- Email: [email protected]
Macroscopic aspects of mixing distance in nonequilibrium thermodynamics
ANNALS
OF PHYSICS
115, 172-190 (1978)
Macroscopic
Aspects of Mixing
Nonequilibrium
Distance in
Thermodynamics
F. SCHL~~GL Chemistry Department, University of Minnesota, Minneapolis, Minnesota 55455 and Institut fiiv Theoretische Physik, Rheinisch- Westfiilisehe Technische Hochschule, Aachen, Germany*
C. ALDEN
MEAD
Chemistry Department, University of Minnesota, Minneapolis, Minnesota 55455
ReceivedOctober11.1977
In this paper,wedescribeseveralfurther developments in the applicationof the principle of increasingmixing character,and the relatedprincipleof decreasing mixing distance, to the time-dependent behaviorof macroscopicsystems.Thereare three main parts to this paper.In the first, somebasicassumptions are discussed which are sufficientfor the validity of the principleof decreasing mixing distance,and the macroscopic formulation thereof.It is shownthat the principleholds,not only for a systemin contactwith a heat bath, but alsoin linearthermodynamics for opensystems with time-independent boundary conditions,and with nonequilibriumsteady-state,rather than equilibrium,solutions. In the secondpart, a newproof is givenfor a resultpreviouslyobtainedby oneof uswhich permitsthe formulationof the principlein termsof observablethermodynamic properties of macroscopic systems.The new proof is more generalthan the old, in that one of the assumptions usedbeforehasbeendropped.Finally, severalsimpleexamples are discussed, which showclearly how this principledirectly yieldsmoredetailedinformationaboutthe time evolutionof a systemthan is affordedby traditional thermodynamics.
1. INTRODUCTION The notion of “mixing character” was introduced by Ruth [l] as a partial ordering of probability distributions over the same set of events; and it was shown that the mixing character of the probability distribution over the microstates in Gibbs’ phase spaceof an isolated systemalways increasesif a master equation is obeyed. This “principle of increasing mixing character” is an essentially stronger restriction on the possible changesof the thermal states of the system than the second law of thermodynamics. The latter can be considered as a special consequenceof this principle. Thus, the secondlaw (increase of entropy for isolated systems)is a necessary,but not a sufficient, condition for increasing mixing character. However, arguments have been * Present,permanentaddress. OOO34916/78/1151-0172$05.00/0 Copyright AU rights
0 1978 by Academic Press, Inc. of reproduction in any form reserved.
172
MIXING
DISTANCE
IN
THERMODYNAMICS
173
given [2, 31 which indicate that, when properly formulated, this principle may be applied to real thermodynamic systems with a generality approaching that of the second law. As shown later on by Ruth and one of the authors [2], a consequence of this principle is that the changes of the thermal states of an open system in a large heat bath are restricted in an analogous way by the decrease of the “mixing distance” of the probability distribution over the microstates in Gibbs’ phase space describing the instantaneous thermal state from the corresponding distribution describing the equilibrium with the heat bath. This theorem is a generalization of the law that the entropy production in the interior of such a system always is positive definite. In a previous paper, one of the authors [3] has derived a criterion for the development in time of thermodynamic variables in a macroscopic system which are allowed by this theorem, thus obtaining a macroscopic thermodynamic theorem. For the development in time of the probability distribution in Gibbs’ phase space of an isolated system, we shall assume the validity of a master equation, i.e., a stochastic equation of motion which is of first order in time and linear with respect to the probability distribution. It then follows that the same has to be true for a system in contact with a large heat bath. Generally the validity of a master equation brings decrease of mixing distance of any solution from the unique stationary solution [2]. Therefore, in that region of linear thermodynamics where the master equation is valid, we can extend the theorem to open systems which are subject to boundary conditions which don’t allow an equilibrium state, replacing the equilibrium state by the steady state corresponding to the stationary solution of the master equation of such a system. This will be made clear in Section 2, where the basic assumptions for the following considerations will be generally formulated and discussed. In Section 3, we give a more general derivation of the macroscopic theorem obtained in Ref. [3], in which the restriction to “many-parameter” systems is dropped. This enables one to apply the theorem without restriction to macroscopic thermodynamic systems, including those characterized by only a small number of parameters. In Section 4, we treat a few examples of this kind, and it is shown that the criterion leads to nontrivial restrictions which are not obtainable from traditional thermodynamics. 2. BASIC For the following, systems:
ASSUMPTIONS
two essential assumptions will be made regarding the physical
(a) validity of a master equation in Gibbs’ phase space; (b) distinct separation between short and long time behavior. The first means the restriction to a certain class of systems. In particular it is fuElled in a regime of linear thermodynamics. The second is a very general restriction to systems for which a description by conventional thermodynamic means is applicable.
174
SCHLdGL
AND
MEAD
For the present, the microstates of the system may be denoted by a discrete subscript 01, corresponding to the division of the phase space into arbitrarily small but finite cells. This is a convenient, though not essential, restriction for the following, and will be abandoned later on. The thermal contact with a large heat bath may be characterized in different ways. Let a: denote the microstates of the system in the heat bath with microstates r. The result that the mixing distance of the probability distribution pal of the system from its equilibrium distribution p ao decreases was obtained in [2] with the help of the requirement that in thermal contact with given pa the mixing character of the probability distribution pra always is maximal. pToris the distribution in the phase space of the system and the bath, which form together an isolated system. This yields the result that pra is independent of r for a given 01if r and 01are compatible. Another characterization of the thermal contact with the heat bath was given earlier by one of the authors [4] by the requirement that the conditional probability p(r 1 a) in (2.1)
be dependent on r, LYonly, and not, for example, on p or the time. It leads to the same result and is fully equivalent with the requirement given in [2]. This leads to the result that the change of po: with time is given by a master equation.
where /l,, is independent of time and of the distribution pa , because the distribution prol of the isolated system certainly is subject to a master equation. (The dot above a letter denotes the time derivative.) If polo is the stationary solution
of the master equation (2.2), then all measures
F(P, ~“1 = c pa”G(p,/p,O) OL
(2.3)
with arbitrary concave functions G(u) never increase in time, as shown in [2]: -qP, P”) < 0.
(2.4)
By a concave function we mean a function G(u) such that the second derivative G (u) is never negative. These measures S already were proposed by Csiszar [5] as a generalization of the relative information measure of p with respect to pa known as “Kullback information” [6] or “information gain” [7] K(P, PO)= C pa Wp,/pmO) a
(2.5)
which is closely related to entropy production [4]. The partial order of the “mixing distance” [2] is defined by the behavior of the totality of all these measures 9 as a generalization of the partial order of the “mixing character” [l]. If (2.4) is fulfilled
MIXING
DISTANCE
IN
THERMODYNAMICS
175
for all concave G(u), then by definition the mixing distance of p from pa decreases. Thus this mixing distance always decreases if the master equation is valid. It may be mentioned that more generally the mixing distance between any pair of solutions of the same master equation decreases [8, 91. The validity of a master equation in Gibbs’ phase space is not restricted to isolated systems and systems in a large heat bath. It also can be postulated for open systems in a region of linear thermodynamics. In the case where the boundary conditions are time independent, yet do not allow a thermal equilibrium state, the stationary solution of the master equation, which is unique, describes a steady nonequilibrium state. We exclude nonergodic systems, in the sense that not all microstates can be reached by a sufficient number of steps starting from a given microstate. Linear thermodynamics is restricted to thermal states which are sufficiently close to an equilibrium state that the dynamical equations can be assumed to be linear in the deviations from equilibrium, to any desired accuracy. It is not sufficient that they are linear for a special choice of quantities describing these deviations. The linearization has to be valid for any choice of these variables. If not only the macroscopic observables, but also the microscopic probabilities, are sufficiently close to their equilibrium values to permit linearization of the equations of motion, then these can be approximated to any desired degree of accuracy by master equationsl. Thus, this strong region of linear thermodynamics implies that the master equation provides a sufficiently accurate description in Gibbs’ phase space, but the converse is not true. It does not follow from the master equation that the stochastic equations over other sets of events obtained, e.g., by elimination of certain state parameters need also be linear. In summary, the master equation is a sufficient condition for the validity of the principle of decreasing mixing distance, and it, in turn, is valid for at least two types of system: isolated systems, or those in contact with a heat bath, regardless of the initial state: and open systems with time-independent boundary conditions and nonequilibrium steady-state solutions, in the region of linear thermodynamics discussed above. Now the assumption (b) will be discussed. The probability distribution pa which gives prescribed mean values (W) of phase space variables A4,” and has maximum entropy S = - C p, In pa (2.6) will be called a “generalized
canonical distribution”: pu = exp[@ + 5 . KJ.
