Untangling complex syste.., p.23
Untangling Complex Systems, page 23
on a well of potential energy.
Around us, we find many examples of ordered structures in space and/or time. Think about
phenomena like convection, the seasonal cycles, all forms of life, a flock of birds or a school of
fishes, and so on. A natural question raises; are all the ordered structures we find in nature due
to self-assembling phenomena? No, indeed. In fact, most of the ordered structures emerge in
out-of-equilibrium conditions. The Fluctuation Theorem tells us that microscopic systems working
in out-of-equilibrium conditions may violate the Second Law of Thermodynamics but for short
timescales. Fluctuation Theorem does not answer our question exhaustively because even macro-
scopic systems may give rise to ordered structures over long timescales. Suffice it to think about the
evolution of a fertilized egg that originates an extremely ordered living being.
We are in out-of-equilibrium conditions when at least one force is maintained not null. If we fix the
value of the force, the system evolves to a stationary state. At the stationary state, the system becomes
a dissipative structure, because it produces entropy inward, but it discharges that entropy outward:
dSsys
diSsys deS
sys
=
+
= 0 [4.3]
dt
dt
dt
with diSsys dt > 0 and deSsys dt < 0. Of course, the system self-organizes at the expense of the entropy of the surrounding environment, which increases:
dS
env > 0. [4.4]
dt
When the force is weak, we are close to equilibrium; the emerging flow depends linearly on the
force. In the linear regime, we can encounter not only stable states (e.g., in phenomena such as diffu-
sion, heat conduction and electrical conduction), but also unstable (exponential growth) and oscilla-
tory states (in electrical circuits). When the force is so strong that we are very far-from-equilibrium,
we abandon the linear regime, and we enter the non-linear regime. In the non-linear regime, we
can have not only stable, unstable and oscillatory states but also aperiodic states that are extremely
sensitive to initial conditions (see Table 4.1).
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TABLE 4.1
All the Possible States in Three Distinct Regimes: (A) at Equilibrium
or Out-of-Equilibrium, in (B) Linear or (C) Non-Linear Regime
States Regimes
Stable
Unstable
Oscillatory
Aperiodic
(A) Equilibrium
X
(B) Out-of-equilibrium, linear
X
X
X
(C) Out-of-equilibrium, non-linear
X
X
X
X
For non-equilibrium systems, we still lack a general criterion of evolution; there are few prin-
ciples in an evolutionary landscape that is still forming.2 The type of evolution and the principle governing the evolution are extremely sensitive to the kind of constraints and variable parameters.
For instance, in the previous chapter, we have seen that when the constraints are k fixed forces F j (with j = 1, 2, …, k), and the variables are the remaining k + 1, k + 2, …, n forces, the Theorem of the Minimum Entropy Production is the criterion dictating the type of evolution. The system will reach
a stationary state wherein the flows corresponding to the constrained forces reach constant values,
whereas the unconstrained forces ( j = k + 1, k + 2, …, n) adjust to make their corresponding flows zero. Such stationary states are stable because they minimize the entropy production. The symmetry properties of the stationary state depend on those of the force: if the force is isotropic, the flow
will be isotropic; if the force is anisotropic, being an n-th order tensor (with n ≥ 1), the flow may be anisotropic too, in agreement with the Curie’s principle and the Onsager’s reciprocal relations. If the
system in its stationary state is temporarily perturbed, it is pushed away from it. But, left to itself,
the system will spontaneously restore the initial stationary state due to deviation-counteracting
feedback actions. The value of the flow in the stationary state depends on the strength of the force.
The stronger the force, the larger the flow. As long as we are in the linear regime and the presence
of stable stationary states, we run across the so-called thermodynamic branch in the diagram of stability
(see Figure 4.2). The transition from the linear to the non-linear regime is marked by a point.
As soon as we reach this point, having F and J as coordinates in the plot of Figure 4.2, the thermoc
c
dynamic branch is no longer stable, and new solutions of stationary state coexist. Such pivotal point
is called bifurcation because from there usually two new paths open. Often, we cannot predict the
path that will be followed and the state that will exist at the end, because the choice, made by the
system, between two possible routes may be a matter of tiny and hardly detectable fluctuations. It is
tough to have any general principle predicting the type of evolution in very-far-from-equilibrium
conditions. In fact, the Second Law of Thermodynamics is a statistical law in character, rather than
being a dynamical law. The consequence is that it tells nothing about the average rate in which the
2 The study of the behavior of Entropy Production P* for out-of-equilibrium systems becomes relevant for a perspective going “Beyond the Second Law” as shown in a recent book by Dewar et al. (2014). For non-equilibrium systems, there
is not just one criterion of evolution, but there are few principles in a landscape that is still forming. Such landscape is somewhat fragmented and the book by Dewar et al. tries to find connections and construct bridges among isolated
principles. The authors rightly suggest that to make sense of such landscape, one must identify three key aspects of each principle: (1) Which are the variables? (2) Which are the constraints? (3) Which entropy function is being maximized or minimized? With this interpretative key on hand, it is not so puzzling to find out that besides the Theorem of Minimum Entropy Production formulated by Prigogine and presented in Chapter 3, there is also the Maximum Entropy Production Principle, formulated by Ziegler, which is applicable when the constraints are the fixed n forces and the variables are the n fluxes (Martyushev and Seleznev 2006).
