Untangling complex syste.., p.84
Untangling Complex Systems, page 84
N ν
( , T) c 2 hν N ν(, T) hν
P ν
( , T) =
=
[12.11]
V
6
c
V
3
Since N ν
( , T ) hν/ V represents the energy density of photons with frequency ν, P ν(, T) = u ν(, T) / 3.
If we consider all the photons, whose frequencies are in the range 0,+∞
, the total pressure will be
∞
∞ u(ν, T)
u ( )
tot T
P( T ) = P(ν , T ) dν =
dν =
∫
∫
[12.12]
3
3
0
0
The total energy density u ( )
tot T depends only on T. The total internal energy of the gas of photons
will be
U = Vu ( )
tot T [12.13]
12 The situation is analogous to that of a swimmer when it hits the wall of a swimming pool and turns his direction of motion.
Complex Systems
429
This last equation demonstrates that the gas of photons is not like an ideal gas because its internal
energy depends not only on T but also V. In fact, if we expand the volume of the cavity, the internal energy of the gas of photons increases. But, if dU > 0 because dV > 0, the thermal radiation must absorb heat. In fact, dU = dq − PdV. The consequent entropy change will be
dU PdV
dS =
+
[12.14]
T
Since U is function of T and V, then
U
U
∂
dU = ∂
dT
dV
+
[12.15]
T
∂
V
∂
V
T
If we introduce equation [12.15] into [12.14], we obtain
1 U
1 U
dS =
∂
dT
P dV [12.16]
T
+
∂
+
T
∂
T V
∂
V
T
Equation [12.16] means that (∂ S ∂ T ) = (1 T )(∂ U ∂ T ) and (∂ S ∂ V ) = (1 T )(∂ U ∂ V ) + P .
V
V
T
T
Knowing that (∂2 ∂ ∂ ) = (∂2
S V T
S ∂ T∂ V ), it follows that
∂ 1 ∂ U
1 ∂ U
P [12.17]
∂
= ∂
+
V T ∂ T
V
∂ T
T
T ∂ V T
V
Equation [12.17] can be rewritten as
1
2
∂ U
1 U
1
2
U
P
= −
∂
[12.18]
2
+ ∂
T
V
∂ T
∂
T
+
∂
V
∂
T
T
∂ V
∂
T
∂ T
T
V
It derives that
∂ U
2
∂ P
= T
[12.19]
∂ V
∂
T T
T
V
known as the Helmholtz equation (Kondepudi and Prigogine 1998). By using equations [12.12] and
[12.13], equation [12.19] transforms in
u
2
( )
tot T
1 u
u
tot
( )
tot T = T −
[12.20]
2
T
+
∂
3
T
3 T
∂ V
The latter can be rearranged in
u
4 u
tot
( )
tot T =
∂
T
[12.21]
T
∂ V
If we separate the two variables appearing in [12.21], and we integrate, we obtain the well-known
Stephan-Boltzmann law:
u ( )
4
tot T = β T [12.22]
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Untangling Complex Systems
wherein β = ξ
=
× −
−
−
4
7 56 10 16
3
4
c
.
Jm K is a constant. If we introduce equation [12.22] into [12.12],
we obtain the equation of state for thermal radiation:
β T 4
P =
[12.23]
3
By introducing the definition of U = Vβ T 4 and equation [12.23] into [12.16], we obtain that (∂ S ∂ V) = (4 3) T 3
β and (∂ S ∂ T ) = 4 T 2
β . After separating the variables and integrating, assum-
T
V
ing that S = 0 when T = 0 and V = 0, the definition of entropy is
S = 4 VT 3
β
[12.24]
3
The other thermodynamic functions are enthalpy H
4
4
T
4
H = U + PV = Vβ
β
T + V
= Vβ T 4 [12.25]
3
3
Helmholtz work function A
4
1
A = U − TS = Vβ T 4 − β VT 4 = − Vβ T 4 [12.26]
3
3
and the Gibbs free energy G
4
4
4
G = H − TS = V T − V T 4
β
β
= 0 [12.27]
3
3
We know that Δ A is proportional to the maximum work that a system can perform, and Δ G to the maxi-
mum work, excluded that of the type “pressure and volume” (Atkins and de Paula 2010). From equations
[12.26] and [12.27], it is evident that, at equilibrium, thermal radiation can perform only work of expan-
sion (i.e., of the type “pressure and volume”). In fact, its chemical potential is µ = 0 (equation [12.27]).