(2.7)
Here the abbreviation c i”K” is used for the summation
over v, which designates the macroscopic
1 A discussion of this is given in Appendix A. 595/IIS/I-12
= 5 . Ma
(2.8) variables (MY)
176
SCHLiiGL
AND MEAD
and their thermodynamic adjoints [, . The subscript 01denotes the value of the phase spacefunctions M, in the microstate 01.In the interpretation of Jaynes [lo] the distribution (2.7) corresponds in an unbiased way to the knowledge of the mean values (MY) only. In thermal equilibrium states the macroscopic variables (MU\ indeed are the only knowledge and therefore the corresponding distribution (2.7) is the appropriate description for thermal equilibrium states in a statistical theory. In extending the scheme to include nonequilibrium states with time- and spacedependent macrovariables (MY) we shall absorb the space coordinates into the set of the subscripts v. The mean values (;M”), and hence also the I&, are thus time dependent. Yet in contrast to the equilibrium statesthe distributions (2.7) do not give the most adequate description in a statistical theory of nonequilibrium phenomena, because they suppress any additional knowledge about the microdynamics of the system. These distributions, which may be called the “accompanying canonical distributions” indeed fail to describe correlations in time, especially of irreversible transport processes[I 1, 121.One can, however, include the essentialknowledge about the Liouville character of microdynamics following a procedure of Mori [ 131,if the system shows a distinct separation into long and short time behavior. This separation can best be expressedby the position of the singularities of the Laplace transforms of autocorrelation functions in time [14]. The distributions which then can be used as an adequate description may be called Mori distributions and are obtained from the accompanying distributions by the transformation which is given by the Liouville dynamics over a time T which is long on the short time scale and short on the long time scale. The resulting distributions also were given, based on different arguments, by McLennan [15] and Zubarev [16]. Not only the (MY) but also another class of quantities belonging to parare equal for the Mori distribution and the accompanying canonical distribution. This is the class of all quantities which are invariant with respect to a canonical transformation in phase space, in particular to reversible dynamics. Let them be called “reversibility invariants” [ 171.To this classin particular belong all measures9 of (2.3). Therefore the assumption (b) for our systemswhich justifies the Mori distribution as a good description and which practically is identical with the applicability of the description by the conventional macroscopic thermal variables gives the following: The principle of decreaseof mixing distance in Gibbs’ phase spacecan be expressed by (2.4) using the accompanying generalized canonical distribution p,pO. 3. THE MACROSCOPIC
CRITERION FOR THE DECREASE OF MIXING
DISTANCE
As in the generalization of the corresponding statement for the mixing character [l], which can be looked on as a special case of mixing distance, the following theorem was proved in Ref. [2]: A sufficient condition for decreaseof the mixing distance of p from p” is that (2.4) is fulfilled for a subclassFA(p, II”) constructed from the concave functions G,(u) with G;(u) = 6(u - A)
(3.1)
MIXING
DISTANCE IN THERMODYNAMICS
177
where h is any real number and S(x) is the Dirac delta-function. If (2.4) is fulfilled for all the functions G, , then it is also fulfilled for an arbitrary concave function G(u). This is easily seen [8] if one writes G”(u)
= 1 dA G”(h) G:(u).
0.2)
The subclass4 represents, in the language of [I] and [2], a “mixing isomorphism”, which gives a more convenient and manageable criterion for the decreaseof mixing distance. This criterion is of practical use, however, only in relatively simple stochastic systems. For general thermodynamic systems, further considerations are necessary to get a criterion for decreasing mixing distance on the Gibbs level which can be formulated in terms of macroscopic quantities only. Such a criterion has been obtained by one of us in a previous paper [3]. Here we give a more general (and, we think, more elegant) proof of this result than was given in Ref. [3]. ln the following, we denote by t the whole set of canonical variables of the system, and by MI(~)the probability density in the &space, i.e., in the Gibbs phase space.We are thus denoting a microstate by a set of continuous variables rather than by a discrete index. Let the steady-state probability distribution be IV,,(<). The deviations of the observables M” from their steady-state average values will be denoted by L”‘.The vu are assumedto have a continuous range of values from minus to plus infinity. The accompanying canonical distribution (2.7) can be written as2 w(f) = IV”(() exp[E + y . r@)].
(3.3)
For fixed We, the D”are phase spacefunctions, and we can introduce the probability density in the space of these functions: W(v) = We(v) exp[B + y . z:],
(3.4)
with Z(y) = --In 1 dr W”(z!) P’.
(3.5)
The Fourier transform of W(v),
satisfies
’ In termsof the continuous variables I, (2.7) takes the form w,(t) = exp[@+ 5 . &f(t)], with the equilibrium distribution being given by w”([) = exp[@’ + 1” . M(f)]. Denoting the steady-state averagesby MO, and defining y = 5 - to, a(f) = M(f) - MO and 8 = @ - @o + y . MO, one immediately obtains (3.3).
178
SCHLijGL
This expression is the generating Thus, we obtain
(pj (dull 0~)
=
[
f &
AND
function
MEAD
for the cumulants
In ri/c(a)] ”
= [- &
of the quantities
= _ !Tfj!$?!! “,=O
ln IQ(~)],=,
(3.8)
Y =
b%(y) -z3$
V.
etc.
(3.9)
An important consequence of (3.7) is that, in the thermodynamic limit, In l%(w) becomes porportional to the size of the system. If V’ is some measure of the size of the system, then we have, as I/ vecomes infinite while keeping average densities constant,
$ [U/V> In W~)l = w(w) where %@(u~)is independent of V. Using (2.3), (2.4) and (3. I), we obtain the following
(3.10)
form for our criterion:
where h(x) is the Heaviside unit function, whose derivative is 6(x). Equation (3.11) must hold for all positive values of h. From (3.3), we see that (w/w”) is a function only of y . u. Using this fact, (3.11) may be rewritten as
where (T = In X - c” takes on all real values. We now make use of the Fourier integral form of the step function (3.13) and also express from (3.7) that
W(v) as a Fourier 1 may,
af&J>
integral with the aid of (3.5). We further
= v(y + iw) - P(y).
note
(3.14)
MlXING
DISTANCE
IN
179
THERMODYNAMICS
Putting all this together, we can transform (3.12) into the form
. [v”(y
$
jw)
-
l)‘,(y)]
e-iw.u
s
<
0
(3.15)
where f is the number of different quantities v”. The v-integration now gives just the f-dimensional delta function, and this in turn makes the w-integration easy. We now find (3.16) where /‘iK = Z,“((l +
iK)Y).
Note that, because of the factor (ui, - v), the integrand no longer has a pole at K = 0. To evaluate the integral (3.16) in the thermodynamic limit, we seek the saddle point of the factor in the integrand
where, because of (3.7), we have
u=h
@(KY)
-
1’KU
=
--s”((l
+
iK)y)
-
8(y)
-
iKU.
(3.17)
To find the saddle point, we require dU/dtc = i(v,, . y - u) = 0.
(3.18)
Now consider the behavior of viM as K moves along the imaginary axis, K = -iq. By inspection of (3.4), one easily convinces oneself that v, . y is a monotonically increasing function of r) which takes on all values physically allowed. Depending on the nature of FVO(v), the “physical range” of 7, that is, the range for which the integrals converge and v, . y is defined may extend from plus to minus infinity or may be restricted in some way. In any case, there is always one and only one 7 in the physical range satisfying (~.I~),so we have found our saddle point. The second derivative of U at the saddle point is d2U/dK2 = - c yu(Av~ Av) yy = g.
UY
(3.19)
g is always negative, so the path of steepest descents is parallel to the real axis, K = --in + q. Because of (3.10), moreover, g and the higher derivatives become proportional to V in the thermodynamic limit. All these derivatives, of course, are
180
SCHLijGL
AND MEAD
expressiblein terms of cumulants at the saddle point. The effect of the higher derivatives in the expansion of CJbecomescomparable with that of g when 4 becomesof the order of unity; by this time, however, becauseof the factor of V in g, the integrand has already become extremely small. As for the rest of the integrand in (3.16), it is a relatively slowly varying function of K, so can be considered constant in the extremely narrow range in which L.c’ is not negligible. In the thermodynamic limit, therefore, we have exactly
=i
2% lP J-p g 1
ena$ c i)“(cn”_ v”). ”
(3.20)
Factoring out inherently positive factors, we find for our criterion: (3.21) It is necessaryand suficient for decreaseof mixing distance that (3.21) be satisfied for all 7 in the physical range. In Ref. [3], the necessity of (3.21) was demonstrated by using convex functions of the form (sgn 7) x”+l/(q + I), which, however, do not in general lead to a mixing isomorphic set of functions, so the sufficiency did not follow in general. The sufficiency was then proved for the thermodynamic limit, but with the additional requirement that the v be arbitrarily large in number. Since our proof does not depend on this last assumption, it is more general than the one given in [3]. Also, the possibility of a bounded physical range was not recognized in [3]. As was also pointed out in [3], the limiting case r] --f 0 is equivalent to taking F =
s
df w ln(MI/r+,“) = K(w, w”).