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Untangling Complex Systems
Linear regime
Non-linear regime
J
Jc
Thermodynamic
branch
Equilibrium
Fc
F
FIGURE 4.2 Sketch representing the evolution of a generic system from the linear to the non-linear regime.
The origin of the graph represents the stable equilibrium condition wherein F = 0 = J.
entropy of the Universe grows, and also about the probability of statistical fluctuations in energy and
mass flow for which, at least momentarily and locally, ( dS
)
tot dt
/
< 0.
The only expression we have seen for the non-linear regime is the Glansdorff-Prigogine stabil-
ity criterion, mentioned in Chapter 3. It asserts that in the stationary state, P* is minimized only with respect to the forces: d P*
F
≤ 0. However, this criterion cannot be applied so widely because it
requires the validity of the local equilibrium approximation, and it does not tell us anything about
how the system evolves and the stationary state that will be reached among many. Therefore, work-
ing in a non-linear regime, out-of-equilibrium systems evolve most often unpredictably. In the case
of chemical reactions, the final state can be either stationary, periodic, or aperiodic. Such states
can be either stable or unstable. The reason why the non-linear regime is so wealthy with surprises
is that in very-far-from-equilibrium conditions there are not only deviation-counteracting but also
deviation-amplifying feedback actions. It is just the fierce fight between negative and positive feed-
back actions, which gives rise to the self-organization phenomena. Self-organization, sometimes
called “Dynamic Self-Assembling” 3 (Whitesides and Grzybowski 2002), is a symmetry breaking
phenomenon because it consists in the spontaneous emergence of order out of less ordered condi-
tions. The order can regard either the time, the space, or both types of coordinates. In non-linear
regime, the Curie’s principle loses its hold. In fact, the effects may have fewer symmetry elements
than their causes. In other words, the flows may be more ordered (or asymmetric) than their forces.
Self-organized systems have also been called “dissipative structures” because, to maintain their
order, they require a constant supply of energy and matter that is dissipated into heat.
The non-linear regime holds another great surprise: Chaos! Chaos must not be confused with the
disorder. Disorder increases at the equilibrium when the entropy is maximized. Chaos is, instead,
a synonym of unpredictability. A dynamic is chaotic when it is aperiodic and susceptible to initial
conditions. Undetectable differences in the initial conditions may have a considerable impact on the
temporal evolution of the system. We will learn more about Chaos in Chapter 10.
The nonlinear regime is breathtaking. Whereas the properties of systems working in the linear
regime are precisely equal to the sum of its parts, in non-linear regime the principle of superposition
fails spectacularly.
The scrutiny of the behavior of a system moving from the equilibrium to non-equilibrium condi-
tions, crossing the linear regime and diving in the non-linear regime is a marvelous journey. It is
3 Dynamic self-assembly or self-organization refers to the emergence of order in out-of-equilibrium conditions. On the other hand, “Static Self-Assembly” or “Equilibrium Self-Assembly” refers to the appearance of stable spatially ordered structures in equilibrium conditions.
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101
Equilibrium
Linear regime
Non-linear
regime
The Pillars
of hercules
FIGURE 4.3 Journey of Odysseus from the coasts of Italy to the Pillars of Hercules. The Pillars of Hercules
are like a bifurcation from the Mediterranean Sea to the Ocean.
comparable to the epic journey set out by Odysseus who, according to Dante in his 26 Canto of
the Inferno of Divine Comedy, wanted to discover the mysteries beyond the Pillars of Hercules
(see Figure 4.3). He convinced his terrorized crew to follow him by saying “Ye were not form’d to live the life of brutes but virtue to pursue and knowledge high.” Maintaining an equilibrium condition is like staying at the seashore. Working in the linear regime is like sailing the Mediterranean
Sea. Arriving at the Pillars of Hercules is like reaching the first bifurcation. Beyond the bifurcation,
there is the oceanic non-linear regime reserving uncountable surprises.
A bifurcation point marks the transition from the linear to the non-linear regime. After entering
the non-linear regime, we may find many other bifurcations.
