. We see this because, if we change the sign of vi in S(A, - vi), we can change the signs of the momenta over which we integrate, leaving the integral invariant. Eq. (19) makes a contribution to Lz which is symmetric with respect to permutation of j and m. Having established this part of Ly, we have to investigate the symmetry of the integral involving Kj,, in eq. (16). The second term on the right in eq. (16) can be written
dxAJS(A - a) i
Fidsfdada’/
[-is(l-
P)L]“(m!)-’
m=O 0
x(1 - P)[g’“‘(a’)/ W(a’)] A$(A
- a’)@S’/aqk) .
(20)
Now AJS(A - a) = iL[AjS(A - a)] - AjiLS(A - a). If this is substituted into eq. (20), the term iLS(A - a) vanishes on integration over a. A typical term of order m in the expansion of the exponential has the form - c 1 dada’ f ds f dx u,S(A - a)(-is)“(m .
!)-‘iL(1 - P) * - * (1 - P)L(l
- P)
k
0
x S(A - a’)A@S’/aT,)C
exp(-S’-
@) c @!)-I( - c
4aS’/an,)
. (21)
R.E. NETTLETON
152
As we have remarked
above,
6(A - a’) commutes
L6(A - a’) = S(A - a’)L - (l/i) c
By repeated
application
left, leaving
a sum of operators
[Aj(a/avi)+
with P. We have
Aj(a/aa~)]6(A
- a’).
in eq. (21) we can move 6(A - a’) to the
of this result
obtained
by substituting
(-l/i)
Cj (A: X3’/aqj +
Aj W/aaj)
for each L in all possible choices of subsets of the L-operators. we call this sum 0 (“‘), the terms in eq. (21 ) for which r 2 1 have the form
da
I I ds
dx (-is)“(m
!)-‘6(A
- u)O’“’
c
Ai(3S’/ar],)C
If
e-@
k
0
This expression is symmetric with respect to the interchange j-n. To consider the terms in eq. (21) for which r = 0, we write them in the form I
2 1 ds 1 dx A;(-is)“(m k
!)-I(1 - P)L . . . L(1 - P)C emS’-@(A’ALaS’/avk
.
0
(23) The product of (1 -P) and L operators is a symmetric product of Hermitian operators and thus itself Hermitian. The x-integral vanishes for m odd, since the integrand is then odd in the particle momenta. Because A and A both commute with P, @(A) commutes with 1 -P and (1- P)L. These circumstances imply that we can interchange Ai and A: in eq. (23) showing that the integral makes a symmetric contribution to Lr. We have now found the principal contributions to LIP’from the second term on the tributions
right
in eq.
(16). In subsection
to Ly from the q,-term,
3.2. Anti-symmetric
3.2, we shall
obtain
additional
con-
as well as to LI;“.
case
We return to eq. (16) which we wish to cast in the form of eq. (6a), thus identifying LI;“. Invoking eq. (lo), we write eq. (16) in the form
d; =
vjg
“‘da +
I
(v, - v,)g”‘G
da.
(24)
153
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
From eq. (11) the second term on the right in eq. (24) is g”‘(vj - qj) da ,
-
which, by the discussion following eq. (11) is of order J exp[-p(u - q)‘](u T)~~u which is small if g is sharply-peaked. It is proportional to third and higher entropy derivatives and to a more negative power of ,U than terms of the same order in F* from the first term of eq. (24). Therefore, we neglect the second term in eq. (24), marking one place where we must continue to invoke the sharp-peakedness part of assumption (4) listed in section 1. This is probably not a drastic approximation, since we shall point out in section 5 that demonstrated failure of the Einstein approximation occurs only when we take into account the coupling to external reservoirs, which is not involved here. Inserting eq. (18) into the remaining term in the right-hand member of eq. (24) and expanding the exponent, we obtain eq. (6a) with LF = K-'c eXp[-S'(a,
T)]
1e-@ W(u)
2 (-)“(m !)-’ mbl
m-l
I
da
vjv,
.
(25)
The term with m = 0 vanishes because W(a) is an even function of vj. We can immediately define L’,3’= -Ly to satisfy the Casimir anti-reciprocity relation. It only remains, then, to recast the first term on the right in eq. (17) in the form of Ly plus a contribution to be added to the second term to yield LI,“‘. Effecting a partial integration in a-space in the first term on the right in eq. (17), we write it in the form
I I da
+
dx c AjAk(aSfla~j)(g(o)l W)a(A - U) = - c Li.z(as/aQ k
P
C ems’2 (aS’/ag) 1 da 1 dx e-“AjAp 2 [ 2 v,(aS’/an,)] m-1 mZ1
X [(-)”
=-
(i
!)-’
+
(-)“-‘((m
- l)!)-‘]S(A - a)
c Lf aS/acu, + C eeS’ c (aS’/ag) P
r
j da e-@W(a)vjvp (c
V,
aS’/aa,)
r
P m-2
(-)“((m - l)!)-‘(m-l>
1).
(26)
R.E. NETTLETON
154
The last term in this expression is symmetric with respect to interchange of j and p. When the last term in eq. (26) is added to the term obtained in eq. (19) and the second term on the right in eq. (16) we find m-2
LI;“’= K-‘C e-S’
I
da
dx vivi e-@ z* (-)“((m
I
x 2 [ vp(as’la~p)(m
-1- 1) -
- l)Y(C
r
v, asm,
>
A~(&S’/aqJ]S(A - a)
P
da
fK-lce-S./
j
ds
c (-is)“(m 1dx vivj Mao
!)-‘6(A - a)O’“’
0
x
2 Al as/aq,
em0C rZl
k
(r!,l(- 2
V,
asmJ
w
ds W(a)K,, em@+ B(G) .
+K-lCe-S~dada’]
(27)
n
The first term in this expression clearly exhibits Onsager symmetry, since the integrand is proportional to vivj, while the symmetry of the second and third terms has been demonstrated in subsection 3.1. If we examine the structure of Ly and Lf’ we see that these expressions, to the order calculated here, are unique. For in Cj Ly aS’/aqj the integrand is proportional to a product of factors v, &S/aq,, and it does not matter which of these we pull out in identifying Ly. Similarly in Zj Ly aS’/arl, each term is a force FT is extracted to define Lr, SUIII K-’ z A,...,F;F; a . . F*p, and whichever the result is the same so long as F: is extracted to define Lz’ = Lr’. Since Ai,,,.,_,,is not symmetric with respect to interchange of any arbitrarily-chosen pair of-its indices, we do not have a7jilaFT= &jJaF;, and so the Onsager symmetry demonstrated here is not of the type proved not to exist in general.
4. Contributions
from G
We now return to the third term in the right-hand member of eq. (16) in order to show that, if there are terms which it contributes to Lf) whose Onsager symmetry cannot readily be established, then these terms are small in comparison with terms of the same order in the forces F* derived in section 3.
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
155
To this end, we observe that G can be expanded in Hermite polynomials, and products thereof, which are functions of the sets {ai - CX~} and {2ri- vi}. Thus G = G, + G,, where G, depends only on Hermite functions of the set {ai - cri}, and the u-dependence resides entirely in G, which may also depend on the u-variables. We shall proceed to show that Onsager symmetry can be established for terms arising from G,, while most of the G, terms, for which the demonstration is more difficult, should be negligible. So, in fact, are the G,-contributions, so that for most systems G can be neglected altogether in calculating the first-moment equations. A typical term in the Go-dependence of eq. (16) can be written 1 dadn’ j dx 6(A - a)(-it>“(n!)-‘A>(1 - P)L * * - (1 - P) c ASS(A - a’) I
x G,(a’)(a/au:)[g’“‘(u’)lW(u’)]
,
where we have set (1 - P)LS(A - a) = - (1 - P)At(Wav)i3(A - a) and have integrated by parts with respect to o. This expression can be rewritten in the form
-jdudu’j
dx S(A - u)AJ[(-it)“(n!)-*I(1 - P)L * * . L(1 - P)
x C Aj(aS’/av,) 6(A - u’)G,(u’>C ems’-@mzo ( - 2 vi tN/l%~,,)~(m !)-’ . I
w (28)
When m = 0, we can carry out the a- and a’-integrations and element of a product of Hermitian operators which is non-zero even number of L’s. Symmetry with respect to the interchange in the discussion of eq. (23). When m 2 1, the discussion is the
obtain a matrix if there are an j-r follows as same as for eq.
