Untangling Complex Systems, page 22
dF
g
M
=
= ( n − )
1 k 3 IPFM + 4
n k 4 MFP = ( 3
n − )
1 k′3 FM + 4
n k′ F
dt
3
4
P
The steady state solution is ( F , F )
P,s
M,s = (0,0). We may imagine it corresponds to a pre-
liminary situation where Pierre and Marie do not know each other because they have never
met, before. Now, imagine that one day they meet each other, casually. They have their first
impact on each other and prove their first reciprocal emotions. Their first meeting is like a
perturbation to the steady state solution ( F , F )
P,s
M,s = (0,0). To predict the course of the love
story, we use the linear stability analysis. The Jacobian is
f
∂
f
∂
F
∂ P
F
∂
( n 1 − )
1 k′1
n 2 k′
J
s
M
s
=
2
=
g
∂
g
∂ n 4 k′
4
( n 3 − )1 k′
3
F
∂ P
F
∂
s
M s
The trace of the Jacobian is tr ( J ) = ( n 1 − )
1 k′1 + ( n 3 − )
1 k′3 and the determinant is
det ( J ) = ( n 1 − )
1 ( n 3 − )
1 k′1 k′3 − n 2 n 4 k′2 k′4. Now, we can play with our imagination and
choose different combinations of the four coefficients n , , , and the four kinetic
1 n 2 n 3 n 4
constants: k ,′1 k ,′2 k ,′3 k′4. The general solutions are bi-exponential functions of the type: F
λ t 1
2
λ t
P = c 1ν1, Pe
+ c 2ν2, Pe
F
λ t 1
2
λ t
M = c 1ν1, M e
+ c 2ν2, Me
where
λ and are the eigenvalues, and ν
ν1
= ( , P and ν
ν2
= ( , P are the eigenvectors. The
, M )
, M )
1
λ 2
1
ν
2
ν
1
2
values of the coefficients c and depend on the initial conditions. Let us examine a few
1
c 2
cases.
Case 1. Two identical passionate lovers:
n
k
k
k
k
1 = 2, ′1 = 2, n 2 = 1, ′2 = 1, n 3 = 2, ′3 = 2, n 4 = 1, ′4 = 1.
tr( J) = 4; det( J) = 3.
Eigenvalues:
λ
1 = 3; λ 2 = 1.
Out-of-Equilibrium Thermodynamics
93
1
1
Eigenvectors:
ν1 = ; ν = .
1
2
1
−
If the initial conditions are F (0)
(0)
P
= +1, FM = +1, we can calculate the values of the coef-
ficients c and :
1
c 2
F ( )
P 0 =
1
+ = c 1 + c
2
F ( )
M 0 =
1
+ = c 1 − c 2
It derives that c
1 and
0. This means that the love between Pierre and Marie will
1 =
c 2 =
grow exponentially towards complete bliss: F
t
( )
( ) 3
P t = FM t = e . It is worthwhile noticing
that for such a couple if the initial impression is negative, the relationship is designated to
fail. For instance, if the initial conditions are F (0)
(0)
P
= −0.1, FM = −0.1, it derives that
c
0. Therefore, the feeling of dislike will grow exponentially.
1 = −0.1 and c 2 =
Case 2. Two identical lovers who are less passionate than those presented in case 1:
n
k
′ = ,
′ = ,
′ = .
1 = 2, ′1 = 1, n 2 = 1, k 2
1 n 3 = 2, k 3 1 n 4 = 1, k 4 1
tr( J) = 2; det( J) = 0.
Eigenvalues:
λ 2,
0.
1 =
λ 2 =
1
1
Eigenvectors:
ν1 = ; ν = .
1
2
1
−
If initially, the two lovers have a good mutual feeling ( F (0)
(0)
P
= +0.1, FM = +0.1), their
love is destined to grow exponentially. However, their love will grow slower than that in
case 1. In fact, F
2 t
( )
( )
P t = FM t = e .
Case 3. One lover is passionate; the other is cautious:
n
2, k′1 =
1, k′2 =
1, k′3 =
1, k′4 =
1 =
1, n 2 =
1, n 3 =
1, n 4 =
1.
tr( J) = 1; det( J) = −1.
1
5
Eigenvalues:
λ1,2 = ±
.
2
1
1
Eigenvectors:
ν
1 =
; ν =
.