TRY EXERCISES 12.6, 12.7, 12.8, 12.9
12.4.2.2 The Fate of the Solar Thermal Radiation and the Climate Change
The sun, with a surface area of ≈6.09 × 1018 m2 (NASA website), emits ≈3.84 × 1026 Joule per sec-
ond in the surrounding space. Its activity is not steady but oscillates with a period of about 11 years.
The exact length of the period can vary. It has been as short as 8 years and as long as 14, and it is
associated with the formation and depletion of black sunspots on its surface. More sunspots deter-
mine an increased solar activity in the level of solar radiation and ejection of solar material. Each
cycle varies dramatically in intensity (NASA website). At the origin of the cycles, there is a flipping
phenomenon of the magnetic north and south poles of the sun.
The amount and fate of solar thermal radiation that reaches the earth regulates the climate of
our planet. The most recent estimate of the average solar irradiance that reaches the outer surface
of the terrestrial atmosphere in one year is (340.2 ± 0.1) Wm−2 (Kopp and Lean 2011). According to
the 5th Assessment Report of the Intergovernmental Panel on Climate Change (IPCC 2013), when
solar radiation crosses terrestrial atmosphere it is partly (22.4%) reflected and scattered (mainly by
clouds and aerosols13) back out into space. Moreover, it is absorbed (23.2%) by the components of
13 Aerosols are colloids made of solid particles or liquid droplets dispersed in air.
Complex Systems
431
the atmosphere. 54.4% of the solar irradiance reaches the terrestrial surface. An average of 7.1% is
reflected back to space. The total solar irradiance reflected back to space (22.4% + 7.1% = 29.5%)
by the earth as a whole is called albedo. The remaining 47.3%(= 54.4%–7.1%) is absorbed. Since
the solar irradiance is not uniformly distributed in space and time, both altitudinal and latitudinal
and longitudinal thermal gradients are generated. These thermal gradients give rise to the convec-
tive motions, winds and oceanic currents. The heat absorbed by water also triggers the water cycle,
whereas the radiation is absorbed particularly by the land becomes a source of infrared thermal
radiation. Such IR radiation is partly reabsorbed by some components of our atmosphere, such as
H O, CO , CH , N O, and so on, causing a greenhouse effect.
2
2
4
2
TRY EXERCISE 12.10
The average temperature of Earth depends on three principal factors:
1. The energy coming from the sun;
2. The chemical composition of the atmosphere;
3. The albedo.
When human activities or volcanic eruptions release greenhouse gases into the atmosphere, they exert
a positive radiative force, which increases the temperature. On the other hand, when aerosols are
unleashed, a significant amount of solar radiation is reflected back to space, and the average terrestrial
temperature decreases (the aerosols exert a negative radiative forcing). According to an ongoing tem-
perature analysis conducted by scientists at NASA’s Goddard Institute for Space Studies (GISTEMP
Team 2017), the average global temperature on Earth has increased by about 0.8°C since 1880. Two-
thirds of the warming has occurred since 1975, at a rate of roughly 0.15°C–0.20°C per decade. There
are two opposite theses about the causes of global warming. The most trusted thesis, proposed by the
Intergovernmental Panel on Climate Change, supports the hypothesis that global warming is due to
the increase of the level of CO in the atmosphere, produced by human activities. In other words, the
2
global warming is anthropogenic. The second thesis (Idso and Singer 2009; Idso et al. 2013), sustained
mainly by the Nongovernmental International Panel on Climate Change, states that global warming is
more likely to be attributable to natural causes than human activities. When our planet warms due to
an increase, for example, in solar activity, there is a positive feedback effect in action. In fact, the con-
sequent melting of the permafrost14 and the greater evaporation of the oceans determine a more massive average content of CO in the atmosphere. Where is the truth? We know that climate is an example
2
of Complex System. Recently, new models, called Earth system models (ESMs), have been proposed,
wherein the climate is studied in terms of the myriad of interrelated physical, chemical, biological, and
socioeconomic processes. These models show that biosphere not only responds to climate change, but
also directly influences the direction and magnitude of climate change (Bonan and Doney 2018). The
dynamics of climate is nonlinear and extremely sensitive to the contour conditions. Therefore, it is not
possible to decide the exact and unequivocal contribution of all the potential factors to global warm-
ing. More powerful computational resources would be useful for better predictions (see Chapter 13).