(20 The case n = 0 in eq. (28) must be discussed separately. This case is the same as eq. (17) with a factor G, introduced into both integrals on the right. By the discussion following eq. (24), the Hermite polynomials in G, should be H, or higher to assure that J (a - a)G,g”’ da = 0 to the order of terms systematically neglected in subsection 3.2. Within this approximation, the first of the two integrals in J qjG&‘O’du vanishes while the second leads to an expression which is eq. (19) with a factor G, inserted. As in that case, we get an Onsagersymmetric contribution to I,!. To discuss the case of G,, we must consider the solution of eq. (5a). As illustrated in ref. 16, it is convenient to express the solution in the form
156
R.E. NE’ITLETON
g = where
$O’[1 + G] )
(29)
2”’ is Gaussian
which,
for simplicity,
we take to be an exponential
of a
sum of squares, and G is a sum of Hermite polynomials and products thereof. We want to use eq. (5a) to investigate the magnitudes of those terms in G which are proportional to powers and products of the forces X$‘/acy and as’/&) and
which
introduce
also contribute a streamlined
g-(‘)- exp[-p(a G=
.. +
where
-a)*-
. wJ,(P”/*(~
j..~(v -
-
B,,H,(pl’*(a
a, u can
to G. To estimate
notation
-
denote
any
pair
it is sufficient
q)*] ,
4)~W2(~
a))H2(p1’*(u
magnitudes
to
and set (304 -
-
7))
+
7))
+
ai, vi, and
B2JJ2W2(u
-
77))
B,H,(/_P2(u
-
7)).
H2 can
represent
. . )
W’b)
a product
of
polynomials HI which are functions e.g. of two variables U, and v2. The coefficients p are proportional to second derivatives of S’ which are large under the assumption that g is sharply-peaked. Since the p’s are all large, they are all taken to be of the same order and replaced by a single symbol. In the notation of eqs. (30a, b), eq. (5a) assumes the form, to leading order
in
each term,
agiat= -(aiau)(qg) - v(aglaa) + K(a2gW)
(31)
Substituting
eqs. (30a, b) and comparing coefficients a)), H,(d’*(u - v)), and H2(p1’*(u - TJ)), we find
4, - (-q/48Kh, B,
-
(q/48K2p
)ti
B,, - (-q2/48K2p ,
of terms in Ho, H,(p”*(a
-
1’2)?j , (32)
B, - (- 1/48Kp “*)?j
From this we find that contributions from G to eq. (16) of order F*” will, with the possible exception of B,,, be of the order of a negative power of p multiplying a term of order F*” in $ and thus negligible if p is sufficiently large. In particular, this holds for second-order terms of the type B,*v?j = we conclude that the coefficient of pU-1B12F*+ By similar arguments, H&“*(u - cz)) in G, should be -B,,, so that those terms in G should also make a negligible contribution to the phenomenological coefficients. However, we have been able to demonstrate Onsager symmetry without neglecting them. It remains to consider the B,, terms which give rise to contributions to Ly whose
Onsager
symmetry
can be demonstrated.
By partial
integration,
as in eq.
ONSAGER RECIPROCITY IN THJ3 NONLINEAR REGIME
1.57
(17) we find that _/ da j dx S(A - u)A~G,(gco)/W) = c j da _/ dx ujt.+w(a)-‘(alaa,)s(A , dx Aj(x)Aj(x)
xGsco) + c r
W-‘(a/at@(A
- u)Gvg(o).
- a)
(33)
When G, = B,,H,(~ “*(a - cy))Hr((p “*(zJ- 7)) we find after partial integration that the first integral on the right in eq. (33) becomes
2 j da ujur(u -
-2~4 +
6, ,,i
(
I
c c W%C ees I da e-@vjtp,(aS’/a~,) u
x
v)g”’ = -2/.~B,,77~ da (U - v)2g’o’(1- S,.)
,
-
C
c
rtj
V,
aslav,
w
(p!)-’
pr1
)p-1(~-~)W(o)+2~~jB,,CeS_CldUqU,e-~W~, r
(34) where S,, = 1 if n = nj, and S,, = 0 otherwise. The second term on the right in eq. (34) exhibits Onsager symmetry with respect to the interchange j-r. To consider the first and third terms, we invert the expansion Fy = aslavi
= C sijvj + C ~~~~~~~~~~ + -.i, k,I i
(354
to get qi = c s,‘F; j
-
c
j,1.m.n
silsirmn2
s&,‘,s,;F;
F;F; + - . - .
W)
P. 9. w
The terms in eqs. (35b) can be grouped to exhibit Onsager symmetry with respect to interchange of an index pair. The B,,-contribution to the second integral on the right in eq. (33) assumes the form dx vjA;[-2p ‘nH& “*(a - a)) + Hl(,u “*(a - a))& x (/.L’~(v - 7)) S’/an,]g’“‘/ W = -2B,,lc.lR c 1 da Vjs,Hl(~“*(u - a))g”) I + 4,
c da u,4,(as’iarl,)~l(~u.‘“(u I I
- ~M4(P
‘%
- rlNg@)
158
R.E. NETTLETON
Xf4(P1’2(U - 77W0)lW - wqs(alau,)[H,(p”12(u - 7&(O)/WI}. This
expression
exhibits
j-r
symmetry.
Similar
can be applied
to
terms in G of the form B,,,H,(~1’2(a - (Y))H,(~“‘(u - n)) for n 2 1. We can extend the foregoing reasoning to higher order in the expansion
of
the third term on the right in eq. (16) in powers partial integration [cf. eq. (21)]
j dada’ j dx 6(A - a)Aj(n!)-‘[-it(1
=-
jdudu’j
reasoning
(36)
of [-it(l
- P)L].
We have after
- P)L]“S(A - u’)G,(a’)[g’“‘(a’)/W(u’)]
dx u,8(A - a)(n!)-‘il[-it(1
- P)L]”
x S(A - u’)G,(u’)[g’“)(a’)/ W(u’)]
dxAjG(A - a)(n!)-‘(-it)“[(l-
= -jdadu’j
x
c
A;@/&$)[ G,g”‘/ W]
For illustration, G,. For this term
C
P&L]“-‘(1 - P)6(A - a’)
we consider
A@~~u;)[G&?
Wa) the contribution
W] = B,,H,(~(~
to eq. (37a) of the B,,-term
- a))
x ( ~/_L”/~A~ - c A$N/~T],)H~(~“~(zJ - v))}g”‘iw. I If we move
the factor
in
6(A - a’) to the left in eq. (37a) we obtain
(37b)
from
the
At-term first a constant of order K multiplying 4. This will contribute a first-order term in F* with coefficient
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
159
from q - CT@‘). Th’is is an approximation analogous to approximation (2) which here is applied to the constant coefficients. Also from the At-term, we get contributions ITT’(F*~) by replacing one of the L-operators by Xi Ai aS’/arl,. We can argue similarly that these are smaller by a factor O(A) than secondorder terms arising from the g(O)-approximation to G. It should also be noted that, because of the factor H,(pl’(u - a)), we are taking an u-derivative, in addition to multiplying by 0(A). Similar reasoning applies to the second term in eq. (37b). Once the factor S(A - a’) has been moved to the left, we can expand g”‘/W in powers of (- Z, o, N/an,,,) and explicitly demonstrate Onsager reciprocity for the higher terms in the expansion, exactly as in eq. (22) since again we have a factor vjv,,, symmetric with respect to i-w. Since g(O) is Gaussian and g(O) only approximately so, we have g(O)= g”‘[l+ G’], where G’ is a polynomial in a - CTand v - n with coefficients which are third- and higher-derivatives of S(a, 7) and powers and products thereof. If G’ gives rise to an integral B’J n(v - n)ng(‘)du - B’,u-(“‘~+~)F*,this is smaller than first-order terms in 7j, provided B’ is not of order p, which we assume true of third and higher entropy derivatives. Similar considerations apply in higher order. Unfortunately, while we have been able to demonstrate Onsager symmetry for most of the contributions of G,, we cannot without much greater effort show that such symmetry is unique. Since G is proportional to 7j, this dependence gives part of the dependence of Ly on F*, and this dependence on F* introduced through the factor + may have symmetry independent of any demonstrated above. However, the argument following eq. (37a) above implies that all the contributions from G!, which are proportional to li are, for a given order in F*, or higher order in A than similar contributions from eq. (21). We can also note that K - (PT))’ where 7 is the relaxation time for n which is short, and ~1 is large. K may, therefore, not be of the order of a negative power of Jo which makes q/K - C7(A2).This could be advanced to justify neglecting contributions to the moment equations which are proportional to q/K as higher order in A than contributions from g (‘). This is like approximation (2) in the earlier linear theories3*5). Then it is probably not a poor approximation to neglect G in the first moment equations for ~5 and 7j, giving coefficients Lf’ which can be determined uniquely as shown in section 3.