/ 2
/
(
)
( 5 −1) 2
− 5 +1 2
The general solution is
F
λ t 1
2
λ t
P = c e
1
+ c 2 e
5 1
λ
5 1
F
t
1
2
λ t
M =
−
c 1
e
c 2
e
−
+
.
2
2
The dynamics of the love relation is strongly dependent on the feelings the protagonists have at
their first encounter. In other words, the evolution is strongly dependent on initial conditions.
Let us imagine that both Pierre and Marie had good feelings: F (0)
(0)
P
= FM = +1. It derives that
+1 = 1
c + 2
c
5 1
5 +1
+1 =
−
1
c
− 2 c
2
2
Finally,
c
(
/
)
(
/
)
1 =
5 + 3 2 5 > 0, c 2 =
5 − 3 2 5 < 0. This result means that their love is
expected to grow exponentially, as occurred in reality. On the other hand, if their first feel-
ings were not good at all, (for instance F (0)
(0)
P
= FM = −1), then their relationship would
94
Untangling Complex Systems
F
2
M
+1.618
1.5
1
+0.618
v1
0.5
−1
+1
FM 0
FP
–0.5
−0.618
–1
–1.5
−1.618
v2
–2
–2 –1.5 –1 –0.5 0
0.5
1
1.5
2
(a)
(b)
FP
FIGURE 3.25 Plot of the eigenvectors ν1 and ν2 in (a) and plot of the phase portrait in (b) for case 3.
have been destined to fail. In fact, we would have c 1 < 0, and both partners would finish
hating each other. There is also a third possibility. The initial condition is a point that lies
on the eigenvector ν
( )
(
)
2 (see Figure 3.25). For instance, F (0)
= −
+ / .
P
= +1 and FM 0
5 1 2
In such a case, it derives that c 1 = 0 and c 2 = 1. The love relationship fades to indiffer-
ence. In Figure 3.25, there is the phase portrait for the system of case 3. Each vector repre-
sents the evolution from a particular initial condition or point of the phase space.
Case 4. Both protagonists are cautious lovers. However, one gets excited by positive messages
coming from the partner, whereas the other gets discouraged:
n
1, k′1 =
1, k′2 =
1, k′3 =
1 =
1, n 2 =
1, n 3 =
1, n 4 = −1, k′4 =1.
tr( J) = 0; det( J) = 1.
Eigenvalues:
λ1,2 = ± i.
1
1
Eigenvectors:
ν1 =
; ν =
.
− 2
i
i
The love story is oscillatory, as shown in Figure 3.26. The more Pierre shows interest in
Marie, the more Marie wants to run away. But when Pierre gets tired and shows discour-
agement, Marie starts to find him strangely attractive. Her messages of appreciation trigger
a warm up in Pierre, who echoes her sentiments. However, the growing attention by Pierre
makes Marie more cautious, and the relationship oscillates. The outcome of this link is a
PM
0.2
0.2
Marie loves Pierre,
They love
but Pierre does not
each other
FX 0.0
F
0.0
M
They dislike
Pierre loves Marie,
each other
but Marie does not
–0.2
–0.2
0
9
18
–0.2
0.0
0.2
(a)
Time
(b)
FP
FIGURE 3.26 Oscillatory trends of both F and F over time in (a) and the cycle of love in (b).
P
M
Out-of-Equilibrium Thermodynamics
95
never-ending cycle of love and hate as depicted in Figure 3.26. At least, for a quarter of the
entire period, they love each other.
Case 5. The two protagonists do not like each other. However, one lover gets excited when
the other sends signs of even small interest, whereas the other lover gets excited when the
partner looks indifferent.
n
0, k′1 =
,
1, k′2 =
0, k′3 =
k
1 =
0.2 n 2 =
1, n 3 =
0 2
. , n 4 = −1, ′4 =1.
tr( J) = −0.4; det( J) = 1.04.
Eigenvalues:
λ 1 = −0.2 + i; λ 2 = −0.2 − i.
1
1
Eigenvectors:
ν1 = ; ν = .
i 2 − i
The relationship wanes after few oscillations like shown in Figure 3.27.
