Untangling Complex Systems, page 17
dlnC
p ( x) = −
= − RJ
k [3.118]
T dx
k
dx
Out-of-Equilibrium Thermodynamics
67
If we separate the variables, x, and lnC , and integrate between 0 and L, we obtain that the total k
entropy production P* is
L
C 2, k
*
*
C
P = p ( x) dx = − RJ
,
1 k
k
dlnCk = RJkln
∫
∫
[3.119]
C 2, k
0
C ,1 k
Since C
,
1,
P* is positive. At the stationary state, the entropy produced inside the system is
k > C 2, k
released outside, and, in fact,
deS
diS
= − RJklnC 1, k + RJklnC 2, = −
[3.120]
dt
k
dt
As in the case of heat conduction, at the stationary state, the total entropy of the system is constant:
it is produced inside and released into the environment.
TRY EXERCISES 3.10 AND 3.11
3.5 THE THEOREM OF MINIMUM ENTROPY
PRODUCTION IN LINEAR REGIME
In Section 3.4, we have ascertained that a system, working out-of-equilibrium and in the linear regime, evolves spontaneously towards a stationary state wherein the forces become linear functions of the
spatial coordinates, whereas the flows become constant over time. When such stationary states are
reached, the entropy production of the system is minimized. This statement is known as the Minimum
Entropy Production Theorem (Nicolis and Prigogine 1977). Now, we demonstrate this theorem in
two situations: (1) in the case of a single force and flow and (2) in the case of many forces and flows.
3.5.1 a single force and flow
For the system described in Section 3.4.2, consisting of a vessel of length L (along the x coordinate) and having two reservoirs at its extremes, containing the k-th species at concentration C , and
1,
C
k
2, k
respectively, the entropy production P* due to the diffusion of the k-th species is
L
µ
*
k / T
P = −
∂
Jk
dx
∫
[3.121]
x
∂
0
If we insert equation [3.41] in [3.121], we obtain
L
2
µ
*
k / T
P =
∂
Lk, d
dx
∫
[3.122]
x
∂
0
To find the minimum of P*, we exploit the Euler–Lagrange equation,13 assuming that f = µ k / T and f = ∂(µ k / T) / x
∂ :
d
2 L
(
)
k
=
, d f
0 [3.123]
dx
13 In the calculus of variation, the Euler–Lagrange equation is useful for solving optimization problems. If In is an integral of the form In
b
= ∫ Λ( x, f( x), f( x)) dx, where the integrand Λ is a function of x, f and its derivative with respect to a
x ( f ( x ) = ( f
∂ x
∂
/ )), it is extremized when it is verified the Euler–Lagrange differential equation d ∂Λ
∂
= Λ.
dx f
∂
f
∂
68
Untangling Complex Systems
It means that 2 L
k, d f is a constant (symbol cost. ),
2 Lk, d ∂µ k
lnCk
2 Lk, dR C
=
∂
2 L
k
k, d R
=
∂
= cost [3.124]
T
x
∂
x
∂
Ck
x
∂
Keeping in mind that L
CkDk
k =
, then the final result is
R
C
∂
2 D
k
k
= cost [3.125]
x
∂
This means that J is constant and it confirms that P* minimizes when C is a linear function of x.
k
k
TRY EXERCISE 3.12
3.5.2 The case of more Than one force and one flow
Imagine having a system with two forces ( F and ) and two flows ( and ), that are coupled. An
1
F 2
J 1
J 2
example can be the thermal diffusion, where F is the thermal gradient, whereas is the concentra-
1
F 2
tion gradient. The total entropy production will be
P* = ( J F + J F ) dV
∫ 1 1 2 2 [3.126]
Assume that only F is maintained at a fixed value, whereas is free to change over time. The
1
F 2
system will evolve to a stationary state characterized by a null value for J and a value different
2
from zero for J . This stationary state minimizes the entropy production of the system. Since we
1
are in linear regime
J 1 = L 11 F 1 + L 12 F 2
[3.127]
J 2 = L 21 F 2 + L 22 F 2
Exploiting the Onsager’s reciprocal relation L
, and inserting the definitions of the flows
12 = L 21
[3.127] into [3.126], we obtain
P* = ∫( L 2
2
11 F 1 + 2 L 12 F 1 F 2 + L 22 F 2 ) dV [3.128]
The function linking P* to the two forces is represented in Figure 3.15: It is an elliptic paraboloid.