4.2 BIFURCATIONS
If, after modifying the contour conditions, we detect changes in the number and/or stability of the
fixed-point solutions for a differential equation (e.g., we assist to a change in the topology of the attrac-
tor), we have found a bifurcation. The simplest bifurcations of fixed-point solutions, which depend on
a single control parameter λ and involve just one variable, are of three types (Strogatz 1994):
dx
saddle-node bifurcation:
= λ − x 2 [4.5]
dt
dx
trans-critical bifurcation:
= λ x − x 2
[4.6]
dt
dx
pitchfork bifurcation:
= λ x − x 3 [4.7]
dt
4.2.1 saddle-node bifurcaTion
The prototypical example of a saddle-node bifurcation is given by the first-order system [4.5].
The fixed points are x 0 = ± λ . When λ < 0, there are no fixed points. When λ = 0, we have one fixed point that is x 0 = 0. When λ > 0, we have two fixed points: one positive and the other negative.
Through the linear stability analysis, we define the properties of the fixed points.
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Untangling Complex Systems
x
0
λ
FIGURE 4.4 Bifurcation diagram for the saddle-node case.
dx
f ( x) =
= f ( x
0 ) + ( x − x 0 ) f ( x 0 ) +… [4.8]
dt
Since f ( x )
0
= 0, equation [4.8] can be rearranged in
dx
∫
(
f x 0 dt [4.9]
x − x 0 ) = ( )∫
After integrating, we obtain:
x x
0
2 0
0
e f ( x t)
x 0 e− x t
=
+
=
+
[4.10]
The solution x 0 = 0 is unstable like the fixed points x 0 = − λ . On the other hand, the fixed points x 0 = + λ represent stable solutions. Figure 4.4 is the bifurcation diagram, and it shows all the possible solutions of fixed points. The solid line represents the stable solutions, whereas the dashed line
represents the unstable fixed points.
TRY EXERCISE 4.1
4.2.2 Trans-criTical bifurcaTion
The normal form of a differential equation originating a trans-critical bifurcation is equation [4.6].
We have two fixed points: x′0 = 0 and x′′0 = λ. If we apply the linear stability analysis (equation [4.9]), we find that
x e t
= λ [4.11]
for x′0 = 0, and
e−λ t
x = λ −
[4.12]
λ
for x′′0 = λ. From equation [4.11], it is evident that the fixed point x′0 = 0 is stable when λ < 0 and unstable when λ ≥ 0. From equation [4.12], it is clear that the fixed point x′′0 = λ is unstable when λ ≤ 0,
whereas it is stable when λ > 0. Therefore, the bifurcation diagram looks like the plot of Figure 4.5.
If we compare Figure 4.5 with Figure 4.4, we notice that in the trans-critical bifurcation diagram the two fixed points do not disappear after the bifurcation, but they swap their stability.
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103
x
0
0
λ
FIGURE 4.5 Bifurcation diagram for the trans-critical bifurcation.
4.2.2.1 From a Lamp to a Laser: An Example of Trans-Critical Bifurcation
A concrete example of trans-critical bifurcation is offered by a simplified model of the laser (Haken
1983; Milonni and Eberly 2010). The word LASER is an acronym that stands for Light Amplification
by Stimulated Emission of Radiation. The electromagnetic radiation emitted by a laser has proper-
ties that are very different from those of the radiation emitted by a lamp. In fact, the laser radiation
is highly monochromatic, polarized, highly coherent (in space and time), highly directional and it
has high brightness.4 The production of laser radiation requires three fundamental ingredients. The first ingredient is the phenomenon of stimulated emission of radiation. For an atomic or molecular
system with two levels, having energies E and , respectively, the absorption of a photon, having
1
E 2
energy equal to hν = E 1 − E 2, determines a jump from the lower to the higher level; the spontaneous decay from level 2 to 1 causes the emission of a photon whose energy is hν = E 2 − E 1. When the system is in its highest level 2, it may also collide elastically with a photon hν and decay to the
lowest level by emitting a second photon that has the same energy and is in phase with the incident
one. The latter is known as a stimulated emission process (see Figure 4.6).
Usually, at room temperature, in the absence of any perturbation, the number of molecules or
atoms at the lowest level (1) is much more abundant than the number of units staying in the higher
level (2) However, the second ingredient required to generate laser radiation is the promotion of an
inversion of the population of the two levels, like shown in Figure 4.7. Such population inversion is feasible by pumping the active materials correctly, 5 either optically or electrically.
The last ingredient to generate laser radiation is the optical cavity. It is a linear device with one
relatively long optical axis (see Figure 4.8). It is delimited by two mirrors perpendicular to the 2
2
2
hν
hν
hν
2 hν
1
1
1
(a)
(b)
(c)
FIGURE 4.6 Schematic representation of the phenomena of absorption of a photon (a), spontaneous emis-
sion of a photon (b), and stimulated emission (c).
4 The brightness of radiation represents its power per unit area, unit bandwidth, and steradian, or just its spectral intensity per steradian.
5 Pumping a system means releasing a lot of energy to the system.