12.4.2.3 Solar Radiation and Life on Earth
Solar radiation plays a fundamental role in life on Earth. In fact, it is a source of both energy
and information for the living beings. What distinguishes a living being from inanimate matter is
that the former is purposeful (or teleonomic) because it has the functions of surviving and repro-
ducing. 15 For accomplishing their tasks, all known living beings are alike because, irrespective of
14 Permafrost is permanently frozen soil, sediment, or rock.
15 The functions of a human being go beyond survival and reproduction. They extend to love for all the other humans, all the creatures, and, according to the religious wisdom, our Creator.
432
Untangling Complex Systems
form, complexity, time or place, they must capture, transform, store and use energy, and collect,
transduce, process, store and send information. According to the theory of autopoiesis (from
the Greek “αυτός” that means “self” and “ποίησις” that means “creation, production”), a living
being is able to self-maintain and self-reproduce (Maturana and Varela 1980). The information
needed to accomplish their purposes is contained in the DNA and in the surrounding environ-
ment; the energy required to live is furnished by the metabolic network. 16 Before the discovery
of the double helix structure of DNA by Watson and Crick (1953), Schrödinger (1944) reasoned
that the genetic material must be an “aperiodic crystal.” In fact, heredity requires structural sta-
bility and scarce chemical reactivity. Hence, the genetic material must be crystalline. However,
standard crystals are based on the periodic repetition of simple units and thus store small amounts
of information. Therefore, Schrödinger thought that the genetic material must be aperiodic in
order to encode something as complex as a cell. Recently, after sequencing the genome of dif-
ferent living beings, it has been discovered that the number of genes is not enough to specify the
entire structure and organization of an organism (Davies 2012). For example, the human genome
consists of about 22,000–25,000 genes. Such a small number of genes is not enough to specify
the structure of a brain that contains ≈1012 neurons and ≈1016 synaptic connections. Therefore,
the genome is not some sort of blueprint for an organism. In fact, a blueprint has a one-to-one
correspondence between the symbolic representation and the actual object. Instead, a genome is
an algorithm (Davies 2012) for building an organism, and a significant amount of information
contained in a cell or an organism derives from its surrounding environment. Genes constrain the
development of a living being, but they are not alone in determining it. For example, the wiring
of a brain develops and changes over years in response to the external stimuli. Every living being
adapts its inner processes to the features of the environment. The cellular activity can be grouped
into three types of modules (sometimes indicated as networks if a cell is described as a network
of networks): signaling, metabolic and genetic modules (see Figure 12.9). These modules do not
work in isolation, but they are interconnected, although each module maintains its own identity.
The discretization of the cell in modules is possible because each module is chemically isolated
(Hartwell et al. 1999). The chemical isolation comes from spatial localization or chemical speci-
ficity. The connections among modules are guaranteed by components that belong to different
modules. A cell reacts in parallel to many concurrent inputs, and its behavior is not just a func-
tion of the values of its inputs but also their variety, the order in which they arrive, their timing,
Metabolic
Genetic
modules
modules
Signaling modules
Environment
FIGURE 12.9 Discretization of the processes taking place in a cell.
16 Viruses are not alive because they cannot self-sustain. Viruses contain genetic information in the form of DNA or RNA, they are open systems, reproduce, and are subjected to genetic mutations, like any other living being. However, viruses require the metabolic machinery of a living host to produce the energy needed for their replication.
Complex Systems
433
and so forth. A living being, both unicellular and multicellular, is a reactive system maintained
constantly out-of-equilibrium. When this condition vanishes, we have death. 17
12.4.2.4 Solar Radiation as an Energy Source for Life on Earth
All the terrestrial ecosystems are, directly or indirectly, sustained by the energy coming from our
closest star: the sun. In fact, solar radiation drives the photosynthesis of carbohydrates in photo-
autotrophs organisms, such as plants, algae, and phytoplankton that “eat” light.
hν
nH2O + nCO2 → nCH2O + nO2 [12.28]