5. Case of heat conduction
in condensed matter
An example of the application of extended non-equilibrium thermodynamics, much discussed’3,‘5,17) in the literature, involves the CattaneoVernotte equation’8*‘9). F or an anisotropic heat conductor, with Ji (i = 1,2,3)
160
R.E. NE-ITLETON
the components
of heat flow, this equation has the form
4 = 2 1’$_5.+ c N&T. i
(39)
I
If the system to be described is a small cube immersed in a much larger liquid or solid phase, the relation of Ji and the symmetry of Mi,. should be described by the Zwanzig formalism2S3) discussed in preceding sections, since J is a fast variable”) which relaxes in a time short in comparison with the time for appreciable heat diffusion into the system. The term in VT arises from coupling to the surrounding medium regarded as a heat bath, and the Onsager-Casimir properties of Nij can be inferred only phenomenologically within the framework of the present paper. We can set J = (A,(x)), where A,(x) is a dynamical function of the classical phase, x, of the system”). Then the equations which correspond, in this case, to eqs. (17) and (5b) above are I
&=-j-dszj-d 0
’ d ” W(“)Kj!f(u,
u’, s)(alaV;)[g(U’)l
W(U’)] ,
k
W(U)Kjk(U, u’, s) =
6(A - u)Aa{exp[-is(1
- P)L]}(l
-
P)A,G(A
(404
- u’) dx, Gob)
where g(v)du is the probability that A, lies within du centred at 2).The term q, in the first term on the right in eq. (16) vanishes16). If we are discussing a crystal, the x-integration in eq. (28b) is limited to configurations consistent with the assumed structure. We replace g + g(O)in eq. (40a) by reasoning analogous to that in section 4 above. The solution for G has been obtained in ref. 16, eq. (16) and, when expressed in terms of F, is proportional to a negative power of p. Then eq. (40a) assumes the form
j/ =
C ‘jk(aS/aJk),
(41)
k
where A, = Akj follows by reasoning analogous to that in subsection 3.1 above. We have
J, =
I
uig’o’(l + G) du =
v,g(‘)du
(42)
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
if g”’ is sharply-peaked,
4=T
161
from the discussion of eq. (24). We then have
c A$(as/q),
where (43b)
The only argument that we have for inserting eq. (43b) into the last term on the right in eq. (39) is the phenomenological one that such a Casimir antireciprocity relation must hold if the irreversible rate of entropy production is to be positive definite. The term linear in VT in eq. (39) implies16) that the equation for g must be augmented by a term linear in VT. When this term is not zero, g, and correlations calculated from it, depart significantly from the Einstein function -exp S’S/K 16). Such departures have no bearing on the sharp-peakedness of g(O)calculated for a closed system, and therefore demonstrated failures of the Einstein function’6’21) are not arguments against neglect of G in eq. (43b).
6. Summary
and conclusions
The calculations described in previous sections have had two objectives. (A) We can eliminate assumption (3) listed in section 1 and give a less ad hoc character to assumption (4). (B) We can provide an answer to speculations to the effect that, because certain proposed generalisations of Onsager reciprocity to the non-linear case have been shown’0-‘3) not to be valid, then no such generalisation can be found. In the original work of Zwanzig, non-linear reciprocity relations were derived for symmetric a-variables, subject to assumptions (1) and (2). The extended non-equilibrium thermodynamics has since been developed in recognition of the fact that, when assumption (1) is invalid, so that phenomenological coefficients depend explicitly on time, we are usually concerned with times short compared with the relaxation times of the a-variables. Under these circumstances, there are T-variables, where 7 = Cw,whose introduction as independent state variables provides for the inertial effects which assert themselves over short times. There should be a wide range of phenomena which are describable using phenomenological coefficients not explicitly time-dependent as long as a sufficient number of independent state variables is employed. Assumption (1) should be reasonable whenever this is the case.
162
R.E. NETnETON
Although
the contributions
G,, have been
shown
from g(O)and some of those from g”‘G, e.g. from
to obey unique
reciprocity
relations
to all orders
forces F and F*, this does not imply that we can, in practice, third order.
Unless
we can neglect
G,, it is difficult
in the
go much beyond
to show that Lc’ is unique.
To neglect G,, we have proposed to drop, as in assumption (2) contributions to Lr which are of higher order in A than similar contributions of the same order in F and F*. However,
the terms
dropped
may be of the order
of terms
of
fourth or higher-order in the forces. Thus we should be able to carry Onsager reciprocity to terms of second- or third-order in the forces, but it is likely to prove impractical to go beyond that. Most phenomenological models, in any case, have been linear in 77, although not necessarily in (Y- LYE.For example, the Cattaneo-Vernotte equation (39)“*“) is generally used in a form linear in heat flow J, and the Maxwell stress-relaxation equation**) is usually linear in the pressure tensor. Landau and Lifshitz6) have derived a Lagrangian formalism, equivalent to ref. 5, where linearity is preserved in the n-variables but not in the set {(Y~- ajo}. There are, accordingly, few examples which show experimentally that it is necessary to go to terms non-linear in the F* forces. If we do get far from equilibrium so that higher-order terms must be included, these contributions are likely to be of the order of effects neglected in the Markovian approximation, and an integral equation with memory kernel will be more
appropriate.
In the formalism of the present paper, the (Y- and n-variables have all been “fast” variables with relaxation times short compared with the time required for appreciable hydrodynamic flow of energy or particles into or out of the system, when the latter is immersed in a much larger fluid. The dynamics is effectively that of a closed system, with a time scale sufficiently long so that explicit t-dependence of K, can be neglected, but short compared with recurrence times. If there exist macroscopic temperature and velocity gradients, additional terms, depending on these gradients, must be added to eq. (5a), as explained in section 5 and exemplified in eq. (39). The distribution g then acquires terms dependent on these gradients, in addition to the terms in g”’ and G16). If “slow” variables such as the density and temperature are added to the set {a;}, a further modification is required, embodied in the projection operator technique of Grabert4) which is consistent with a canonical equilibrium distribution, instead of the microcanonical distribution appropriate to a closed system. These modifications permit us to include correlations in the fluctuations of slow and fast variables without materially affecting the phenomenology of the latter23). One achievement of the foregoing calculations is to elucidate the question of utility of the Einstein function. g(O) is a generalised Einstein approximation to which terms of third- and higher-order in the Taylor expansion of S about S(cr, r)) have been added. The arguments of section 4 indicate that g(O)should
ONSAGER
RECIPROCITY
IN THE
NONLINEAR
REGIME
163
in most cases be an adequate approximation to g for purposes of calculating the first moments of eq. @a), although not in general for calculating correlation functions.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)
S.R. de Groot, Thermodynamics of Irreversible Processes (North-Holland, Amsterdam, 1951). R. Zwanzig, J. Chem. Phys. 33 (1960) 1338. R. Zwanzig, Phys. Rev. 124 (1961) 983. H. Grabert, Projection Operator Techniques in Non-Equilibrium Statistical Mechanics (Springer, Berlin, 1982). R.E. Nettleton, J. Chem. Phys. 48 (1964) 112. L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon, London, 1958). D.G. Sannikov, Soviet Phys. JETP 14 (1962) 98. D.G. Sannikov, Soviet Phys. Solid State 4 (1962) 1187. R.E. Nettleton, Phys. Lett. 24A (1967) 231. D. Edelen, Arch. Rat. Mech. Anal. 51 (1973) 218. D. Edelen, Int. J. Eng. Sci. 12 (1974) 121. D. Edelen, J. Non-Eq. Thermodyn. 2 (1977) 205. G. Lebon, Int. J. Eng. Sci. 18 (1980) 727. D. Jou and J.M. Rubi, Phys. Lett. 72A (1979) 78. D. Jou, J.E. Llebot and J. Casas-Vasquez, Physica 109A (1981) 208. R.E. Nettleton, J. Phys. A: Math. Gen. 17 (1984) 2150. R.E. Nettleton, Phys. Fluids 3 (1960) 216. P. Vernotte, Compt. Rend. Acad. Sci. Paris 247 (1958) 3154. C. Cattaneo, Compt. Rend. Acad. Sci. Paris 247 (1958) 431. S.A. Rice and P. Gray, The Statistical Mechanics of Simple Liquids (Interscience, New York, 1%5), p. 382. D. Jou and T. Careta, J. Phys. A: Math. Gen. 15 (1982) 3195. J.C. Maxwell, Trans. Roy. Sot. (London) 157 (1867) 99. R.E. Nettleton, J. Chem. Phys. 81 (1984) 2458.
