Untangling complex syste.., p.8
Untangling Complex Systems, page 8
that must be won. Moreover, it describes the main characteristics of Complex Systems. Chapter 13
proposes strategies to untangle Complex Systems and try to win the Complexity Challenges.
Lastly, this book is completed by five appendices that are useful tools for students and researchers.
Appendices A and C face the numerical solution of differential equations and the Fourier transform
of time series, respectively. Appendix B is a short introduction to the Maximum Entropy Method.
The last two Appendices give instructions on how to deal with experimental (D) and computational
(E) errors.
Finally, with this book, I want to achieve another goal. I want to stimulate public and private
funding agencies to sustain interdisciplinary projects and to not be afraid of providing money to
Dionysian scientists engaged in “Untangling Complexity.” As the Hungarian biochemist Albert
Szent-Györgyi (who won the Nobel Prize in Physiology or Medicine in 1937) said in his funny
letter to Science (1972), scientists can be divided into Apollonians and Dionysians. An Apollonian
“tends to develop established lines to perfection, while a Dionysian rather relies on intuition and
is more likely to open new, unexpected alleys of research.” The Apollonian sees the future lines of
his research and has no difficulty writing a systematic and classical project. Not so the Dionysian.
He knows only the direction in which he wants to go; he wants to explore the unknown, and he
has no idea what is going to discover. In fact, intuition, which guides a Dionysian, is “a sort of
subconscious reasoning, only the end result of which becomes conscious.” Therefore, “defining
the unknown or writing down the subconscious is a contradiction in absurdum.” Hopefully, this
book will be useful for preparing the next generations of both Dionysian and Apollonian scien-
tists. The former will make the awaited inductive jumps for the formulations of axioms regarding
Complexity. The latter will deduce systematically all the theorems and propositions necessary
to predict the behavior of Complex Systems and will perform the experiments for confirming
the new theories. Together, they will contribute to finding innovative solutions to the Complexity
Challenges.
20
Untangling Complex Systems
1.4 KEY QUESTIONS
• What promotes the development of science and technology?
• What are the two gateway events in the journey of humankind to discovering nature?
• What are the principal achievements of the “Practical Period”?
• Why was philosophy born in ancient Greece?
• Which are the intellectual and spiritual pillars inspiring philosophy?
• Who paved the way for the experimental method?
• When and where did the experimental method sprout?
• What are the four epistemological pillars established by the formulation of Classical
Physics?
• Which theories induced to debunk the four epistemological pillars?
• How does the scientific knowledge evolve?
• What are Natural Complex Systems?
• What are the “Natural Complexity Challenges”?
1.5 KEY WORDS
Philo-physicists; Physical Technology and Social Technology; Gateway Events; Philosophy; Etiology;
Ockham’s razor; Reductionism; Uniformitarianism; Determinism; Mechanism; Completeness and
Consistency of Mathematics; Solvability and Tractability of a Computational Problem; Induction;
Deduction; Abduction.
1.6 HINTS FOR FURTHER READING
• More information about the origin of languages can be found in the book by Cavalli-Sforza
(2001) and the book by Carruthers and Chamberlain (2000).
• Norbert Wiener (1894–1964) was convinced that to understand Complex Systems it is
necessary to focus on the concepts of information, communication, feedback, control, and
purpose or “teleology.” He is considered the originator of cybernetics that is the discipline
of control and communication theory (Wiener 1948).
• Among other attempts at formulating a general theory for Complex Systems, it is worth-
while mentioning the General System Theory by the biologist Ludwig von Bertalanffy
(1969), Synergetics by the physicist Hermann Hacken (2004), Self-Organized Criticality
by the physicist Per Bak (Bak et al. 1987) and the theory of Local Activity by the electrical
engineer Leon Chua (Mainzer and Chua 2013).
• Examples of books on Complexity, which include philosophical considerations, are those
by Prigogine and Stengers (1984), Mainzer (2007), Capra and Luisi (2014).
Reversibility or Irreversibility?
2 That Is the Question!
If you do not ask me what time is, I know it. If you ask me, I do not know.
Augustine of Hippo (354–430 AD)
Life can only be understood backwards, but it must be lived forwards
Søren Kierkegaard (1813–1855 AD)
2.1 INTRODUCTION
A “Really Big Question” (RBQ) in philosophy is the one spoken by Hamlet in the homonymous
play written by William Shakespeare (1564 – 1616 AD):
“To be or not to be? That is the question.”
Analogously, a “Really Big Question” in science is the following:
“Reversibility or Irreversibility? That is the question.”
In other words, are the transformations in nature reversible or irreversible?