Of course, we may consider many more situations. To draw the phase portraits for the
love affairs, you may use the following MATLAB code:
[ y 1, y 2] = meshgrid(−2:0.2:2, −2:0.2:2);
n 1 = 2;
n 2 = 1;
n 3 = 0;
n 4 = −1;
k 1 = 1;
k 2 = 1;
k 3 = 0.5;
k 4 = 0.5;
y 1dot = ( n 1 − 1)* k 1* y 1 + n 2* k 2* y 2;
y 2dot = ( n 3 − 1)* k 3* y 2 + n 4* k 4* y 1;
quiver( y 1, y 2, y 1dot, y 2dot)
To calculate the time evolution of Pierre’s and Marie’s feelings, you may use the following func-
tion file in MATLAB:
function dy = love( t, y)
dy = zeros(2,1);
n 1 = 0;
n 2 = 1;
n 3 = 0;
n 4 = −1;
k 1 = 0.2;
0.2
P
0.1
M
0.0
0.0
FX
FM –0.1
–0.2
–0.2
0
10
20
30
–0.1
0.0
0.1
0.2
(a)
Time
(b)
FP
FIGURE 3.27 Trends of both F and F over time in (a) and the damped cycles in (b).
P
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96
Untangling Complex Systems
k 2 = 1;
k 3 = 0.2;
k 4 = 1;
dy(1) = ( n 1 − 1)* k 1* y(1)+ n 2* k 2* y(2);
dy(2) = ( n 3 − 1)* k 3* y(2)+ n 4* k 4* y(1);
The command line to calculate the evolution in the time window [0 250] and from the initial
conditions F (0)
(0)
P
= −0.2, FM = 0.2, will be of the type:
[ t, y] = ode45(‘love’, [0 250], [−0.2 0.2]);
An Amazing Scientific Voyage
4 From Equilibrium up to
Self-Organization through
Bifurcations
Ye were not form’d to live the life of brutes but virtue to pursue and knowledge high.
Odysseus in 26 Canto of the Inferno by Dante
4.1 INTRODUCTION
All the macroscopic transformations that occur in our universe abide by the Second Law of
Thermodynamics. It tells us that the total entropy of the Universe either grows or remains constant if
either a spontaneous or a reversible process occurs, respectively. The entropy of the Universe is the
sum of two contributions: the entropy of the system ( S ) wherein the transformation takes place, and
sys
that of the environment ( S ). An irreversible or spontaneous transformation ceases when it reaches
env
the equilibrium. When a system arrives at the equilibrium, the entropy of the Universe is maximized:
dStot
dSsys dS
env
=
+
= 0 [4.1]
dt
dt
dt
The entropy of the system consists of two contributions:
dSsys
diSsys deS
sys
=
+
= 0 [4.2]
dt
dt
dt
with both diSsys dt = 0 and deSsys dt = − dSenv dt = 0.
After reaching the equilibrium, the number of all the possible microscopic configurations are
maximized, and our information about the microscopic properties of the system is minimized. This
loss of knowledge is usually expressed by saying that disorder has increased. If the system is iso-
lated, the disorder regards only the system. On the other hand, if the system is closed, the disorder
does not necessarily store entirely inside of it. In fact, a closed system that reaches the equilibrium,
can be the theater of self-assembly phenomena, like the formation of crystals, block copolymer
assemblies, and ordered supramolecular structures, such as micelles and micro-emulsions, and oth-
ers (for a list of many examples of self-assembled systems see Grzybowski et al. 2009). At equilib-
rium, all the forces are null, and the system stays indefinitely in that state unless it is temporarily
perturbed. Even if the system is temporarily pushed away from the equilibrium, it spontaneously
recovers its initial state, because the equilibrium is stable. At the equilibrium, the Gibbs free energy
of the system1 (see Figure 4.1) is at its minimum, and whenever the system is moved away from it,
1 We consider Gibbs free energy G when temperature, pressure, and the number of particles are maintained constant.
When temperature, volume, and the number of particles are held constant, the Helmholtz free energy reaches a minimum
at the equilibrium. By the way, at the equilibrium, the power to do work is minimized.