ONSAGER
RRCR’ROCITY
IN THE NONLINEAR
REGIME
R.E. NEmTON Department of Physics, University of the Witwatersrand, Johannesburg, South Africa
Received 28 September 1984
A system is assumed, in the framework of extended non-equilibrium thermodynamics, to be described by state variables ai which are ensemble averages of dynamical functions, Ai, and their time-derivatives ui = o;i. A Fokker-Planck type equation is derived by the Zwanzig projection operator technique for g(a. o), the amplitude of the probability that A = a and A = a. The first moments of this equation give the phenomenological equations which can be cast in OnsagerCasimir canonical form, with forces and coefficients exhibiting reciprocity and non-linear in both (Iand n-variables. It is necessary to assume that g is sharply peaked in equilibrium and that a Markovian approximation holds for the particular sets of variables and time-scale chosen. The arguments are illustrated by applying them to heat conduction in a liquid or solid.
1. Introduction
In the traditional approach’) to non-equilibrium thermodynamics, we specify a non-equilibrium state by a finite set {(Ye}of time-dependent variables, n in number, and introduce an entropy S({cu,}).If the rates, bi, are cast in the form
then Lii = Lji in zero magnetic field provided the [Y~are all symmetric or all anti-symmetric under time-reversal, and the equations are linear in the set {(Y~ - aiO}, where the {ai,,} are the values assumed by the variables in equilibrium. Under these circumstances Lij = Lt({aio}). The conclusion that reciprocity relations are not limited to the linearized form of eq. (1) is reached in a derivation of the phenomenological equations from the classical Liouville equation given by Zwanzig2”) and more recently extended by Grabert4). In this approach, the ‘Ye are averages (Ai), in a time-dependent ensemble, of dynamical functions Ai which depend on the phase x of the system. By operating with a projection operator, one obtains from the Liouville equation a Fokker-Planck-type equation obeyed by the 037%4371/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
144
R.E. NETnETON
function values
g({a,})
which
is the
amplitude
a,. Eqs. (1) are the first moments
to hold
with
L,
and
W&X,
for the
probability
of the equation
non-linear
functions
that
A, assume
for g. They are shown
of {oi - (Y;,,} and
L,,= L,
provided: 1) The equations
memory
kernel
decays
fast enough
can be used in Markovian
2) Contributions
o(A’)
so that
the phenomenological
approximation.
to eqs. (1) are neglected,
where
A, = iLA,,
and L is
the self-adjoint Liouville operator. An extension of the Zwanzig derivation has been made5) for the case where the set {Ai}is assumed to be even under time-reversal and is augmented by the set {A,}. Under these circumstances, to the set of state variables {a,} we now add the set {vi = bi}. This extension was made5) to serve as a basis for what is “extended non-equilibrium thermodynamics”, in which the now called phenomenological equations have inertial terms, 7ji. It was possible to demonstrate Onsager-Casimir reciprocity in this extension provided assumptions (1) and (2) were retained, and we make the additional restrictions: 3) Only terms linear in the v-variables are kept. 4) g is sharply peaked and given by the Einstein function, g - exp ~*S/K, where 6*S represents quadratic terms in the expansion of S(a, u) about S(cy, 7). S(a, v) is the entropy when A = a,A = u. The possible non-linearity in {(Y~- (Y,} is retained. Assumption equivalent to taking dissipative forces linear in velocity. This restricted the phenomenological
equations
derive an equivalent Lagrangian of-motion of continuous media.
has been formalism
used
by Landau
used7,*) in obtaining
and
(3) is form of
Lifshitz6)
to
the equations-
Because many mechanical models have dissipative forces linear in velocity, the references cited above3X5) establish reciprocity in a variety of nonlinear problems of interest. It has been shown’) that if the equations for the Crj are inferred from an intuitive or phenomenological model, they can be cast uniquely in Onsager-Casimir canonical form, so long as they are linear in the n-variables,
but not necessarily
in the {(Y;- Q},
and so reciprocity
can be of
practical use in such problems. For example, some coefficients not calculable from a simple molecular model can be expressed in terms of others more accessible to model calculations. Nevertheless, more general cases may be encountered in a theory of states far from equilibrium, and so we should like to remove as many as possible of the restrictions (l)-(4) listed above. Various generalisations of Onsager reciprocal relations to the nonlinear regime have been proposed and shown to fai11@‘2,‘3 ), and therefore we desire to ascertain whether a valid generalisation does indeed exist. The answer to this problem lies in the equation derived5) for the fluctuation distribution function g(a, v), defined by
145
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
g(U,v> = f(x) dX n S(Ai I
(2)
- ai)S(Ai - pi) )
with 6 a Dirac delta and f(x) the solution of the classical Liouville equation i aflat = Lf.
(3)
g IIi da,dv, is the probability that A, has value between ai and Ui+ dUi and Ai between ui and ui + dv, (all i). Use of eq. (3) implies that a closed system is assumed or, alternatively, we postulate that the variables we are discussing are “fast” and relax over a time short compared to the time required for appreciable amounts of energy or particles to diffuse into or out of the system. This means that the x-integration in eq. (2) is taken over an energy shell. As we shall point out in section 5, the phenomenological equations may have additional terms arising from coupling to the surroundings, and the OnsagerCasimir symmetry of this coupling can be inferred in general only by phenomenological reasoning. To obtain an equation for g, we define a projection operator, P, such that for arbitrary B(x) PB(x) = [W(A)]-’
1 B(x’)6(A(x)
W(A) = j S(A(x)
- A(x’)) dx' ,
S(A(x)-
A(x’)) = fl S(A,(x)
- A(x’)) dx' ,
W)
- A,(x’))S(&x)-
Ai(
.
(4c)
With this definition, we have
g(4 v) = WG Mfl,4E&&” *
(44
The equation derived for g by operating with P on the Liouville equation is [ref. 5, eq. (S)], with the definition A> = (iL)‘Aj:
(a’at)g(u9t, = - i: {(alauj)[ujg(U,t)] + (a/auj)[qjg(U,
t)])
j=l
+ ?+ i j=l k=l
j ds J du’ (a/auj)[ 0
W(u)Kjk(U,
U’, S)](a/aV;)[g(U’~
t - S) W(U’)-‘1 (54
)
146
R.E. NETILETON
K,,(a, a’, s) = (Aj{exp[ - is(1 - P)L]}(l (G; u)=PGI,=, qj=(A;;
, a).
- P)AiG(A - a’); a),
(5b) PC) (54
In these equations we have written g(u) for g({u,}, {2ri})and da for Iii da,du,. Also “A = a” means both Ai = a, and Ai = ui (all i). We shall use this abbreviated notation’extensively in what follows. The phenomenological equations obeyed by the variables LY~ and ni = cUiare obtained from the first moments of eq. @a). In the following sections, we want to show that these moment equations can be cast in the form ki =
c LFF; ,
(64
where F, = aSIaa;.and FT = aS/avj for S the entropy. This is to be done in such a way that the Lf’ are, in general, nonlinear in the n-variables and in the set {ai - a,}, while Lz’ = LF’ and Li,” = -L$’ in zero magnetic field. We can do this without approximation (3) and with a weakened version of approximation (4). To show uniqueness of LI;“, we shall neglect some contributions to the constants which are of higher order in A. This is analogous to a weakened form of approximation (2). In subsection 2.1 we propose an ansatz for g consistent with the requirement that it reduce to the Einstein function in equilibrium. In subsection 2.2, we derive the equations for the first moments, {ai} and {vi}, of g by taking moments of eq. (5a). The rates Criand 7ji depend on g(O),a generalized Einstein function, and on an additive correction g”‘G assumed in the ansatz for g. Section 3 regroups the contributions from g(O)to the moment equations so that they assume the form of eqs. (6a, b), with 3.1 demonstrating the symmetry of the principal contributions to LI;” and 3.2 the antisymmetry relation. Section 4 takes up the contributions of the additive correction g”‘G. These additional contributions are cast in Onsager-symmetric form, with neglect of terms involving fourth-order fluctuations and third and higher-order derivatives of the entropy and also terms smaller by a power of p, the second derivative of S, or of A than other terms of the same order in F*. Such terms should be small if g is sharply-peaked. The calculations are exemplified in section 5 for the special case where the A-functions are the components of the heat flux, and Onsager symmetry is demonstrated for that case. A discussion is given in section 6, which mentions extensions to include cases of coupling to external
147
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
resevoirs and the inclusion of “slow” variables such as temperature energy in an open system.