If we reflect on our life, we do not doubt to answer that the events are irreversible. Suffice to think
about our unrepeatable experiences of everyday life and to consider the unequivocal existence of
an arrow of time in the evolution of the universe and the history of the humankind. However, if we
study the fundamental laws of physics, we discover that most of them tell us that transformations in
nature are reversible. Physics can be divided into four main disciplines:
• Classical physics that deals with the mechanics of macroscopic bodies having speeds
much smaller than that of light (being about c ≈ 3×108 m/s)
• Quantum physics, whose purpose is the description of the mechanics of microscopic bodies
• Relativistic physics, concerning the motion of both macroscopic and microscopic entities
whose velocities approach that of light
• Thermodynamics, dealing with the transfer of energy from one place to another and from
one form to another
All these different disciplines of physics are based on three Conservation Laws:
1. Conservation of Mass
2. Conservation of Energy
3. Conservation of Charge
The famous Einstein’s formula,
E = mc 2 [2.1]
21
22
Untangling Complex Systems
reveals that the Conservation of Mass is included in the Conservation of Energy.1 Classical,
Quantum, and Relativistic physics present only reversible processes, i.e., transformations wherein it
is possible to go from one state A to another state B, but also from B to A, without limitations. The
time-reversibility or, we could even say, the time-reversal symmetry of physics is a consequence of
the Conservation Laws.
The puzzle is that we continuously ascertain the irreversibility. If a glass falls from our hands,
it shatters irreversibly; wear and tear of objects are irreversible; aging and dying are exhibited by
living beings, but also by inanimate objects such as the stars in the universe.
Thermodynamics is the only discipline that does not ignore the irreversibility observed in nature
and tries to explain it. As Ilya Prigogine (Nobel laureate in chemistry in 1977 for his work on dissipa-
tive structures and irreversibility) and Kondepudi (1998) claimed: “Science has no final formulation.
And it is moving away from a static geometrical picture towards a description in which evolution
and history play essential roles. For this new description of nature, thermodynamics is basic.” Even
Einstein (Schilpp 1979) had high regard for Thermodynamics: “A theory is more impressive the
greater the simplicity of its premises is, the more different kinds of things it relates, and the more
extended its area of applicability. Therefore, the deep impression which classical thermodynamics
made upon me. It is the only physical theory of universal content concerning which I am convinced
that, within the framework of the applicability of its basic concepts, it will never be overthrown.”
2.2 THE THERMODYNAMIC APPROACH
According to the thermodynamic approach, the description of a natural phenomenon requires the
partition of the universe into two parts: a “system” that is the theatre of the transformation, and what
surrounds the system, which is called “environment.”
The system can interact or not with the environment; it depends on the properties of its boundar-
ies. In fact, a system can be
• Open when it exchanges matter and energy with its environment
• Closed when it exchanges energy but not matter with the environment
• Adiabatic when it transfers neither matter nor heat
• Isolated when it exchanges neither matter nor energy
At every instant, the thermodynamic state of a system can be described as determining the values of
macroscopic variables such as temperature T, pressure P, volume V, surface Ar, chemical composition. When we define the chemical composition of a system, we need to specify the compounds that are
present (labeled as i = 1, …, n) and their concentrations: χ i (∀ i = …
1, , n). If there is the same value of
T, P, and χ i i
∀ = …
1, , n, in every point of the system, then it means that the system is at equilibrium.
Otherwise, it is out of equilibrium. Table 2.1 reports two states, labeled as A and B, regarding four different systems. Read it, carefully. Now, I ask you: “Is it possible that state A precedes state B for
all the systems? Is it possible the opposite, i.e., state B precedes state A?”
According to the First Principle of Thermodynamics, which applies the conservation of energy, it
is possible that state A comes first, but also the opposite, i.e., state B precedes state A. However, our
daily experience induces us to admit that states A surely come first for all the four systems listed in
Table 2.1. State A evolves spontaneously to state B, but not vice versa. It means that there is an arrow of time: A→B. The direction of the spontaneous evolution can be predicted merely by considering
the thermodynamic properties of the two states involved in the transformation. Therefore, there will
be a state function distinguishing between spontaneous and nonspontaneous changes.
1 Unequivocal experimental proofs of the mass-energy equivalence come from nuclear reactions.
Reversibility or Irreversibility? That Is the Question!