97
98
Untangling Complex Systems
G
ξ
FIGURE 4.1 Profile of the Gibbs free energy ( G) as a function of a hypothetical coordinate ξ for either an isolated or an adiabatic or a closed system. The black arrows represent temporary perturbations pushing the
system out-of-equilibrium; the slight grey curves represent the deviation-counteracting feedback actions driv-
ing the system back to equilibrium (see tick grey arrows).
deviation-counteracting feedback actions draw the system back to the minimum as if it were a ball
g
M
=
= ( n − )
1 k 3 IPFM + 4
n k 4 MFP = ( 3
n − )
1 k′3 FM + 4
n k′ F
dt
3
4
P
The steady state solution is ( F , F )
P,s
M,s = (0,0). We may imagine it corresponds to a pre-
liminary situation where Pierre and Marie do not know each other because they have never
met, before. Now, imagine that one day they meet each other, casually. They have their first
impact on each other and prove their first reciprocal emotions. Their first meeting is like a
perturbation to the steady state solution ( F , F )
P,s
M,s = (0,0). To predict the course of the love
story, we use the linear stability analysis. The Jacobian is
f
∂
f
∂
F
∂ P
F
∂
( n 1 − )
1 k′1
n 2 k′
J
s
M
s
=
2
=
g
∂
g
∂ n 4 k′
4
( n 3 − )1 k′
3
F
∂ P
F
∂
s
M s
The trace of the Jacobian is tr ( J ) = ( n 1 − )
1 k′1 + ( n 3 − )
1 k′3 and the determinant is
det ( J ) = ( n 1 − )
1 ( n 3 − )
1 k′1 k′3 − n 2 n 4 k′2 k′4. Now, we can play with our imagination and
choose different combinations of the four coefficients n , , , and the four kinetic
1 n 2 n 3 n 4
constants: k ,′1 k ,′2 k ,′3 k′4. The general solutions are bi-exponential functions of the type: F
λ t 1
2
λ t
P = c 1ν1, Pe
+ c 2ν2, Pe
F
λ t 1
2
λ t
M = c 1ν1, M e
+ c 2ν2, Me
where
λ and are the eigenvalues, and ν
ν1
= ( , P and ν
ν2
= ( , P are the eigenvectors. The
, M )
, M )
1
λ 2
1
ν
2
ν
1
2
values of the coefficients c and depend on the initial conditions. Let us examine a few
1
c 2
cases.
Case 1. Two identical passionate lovers:
n
k
k
k
k
1 = 2, ′1 = 2, n 2 = 1, ′2 = 1, n 3 = 2, ′3 = 2, n 4 = 1, ′4 = 1.
tr( J) = 4; det( J) = 3.
Eigenvalues:
λ
1 = 3; λ 2 = 1.
Out-of-Equilibrium Thermodynamics
93
1
1
Eigenvectors:
ν1 = ; ν = .
1
2
1
−
If the initial conditions are F (0)
(0)
P
= +1, FM = +1, we can calculate the values of the coef-
ficients c and :
1
c 2
F ( )
P 0 =
1
+ = c 1 + c
2
F ( )
M 0 =
1
+ = c 1 − c 2
It derives that c
1 and
0. This means that the love between Pierre and Marie will
1 =
c 2 =
grow exponentially towards complete bliss: F
t
( )
( ) 3
P t = FM t = e . It is worthwhile noticing
that for such a couple if the initial impression is negative, the relationship is designated to
fail. For instance, if the initial conditions are F (0)
(0)
P
= −0.1, FM = −0.1, it derives that
c
0. Therefore, the feeling of dislike will grow exponentially.
1 = −0.1 and c 2 =
Case 2. Two identical lovers who are less passionate than those presented in case 1:
n
k
′ = ,
′ = ,
′ = .
1 = 2, ′1 = 1, n 2 = 1, k 2
1 n 3 = 2, k 3 1 n 4 = 1, k 4 1
tr( J) = 2; det( J) = 0.
Eigenvalues:
λ 2,
0.
1 =
λ 2 =
1
1
Eigenvectors:
ν1 = ; ν = .
1
2
1
−
If initially, the two lovers have a good mutual feeling ( F (0)
(0)
P
= +0.1, FM = +0.1), their
love is destined to grow exponentially. However, their love will grow slower than that in
case 1. In fact, F
2 t
( )
( )
P t = FM t = e .
Case 3. One lover is passionate; the other is cautious:
n
2, k′1 =
1, k′2 =
1, k′3 =
1, k′4 =
1 =
1, n 2 =
1, n 3 =
1, n 4 =
1.
tr( J) = 1; det( J) = −1.
1
5
Eigenvalues:
λ1,2 = ±
.
2
1
1
Eigenvectors:
ν
1 =
; ν =
.