(3.22)
This is the Kullback information or information gain [lo]. Its decrease, a special consequenceof decreaseof mixing distance, leads to the positivity of entropy production [4] if wo is the equilibrium with t a heat bath: P = -X(w,
w”) = -y
* d 2 0.
In this case, the vy are “fluxes”, and --yy the corresponding “forces”.
4. EXAMPLES In the following, open systemsin a heat bath shall be discussed.This means that they are in thermal contact with their surroundings, which are so large that their temperature, chemical potential and other intensive parameters remain unchanged. The steady state is the thermal equilibrium with the surroundings.
MIXING
DISTANCE
IN
181
THERMODYNAMICS
The more general form of the macroscopic criterion obtained in the last section permits its application also to systems characterized by a small number of parameters, and our examples will be of this type. 4.1. Chemical Reactions in Ideal Mixtures
We consider chemical reactions in an ideal mixture which occur practically isothermally, and so slowly, compared to any diffusion process, that the molar concentration n, of any species v remains homogeneous in space. Let nyo be the equilibrium concentration and dpy be the difference between the chemical potential of species v and its equilibrium value. The possible changes of dp., are restricted by the chemical reaction equations and therefore not independent one from another. We can, however, consider separately the question of which restrictions are given by the requirement of decreasing mixing distance and apply the criterion (3.21) as if the n, were independent. In this way, we find restrictions which must hold for all such mixtures. Afterwards the chemical reaction equations give additional individual restrictions for the special system. Only the general restrictions shall be discussed in the following. The generalized canonical distribution in this case becomes the grand canonical distribution and we get in particular vV=n ” -.-noY >
yy = (l/T)4
(4.1)
= ln(nJny”).
(4.2)
In the last equation the gas constant is set equal to one by appropriate choice of units. This equation directly leads to (nJ, = n,(nJn,0P.
(4.3)
(~9, - 0” = (4, - n, ,
(4.4)
Since
the criterion (3.21) here gives +- 1 [(n,), - n,] 3 = 1rl c [(S)” n, Y Y
-
11 % G 0.
As E(y) and thus the grand canonical partition function of the ideal mixture converges for all real values & , the physical region of 77is the whole real axis. A relation equivalent to (4.5) was obtained in Ref. [2], albeit in a rather ad hoc manner. First we shall discuss the case where only two concentrations n, , n2 are variable. All other participating chemical components may be fixed in concentration by exchange with surroundings of the reaction. Then the criterion reads (4.6)
182
SCHLiiGL
AND
MEAD
Thus, for any given initial values of (n, , IQ, each real value of 7) restricts the direction of (ri, , ti,) to within a half-plane. Requiring that (4.6) hold for two or more different values of 17will restrict (ti r , tiie) to the intersection of two half-planes, i.e., to within an angle < 7~. In particular, we consider the limits of the physical range, 7 = *co, and obtain: (a) For 1 < n&O
< n2/n20:
7ii,< 0, rii, + ?ii,< 0
(4.7)
(b) For nI/nlo < 1 < n,/n,“: rii, < 0, tii, > 0 (c) For nl/nlo < L&O
< 1:
tii, + fi, 3 0, 7i, > 0
(4.9)
and corresponding inequalities if we exchange n1 and n2 . These restrictions to certain angles for the direction (ri, , tiJ of motion are given in Fig. 1 and are in any case sharper than the traditional restriction obtained from the positivity of entropy production:
n?
“1
1. Comparison of the requirements imposed by decreasing mixing distance with those imposed by the second law for the chemical reaction system discussed in Section 4.1. At each point on the closed curve, the second law (positivity of entropy production) allows motion in any direction toward the interior of the curve. Decrease of mixing distance is fulfilled only for motion in directions within the angle enclosed by the arrows. These angles are different for the six points shown, as discussed in this text. The dotted line corresponds to n&z,O = n2/n2”. The equilibrium point is the intersection of the three straight lines. FIG.
MIXING
DISTANCE
IN
THERMODYNAMICS
183
A more detailed analysis (see Appendix B) shows that in this special system the limits of the physical region already give the sharpest restrictions obtainable, i.e., no other value of 71gives a restriction which makes the allowed angles smaller. It should be mentioned that this system is formally related to the case of complete disjunction of only three events if one asks whether an arbitrarily chosen molecule belongs to species 1, 2 or neither one. It therefore shows similar features with this disjunction in respect to the discussion of the conditions for decreasing mixing character as given by Lesche [18], and Ruth and Lesche [19]. The physical interpretation, however, is quite different in the two cases. In the more general case off components, it is convenient to use the subscripts v in such a way that the ratios u, = n”/n,” are rearranged
(4.11)
in an increasing order u~uf.
(4.12)
Then the limits &co for 7 give: (a) 1 < u1 : f rif
<
1%
0,
(4.13)
GO;
v=l
(b) u1 < 1 < u, : ?ii, < 0, ti, 3 0
(4.14)
(c) 24, < 1: f 2
fi,
b
0,
ri,
>,
(4.15)
0.
This shows that, in particular, the concentration with the largest value u > 1 always decreases and that with the smallest u < 1 always increases. This is a result that is not obtained through the traditional condition of positivity of entropy production. The intermediate values of 7 would give still more restrictions, the analysis of which, however, is more complicated.
4.2. Slow Exchange of Heat and Matter with Surroundings at Constant Volume We consider a substance enclosed in a box of fixed volume in a heat bath. An exchange of heat and matter with the bath shall be possible yet is assumed to be so slow that all intermediate thermal states of the system are practically homogeneous in space. With internal energy U and mole number N of the substance in the system we choose VI = uuo (4.16) v2 = N - No,
(4.17)
184
SCHLiiGL
AND
MEAD
where the superscript zero here and in the following examples always characterizes the equilibrium value. The generalized canonical distribution for given mean values U, N is the grand canonical ensemble. Accordingly, with temperature T and chemical potential p we get ,=-f+l
(4.18)
TU
(4.19) 1 T,
L i
1n
T
1 =
-T-w1
=
f
(4.20)
+
qy.2.
We can eliminate v and write (4.22) We shall use the criterion (3.21) in the form
only for the limits of the physical region of 7, becausewe then get very general results independent of individual features of the substance.The limits are given by vanishing T,, when 77has opposite sign to y, and by infinite T, when 71has the same sign as y1 . For infinite T, we get finite (p/T), . The substancebehaves for such values of the thermal variables like an ideal gas. U, is iniinite and < Unlu.
(4.24)
sgnj, = -sgny,.
(4.25)
&IN The criterion gives
For vanishing T, we get Ps
=
(4.26)
-Y2IY1.
Energy U,, goes to zero. If y2/y1 is positive, N,, goes to zero as well and we get sgn(Uj, + W2) = -sgn
y1 .
(4.27a)
If, however, y2/y1 is negative, N,, assumesa nonvanishing value NY , which dependson y21y1, and we get sgn[(N,, - N) j2 - UfJ = -sgn y2 .
(4.27b)
MIXING
185
DISTANCE IN THERMODYNAMICS
It should be stressed that we have assumedthat ,U can take any value, positive or negative. This is true, at least, for all “classical” substances.We have to exclude for
instance a Bose-Einstein gas. The criterion thus gives the restriction (rI , y?)-plane
to certain
of the possible angles as represented in Fig. 2.
directions
(fl , &)
in the
FIG. 2. Restrictions on the time development in the (n , r&plane of the system of Section 4.2 imposed by decreasing mixing distance, using only the limits of the physical range of 7. The direction of the motion must lie within the angle enclosed by the arrows at each of the four points shown. The equilibrium point is at the origin.