23
TABLE 2.1
Examples of Four Systems in Two Different States: A and B
System
State A
State B
Two identical pieces of Al, labeled
The two pieces have
The two pieces have the same
as 1 and 2, are maintained in an
different temperatures:
temperature: T 1 = T 2 = 35°C
isolated container
T 1 = 30°C, T 2 = 40°C
An isolated container is divided
V contains pure water;
and contain water and sugar
1
V 1
V 2
into two identical volumes:
V contains water with
at
2
C = 0.5 M
V
sugar at
1 = V 2 = V
C = 1 M
A reactor having isolating walls
Water and Na are inside
NaOH, H , and water are inside
2(g)
the reactor at T
the reactor at T
A
B > TA
A computer
N bitsa are stored in
(N/2) bits are stored in the memory
memory
and (N/2) bits are present as (N/2)
kTln 2 of thermal energy
a One bit (Binary unIT) is the basic unit of extent of information.
This state function is entropy, which is introduced in the Second Principle of Thermodynamics. There
are different definitions of entropy (its symbol is S). I present some of them in the next three paragraphs.
2.2.1 The classical definiTion of enTroPy
During the Industrial Revolution, the first definition of entropy was formulated by the German
physicist Rudolf Clausius in the nineteenth century. At that time, entrepreneurs and scientists were
striving to optimize the efficiencies of heat engines. A heat engine (see Figure 2.1) is a machine that transforms heat into work. The engine takes heat q from a hot reservoir. Part of this heat is
h
transformed into work ( w). Another part ( q − w
) is squandered as heat in a cold reservoir. The
h
= qc
efficiency of the thermal machine is
w
qh q
η =
=
− c . [2.2]
qh
qh
Equation [2.2] does not take into account the possible reduction of the efficiency due to frictions or
other defects functioning in the engine.
If there is a thermal gradient between the two reservoirs (Figure 2.1), then it is still possible to carry out work. William Thomson (Thomson 1852) extrapolated this evidence to a universal scale.
Cold reservoir ( Tc)
qc
Heat
w
engine
qh
Hot reservoir ( Th)
FIGURE 2.1 Scheme illustrating the working principle of a thermal machine.
24
Untangling Complex Systems
He proposed the idea of the heat death of the Universe. The Universe is asymptotically approaching
a state where all energy is evenly distributed, and its temperature is uniform. In this state, it is not
possible to perform any kind of work and any activity is suppressed.
According to Clausius’ definition, the entropy of a system increases if it absorbs heat, and the
lower the temperature, the larger the increment of S:
dq
dSsys =
. [2.3]
T rev
In [2.3], dq is the differential heat absorbed by the system in a reversible ( rev) manner. Any reversible transformation is an ideal process because it is a time sequence of equilibrium states. Only
extremely slow transformations can approach the idea of thermodynamic reversibility. A reversible
transformation looks like the slow motion of a sloth. It is convenient to refer to reversible transfor-
mations because we can describe them by using the principle of equilibrium thermodynamics.
The entropy of the universe increases in any irreversible or spontaneous transformation, whereas
it remains constant in any reversible process. In mathematical terms, this is stated as:
dSsys + dSenv = dSuniv ≥ 0. [2.4]
The inequality [2.4] is the mathematical formulation of the Second Principle of Thermodynamics.
The entropy of the Universe (which is an isolated system) unrelentingly grows due to irreversible pro-
cesses, whereas it does not change if only reversible events occur. An isolated system evolves towards
the equilibrium where the entropy of the system is maximized. If this system is not perturbed anymore,
it will not change its macroscopic properties over time: it has exhausted all its capacity for change.
As far as the four systems of Table 2.1 are concerned, the spontaneous transformations are those
from state A to state B. They are the following ones: (1) the flow of heat from the warmer to the
colder piece of Al; (2) the diffusion of sugar from the concentrated solution to pure water; (3) the
partial degradation of chemical energy into heat; (4) the degradation of potential energy of bits
into heat. Each of these four processes determines a maximization of the entropy of the universe.
The reverse transformations, B→A, are not allowed, and in fact, they are in contradiction with the
Second Principle of Thermodynamics.
2.2.2 The sTaTisTical definiTion of enTroPy
In the second half of the nineteenth century, James C. Maxwell (Scottish mathematician and
physicist), Ludwig Boltzmann (Austrian mathematician and physicist), and Josiah Gibbs
(American engineer, chemist, and physicist) interpreted the laws of classical thermodynamics
through the laws of statistical mechanics applied to gases. In statistical mechanics, physical
variables, referred to macroscopic systems, are related to the properties of their constitutive
elements, i.e., atoms and molecules. For instance, the temperature of a gas is a macroscopic
parameter related to the kinetic energy of the atoms. If the atoms move faster in average, i.e.,
their average kinetic energy increases, then the temperature of the gas rises. Statistical mechan-
ics combines probability theory (essential for dealing with large populations) with the laws of
classical or quantum mechanics.
The features of an ensemble of many atoms or molecules in an isolated system are described
through the concept of microstates. The definition of entropy according to the statistical thermody-