/ 2
/
(
)
( 5 −1) 2
− 5 +1 2
The general solution is
F
λ t 1
2
λ t
P = c e
1
+ c 2 e
5 1
λ
5 1
F
t
1
2
λ t
M =
−
c 1
e
c 2
e
−
+
.
2
2
The dynamics of the love relation is strongly dependent on the feelings the protagonists have at
their first encounter. In other words, the evolution is strongly dependent on initial conditions.
Let us imagine that both Pierre and Marie had good feelings: F (0)
(0)
P
= FM = +1. It derives that
+1 = 1
c + 2
c
5 1
5 +1
+1 =
−
1
c
− 2 c
2
2
Finally,
c
(
/
)
(
/
)
1 =
5 + 3 2 5 > 0, c 2 =
5 − 3 2 5 < 0. This result means that their love is
expected to grow exponentially, as occurred in reality. On the other hand, if their first feel-
ings were not good at all, (for instance F (0)
(0)
P
= FM = −1), then their relationship would
94
Untangling Complex Systems
F
2
M
+1.618
1.5
1
+0.618
v1
0.5
−1
+1
FM 0
FP
–0.5
−0.618
–1
–1.5
−1.618
v2
–2
–2 –1.5 –1 –0.5 0
0.5
1
1.5
2
(a)
(b)
FP
FIGURE 3.25 Plot of the eigenvectors ν1 and ν2 in (a) and plot of the phase portrait in (b) for case 3.
have been destined to fail. In fact, we would have c 1 < 0, and both partners would finish
hating each other. There is also a third possibility. The initial condition is a point that lies
on the eigenvector ν
( )
(
)
2 (see Figure 3.25). For instance, F (0)
= −
+ / .
P
= +1 and FM 0
5 1 2
In such a case, it derives that c 1 = 0 and c 2 = 1. The love relationship fades to indiffer-
ence. In Figure 3.25, there is the phase portrait for the system of case 3. Each vector repre-
sents the evolution from a particular initial condition or point of the phase space.
Case 4. Both protagonists are cautious lovers. However, one gets excited by positive messages
coming from the partner, whereas the other gets discouraged:
n
1, k′1 =
1, k′2 =
1, k′3 =
1 =
1, n 2 =
1, n 3 =
1, n 4 = −1, k′4 =1.
tr( J) = 0; det( J) = 1.
Eigenvalues:
λ1,2 = ± i.
1
1
Eigenvectors:
ν1 =
; ν =
.
− 2
i
i
The love story is oscillatory, as shown in Figure 3.26. The more Pierre shows interest in
Marie, the more Marie wants to run away. But when Pierre gets tired and shows discour-
agement, Marie starts to find him strangely attractive. Her messages of appreciation trigger
a warm up in Pierre, who echoes her sentiments. However, the growing attention by Pierre
makes Marie more cautious, and the relationship oscillates. The outcome of this link is a
PM
0.2
0.2
Marie loves Pierre,
They love
but Pierre does not
each other
FX 0.0
F
0.0
M
They dislike
Pierre loves Marie,
each other
but Marie does not
–0.2
–0.2
0
9
18
–0.2
0.0
0.2
(a)
Time
(b)
FP
FIGURE 3.26 Oscillatory trends of both F and F over time in (a) and the cycle of love in (b).
P
M
Out-of-Equilibrium Thermodynamics
95
never-ending cycle of love and hate as depicted in Figure 3.26. At least, for a quarter of the
entire period, they love each other.
Case 5. The two protagonists do not like each other. However, one lover gets excited when
the other sends signs of even small interest, whereas the other lover gets excited when the
partner looks indifferent.
n
0, k′1 =
,
1, k′2 =
0, k′3 =
k
1 =
0.2 n 2 =
1, n 3 =
0 2
. , n 4 = −1, ′4 =1.
tr( J) = −0.4; det( J) = 1.04.
Eigenvalues:
λ 1 = −0.2 + i; λ 2 = −0.2 − i.
1
1
Eigenvectors:
ν1 = ; ν = .
i 2 − i
The relationship wanes after few oscillations like shown in Figure 3.27.