4.3. Paramagnet We consider a paramagnet whose magnetization M is a unique function of the ratio H/T of the magnetic field H to the absolute temperature T. Thus, asa consequence of the second law, the internal energy U is a function of T only, and independent of M. The generalized canonical distribution for given mean values U, M is the “magnetic field ensembIe”. We therefore put 211= u - uo
(4.28)
v,=M-M”O
(4.29) (4.30)
y1 is given by (4.18) and yZ becomesequal to (4.19) if p is replaced by H. As H like p can assumeany real value, the physical region for 7 is the same as in the preceding example. Internal energy U and M refer to unit volume. U is the total internal energy, not just the magnetic contribution. The criterion (3.21) has the form (4.23) if we replace N, N, by M, M,, . Independent of individual features of the paramagnet we can assume that U vanishes or becomes infinite simultaneously with T. The magnetization M vanishes with H/T and assumesa saturation value fM, for infinite H/T with the same sign as H.
186
SCHLijGL
AND
MEAD
For the limits of the physical region, the criterion For infinite T, , sgnj, = -sgn yl; for vanishing
leads to the following
results: (4.31)
T, ,
smWl + W h MS)&I = - sgny1
(4.32)
where the upper sign holds if y2/y1 is positive, the lower sign of y2/y1 is negative. The restrictions of the possible directions (& , pZ) are of the same kind as shown in Fig. 2. 4.4. Substance in a Cylinder with Piston We consider an arbitrary substance in a chamber with a piston the motion of which is slow enough that the pressure and temperature in the interior are practically always homogeneous in space. The entire apparatus is taken to be in contact with a heat bath with constant temperature To and pressure PO. The generalized canonical distribution for given mean values of energy U and volume V is the pressure ensemble. Accordingly, we take 2.Q= vP
y2=-7+p
V”
(4.33)
PO
(4.34)
v1 , y1 are the same as in the two preceding examples. We can replace p by -P in (4.19) to get (4.34). Yet P is restricted to positive values. Therefore this example shows a new feature compared to the preceding ones, as yet another condition restricts the physical region of 7. Now not only T, has to be positive but also P =f+($-f)?. c-1 T n In determining cases:
(4.35)
the physical region of T,, , P,, , we have to distinguish
three different
(a) If y2/y1 > P, then at one limit of the physical region T,, vanishes and P, has the value y2/y1 . This means that U, is zero and V, is finite, or zero in the case of an ideal gas. We set V, = V,, . The criterion (3.21) has the form (4.23) if we replace N, N, by V, V, . It gives for this limit: sgn[Uj,
+ (V -
V,) f21 = -sgn
yl.
(4.36)
The other limit belongs to finite T,, and vanishing P,, . For these values of temperature and pressure the substance becomes an ideal gas. Therefore U,, is finite and V,, infinite. The criterion gives then sgnj,
= -sgn
y2.
(4.37)
MIXING
DISTANCE
IN
THERMODYNAMICS
187
(b) If 0 < y2/y1 < P, one limit belongs to vanishing T, and finite P,, as before and gives again (4.36). The other limit, however, belongs to infinite T,, and finite (P/T), . It belongs again to ideal gas behavior, but now with infinite U, and finite V, . The criterion then gives
sgnj,
= -sgny,.
(4.38)
(c) If ye/y1 < 0, one limit belongs to infinite T, and finite (P/T), giving again (4.38), the other limit belongs to finite T, and vanishing P, and gives (4.37). The restrictions in Fig. 3.
of (& , &J to angles in the (rl , y.J-plane for this system is shown
FIG. 3. Analogous to the preceding figure. Typical angles are shown for six regions ranged about the equilibrium point. The dotted line corresponds to ye/y1 = P, and the equilibrium point is at the origin.
In all these examples the criterion (3.21) was applied only for the limiting values q of the physical region. We therefore got necessary but not sufficient conditions for decreasing mixing character on the Gibbs’ level. Nevertheless, these conditions are nontrivial new restrictions for the possiblechangesof the systemin time in comparison to the traditional thermodynamical laws. They are restrictions of the directions of motion in a diagram of the macroscopic variables to certain angleswhich are smaller than rr.
188
SCHLijCL
AND
MEAD
5. CONCLUSION The validity of master equations on the Gibbs’ level is justified for at least two important classesof thermodynamic nonequilibrium processes.The first classincludes the changesin time of a system in a heat bath. The secondclassconsistsof all processes jn linear thermodynamics. As shown in Ref. [2], then, the mixing distance of probability distributions on the Gibbs’ level from the equilibrium state, or from a steady state, always decreases.For macroscopic systems, a criterion is fulfilled which first was given in Ref. [3] and for which a more general proof has been given here. This criterion is expressedby means of macroscopic thermal variables only. Some examples of systems in a heat bath are discussed. The thermodynamic criterion is used for the limiting values of the physical region of the test parameterT which occurs in the criterion. It gives new conditions on the possible processeswhich are essentially more restrictive than the traditional thermodynamic laws. Thus these examples demonstrate that the criterion leads to independent new statements in thermodynamics.
APPENDIX
A: LINEAR
THERMODYNAMICS
AND LINEAR
MASTER
EQUATION
Let the microstate probabilities be given by poL, their steady-state values by p;‘, and let 6, =pol - pI!” . Let the underlying equation of motion for the pe be given by
where L,(p) simply denotes some function of all the p’s. Of course, L,(p”) = 0. If one is sufficiently close to the steady state, therefore, (A.l) may be expanded as (A.21 If the 6, are all sufficiently small, the nonlinear terms become negligible, and (A.2) may be replaced by its linear part: j, = 8, = &&A
= &L&(P,
- PsO)-
Defining -L = 43L’BL,13P,B0 and we finally obtain
which has the form of a master equation.
(A.3)
MIXING
APPENDIX
B:
DISTANCE IN THERMODYNAMICS PROOF OF ASSERTION MADE
189
IN SECTION 4.2
It will be shown that, in the two-dimensional example of chemical reactions in an ideal mixture given in 4.1 the totality of all values 77give no sharper restriction than already given by the limits of the physical region. With II, = n”/n,”
(B-1)
.f(u) = UT
03.2) (B.3)
we write the criterion (4.6) in the form
t
[f(zfp) - I](?& - atii,) < 0.
(B.4)
For a given pair of values n, , n2 the coefficient N is a function of q. Two values of n give in general an angle larger than 7~for the forbidden direction of (tiI , K&).We have to show that 01is a monotonic function of n to conclude that this angle taken for the limits of the physical region includes the angles for any other pair of values 7. That means that the limits give the sharpest restriction. We can write
where g(u) _
f(u) f(u) - 1 In z4
is a monotonically increasing function of u as (B.7) is positively definite. Then (A.5) shows indeed that In 1iy./ and thus 01is a monotonic function of 77.
ACKNOWLEDGMENTS One of us (F.S.) wishes to express his thanks to the University of Minnesota for hospitality and financial support during the time when this work was carried out. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research.
190
SCHLijCL
AND
MEAD
REFERENCES
1. E. RUCH,
Them. Chim. Acta 38 (1975), 167. E. RUCH AND A. MEAD, Theor. Chim. Acta 41 (1976), 95. A. MEAD, J. Chem. Physics 66 (1977) 459. F. SCHL~L, 2. Physik 191 (1966), 81. 1. CSISZAR, Stud. Sri. Math. Hung. 2 (1967), 299. S. KULLBACH. “Information Theory and Statics,” Wiley, New York, 1951. 7. A. RBNYI, “Wahrscheinlichkeitsrechnung,” VEB Deutscher Verlag der Wissenschaffen, 1966. 8. F. SCHLijcL, 2. Physik B 25 (1976), 411. 9. E. RUCH, R. SCHRANNER AND T. H. SELIGMAN, J. Chem. Physics 69 (1978), 386. 10. E. T. JAYNES, Phys. Rev. 106 (1957), 620. 11. R. BAUSCH, 2. Physik 244 (1971), 190. 12. H. K. JANSSEN, 2. Physik 253 (1972), 176. 13. H. MORI, J. Phys. Sot. Japnn 11 (1956), 1029. 14. R. BESSENRODT, 2. Physik 235 (1970), 110; 238 (1970), 258. 15. Y. A. MCLENNAN, in “Advances in Chemical Physics” (I. Prigogine, Ed.), Interscience, 2. 3. 4. 5. 6.
16. 17. 18. 19.
York, 1963. D. N. ZUBAREV, Soviet Phys. Doklad. 10 (1965), 526. F. SCHL~~GL, 2. Physik B20 (1975), 177. B. LESCHE, preprint. E. RUCH AND B. LESCHE, J. Chem. Physics 69 (1978), 393.