or internal
2. First moment of the fluctuation distribution We proceed here to multiply eq. (5a) by uj and vi and integrate over the space of a- and v-variables. The right-hand side depends on g, and so it falls into two parts, stemming respectively from the additive terms g(O)and g”‘G in the ansatz of subsection 2.1. The explicit form of these contributions, which will be maneuvered into canonical form in sections 3 and 4, will be written down in subsection 2.2. 2.1. Ansatz for g In an equilibrium energy-shell ensemble, g(a) is proportional to W(a) = where we use the single argument “a” to represent both the aand v-variables. This proportionality holds since W(a) is the number of configurations consistent with A = a and A = v. This implies g(u) is given by the Einstein function in equilibrium, since w(a) = exp(S(u)/K). If we expand S about the values c+, and vi0 = 0 which maximise it in equilibrium, we have a sharp maximum, with W(A)(A,o,A,V
s = So(ao)+ szs
(7)
)
where the first-order terms vanish for expansion about a maximum. Because of the sharpness of the maximum, the higher terms are neglected in first approximation. In a non-equilibrium state characterised by (Y# (Ye, n f 0, a function having the desirable property of being sharply-peaked and reducing to the Einstein function in equilibrium is obtained if we replace a - (Ye-+a - (Y and v + v - n in S’S and set g - exp(S’S/K) following this replacement. This assumption has been used’4T’5)an d sh own’“) to be the first approximation to the solution of eq. (5a) in the case where g describes fluctuations in the heat flow. Another generalisation of the Einstein distribution, which reduces to it in equilibrium and agrees with exp(S’S/K) to second order, is g”‘(a) = C(a,
7) exp{S’(a) - S’((Y,7) -
where S’ =
S/K
2 [(Ui - ai)(3S'laai) + (Vi - 77i)(aS’la77i)l~
and the single argument
9
(8)
“a” denotes both a and v. C(a, 7)
148
R.E. NETTLETON
normalises expansion
g”’ to unity. Eq. (8) merely ads to S’S the higher-order terms in the of S(a, v) about S((Y, v). A suitable ansatz for g is then
g(a) = s’“‘(4P+ G(a)] , where,
I
since g(O)is normalized
(9 to unity,
g"'G da = 0,
with, as before,
da = llj du,dv,.
The function G(u) is to be determined to satisfy eq. (5a) subject to eq. (10). G must vanish in equilibrium, and so is a sum of powers and products of the forces {E;;} and {FT}, with coefficients which are polynomials in a - LYand v - 7. An inconsistency arises if we assume G = 0. To see this, note that
I
(v-n)gdu=O.
(11)
Since S(u) is a sum of even powers of v, its expansion will have terms of the type ~(v - r])3 which means that g(O)will not satisfy eq. (11). The contribution of G to the first moments of eq. (5a) involves moments of g(O)which are small if the latter is sharply-peaked. Therefore, we shall discuss the contributions to the Lr’ coefficients in section 4 as a correction believed
the main contributions
derived
in section
G-dependent to what are
3.
2.2. Moment equations Multiply
d; =
eq. (5a) by a, and integrate
I
over a and v. We find immediately
v,g da = nj.
that
(12)
This result shows that bj depends on variables odd with respect to time reversal, so that it can be expanded in forces from the set {as/an,}. We shall show in subsection 3.2 that the contribution from g”’ to the integral in eq. (12) yield a matrix Ly (cf. eq. (6a)) which satisfies the Casimir antireciprocity relation. If we multiply
eq. (5a) by v, and integrate,
the last term on the right yields
149
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
-C]dudu’j
ds W(u)Kjk(u, a’, s)(a/au;)[gco’/ W(u')][l
k
+
G(u’)]
0
=x
j ds
jdudu’ k
W(a)Kjk[g'"'/W(a')]as'/arlk
0
+F jduduj
ds w(a)G(a’)(aiaU;)Kjk(a,
d, &fV
w(d)]
(13)
0
to v; and have integrated by parts the term proportional From eq. (5b) for Kjk we find that
to G.
dx 6(A - a)Aj{exp[ - is(1 - P)LJ}(l - P) 2 A:(Wv;)S(A
=
- a’)
k
=
I
dx 6(A - u)Aj{exp[-is(l-
x C [A:(alav;)
P)L]}(l-
P)
- U)
+ A,(alaa;)]s(A
k
=I
dx S(A - a)A){exp[-is(1
- P)L]}(l
= (a/as) 1 dx 6(A - u)A:{exp[-is(lHere x
we have
added
a term
S(A - a’) = 0, because (1 - P)S(A
- P)iLG(A - a’)
P)L]}S(A
(1 - P)A,(a/a+S(A - a’) = 0.
- a’). - a’) = v,(a/au;)(l
(14) -P)
Introducing eq. (14) into the second term on the right in eq. (13) we find that the latter assumes the form - 1 da I‘ dx [g(O)/W(u)]G(u)A;S(A x
{exp[-it(1
- P)L]}S(A
- a) + 1 dada’ j dx 6(A - u)A;
- a’)G(u’)[g@)(u’)/
W(u’)] .
The first term in eq. (15) cancels the contribution
(15)
from G to the integral
150
R.E. NETI’LETON
J q,g da arising the moment
from the second
term on the right in eq. @a). We are left with
equation
rjj = 1 qjg”‘(u) da + T j dada’i
ds W(a)Kj,[g”),W(a’),aS’,arlk
dx 6(A - a)A:{exp[-it(1 ’
+jdadn’l
x G(a’)[g”)(a’)/W(a’)]
- P)L]}G(A
- a’)
.
In subsection 3.2 we consider to the matrix in eq. (6a) plus (16) give rise to LF), with the and the G-dependent term since most of its contributions in the forces stemming from
(16)
the qj-term in eq. (16) and show that it gives rise contributions to Ly. The remaining terms in eq. first contribution being discussed in subsection 3.1 in section 4. The latter is considered separately, are small compared to terms of the same order g(O).
3. Reciprocity relations with G neglected We proceed
in this section
to regroup
contributions
to the first two terms on
the right in eq. (16) in order to show that these terms can be cast uniquely in a form exhibiting Onsager-Casimir reciprocity, with one application of approximation (4). L1;4’stems from both the terms we are considering in eq. (16) whereas LF comes only from the q,-term. To investigate the latter, we write, with the aid of a partial x-integration [cf. ref. 5, eq. (23)]:
I
qjgco’du =
=
dx (g’“‘/W(u))A;&4
I
da
+ J
A partial
integration
I
dx c
- u)
A,F&(~‘~‘/ W)(a/au,)S(A - a)
k
/ da Ji dx C A,A:(g(O)/W)(aiav,)s(~ k in a- and v-space
causes
- U) .
(17)
the first term on the right to be
proportional to the forces ~S’/&X, and the second to &S’/a~,. We start with the latter and determine its contribution to Ly. There is an additional contribution to L1;4’from the first term on the right in eq. (17) and we shall take that up in subsection 3.2.
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
151
3.1. Symmetriccase Define @(a, q, a) so that g”‘(a)/ W(a) = C exp[-S’(cu, v)] exp( - [ @ + 2 t;(aS~/~ll,)]}
.
(18)
j
Putting this into the second term on the right in eq. (17) and carrying out a partial integration in u-space, we have C
da
I
I
dx exp[-S’-
x c (-r(p!)-‘(
@] C A~(aS’/a~,)uj r
c u,,aS’k~q,)~ S(A - a)
p=l
n
=Ccfdaf x c
dx exp[-S’-
~#S’la~,)
I
@]Vjn,(aS’/aTm)
c (-)p(p!)-@
u,,aS’/avn)p-l~(A - a). n
PL1
The term with p = 0 does not occur because J A,S(A - a) dx is an even function of u. We see this because, if we change the sign of vi in S(A, - vi), we can change the signs of the momenta over which we integrate, leaving the integral invariant. Eq. (19) makes a contribution to Lz which is symmetric with respect to permutation of j and m. Having established this part of Ly, we have to investigate the symmetry of the integral involving Kj,, in eq. (16). The second term on the right in eq. (16) can be written
dxAJS(A - a) i
Fidsfdada’/
[-is(l-
P)L]“(m!)-’
m=O 0
x(1 - P)[g’“‘(a’)/ W(a’)] A$(A
- a’)@S’/aqk) .