Of course, we may consider many more situations. To draw the phase portraits for the
love affairs, you may use the following MATLAB code:
[ y 1, y 2] = meshgrid(−2:0.2:2, −2:0.2:2);
n 1 = 2;
n 2 = 1;
n 3 = 0;
n 4 = −1;
k 1 = 1;
k 2 = 1;
k 3 = 0.5;
k 4 = 0.5;
y 1dot = ( n 1 − 1)* k 1* y 1 + n 2* k 2* y 2;
y 2dot = ( n 3 − 1)* k 3* y 2 + n 4* k 4* y 1;
quiver( y 1, y 2, y 1dot, y 2dot)
To calculate the time evolution of Pierre’s and Marie’s feelings, you may use the following func-
tion file in MATLAB:
function dy = love( t, y)
dy = zeros(2,1);
n 1 = 0;
n 2 = 1;
n 3 = 0;
n 4 = −1;
k 1 = 0.2;
0.2
P
0.1
M
0.0
0.0
FX
FM –0.1
–0.2
–0.2
0
10
20
30
–0.1
0.0
0.1
0.2
(a)
Time
(b)
FP
FIGURE 3.27 Trends of both F and F over time in (a) and the damped cycles in (b).
P
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Untangling Complex Systems
k 2 = 1;
k 3 = 0.2;
k 4 = 1;
dy(1) = ( n 1 − 1)* k 1* y(1)+ n 2* k 2* y(2);
dy(2) = ( n 3 − 1)* k 3* y(2)+ n 4* k 4* y(1);
The command line to calculate the evolution in the time window [0 250] and from the initial
conditions F (0)
(0)
P
= −0.2, FM = 0.2, will be of the type:
[ t, y] = ode45(‘love’, [0 250], [−0.2 0.2]);
An Amazing Scientific Voyage
4 From Equilibrium up to
Self-Organization through
Bifurcations
Ye were not form’d to live the life of brutes but virtue to pursue and knowledge high.
Odysseus in 26 Canto of the Inferno by Dante
4.1 INTRODUCTION
All the macroscopic transformations that occur in our universe abide by the Second Law of
Thermodynamics. It tells us that the total entropy of the Universe either grows or remains constant if
either a spontaneous or a reversible process occurs, respectively. The entropy of the Universe is the
sum of two contributions: the entropy of the system ( S ) wherein the transformation takes place, and
sys
that of the environment ( S ). An irreversible or spontaneous transformation ceases when it reaches
env
the equilibrium. When a system arrives at the equilibrium, the entropy of the Universe is maximized:
dStot
dSsys dS
env
=
+
= 0 [4.1]
dt
dt
dt
The entropy of the system consists of two contributions:
dSsys
diSsys deS
sys
=
+
= 0 [4.2]
dt
dt
dt
with both diSsys dt = 0 and deSsys dt = − dSenv dt = 0.
After reaching the equilibrium, the number of all the possible microscopic configurations are
maximized, and our information about the microscopic properties of the system is minimized. This
loss of knowledge is usually expressed by saying that disorder has increased. If the system is iso-
lated, the disorder regards only the system. On the other hand, if the system is closed, the disorder
does not necessarily store entirely inside of it. In fact, a closed system that reaches the equilibrium,
can be the theater of self-assembly phenomena, like the formation of crystals, block copolymer
assemblies, and ordered supramolecular structures, such as micelles and micro-emulsions, and oth-
ers (for a list of many examples of self-assembled systems see Grzybowski et al. 2009). At equilib-
rium, all the forces are null, and the system stays indefinitely in that state unless it is temporarily
perturbed. Even if the system is temporarily pushed away from the equilibrium, it spontaneously
recovers its initial state, because the equilibrium is stable. At the equilibrium, the Gibbs free energy
of the system1 (see Figure 4.1) is at its minimum, and whenever the system is moved away from it,
1 We consider Gibbs free energy G when temperature, pressure, and the number of particles are maintained constant.
When temperature, volume, and the number of particles are held constant, the Helmholtz free energy reaches a minimum
at the equilibrium. By the way, at the equilibrium, the power to do work is minimized.
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98
Untangling Complex Systems
G
ξ
FIGURE 4.1 Profile of the Gibbs free energy ( G) as a function of a hypothetical coordinate ξ for either an isolated or an adiabatic or a closed system. The black arrows represent temporary perturbations pushing the
system out-of-equilibrium; the slight grey curves represent the deviation-counteracting feedback actions driv-
ing the system back to equilibrium (see tick grey arrows).
deviation-counteracting feedback actions draw the system back to the minimum as if it were a ball