Berlin
New
OF PHYSICS
115, 172-190 (1978)
Macroscopic
Aspects of Mixing
Nonequilibrium
Distance in
Thermodynamics
F. SCHL~~GL Chemistry Department, University of Minnesota, Minneapolis, Minnesota 55455 and Institut fiiv Theoretische Physik, Rheinisch- Westfiilisehe Technische Hochschule, Aachen, Germany*
C. ALDEN
MEAD
Chemistry Department, University of Minnesota, Minneapolis, Minnesota 55455
ReceivedOctober11.1977
In this paper,wedescribeseveralfurther developments in the applicationof the principle of increasingmixing character,and the relatedprincipleof decreasing mixing distance, to the time-dependent behaviorof macroscopicsystems.Thereare three main parts to this paper.In the first, somebasicassumptions are discussed which are sufficientfor the validity of the principleof decreasing mixing distance,and the macroscopic formulation thereof.It is shownthat the principleholds,not only for a systemin contactwith a heat bath, but alsoin linearthermodynamics for opensystems with time-independent boundary conditions,and with nonequilibriumsteady-state,rather than equilibrium,solutions. In the secondpart, a newproof is givenfor a resultpreviouslyobtainedby oneof uswhich permitsthe formulationof the principlein termsof observablethermodynamic properties of macroscopic systems.The new proof is more generalthan the old, in that one of the assumptions usedbeforehasbeendropped.Finally, severalsimpleexamples are discussed, which showclearly how this principledirectly yieldsmoredetailedinformationaboutthe time evolutionof a systemthan is affordedby traditional thermodynamics.
1. INTRODUCTION The notion of “mixing character” was introduced by Ruth [l] as a partial ordering of probability distributions over the same set of events; and it was shown that the mixing character of the probability distribution over the microstates in Gibbs’ phase spaceof an isolated systemalways increasesif a master equation is obeyed. This “principle of increasing mixing character” is an essentially stronger restriction on the possible changesof the thermal states of the system than the second law of thermodynamics. The latter can be considered as a special consequenceof this principle. Thus, the secondlaw (increase of entropy for isolated systems)is a necessary,but not a sufficient, condition for increasing mixing character. However, arguments have been * Present,permanentaddress. OOO34916/78/1151-0172$05.00/0 Copyright AU rights
0 1978 by Academic Press, Inc. of reproduction in any form reserved.
172
MIXING
DISTANCE
IN
THERMODYNAMICS
173
given [2, 31 which indicate that, when properly formulated, this principle may be applied to real thermodynamic systems with a generality approaching that of the second law. As shown later on by Ruth and one of the authors [2], a consequence of this principle is that the changes of the thermal states of an open system in a large heat bath are restricted in an analogous way by the decrease of the “mixing distance” of the probability distribution over the microstates in Gibbs’ phase space describing the instantaneous thermal state from the corresponding distribution describing the equilibrium with the heat bath. This theorem is a generalization of the law that the entropy production in the interior of such a system always is positive definite. In a previous paper, one of the authors [3] has derived a criterion for the development in time of thermodynamic variables in a macroscopic system which are allowed by this theorem, thus obtaining a macroscopic thermodynamic theorem. For the development in time of the probability distribution in Gibbs’ phase space of an isolated system, we shall assume the validity of a master equation, i.e., a stochastic equation of motion which is of first order in time and linear with respect to the probability distribution. It then follows that the same has to be true for a system in contact with a large heat bath. Generally the validity of a master equation brings decrease of mixing distance of any solution from the unique stationary solution [2]. Therefore, in that region of linear thermodynamics where the master equation is valid, we can extend the theorem to open systems which are subject to boundary conditions which don’t allow an equilibrium state, replacing the equilibrium state by the steady state corresponding to the stationary solution of the master equation of such a system. This will be made clear in Section 2, where the basic assumptions for the following considerations will be generally formulated and discussed. In Section 3, we give a more general derivation of the macroscopic theorem obtained in Ref. [3], in which the restriction to “many-parameter” systems is dropped. This enables one to apply the theorem without restriction to macroscopic thermodynamic systems, including those characterized by only a small number of parameters. In Section 4, we treat a few examples of this kind, and it is shown that the criterion leads to nontrivial restrictions which are not obtainable from traditional thermodynamics. 2. BASIC For the following, systems:
ASSUMPTIONS
two essential assumptions will be made regarding the physical
(a) validity of a master equation in Gibbs’ phase space; (b) distinct separation between short and long time behavior. The first means the restriction to a certain class of systems. In particular it is fuElled in a regime of linear thermodynamics. The second is a very general restriction to systems for which a description by conventional thermodynamic means is applicable.
174
SCHLdGL
AND
MEAD
For the present, the microstates of the system may be denoted by a discrete subscript 01, corresponding to the division of the phase space into arbitrarily small but finite cells. This is a convenient, though not essential, restriction for the following, and will be abandoned later on. The thermal contact with a large heat bath may be characterized in different ways. Let a: denote the microstates of the system in the heat bath with microstates r. The result that the mixing distance of the probability distribution pal of the system from its equilibrium distribution p ao decreases was obtained in [2] with the help of the requirement that in thermal contact with given pa the mixing character of the probability distribution pra always is maximal. pToris the distribution in the phase space of the system and the bath, which form together an isolated system. This yields the result that pra is independent of r for a given 01if r and 01are compatible. Another characterization of the thermal contact with the heat bath was given earlier by one of the authors [4] by the requirement that the conditional probability p(r 1 a) in (2.1)
be dependent on r, LYonly, and not, for example, on p or the time. It leads to the same result and is fully equivalent with the requirement given in [2]. This leads to the result that the change of po: with time is given by a master equation.
where /l,, is independent of time and of the distribution pa , because the distribution prol of the isolated system certainly is subject to a master equation. (The dot above a letter denotes the time derivative.) If polo is the stationary solution
of the master equation (2.2), then all measures
F(P, ~“1 = c pa”G(p,/p,O) OL
(2.3)
with arbitrary concave functions G(u) never increase in time, as shown in [2]: -qP, P”) < 0.
(2.4)
By a concave function we mean a function G(u) such that the second derivative G (u) is never negative. These measures S already were proposed by Csiszar [5] as a generalization of the relative information measure of p with respect to pa known as “Kullback information” [6] or “information gain” [7] K(P, PO)= C pa Wp,/pmO) a
(2.5)
which is closely related to entropy production [4]. The partial order of the “mixing distance” [2] is defined by the behavior of the totality of all these measures 9 as a generalization of the partial order of the “mixing character” [l]. If (2.4) is fulfilled
MIXING
DISTANCE
IN
THERMODYNAMICS
175
for all concave G(u), then by definition the mixing distance of p from pa decreases. Thus this mixing distance always decreases if the master equation is valid. It may be mentioned that more generally the mixing distance between any pair of solutions of the same master equation decreases [8, 91. The validity of a master equation in Gibbs’ phase space is not restricted to isolated systems and systems in a large heat bath. It also can be postulated for open systems in a region of linear thermodynamics. In the case where the boundary conditions are time independent, yet do not allow a thermal equilibrium state, the stationary solution of the master equation, which is unique, describes a steady nonequilibrium state. We exclude nonergodic systems, in the sense that not all microstates can be reached by a sufficient number of steps starting from a given microstate. Linear thermodynamics is restricted to thermal states which are sufficiently close to an equilibrium state that the dynamical equations can be assumed to be linear in the deviations from equilibrium, to any desired accuracy. It is not sufficient that they are linear for a special choice of quantities describing these deviations. The linearization has to be valid for any choice of these variables. If not only the macroscopic observables, but also the microscopic probabilities, are sufficiently close to their equilibrium values to permit linearization of the equations of motion, then these can be approximated to any desired degree of accuracy by master equationsl. Thus, this strong region of linear thermodynamics implies that the master equation provides a sufficiently accurate description in Gibbs’ phase space, but the converse is not true. It does not follow from the master equation that the stochastic equations over other sets of events obtained, e.g., by elimination of certain state parameters need also be linear. In summary, the master equation is a sufficient condition for the validity of the principle of decreasing mixing distance, and it, in turn, is valid for at least two types of system: isolated systems, or those in contact with a heat bath, regardless of the initial state: and open systems with time-independent boundary conditions and nonequilibrium steady-state solutions, in the region of linear thermodynamics discussed above. Now the assumption (b) will be discussed. The probability distribution pa which gives prescribed mean values (W) of phase space variables A4,” and has maximum entropy S = - C p, In pa (2.6) will be called a “generalized
canonical distribution”: pu = exp[@ + 5 . KJ.