(20)
Now AJS(A - a) = iL[AjS(A - a)] - AjiLS(A - a). If this is substituted into eq. (20), the term iLS(A - a) vanishes on integration over a. A typical term of order m in the expansion of the exponential has the form - c 1 dada’ f ds f dx u,S(A - a)(-is)“(m .
!)-‘iL(1 - P) * - * (1 - P)L(l
- P)
k
0
x S(A - a’)A@S’/aT,)C
exp(-S’-
@) c @!)-I( - c
4aS’/an,)
. (21)
R.E. NETTLETON
152
As we have remarked
above,
6(A - a’) commutes
L6(A - a’) = S(A - a’)L - (l/i) c
By repeated
application
left, leaving
a sum of operators
[Aj(a/avi)+
with P. We have
Aj(a/aa~)]6(A
- a’).
in eq. (21) we can move 6(A - a’) to the
of this result
obtained
by substituting
(-l/i)
Cj (A: X3’/aqj +
Aj W/aaj)
for each L in all possible choices of subsets of the L-operators. we call this sum 0 (“‘), the terms in eq. (21 ) for which r 2 1 have the form
da
I I ds
dx (-is)“(m
!)-‘6(A
- u)O’“’
c
Ai(3S’/ar],)C
If
e-@
k
0
This expression is symmetric with respect to the interchange j-n. To consider the terms in eq. (21) for which r = 0, we write them in the form I
2 1 ds 1 dx A;(-is)“(m k
!)-I(1 - P)L . . . L(1 - P)C emS’-@(A’ALaS’/avk
.
0
(23) The product of (1 -P) and L operators is a symmetric product of Hermitian operators and thus itself Hermitian. The x-integral vanishes for m odd, since the integrand is then odd in the particle momenta. Because A and A both commute with P, @(A) commutes with 1 -P and (1- P)L. These circumstances imply that we can interchange Ai and A: in eq. (23) showing that the integral makes a symmetric contribution to Lr. We have now found the principal contributions to LIP’from the second term on the tributions
right
in eq.
(16). In subsection
to Ly from the q,-term,
3.2. Anti-symmetric
3.2, we shall
obtain
additional
con-
as well as to LI;“.
case
We return to eq. (16) which we wish to cast in the form of eq. (6a), thus identifying LI;“. Invoking eq. (lo), we write eq. (16) in the form
d; =
vjg
“‘da +
I
(v, - v,)g”‘G
da.
(24)
153
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
From eq. (11) the second term on the right in eq. (24) is g”‘(vj - qj) da ,
-
which, by the discussion following eq. (11) is of order J exp[-p(u - q)‘](u T)~~u which is small if g is sharply-peaked. It is proportional to third and higher entropy derivatives and to a more negative power of ,U than terms of the same order in F* from the first term of eq. (24). Therefore, we neglect the second term in eq. (24), marking one place where we must continue to invoke the sharp-peakedness part of assumption (4) listed in section 1. This is probably not a drastic approximation, since we shall point out in section 5 that demonstrated failure of the Einstein approximation occurs only when we take into account the coupling to external reservoirs, which is not involved here. Inserting eq. (18) into the remaining term in the right-hand member of eq. (24) and expanding the exponent, we obtain eq. (6a) with LF = K-'c eXp[-S'(a,
T)]
1e-@ W(u)
2 (-)“(m !)-’ mbl
m-l
I
da
vjv,
.
(25)
The term with m = 0 vanishes because W(a) is an even function of vj. We can immediately define L’,3’= -Ly to satisfy the Casimir anti-reciprocity relation. It only remains, then, to recast the first term on the right in eq. (17) in the form of Ly plus a contribution to be added to the second term to yield LI,“‘. Effecting a partial integration in a-space in the first term on the right in eq. (17), we write it in the form
I I da
+
dx c AjAk(aSfla~j)(g(o)l W)a(A - U) = - c Li.z(as/aQ k
P
C ems’2 (aS’/ag) 1 da 1 dx e-“AjAp 2 [ 2 v,(aS’/an,)] m-1 mZ1
X [(-)”
=-
(i
!)-’
+
(-)“-‘((m
- l)!)-‘]S(A - a)
c Lf aS/acu, + C eeS’ c (aS’/ag) P
r
j da e-@W(a)vjvp (c
V,
aS’/aa,)
r
P m-2
(-)“((m - l)!)-‘(m-l>
1).
(26)
R.E. NETTLETON
154
The last term in this expression is symmetric with respect to interchange of j and p. When the last term in eq. (26) is added to the term obtained in eq. (19) and the second term on the right in eq. (16) we find m-2
LI;“’= K-‘C e-S’
I
da
dx vivi e-@ z* (-)“((m
I
x 2 [ vp(as’la~p)(m
-1- 1) -
- l)Y(C
r
v, asm,
>
A~(&S’/aqJ]S(A - a)
P
da
fK-lce-S./
j
ds
c (-is)“(m 1dx vivj Mao
!)-‘6(A - a)O’“’
0
x
2 Al as/aq,
em0C rZl
k
(r!,l(- 2
V,
asmJ
w
ds W(a)K,, em@+ B(G) .
+K-lCe-S~dada’]
(27)
n
The first term in this expression clearly exhibits Onsager symmetry, since the integrand is proportional to vivj, while the symmetry of the second and third terms has been demonstrated in subsection 3.1. If we examine the structure of Ly and Lf’ we see that these expressions, to the order calculated here, are unique. For in Cj Ly aS’/aqj the integrand is proportional to a product of factors v, &S/aq,, and it does not matter which of these we pull out in identifying Ly. Similarly in Zj Ly aS’/arl, each term is a force FT is extracted to define Lr, SUIII K-’ z A,...,F;F; a . . F*p, and whichever the result is the same so long as F: is extracted to define Lz’ = Lr’. Since Ai,,,.,_,,is not symmetric with respect to interchange of any arbitrarily-chosen pair of-its indices, we do not have a7jilaFT= &jJaF;, and so the Onsager symmetry demonstrated here is not of the type proved not to exist in general.
4. Contributions
from G
We now return to the third term in the right-hand member of eq. (16) in order to show that, if there are terms which it contributes to Lf) whose Onsager symmetry cannot readily be established, then these terms are small in comparison with terms of the same order in the forces F* derived in section 3.
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
155
To this end, we observe that G can be expanded in Hermite polynomials, and products thereof, which are functions of the sets {ai - CX~} and {2ri- vi}. Thus G = G, + G,, where G, depends only on Hermite functions of the set {ai - cri}, and the u-dependence resides entirely in G, which may also depend on the u-variables. We shall proceed to show that Onsager symmetry can be established for terms arising from G,, while most of the G, terms, for which the demonstration is more difficult, should be negligible. So, in fact, are the G,-contributions, so that for most systems G can be neglected altogether in calculating the first-moment equations. A typical term in the Go-dependence of eq. (16) can be written 1 dadn’ j dx 6(A - a)(-it>“(n!)-‘A>(1 - P)L * * - (1 - P) c ASS(A - a’) I
x G,(a’)(a/au:)[g’“‘(u’)lW(u’)]
,
where we have set (1 - P)LS(A - a) = - (1 - P)At(Wav)i3(A - a) and have integrated by parts with respect to o. This expression can be rewritten in the form
-jdudu’j
dx S(A - u)AJ[(-it)“(n!)-*I(1 - P)L * * . L(1 - P)
x C Aj(aS’/av,) 6(A - u’)G,(u’>C ems’-@mzo ( - 2 vi tN/l%~,,)~(m !)-’ . I
w (28)
When m = 0, we can carry out the a- and a’-integrations and element of a product of Hermitian operators which is non-zero even number of L’s. Symmetry with respect to the interchange in the discussion of eq. (23). When m 2 1, the discussion is the
obtain a matrix if there are an j-r follows as same as for eq.
