Untangling Complex Systems, page 12
surrounding. Even in our scientific laboratories, out-of-equilibrium systems are typical. Open sys-
tems are constantly far-from-equilibrium. On the other hand, we may assume that closed, adiabatic
and isolated systems reach an equilibrium state when they are not perturbed, and the effect of the
gravitational field of the earth can be overlooked.
Out-of-equilibrium systems exhibit the power of self-organizing. How is it possible? Do they
violate the Second Law of Thermodynamics?
3.2 DEFINITION OF THE ENTROPY CHANGE FOR
AN OUT-OF-EQUILIBRIUM SYSTEM
To define the entropy change ( dS) for an out-of-equilibrium system in the time interval dt, we dis-
tinguish two contributions (Prigogine 1968):
dS = diS + deS [3.1]
wherein diS is the change of entropy due to processes occurring inside the system as if it were iso-
lated, whereas deS is the change of entropy due to the exchange of energy and/or matter with the
environment. The Second Law of Thermodynamics allows us to state that
diS ≥ 0 [3.2]
41
42
Untangling Complex Systems
The term diS is positive when it refers to irreversible events, whereas it is null for reversible transfor-
mations. On the other hand, deS can be positive, negative, and null. Equations [3.1] and [3.2] hold for
any “macroscopic portion” of the universe. A portion of the universe is considered “macroscopic”
if it contains a number of structural units (for instance molecules) so large to allow to overlook
microscopic fluctuations. The fluctuations are due to the random motion of the structural units and
their interactions with the surrounding environment. For an extensive thermodynamic variable X,
the fluctuations become negligible if they determine a change δX, which is very small compared
to the value of X. The value of X is proportional to the number of structural units, N, whereas δX, in agreement with the law of large numbers, is proportional to N 1/2 (being the root-mean-square).
Therefore, the relative effect of the microscopic fluctuations (δ X / X ) will be inversely proportional to the square root of the number of particles:
δ X
= 1 [3.3]
X
N
If we want to overlook the fluctuations, we must consider “macroscopic portions” of the universe
containing many structural units ( N). However, when we deal with an out-of-equilibrium system,
distinct parts of it may have a different temperature, pressure, chemical composition, et cetera.
When we describe the behavior of the system, it is convenient to partition it into smaller subsystems
that are in internal equilibrium, i.e., have uniform values of their intensive thermodynamic vari-
ables. This approach is called “approximation of local equilibrium.”
In Figure 3.1, there is a schematic representation of a generic out-of-equilibrium system. Inside
this system, there is not a unique value of T, P, and a uniform distribution of the chemicals. The
intensive variables, T, P, and μ (the chemical potential of the k-th species) depend on the spatial k
coordinates ( x, y, z) and the time ( t): T ( x, y, z, t), P( x, y, z, t),µ (
)
k x, y, z, t . To describe the evolution
of the entire system, we partition it in a certain number of smaller “macroscopic” regions (those
labeled by different capital letters in Figure 3.1), within which the conditions of internal equilibrium are verified. The extensive variables of these subsystems can be transformed into densities, dividing
them by the volume of each portion. Instead of having the total internal energy U, the total entropy
S, and the total moles of k, and so on, we will have the density of internal energy ( u = U/ V), the density of entropy ( s = S/ V), the molar concentration of k ( Ck = nk v
/ ), which are intensive variables
that depend on time and spatial coordinates: u( x, y, z, t ), s( x, y, z, t ), C (
)
k x, y, z, t .
Within each subsystem of Figure 3.1, it is possible to apply equations [3.1] and [3.2]. This
approach is known as “local formulation” of the Second Principle of Thermodynamics. For the
entire system, we can write
diStot = diSA + diSB +…+ diSH ≥ 0 [3.4]
H
G
F
E
A
D
C
B
FIGURE 3.1 Partition of the out-of-equilibrium system in eight macroscopic subsystems. The partition must
be performed in such a way that each subsystem is in internal equilibrium, and the microscopic fluctuations
can be overlooked.
Out-of-Equilibrium Thermodynamics
43
The terms d S will be positive or null for each subsystem. It may happen that inside the same
i
subsystem there are transformations reducing entropy, which are “coupled” with transformations
producing entropy. The overall internal balance will determine an entropy growth in agreement
with the Second Law of Thermodynamics. It may also occur that two or more subsystems are
combined by a common transformation: in some of them, the internal entropy decreases, whereas
into the others increases of a larger amount, such that the second principle is not violated.