P∗
F 1
F 2
FIGURE 3.15 Dependence of P* on the two forces F and in linear regime and when the Onsager’s recip-1
F 2
rocal relations are applicable. Note that the equilibrium state for this system is located in a unique point of the
graph, i.e., in the origin of the axes, where both F and are null.
1
F 2
Out-of-Equilibrium Thermodynamics
69
To find the minimum for P* with respect to F , being fixed, we calculate ∂ P* :
2
F 1
∂ F 2
∂ P*
= (2 L 12 F 1 + 2 L 22 F 2 ) dV = 0 [3.129]
∂ F 2 ∫
In fact, the integrand is 2 J . The derivative of
is null when is null. This
2
P* with respect to F 2
J 2
condition represents a minimum of P* because
∂2 P*
= 2 L dV > 0 [3.130]
∂ 2
22
2
F
∫
In the case of thermal diffusion, if the thermal gradient is fixed, the system evolves up to reach the
stationary state characterized by constant heat flow and the absence of the flow of matter.
The result achieved can be generalized to an arbitrary number of flows and forces. When we have
a system with n forces, and n flows, if just the first m < n forces are fixed, the system will evolve to a stationary state where the flows J , J , …, J are null, and the entropy production of the system m+1
m+2
n
is minimized.
In linear regime, a system changes over time in such a way that
dP*
d diS
=
dt
dt
≤ 0 [3.131]
dt
The time derivative of P* becomes null at the stationary state, which corresponds to a minimum
for P*. Since P* minimizes, the stationary state in linear regime is stable. In fact, the function dP*
( )
dt
looks like the free energy G for a system that evolves towards an equilibrium state.
3.6 EVOLUTION OF OUT-OF-EQUILIBRIUM SYSTEMS IN NONLINEAR REGIME
When strong thermodynamic forces are present, they bring a system very far from equilibrium.
These strong forces give rise to flows, which are in nonlinear relation with their causes. It is in the
nonlinear regime that a system can exhibit several amazing dynamical evolutions, which depend
on the initial and contour conditions, as we will discover in the rest of this book. Now, we focus on
chemical reactions.
3.6.1 chemical reacTions
Let us consider an elementary chemical reaction that occurs as it is written in equation [3.132]. It
involves four species, A, B, C, and D that react in a system showing ideal behavior; a, b, c, and d are the stoichiometric coefficients.14
aA + bB = cC + dD [3.132]
The rate of the forward elementary step is
v
a
b
f = k f A
B
[3.133]
14 For an ideal chemical system, the activity or the fugacity of a species is equivalent to its concentration or pressure, respectively. See also Box 3.1 in this chapter.
70
Untangling Complex Systems
whereas the rate of the backward step is
v
c
d
b = kb C
D
[3.134]
and k f and kb are the kinetic constants of the forward and backward elementary steps, respectively.
The net reaction rate is
c
d
a
b
kb
C
D
v = vf − vb = kf
A B 1−
[3.135]
a
b
kf
A B
When the reaction is at equilibrium, v
and v
f = vb
= 0, i.e.,
c
d
k
C
D
f
eq
eq
=
= K [3.136]
k
a
b
b
A
B
eq
eq
K is the equilibrium constant. Therefore,
Q
v = vf 1− [3.137]
K
where Q is the reaction quotient:
C c D d
Q =
[3.138]
A a B b
The chemical affinity A of the elementary reaction is
A = aµ A + bµ B − cµ C − dµ D [3.139]
A a B b
A = a 0
0
0
0
µ
µ
µ
µ
A + b B − c C − d D + RTln
[3.140]
C c D d
A = RTlnK − RTlnQ [3.141]
From equation [3.141], it derives that
Q
A
−
e RT
=
[3.142]
K
Introducing the latter equation in [3.137], we obtain
− A
v = v
RT
f 1 − e
[3.143]
From equation [3.143], we infer that the chemical flow—the rate of the reaction—is in a nonlinear
relationship with the chemical force, which has the ratio ( A/T). However, equation [3.143] is not
a pure thermodynamic definition of the rate because the kinetic term v appears on the right side
f
Out-of-Equilibrium Thermodynamics
71
of [3.143]. When A >> RT, v ≈ v , and v , like any other reaction rate, may depend on some non-f
f
thermodynamic factors, such as the presence of a catalyst or an inhibitor, etc.