(20 The case n = 0 in eq. (28) must be discussed separately. This case is the same as eq. (17) with a factor G, introduced into both integrals on the right. By the discussion following eq. (24), the Hermite polynomials in G, should be H, or higher to assure that J (a - a)G,g”’ da = 0 to the order of terms systematically neglected in subsection 3.2. Within this approximation, the first of the two integrals in J qjG&‘O’du vanishes while the second leads to an expression which is eq. (19) with a factor G, inserted. As in that case, we get an Onsagersymmetric contribution to I,!. To discuss the case of G,, we must consider the solution of eq. (5a). As illustrated in ref. 16, it is convenient to express the solution in the form
156
R.E. NE’ITLETON
g = where
$O’[1 + G] )
(29)
2”’ is Gaussian
which,
for simplicity,
we take to be an exponential
of a
sum of squares, and G is a sum of Hermite polynomials and products thereof. We want to use eq. (5a) to investigate the magnitudes of those terms in G which are proportional to powers and products of the forces X$‘/acy and as’/&) and
which
introduce
also contribute a streamlined
g-(‘)- exp[-p(a G=
.. +
where
-a)*-
. wJ,(P”/*(~
j..~(v -
-
B,,H,(pl’*(a
a, u can
to G. To estimate
notation
-
denote
any
pair
it is sufficient
q)*] ,
4)~W2(~
a))H2(p1’*(u
magnitudes
to
and set (304 -
-
7))
+
7))
+
ai, vi, and
B2JJ2W2(u
-
77))
B,H,(/_P2(u
-
7)).
H2 can
represent
. . )
W’b)
a product
of
polynomials HI which are functions e.g. of two variables U, and v2. The coefficients p are proportional to second derivatives of S’ which are large under the assumption that g is sharply-peaked. Since the p’s are all large, they are all taken to be of the same order and replaced by a single symbol. In the notation of eqs. (30a, b), eq. (5a) assumes the form, to leading order
in
each term,
agiat= -(aiau)(qg) - v(aglaa) + K(a2gW)
(31)
Substituting
eqs. (30a, b) and comparing coefficients a)), H,(d’*(u - v)), and H2(p1’*(u - TJ)), we find
4, - (-q/48Kh, B,
-
(q/48K2p
)ti
B,, - (-q2/48K2p ,
of terms in Ho, H,(p”*(a
-
1’2)?j , (32)
B, - (- 1/48Kp “*)?j
From this we find that contributions from G to eq. (16) of order F*” will, with the possible exception of B,,, be of the order of a negative power of p multiplying a term of order F*” in $ and thus negligible if p is sufficiently large. In particular, this holds for second-order terms of the type B,*v?j = we conclude that the coefficient of pU-1B12F*+ By similar arguments, H&“*(u - cz)) in G, should be -B,,, so that those terms in G should also make a negligible contribution to the phenomenological coefficients. However, we have been able to demonstrate Onsager symmetry without neglecting them. It remains to consider the B,, terms which give rise to contributions to Ly whose
Onsager
symmetry
can be demonstrated.
By partial
integration,
as in eq.
ONSAGER RECIPROCITY IN THJ3 NONLINEAR REGIME
1.57
(17) we find that _/ da j dx S(A - u)A~G,(gco)/W) = c j da _/ dx ujt.+w(a)-‘(alaa,)s(A , dx Aj(x)Aj(x)
xGsco) + c r
W-‘(a/at@(A
- u)Gvg(o).
- a)
(33)
When G, = B,,H,(~ “*(a - cy))Hr((p “*(zJ- 7)) we find after partial integration that the first integral on the right in eq. (33) becomes
2 j da ujur(u -
-2~4 +
6, ,,i
(
I
c c W%C ees I da e-@vjtp,(aS’/a~,) u
x
v)g”’ = -2/.~B,,77~ da (U - v)2g’o’(1- S,.)
,
-
C
c
rtj
V,
aslav,
w
(p!)-’
pr1
)p-1(~-~)W(o)+2~~jB,,CeS_CldUqU,e-~W~, r
(34) where S,, = 1 if n = nj, and S,, = 0 otherwise. The second term on the right in eq. (34) exhibits Onsager symmetry with respect to the interchange j-r. To consider the first and third terms, we invert the expansion Fy = aslavi
= C sijvj + C ~~~~~~~~~~ + -.i, k,I i
(354
to get qi = c s,‘F; j
-
c
j,1.m.n
silsirmn2
s&,‘,s,;F;
F;F; + - . - .
W)
P. 9. w
The terms in eqs. (35b) can be grouped to exhibit Onsager symmetry with respect to interchange of an index pair. The B,,-contribution to the second integral on the right in eq. (33) assumes the form dx vjA;[-2p ‘nH& “*(a - a)) + Hl(,u “*(a - a))& x (/.L’~(v - 7)) S’/an,]g’“‘/ W = -2B,,lc.lR c 1 da Vjs,Hl(~“*(u - a))g”) I + 4,
c da u,4,(as’iarl,)~l(~u.‘“(u I I
- ~M4(P
‘%
- rlNg@)
158
R.E. NETTLETON
Xf4(P1’2(U - 77W0)lW - wqs(alau,)[H,(p”12(u - 7&(O)/WI}. This
expression
exhibits
j-r
symmetry.
Similar
can be applied
to
terms in G of the form B,,,H,(~1’2(a - (Y))H,(~“‘(u - n)) for n 2 1. We can extend the foregoing reasoning to higher order in the expansion
of
the third term on the right in eq. (16) in powers partial integration [cf. eq. (21)]
j dada’ j dx 6(A - a)Aj(n!)-‘[-it(1
=-
jdudu’j
reasoning
(36)
of [-it(l
- P)L].
We have after
- P)L]“S(A - u’)G,(a’)[g’“‘(a’)/W(u’)]
dx u,8(A - a)(n!)-‘il[-it(1
- P)L]”
x S(A - u’)G,(u’)[g’“)(a’)/ W(u’)]
dxAjG(A - a)(n!)-‘(-it)“[(l-
= -jdadu’j
x
c
A;@/&$)[ G,g”‘/ W]
For illustration, G,. For this term
C
P&L]“-‘(1 - P)6(A - a’)
we consider
A@~~u;)[G&?
Wa) the contribution
W] = B,,H,(~(~
to eq. (37a) of the B,,-term
- a))
x ( ~/_L”/~A~ - c A$N/~T],)H~(~“~(zJ - v))}g”‘iw. I If we move
the factor
in
6(A - a’) to the left in eq. (37a) we obtain
(37b)
from
the
At-term first a constant of order K multiplying 4. This will contribute a first-order term in F* with coefficient
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
159
from q - CT@‘). Th’is is an approximation analogous to approximation (2) which here is applied to the constant coefficients. Also from the At-term, we get contributions ITT’(F*~) by replacing one of the L-operators by Xi Ai aS’/arl,. We can argue similarly that these are smaller by a factor O(A) than secondorder terms arising from the g(O)-approximation to G. It should also be noted that, because of the factor H,(pl’(u - a)), we are taking an u-derivative, in addition to multiplying by 0(A). Similar reasoning applies to the second term in eq. (37b). Once the factor S(A - a’) has been moved to the left, we can expand g”‘/W in powers of (- Z, o, N/an,,,) and explicitly demonstrate Onsager reciprocity for the higher terms in the expansion, exactly as in eq. (22) since again we have a factor vjv,,, symmetric with respect to i-w. Since g(O) is Gaussian and g(O) only approximately so, we have g(O)= g”‘[l+ G’], where G’ is a polynomial in a - CTand v - n with coefficients which are third- and higher-derivatives of S(a, 7) and powers and products thereof. If G’ gives rise to an integral B’J n(v - n)ng(‘)du - B’,u-(“‘~+~)F*,this is smaller than first-order terms in 7j, provided B’ is not of order p, which we assume true of third and higher entropy derivatives. Similar considerations apply in higher order. Unfortunately, while we have been able to demonstrate Onsager symmetry for most of the contributions of G,, we cannot without much greater effort show that such symmetry is unique. Since G is proportional to 7j, this dependence gives part of the dependence of Ly on F*, and this dependence on F* introduced through the factor + may have symmetry independent of any demonstrated above. However, the argument following eq. (37a) above implies that all the contributions from G!, which are proportional to li are, for a given order in F*, or higher order in A than similar contributions from eq. (21). We can also note that K - (PT))’ where 7 is the relaxation time for n which is short, and ~1 is large. K may, therefore, not be of the order of a negative power of Jo which makes q/K - C7(A2).This could be advanced to justify neglecting contributions to the moment equations which are proportional to q/K as higher order in A than contributions from g (‘). This is like approximation (2) in the earlier linear theories3*5). Then it is probably not a poor approximation to neglect G in the first moment equations for ~5 and 7j, giving coefficients Lf’ which can be determined uniquely as shown in section 3.