(2.7)
Here the abbreviation c i”K” is used for the summation
over v, which designates the macroscopic
1 A discussion of this is given in Appendix A. 595/IIS/I-12
= 5 . Ma
(2.8) variables (MY)
176
SCHLiiGL
AND MEAD
and their thermodynamic adjoints [, . The subscript 01denotes the value of the phase spacefunctions M, in the microstate 01.In the interpretation of Jaynes [lo] the distribution (2.7) corresponds in an unbiased way to the knowledge of the mean values (MY) only. In thermal equilibrium states the macroscopic variables (MU\ indeed are the only knowledge and therefore the corresponding distribution (2.7) is the appropriate description for thermal equilibrium states in a statistical theory. In extending the scheme to include nonequilibrium states with time- and spacedependent macrovariables (MY) we shall absorb the space coordinates into the set of the subscripts v. The mean values (;M”), and hence also the I&, are thus time dependent. Yet in contrast to the equilibrium statesthe distributions (2.7) do not give the most adequate description in a statistical theory of nonequilibrium phenomena, because they suppress any additional knowledge about the microdynamics of the system. These distributions, which may be called the “accompanying canonical distributions” indeed fail to describe correlations in time, especially of irreversible transport processes[I 1, 121.One can, however, include the essentialknowledge about the Liouville character of microdynamics following a procedure of Mori [ 131,if the system shows a distinct separation into long and short time behavior. This separation can best be expressedby the position of the singularities of the Laplace transforms of autocorrelation functions in time [14]. The distributions which then can be used as an adequate description may be called Mori distributions and are obtained from the accompanying distributions by the transformation which is given by the Liouville dynamics over a time T which is long on the short time scale and short on the long time scale. The resulting distributions also were given, based on different arguments, by McLennan [15] and Zubarev [16]. Not only the (MY) but also another class of quantities belonging to parare equal for the Mori distribution and the accompanying canonical distribution. This is the class of all quantities which are invariant with respect to a canonical transformation in phase space, in particular to reversible dynamics. Let them be called “reversibility invariants” [ 171.To this classin particular belong all measures9 of (2.3). Therefore the assumption (b) for our systemswhich justifies the Mori distribution as a good description and which practically is identical with the applicability of the description by the conventional macroscopic thermal variables gives the following: The principle of decreaseof mixing distance in Gibbs’ phase spacecan be expressed by (2.4) using the accompanying generalized canonical distribution p,pO. 3. THE MACROSCOPIC
CRITERION FOR THE DECREASE OF MIXING
DISTANCE
As in the generalization of the corresponding statement for the mixing character [l], which can be looked on as a special case of mixing distance, the following theorem was proved in Ref. [2]: A sufficient condition for decreaseof the mixing distance of p from p” is that (2.4) is fulfilled for a subclassFA(p, II”) constructed from the concave functions G,(u) with G;(u) = 6(u - A)
(3.1)
MIXING
DISTANCE IN THERMODYNAMICS
177
where h is any real number and S(x) is the Dirac delta-function. If (2.4) is fulfilled for all the functions G, , then it is also fulfilled for an arbitrary concave function G(u). This is easily seen [8] if one writes G”(u)
= 1 dA G”(h) G:(u).
0.2)
The subclass4 represents, in the language of [I] and [2], a “mixing isomorphism”, which gives a more convenient and manageable criterion for the decreaseof mixing distance. This criterion is of practical use, however, only in relatively simple stochastic systems. For general thermodynamic systems, further considerations are necessary to get a criterion for decreasing mixing distance on the Gibbs level which can be formulated in terms of macroscopic quantities only. Such a criterion has been obtained by one of us in a previous paper [3]. Here we give a more general (and, we think, more elegant) proof of this result than was given in Ref. [3]. ln the following, we denote by t the whole set of canonical variables of the system, and by MI(~)the probability density in the &space, i.e., in the Gibbs phase space.We are thus denoting a microstate by a set of continuous variables rather than by a discrete index. Let the steady-state probability distribution be IV,,(<). The deviations of the observables M” from their steady-state average values will be denoted by L”‘.The vu are assumedto have a continuous range of values from minus to plus infinity. The accompanying canonical distribution (2.7) can be written as2 w(f) = IV”(() exp[E + y . r@)].
(3.3)
For fixed We, the D”are phase spacefunctions, and we can introduce the probability density in the space of these functions: W(v) = We(v) exp[B + y . z:],
(3.4)
with Z(y) = --In 1 dr W”(z!) P’.
(3.5)
The Fourier transform of W(v),
satisfies
’ In termsof the continuous variables I, (2.7) takes the form w,(t) = exp[@+ 5 . &f(t)], with the equilibrium distribution being given by w”([) = exp[@’ + 1” . M(f)]. Denoting the steady-state averagesby MO, and defining y = 5 - to, a(f) = M(f) - MO and 8 = @ - @o + y . MO, one immediately obtains (3.3).
178
SCHLijGL
This expression is the generating Thus, we obtain
(pj (dull 0~)
=
[
f &
AND
function
MEAD
for the cumulants
In ri/c(a)] ”
= [- &
of the quantities
= _ !Tfj!$?!! “,=O
ln IQ(~)],=,
(3.8)
Y =
b%(y) -z3$
V.
etc.
(3.9)
An important consequence of (3.7) is that, in the thermodynamic limit, In l%(w) becomes porportional to the size of the system. If V’ is some measure of the size of the system, then we have, as I/ vecomes infinite while keeping average densities constant,
$ [U/V> In W~)l = w(w) where %@(u~)is independent of V. Using (2.3), (2.4) and (3. I), we obtain the following
(3.10)
form for our criterion:
where h(x) is the Heaviside unit function, whose derivative is 6(x). Equation (3.11) must hold for all positive values of h. From (3.3), we see that (w/w”) is a function only of y . u. Using this fact, (3.11) may be rewritten as
where (T = In X - c” takes on all real values. We now make use of the Fourier integral form of the step function (3.13) and also express from (3.7) that
W(v) as a Fourier 1 may,
af&J>
integral with the aid of (3.5). We further
= v(y + iw) - P(y).
note
(3.14)
MlXING
DISTANCE
IN
179
THERMODYNAMICS
Putting all this together, we can transform (3.12) into the form
. [v”(y
$
jw)
-
l)‘,(y)]
e-iw.u
s
<
0
(3.15)
where f is the number of different quantities v”. The v-integration now gives just the f-dimensional delta function, and this in turn makes the w-integration easy. We now find (3.16) where /‘iK = Z,“((l +
iK)Y).
Note that, because of the factor (ui, - v), the integrand no longer has a pole at K = 0. To evaluate the integral (3.16) in the thermodynamic limit, we seek the saddle point of the factor in the integrand
where, because of (3.7), we have
u=h
@(KY)
-
1’KU
=
--s”((l
+
iK)y)
-
8(y)
-
iKU.
(3.17)
To find the saddle point, we require dU/dtc = i(v,, . y - u) = 0.
(3.18)
Now consider the behavior of viM as K moves along the imaginary axis, K = -iq. By inspection of (3.4), one easily convinces oneself that v, . y is a monotonically increasing function of r) which takes on all values physically allowed. Depending on the nature of FVO(v), the “physical range” of 7, that is, the range for which the integrals converge and v, . y is defined may extend from plus to minus infinity or may be restricted in some way. In any case, there is always one and only one 7 in the physical range satisfying (~.I~),so we have found our saddle point. The second derivative of U at the saddle point is d2U/dK2 = - c yu(Av~ Av) yy = g.
UY
(3.19)
g is always negative, so the path of steepest descents is parallel to the real axis, K = --in + q. Because of (3.10), moreover, g and the higher derivatives become proportional to V in the thermodynamic limit. All these derivatives, of course, are
180
SCHLijGL
AND MEAD
expressiblein terms of cumulants at the saddle point. The effect of the higher derivatives in the expansion of CJbecomescomparable with that of g when 4 becomesof the order of unity; by this time, however, becauseof the factor of V in g, the integrand has already become extremely small. As for the rest of the integrand in (3.16), it is a relatively slowly varying function of K, so can be considered constant in the extremely narrow range in which L.c’ is not negligible. In the thermodynamic limit, therefore, we have exactly
=i
2% lP J-p g 1
ena$ c i)“(cn”_ v”). ”
(3.20)
Factoring out inherently positive factors, we find for our criterion: (3.21) It is necessaryand suficient for decreaseof mixing distance that (3.21) be satisfied for all 7 in the physical range. In Ref. [3], the necessity of (3.21) was demonstrated by using convex functions of the form (sgn 7) x”+l/(q + I), which, however, do not in general lead to a mixing isomorphic set of functions, so the sufficiency did not follow in general. The sufficiency was then proved for the thermodynamic limit, but with the additional requirement that the v be arbitrarily large in number. Since our proof does not depend on this last assumption, it is more general than the one given in [3]. Also, the possibility of a bounded physical range was not recognized in [3]. As was also pointed out in [3], the limiting case r] --f 0 is equivalent to taking F =
s
df w ln(MI/r+,“) = K(w, w”).
(3.22)
This is the Kullback information or information gain [lo]. Its decrease, a special consequenceof decreaseof mixing distance, leads to the positivity of entropy production [4] if wo is the equilibrium with t a heat bath: P = -X(w,
w”) = -y
* d 2 0.
In this case, the vy are “fluxes”, and --yy the corresponding “forces”.
4. EXAMPLES In the following, open systemsin a heat bath shall be discussed.This means that they are in thermal contact with their surroundings, which are so large that their temperature, chemical potential and other intensive parameters remain unchanged. The steady state is the thermal equilibrium with the surroundings.
MIXING
DISTANCE
IN
181
THERMODYNAMICS
The more general form of the macroscopic criterion obtained in the last section permits its application also to systems characterized by a small number of parameters, and our examples will be of this type. 4.1. Chemical Reactions in Ideal Mixtures
We consider chemical reactions in an ideal mixture which occur practically isothermally, and so slowly, compared to any diffusion process, that the molar concentration n, of any species v remains homogeneous in space. Let nyo be the equilibrium concentration and dpy be the difference between the chemical potential of species v and its equilibrium value. The possible changes of dp., are restricted by the chemical reaction equations and therefore not independent one from another. We can, however, consider separately the question of which restrictions are given by the requirement of decreasing mixing distance and apply the criterion (3.21) as if the n, were independent. In this way, we find restrictions which must hold for all such mixtures. Afterwards the chemical reaction equations give additional individual restrictions for the special system. Only the general restrictions shall be discussed in the following. The generalized canonical distribution in this case becomes the grand canonical distribution and we get in particular vV=n ” -.-noY >
yy = (l/T)4
(4.1)
= ln(nJny”).
(4.2)
In the last equation the gas constant is set equal to one by appropriate choice of units. This equation directly leads to (nJ, = n,(nJn,0P.
(4.3)
(~9, - 0” = (4, - n, ,
(4.4)
Since
the criterion (3.21) here gives +- 1 [(n,), - n,] 3 = 1rl c [(S)” n, Y Y
-
11 % G 0.
As E(y) and thus the grand canonical partition function of the ideal mixture converges for all real values & , the physical region of 77is the whole real axis. A relation equivalent to (4.5) was obtained in Ref. [2], albeit in a rather ad hoc manner. First we shall discuss the case where only two concentrations n, , n2 are variable. All other participating chemical components may be fixed in concentration by exchange with surroundings of the reaction. Then the criterion reads (4.6)
182
SCHLiiGL
AND
MEAD
Thus, for any given initial values of (n, , IQ, each real value of 7) restricts the direction of (ri, , ti,) to within a half-plane. Requiring that (4.6) hold for two or more different values of 17will restrict (ti r , tiie) to the intersection of two half-planes, i.e., to within an angle < 7~. In particular, we consider the limits of the physical range, 7 = *co, and obtain: (a) For 1 < n&O
< n2/n20:
7ii,< 0, rii, + ?ii,< 0
(4.7)
(b) For nI/nlo < 1 < n,/n,“: rii, < 0, tii, > 0 (c) For nl/nlo < L&O
< 1:
tii, + fi, 3 0, 7i, > 0
(4.9)
and corresponding inequalities if we exchange n1 and n2 . These restrictions to certain angles for the direction (ri, , tiJ of motion are given in Fig. 1 and are in any case sharper than the traditional restriction obtained from the positivity of entropy production:
n?
“1
1. Comparison of the requirements imposed by decreasing mixing distance with those imposed by the second law for the chemical reaction system discussed in Section 4.1. At each point on the closed curve, the second law (positivity of entropy production) allows motion in any direction toward the interior of the curve. Decrease of mixing distance is fulfilled only for motion in directions within the angle enclosed by the arrows. These angles are different for the six points shown, as discussed in this text. The dotted line corresponds to n&z,O = n2/n2”. The equilibrium point is the intersection of the three straight lines. FIG.
MIXING
DISTANCE
IN
THERMODYNAMICS
183
A more detailed analysis (see Appendix B) shows that in this special system the limits of the physical region already give the sharpest restrictions obtainable, i.e., no other value of 71gives a restriction which makes the allowed angles smaller. It should be mentioned that this system is formally related to the case of complete disjunction of only three events if one asks whether an arbitrarily chosen molecule belongs to species 1, 2 or neither one. It therefore shows similar features with this disjunction in respect to the discussion of the conditions for decreasing mixing character as given by Lesche [18], and Ruth and Lesche [19]. The physical interpretation, however, is quite different in the two cases. In the more general case off components, it is convenient to use the subscripts v in such a way that the ratios u, = n”/n,” are rearranged
(4.11)
in an increasing order u~
(4.12)
Then the limits &co for 7 give: (a) 1 < u1 : f rif
<
1%
0,
(4.13)
GO;
v=l
(b) u1 < 1 < u, : ?ii, < 0, ti, 3 0
(4.14)
(c) 24, < 1: f 2
fi,
b
0,
ri,
>,
(4.15)
0.
This shows that, in particular, the concentration with the largest value u > 1 always decreases and that with the smallest u < 1 always increases. This is a result that is not obtained through the traditional condition of positivity of entropy production. The intermediate values of 7 would give still more restrictions, the analysis of which, however, is more complicated.
4.2. Slow Exchange of Heat and Matter with Surroundings at Constant Volume We consider a substance enclosed in a box of fixed volume in a heat bath. An exchange of heat and matter with the bath shall be possible yet is assumed to be so slow that all intermediate thermal states of the system are practically homogeneous in space. With internal energy U and mole number N of the substance in the system we choose VI = uuo (4.16) v2 = N - No,
(4.17)
184
SCHLiiGL
AND
MEAD
where the superscript zero here and in the following examples always characterizes the equilibrium value. The generalized canonical distribution for given mean values U, N is the grand canonical ensemble. Accordingly, with temperature T and chemical potential p we get ,=-f+l
(4.18)
TU
(4.19) 1 T,
L i
1n
T
1 =
-T-w1
=
f
(4.20)
+
qy.2.
We can eliminate v and write (4.22) We shall use the criterion (3.21) in the form
only for the limits of the physical region of 7, becausewe then get very general results independent of individual features of the substance.The limits are given by vanishing T,, when 77has opposite sign to y, and by infinite T, when 71has the same sign as y1 . For infinite T, we get finite (p/T), . The substancebehaves for such values of the thermal variables like an ideal gas. U, is iniinite and < Unlu.
(4.24)
sgnj, = -sgny,.
(4.25)
&IN The criterion gives
For vanishing T, we get Ps
=
(4.26)
-Y2IY1.
Energy U,, goes to zero. If y2/y1 is positive, N,, goes to zero as well and we get sgn(Uj, + W2) = -sgn
y1 .
(4.27a)
If, however, y2/y1 is negative, N,, assumesa nonvanishing value NY , which dependson y21y1, and we get sgn[(N,, - N) j2 - UfJ = -sgn y2 .
(4.27b)
MIXING
185
DISTANCE IN THERMODYNAMICS
It should be stressed that we have assumedthat ,U can take any value, positive or negative. This is true, at least, for all “classical” substances.We have to exclude for
instance a Bose-Einstein gas. The criterion thus gives the restriction (rI , y?)-plane
to certain
of the possible angles as represented in Fig. 2.
directions
(fl , &)
in the
FIG. 2. Restrictions on the time development in the (n , r&plane of the system of Section 4.2 imposed by decreasing mixing distance, using only the limits of the physical range of 7. The direction of the motion must lie within the angle enclosed by the arrows at each of the four points shown. The equilibrium point is at the origin.