3.6.2 The glansdorff-Prigogine sTabiliTy criTerion
How do systems evolve when they work in nonlinear regime?
So far, we have learned that, in general, for an out-of-equilibrium system, the entropy production
is P* = ∑ J . The second principle of thermodynamics lets us state that P* ≥ 0. P* is a function k k Fk
of both the flows and the forces. An infinitesimal variation of P*, dP*, can be due to changes in both flows and forces:
dP*
J
*
*
=
kdFk +
FkdJk = dFP + dJP
∑
∑
[3.144]
k
k
In linear regime, Jk = ∑ L
and L = L . Therefore,
j
k, j Fj
k,j
j,k
d *
*
F P =
Lk
=
∑∑
∑∑ ( )=
=
, j FjdFk
Fjd Lj, kFk
FjdJj dJP
∑
[3.145]
k
j
k
j
j
From Section 3.5, we know that dP* ≤
0. We, finally, infer that
1
d *
*
*
F P = dJ P =
dP ≤ 0 [3.146]
2
Equation [3.146] is strictly valid only in linear regime (de Groot and Mazur 1984).
When a system works in nonlinear regime, its evolution is described by the following inequality
d *
F P ≤
0 [3.147]
Equation [3.147], known as Glansdorff-Prigogine stability criterion, becomes the universal prin-
ciple of evolution for out-of-equilibrium systems, which is verified when the contour conditions do
not change over time (Glansdorff and Prigogine 1954). Let us prove the relation [3.147] in the case
of chemical transformations. Imagine having a system wherein ρ chemical reactions occur. This
system is in physical contact with reservoirs, each containing a specific compound at a constant
temperature, pressure, and chemical potential. The reactor is separated from the reservoirs through
semi-permeable membranes. These membranes allow the transfer of compounds contained into the
reservoirs, but not of the others. For each species k inside the reactor, one of the following two con-
ditions are true: either its chemical potential is constant and fixed by the composition of its reservoir
( μ constant) or its molecules cannot cross any membrane ( d n
k
e k = 0). The reactor is an open system
that is maintained far from equilibrium. The equation defining the time variation of the moles for
the k-th species is
dnk
den
k
=
+ ∑ v , v [3.148]
dt
dt
k ρ ρ
ρ
If we multiply both terms of equation [3.148] for µ k , i.e., the time derivative of the k-th chemical potential, we obtain
dn
d n
µ
k
e k
k
= µ k
+
ν k,ρ µ kvρ
∑
[3.149]
dt
dt
ρ
72
Untangling Complex Systems
The first term on the right side of equation [3.149] is null, because either the k-th molecules do not
cross the membranes denk
( = 0) or their chemical potential is constant over time ( µ
dt
k = 0). Summing
up the relations of the type [3.149] for each chemical component, we achieve
dn
k
µ k
=
ν k,ρ µ k
∑
∑∑ v [3.150]
dt
ρ
k
k
ρ
Reminding the definition of the affinity for the ρ-th reaction, Aρ = −∑ ν , µ , we can rearrange the
k k ρ k
previous equation, [3.150] in the following form
dµ dn ′ dn
dAρ
k
k
k
∑∑ =−∑ vρ [3.151]
dn dt dt
dt
k
k
k
′
′
ρ
When T and P are constant, and it is possible to apply the approximation of local equilibrium, 15 from equation [3.151], it results that −∑
v
ρ ( dAρ
0. 16 Therefore,
dt ) ρ ≥
1