5. Case of heat conduction
in condensed matter
An example of the application of extended non-equilibrium thermodynamics, much discussed’3,‘5,17) in the literature, involves the CattaneoVernotte equation’8*‘9). F or an anisotropic heat conductor, with Ji (i = 1,2,3)
160
R.E. NE-ITLETON
the components
of heat flow, this equation has the form
4 = 2 1’$_5.+ c N&T. i
(39)
I
If the system to be described is a small cube immersed in a much larger liquid or solid phase, the relation of Ji and the symmetry of Mi,. should be described by the Zwanzig formalism2S3) discussed in preceding sections, since J is a fast variable”) which relaxes in a time short in comparison with the time for appreciable heat diffusion into the system. The term in VT arises from coupling to the surrounding medium regarded as a heat bath, and the Onsager-Casimir properties of Nij can be inferred only phenomenologically within the framework of the present paper. We can set J = (A,(x)), where A,(x) is a dynamical function of the classical phase, x, of the system”). Then the equations which correspond, in this case, to eqs. (17) and (5b) above are I
&=-j-dszj-d 0
’ d ” W(“)Kj!f(u,
u’, s)(alaV;)[g(U’)l
W(U’)] ,
k
W(U)Kjk(U, u’, s) =
6(A - u)Aa{exp[-is(1
- P)L]}(l
-
P)A,G(A
(404
- u’) dx, Gob)
where g(v)du is the probability that A, lies within du centred at 2).The term q, in the first term on the right in eq. (16) vanishes16). If we are discussing a crystal, the x-integration in eq. (28b) is limited to configurations consistent with the assumed structure. We replace g + g(O)in eq. (40a) by reasoning analogous to that in section 4 above. The solution for G has been obtained in ref. 16, eq. (16) and, when expressed in terms of F, is proportional to a negative power of p. Then eq. (40a) assumes the form
j/ =
C ‘jk(aS/aJk),
(41)
k
where A, = Akj follows by reasoning analogous to that in subsection 3.1 above. We have
J, =
I
uig’o’(l + G) du =
v,g(‘)du
(42)
ONSAGER RECIPROCITY IN THE NONLINEAR REGIME
if g”’ is sharply-peaked,
4=T
161
from the discussion of eq. (24). We then have
c A$(as/q),
where (43b)
The only argument that we have for inserting eq. (43b) into the last term on the right in eq. (39) is the phenomenological one that such a Casimir antireciprocity relation must hold if the irreversible rate of entropy production is to be positive definite. The term linear in VT in eq. (39) implies16) that the equation for g must be augmented by a term linear in VT. When this term is not zero, g, and correlations calculated from it, depart significantly from the Einstein function -exp S’S/K 16). Such departures have no bearing on the sharp-peakedness of g(O)calculated for a closed system, and therefore demonstrated failures of the Einstein function’6’21) are not arguments against neglect of G in eq. (43b).
6. Summary
and conclusions
The calculations described in previous sections have had two objectives. (A) We can eliminate assumption (3) listed in section 1 and give a less ad hoc character to assumption (4). (B) We can provide an answer to speculations to the effect that, because certain proposed generalisations of Onsager reciprocity to the non-linear case have been shown’0-‘3) not to be valid, then no such generalisation can be found. In the original work of Zwanzig, non-linear reciprocity relations were derived for symmetric a-variables, subject to assumptions (1) and (2). The extended non-equilibrium thermodynamics has since been developed in recognition of the fact that, when assumption (1) is invalid, so that phenomenological coefficients depend explicitly on time, we are usually concerned with times short compared with the relaxation times of the a-variables. Under these circumstances, there are T-variables, where 7 = Cw,whose introduction as independent state variables provides for the inertial effects which assert themselves over short times. There should be a wide range of phenomena which are describable using phenomenological coefficients not explicitly time-dependent as long as a sufficient number of independent state variables is employed. Assumption (1) should be reasonable whenever this is the case.
162
R.E. NETnETON
Although
the contributions
G,, have been
shown
from g(O)and some of those from g”‘G, e.g. from
to obey unique
reciprocity
relations
to all orders
forces F and F*, this does not imply that we can, in practice, third order.
Unless
we can neglect
G,, it is difficult
in the
go much beyond
to show that Lc’ is unique.
To neglect G,, we have proposed to drop, as in assumption (2) contributions to Lr which are of higher order in A than similar contributions of the same order in F and F*. However,
the terms
dropped
may be of the order
of terms
of
fourth or higher-order in the forces. Thus we should be able to carry Onsager reciprocity to terms of second- or third-order in the forces, but it is likely to prove impractical to go beyond that. Most phenomenological models, in any case, have been linear in 77, although not necessarily in (Y- LYE.For example, the Cattaneo-Vernotte equation (39)“*“) is generally used in a form linear in heat flow J, and the Maxwell stress-relaxation equation**) is usually linear in the pressure tensor. Landau and Lifshitz6) have derived a Lagrangian formalism, equivalent to ref. 5, where linearity is preserved in the n-variables but not in the set {(Y~- ajo}. There are, accordingly, few examples which show experimentally that it is necessary to go to terms non-linear in the F* forces. If we do get far from equilibrium so that higher-order terms must be included, these contributions are likely to be of the order of effects neglected in the Markovian approximation, and an integral equation with memory kernel will be more
appropriate.
In the formalism of the present paper, the (Y- and n-variables have all been “fast” variables with relaxation times short compared with the time required for appreciable hydrodynamic flow of energy or particles into or out of the system, when the latter is immersed in a much larger fluid. The dynamics is effectively that of a closed system, with a time scale sufficiently long so that explicit t-dependence of K, can be neglected, but short compared with recurrence times. If there exist macroscopic temperature and velocity gradients, additional terms, depending on these gradients, must be added to eq. (5a), as explained in section 5 and exemplified in eq. (39). The distribution g then acquires terms dependent on these gradients, in addition to the terms in g”’ and G16). If “slow” variables such as the density and temperature are added to the set {a;}, a further modification is required, embodied in the projection operator technique of Grabert4) which is consistent with a canonical equilibrium distribution, instead of the microcanonical distribution appropriate to a closed system. These modifications permit us to include correlations in the fluctuations of slow and fast variables without materially affecting the phenomenology of the latter23). One achievement of the foregoing calculations is to elucidate the question of utility of the Einstein function. g(O) is a generalised Einstein approximation to which terms of third- and higher-order in the Taylor expansion of S about S(cr, r)) have been added. The arguments of section 4 indicate that g(O)should
ONSAGER
RECIPROCITY
IN THE
NONLINEAR
REGIME
163
in most cases be an adequate approximation to g for purposes of calculating the first moments of eq. @a), although not in general for calculating correlation functions.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)
S.R. de Groot, Thermodynamics of Irreversible Processes (North-Holland, Amsterdam, 1951). R. Zwanzig, J. Chem. Phys. 33 (1960) 1338. R. Zwanzig, Phys. Rev. 124 (1961) 983. H. Grabert, Projection Operator Techniques in Non-Equilibrium Statistical Mechanics (Springer, Berlin, 1982). R.E. Nettleton, J. Chem. Phys. 48 (1964) 112. L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon, London, 1958). D.G. Sannikov, Soviet Phys. JETP 14 (1962) 98. D.G. Sannikov, Soviet Phys. Solid State 4 (1962) 1187. R.E. Nettleton, Phys. Lett. 24A (1967) 231. D. Edelen, Arch. Rat. Mech. Anal. 51 (1973) 218. D. Edelen, Int. J. Eng. Sci. 12 (1974) 121. D. Edelen, J. Non-Eq. Thermodyn. 2 (1977) 205. G. Lebon, Int. J. Eng. Sci. 18 (1980) 727. D. Jou and J.M. Rubi, Phys. Lett. 72A (1979) 78. D. Jou, J.E. Llebot and J. Casas-Vasquez, Physica 109A (1981) 208. R.E. Nettleton, J. Phys. A: Math. Gen. 17 (1984) 2150. R.E. Nettleton, Phys. Fluids 3 (1960) 216. P. Vernotte, Compt. Rend. Acad. Sci. Paris 247 (1958) 3154. C. Cattaneo, Compt. Rend. Acad. Sci. Paris 247 (1958) 431. S.A. Rice and P. Gray, The Statistical Mechanics of Simple Liquids (Interscience, New York, 1%5), p. 382. D. Jou and T. Careta, J. Phys. A: Math. Gen. 15 (1982) 3195. J.C. Maxwell, Trans. Roy. Sot. (London) 157 (1867) 99. R.E. Nettleton, J. Chem. Phys. 81 (1984) 2458.