When we apply the approximation of local equilibrium, we overlook the flows between pairs
of subsystems. Such flows are generated by the inevitable gradients that are present between
subsystems. They are negligible as far as the overall system is not so far from equilibrium condi-
tion. On the other hand, if the system is very far-from-equilibrium, it is not fair to overlook the
gradients and the respective fluxes. In these situations, we need to apply the theory of “Extended
Thermodynamics” (Jou et al. 2010).
Finally, the equations [3.1] and [3.2], which we have introduced at the beginning of this
paragraph, are postulates that are indirectly confirmed by their power of describing the behavior
of out-of-equilibrium systems. To become familiar with them, we are now dealing with some
examples of irreversible transformations, the first of which is the conduction of heat.
3.2.1 heaT conducTion
Imagine having a system consisting of two metallic cylinders, each having a certain tempera-
ture (Figure 3.2): T for the block A and T for block B, with T
. If the surrounding envi-
A
B
A ≠ TB
ronment has another T , different from those of the two metals, we must consider three distinct
env
flows of heat:1
1. Either the heat flow from A to B ( diqB > 0) or the heat flow from B to A ( diqA > 0)
2. The heat flow from the environment to A ( deqA > 0)
3. The heat flow from the environment to B ( deqB > 0)
The overall entropy change for the system is given by
dStot = dSA + dSB [3.5]
dStot = diSA + deSA + diSB + deSB [3.6]
deqB
B
diqB
diqA
A
deqA
FIGURE 3.2 A system consisting of two cylinders, A and B, at two different uniform temperatures, T and A
T , which exchange heat between them and the environment.
B
1 In this book, we choose the convention according to which dq is positive when the heat is absorbed by the system, and dw is positive when the work is performed on the system.
44
Untangling Complex Systems
diqA deqA diqB deq
dS
B
tot =
+
+
+
[3.7]
TA
TA
TB
TB
From the energy conservation law, it derives that diqA = − diqB = dq. Therefore,
1
1
diStot = dq
−
≥ 0 [3.8]
TA TB
From equation [3.8], it derives that if A is warmer than B ( T
), heat is released from A to B,
A > TB
then dq < 0. On the other hand, if T
, the warmer B will release heat to the colder A, then
B > TA
dq > 0. Of course, if A and B are at the same T, there is no net exchange of heat, then dq = 0. If we divide both terms of equation [3.8] by the unit of time, we achieve the definition of entropy production P* (remember equation [2.19]):
*
diStot
dq 1
1
P =
=
−
[3.9]
dt
dt
TA TB
It is evident that the entropy production corresponds to the product between two distinct terms:
(1) the flow of heat, which is the rate of the irreversible process; and (2) (1/ T
), which is the
A − 1/ TB
cause, or the “force” of the irreversible process since its sign rules the direction of the flow.
If the thermal gradient is not maintained, as the heat flows from the warmer to the colder block,
the difference between the two temperatures eventually vanishes. As soon as the two subsystems
have the same T, the system is at equilibrium, and there is no net exchange of heat between the
two blocks.
3.2.2 chemical reacTions
Another example of an out-of-equilibrium system is schematically depicted in Figure 3.3. It is an
open system exchanging energy and matter with its environment, performing mechanical work
(because the boundaries are assumed to be flexible) and hosting a reaction in its interior. The inter-
nal energy change is
dU = TdS − PdV +
µ kdink +
µ kdenk
∑
∑
[3.10]
k
k
wherein dink and denk are the changes in the moles of the k-th species due to the internal chemical reaction and the influx of matter, respectively.
aA + bB → cC + dD
denk
FIGURE 3.3 Scheme of an open system hosting a chemical reaction and exchanging matter with the
environment.
Out-of-Equilibrium Thermodynamics
45
The last two terms of equation [3.10] represent chemical work. The symbol µ k is the chemical
potential of the k-th species.
BOX 3.1 CHEMICAL POTENTIAL
The chemical potential for a k-th species is
µ
G
k =
∂
n
∂ k T, P, nj≠ k
It represents how the Gibbs free energy of the system varies when the number of moles of
k-th species changes at T, P constant and when the moles of all the other species present
in the system do not change. In the case of just one species, μ is the molar free energy at
k
specific T and P, which is the chemical energy available to carry out work. The equation
that links the chemical potential of a species to its concentration or pressure is
µ
0
k = µ k + RTlnCk
µ
0
( 0
/
)
k = µ k + RTln
k
P P
wherein µ0 k is the chemical potential in standard conditions (i.e., at P 0 = 1 atm and temperature T), and C and P are the concentration and pressure of k-th species, respectively. The previous
k
k
equations are valid when the solid, liquid or gaseous solutions show ideal behaviors. In the
case of real solutions, the concentration must be substituted by the activity ak = γ kCk and the
pressure by the fugacity fk = γ kPk in the definition of µ k . The coefficients γ of activity or k
fugacity quantify how much the real solution deviates from the ideal case.