4.3. Paramagnet We consider a paramagnet whose magnetization M is a unique function of the ratio H/T of the magnetic field H to the absolute temperature T. Thus, asa consequence of the second law, the internal energy U is a function of T only, and independent of M. The generalized canonical distribution for given mean values U, M is the “magnetic field ensembIe”. We therefore put 211= u - uo
(4.28)
v,=M-M”O
(4.29) (4.30)
y1 is given by (4.18) and yZ becomesequal to (4.19) if p is replaced by H. As H like p can assumeany real value, the physical region for 7 is the same as in the preceding example. Internal energy U and M refer to unit volume. U is the total internal energy, not just the magnetic contribution. The criterion (3.21) has the form (4.23) if we replace N, N, by M, M,, . Independent of individual features of the paramagnet we can assume that U vanishes or becomes infinite simultaneously with T. The magnetization M vanishes with H/T and assumesa saturation value fM, for infinite H/T with the same sign as H.
186
SCHLijGL
AND
MEAD
For the limits of the physical region, the criterion For infinite T, , sgnj, = -sgn yl; for vanishing
leads to the following
results: (4.31)
T, ,
smWl + W h MS)&I = - sgny1
(4.32)
where the upper sign holds if y2/y1 is positive, the lower sign of y2/y1 is negative. The restrictions of the possible directions (& , pZ) are of the same kind as shown in Fig. 2. 4.4. Substance in a Cylinder with Piston We consider an arbitrary substance in a chamber with a piston the motion of which is slow enough that the pressure and temperature in the interior are practically always homogeneous in space. The entire apparatus is taken to be in contact with a heat bath with constant temperature To and pressure PO. The generalized canonical distribution for given mean values of energy U and volume V is the pressure ensemble. Accordingly, we take 2.Q= vP
y2=-7+p
V”
(4.33)
PO
(4.34)
v1 , y1 are the same as in the two preceding examples. We can replace p by -P in (4.19) to get (4.34). Yet P is restricted to positive values. Therefore this example shows a new feature compared to the preceding ones, as yet another condition restricts the physical region of 7. Now not only T, has to be positive but also P =f+($-f)?. c-1 T n In determining cases:
(4.35)
the physical region of T,, , P,, , we have to distinguish
three different
(a) If y2/y1 > P, then at one limit of the physical region T,, vanishes and P, has the value y2/y1 . This means that U, is zero and V, is finite, or zero in the case of an ideal gas. We set V, = V,, . The criterion (3.21) has the form (4.23) if we replace N, N, by V, V, . It gives for this limit: sgn[Uj,
+ (V -
V,) f21 = -sgn
yl.
(4.36)
The other limit belongs to finite T,, and vanishing P,, . For these values of temperature and pressure the substance becomes an ideal gas. Therefore U,, is finite and V,, infinite. The criterion gives then sgnj,
= -sgn
y2.
(4.37)
MIXING
DISTANCE
IN
THERMODYNAMICS
187
(b) If 0 < y2/y1 < P, one limit belongs to vanishing T, and finite P,, as before and gives again (4.36). The other limit, however, belongs to infinite T,, and finite (P/T), . It belongs again to ideal gas behavior, but now with infinite U, and finite V, . The criterion then gives
sgnj,
= -sgny,.
(4.38)
(c) If ye/y1 < 0, one limit belongs to infinite T, and finite (P/T), giving again (4.38), the other limit belongs to finite T, and vanishing P, and gives (4.37). The restrictions in Fig. 3.
of (& , &J to angles in the (rl , y.J-plane for this system is shown
FIG. 3. Analogous to the preceding figure. Typical angles are shown for six regions ranged about the equilibrium point. The dotted line corresponds to ye/y1 = P, and the equilibrium point is at the origin.
In all these examples the criterion (3.21) was applied only for the limiting values q of the physical region. We therefore got necessary but not sufficient conditions for decreasing mixing character on the Gibbs’ level. Nevertheless, these conditions are nontrivial new restrictions for the possiblechangesof the systemin time in comparison to the traditional thermodynamical laws. They are restrictions of the directions of motion in a diagram of the macroscopic variables to certain angleswhich are smaller than rr.
188
SCHLijCL
AND
MEAD
5. CONCLUSION The validity of master equations on the Gibbs’ level is justified for at least two important classesof thermodynamic nonequilibrium processes.The first classincludes the changesin time of a system in a heat bath. The secondclassconsistsof all processes jn linear thermodynamics. As shown in Ref. [2], then, the mixing distance of probability distributions on the Gibbs’ level from the equilibrium state, or from a steady state, always decreases.For macroscopic systems, a criterion is fulfilled which first was given in Ref. [3] and for which a more general proof has been given here. This criterion is expressedby means of macroscopic thermal variables only. Some examples of systems in a heat bath are discussed. The thermodynamic criterion is used for the limiting values of the physical region of the test parameterT which occurs in the criterion. It gives new conditions on the possible processeswhich are essentially more restrictive than the traditional thermodynamic laws. Thus these examples demonstrate that the criterion leads to independent new statements in thermodynamics.
APPENDIX
A: LINEAR
THERMODYNAMICS
AND LINEAR
MASTER
EQUATION
Let the microstate probabilities be given by poL, their steady-state values by p;‘, and let 6, =pol - pI!” . Let the underlying equation of motion for the pe be given by
where L,(p) simply denotes some function of all the p’s. Of course, L,(p”) = 0. If one is sufficiently close to the steady state, therefore, (A.l) may be expanded as (A.21 If the 6, are all sufficiently small, the nonlinear terms become negligible, and (A.2) may be replaced by its linear part: j, = 8, = &&A
= &L&(P,
- PsO)-
Defining -L = 43L’BL,13P,B0 and we finally obtain
which has the form of a master equation.
(A.3)
MIXING
APPENDIX
B:
DISTANCE IN THERMODYNAMICS PROOF OF ASSERTION MADE
189
IN SECTION 4.2
It will be shown that, in the two-dimensional example of chemical reactions in an ideal mixture given in 4.1 the totality of all values 77give no sharper restriction than already given by the limits of the physical region. With II, = n”/n,”
(B-1)
.f(u) = UT
03.2) (B.3)
we write the criterion (4.6) in the form
t
[f(zfp) - I](?& - atii,) < 0.
(B.4)
For a given pair of values n, , n2 the coefficient N is a function of q. Two values of n give in general an angle larger than 7~for the forbidden direction of (tiI , K&).We have to show that 01is a monotonic function of n to conclude that this angle taken for the limits of the physical region includes the angles for any other pair of values 7. That means that the limits give the sharpest restriction. We can write
where g(u) _
f(u) f(u) - 1 In z4
is a monotonically increasing function of u as (B.7) is positively definite. Then (A.5) shows indeed that In 1iy./ and thus 01is a monotonic function of 77.
ACKNOWLEDGMENTS One of us (F.S.) wishes to express his thanks to the University of Minnesota for hospitality and financial support during the time when this work was carried out. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research.
190
SCHLijCL
AND
MEAD
REFERENCES
1. E. RUCH,
Them. Chim. Acta 38 (1975), 167. E. RUCH AND A. MEAD, Theor. Chim. Acta 41 (1976), 95. A. MEAD, J. Chem. Physics 66 (1977) 459. F. SCHL~L, 2. Physik 191 (1966), 81. 1. CSISZAR, Stud. Sri. Math. Hung. 2 (1967), 299. S. KULLBACH. “Information Theory and Statics,” Wiley, New York, 1951. 7. A. RBNYI, “Wahrscheinlichkeitsrechnung,” VEB Deutscher Verlag der Wissenschaffen, 1966. 8. F. SCHLijcL, 2. Physik B 25 (1976), 411. 9. E. RUCH, R. SCHRANNER AND T. H. SELIGMAN, J. Chem. Physics 69 (1978), 386. 10. E. T. JAYNES, Phys. Rev. 106 (1957), 620. 11. R. BAUSCH, 2. Physik 244 (1971), 190. 12. H. K. JANSSEN, 2. Physik 253 (1972), 176. 13. H. MORI, J. Phys. Sot. Japnn 11 (1956), 1029. 14. R. BESSENRODT, 2. Physik 235 (1970), 110; 238 (1970), 258. 15. Y. A. MCLENNAN, in “Advances in Chemical Physics” (I. Prigogine, Ed.), Interscience, 2. 3. 4. 5. 6.
16. 17. 18. 19.
York, 1963. D. N. ZUBAREV, Soviet Phys. Doklad. 10 (1965), 526. F. SCHL~~GL, 2. Physik B20 (1975), 177. B. LESCHE, preprint. E. RUCH AND B. LESCHE, J. Chem. Physics 69 (1978), 393.
Berlin
New