The changes in the moles of the different species involved in a chemical reaction can be expressed
as dink =ν kdξ, where ν k is the stoichiometric coefficient of the k-th compound (being positive for a product and negative for a reagent), whereas dξ is the change in the extent of reaction. Therefore:
dw =
ν kµ kdξ +
µ
[3.11]
kdenk
∑
∑
k
k
The total variation of entropy for the system of Figure 3.3 can be split in its two contributions, diS
and deS, according to equation [3.1], and we obtain:
dU
pdV
1
deS =
+
−
µ kden
∑
[3.12]
T
T
T
k
k
1
diS = −
ν kµ kd
∑ ξ [3.13]
T k
The definition of the chemical work associated with the chemical reaction can be further simplified,
introducing the concept of “chemical affinity” A (proposed for the first time by Théophile De Donder):2
A = −
k k
∑ν µ [3.14]
k
2 Théophile De Donder (Bruxelles 1872–1957) was a mathematician and physicist, founder of the Belgian school of thermodynamics.
46
Untangling Complex Systems
Reagents
Products
Potential energy
Chemical potential
FIGURE 3.4 Analogy between a ball sliding over a potential well and a chemical reaction wherein the
reagents at high chemical potentials convert to the products at lower potentials.
The chemical affinity A is the difference between the sum of the chemical potentials of the reagents
and the sum of the chemical potentials of the products; wherein every chemical potential is mul-
tiplied by its stoichiometric coefficient. When A > 0 (it means the sum of the chemical potentials
of the reagents is larger than that of the products), the reagents spontaneously transform to the
products. The species at higher potentials transform into other species having lower potentials. This
behavior is similar to the phenomenon of a ball that slides along a well from a point at high potential
energy to the minimum of the well (see Figure 3.4).
Introducing the definition of A, from equation [3.14], into equation [3.13], we obtain:
A
diS = dξ [3.15]
T
Therefore, the entropy production for the system of Figure 3.3 is
ξ
*
diS
A d
P =
=
≥ 0 [3.16]
dt
T dt
Analogously to what we learnt in the case of heat conduction, the entropy production in [3.16]
results the product of two terms: (1) ( dξ d
/ t ) = v, which represents the overall rate of the reaction,
and can be conceived as a “flow” of matter; (2) A/T, which rules the direction and the extent of the
flow of matter, i.e., it is the force generating the flow. From equation [3.16], it derives that if A > 0, then v > 0, i.e., the reagents convert to the products. On the other hand, if A < 0, then v < 0, i.e., the products transform into the reagents. Finally, if A = 0, then v = 0. This latter condition corresponds to the state of chemical equilibrium, wherein the rate of reagents-to-products transformation is
equal to the rate of the products-to-reagents transformation.
If more than one chemical reaction proceeds inside the system, the overall entropy production
will be the sum of the contributions of each reaction:
*
diS
Aj d
P
j
=
=
≥
∑ ξ 0 [3.17]
dt
T dt
j
Usually, each reaction gives a positive contribution to the entropy production. However, there are
exceptions. For instance, in case of two simultaneous reactions occurring inside the same system,
it may happen that:
A ξ
ξ
ξ
ξ
1 d 1
A 2 d 2
A 1 d 1 A 2 d
2
> 0 and
< 0, but
+
> 0 [3.18]
T dt
T dt
T dt
T dt
Equation [3.18] shows that one reaction goes unexpectedly uphill, whereas the other goes down-
hill. However, the sum of the entropy production due to the two reactions is positive. Situations
Out-of-Equilibrium Thermodynamics
47
similar to that of equation [3.18] are possible when the two reactions are coupled, i.e., they have
two or more species in common. 3
3.2.3 diffusion
Let us imagine having a tank divided into two portions, A and B, containing only the k-th species
but at two different concentrations: C in A and C in B (see Figure 3.5).
k,A
k,B
If the system is closed, delimited by rigid walls, and at a uniform temperature, the only irrevers-
ible process occurring inside it is the diffusion of k-th molecules. The internal change of entropy
will be given by:
