Untangling Complex Systems, page 88
• Which are the factors affecting the climate of the Earth?
• What is life?
• How does life exploit solar radiation as a source of energy?
• How does life exploit solar radiation as a source of information?
• Where does the complexity of human vision come from?
• What is an emergent property and how does it originate?
• How many kinds of emergent properties do you know?
12.6 KEY WORDS
Natural Complex Systems; Computational Complexity; P, NP, and NP-complete Problems; Regular
Networks; Small-World Networks; Random Networks; Scale-free Networks; Modular Networks;
Hierarchical Networks; Thermal radiation; Biomolecular Information System; Neural Information
System; Immune Information System; Social Information System; Fuzzy set; Bayesian probability;
Multiplicity; Interconnection; Integration; Emergent properties.
Complex Systems
447
12.7 HINTS FOR FURTHER READING
• The subject of Computational Complexity can be studied more deeply by reading the book
by Garey and Johnson (1979) and that by Papadimitriou (1994). The description of some of
the principal NP problems can also be found in (Lewis and Papadimitriou, 1978).
• For those who want to deepen their understanding of networks, I suggest the book by
Caldarelli and Catanzaro (2012) and the recent book by Barabási (2016), which is freely
available at the link http://barabasi.com/networksciencebook/. Recent exciting investigations on the phenomena of synchronization in large networks are presented in a special
issue of the journal Chaos, edited by Duane et al. (2017).
• Whoever is interested particularly about Complexity in ecology, I suggest the paper by
Montoya et al. (2006) and that by Sugihara et al. (2012).
• The books by Haken (2006) and Walker et al. (2017) are relevant for whom wants to learn
more about the role of information in living beings and Complex Systems.
• A program for large network analysis is available at http://mrvar.fdv.uni-lj.si/pajek/
• A list of databases for biochemical networks is available in (Subramanian et al. 2015).
12.8 EXERCISES
12.1. Compare the time required to solve three polynomial problems of the type O(N1), O(N2),
O(N3), and two exponential problems of the type O(2N) and O(NN), for N = 10, 20, 100,
and 500. Image using the Chinese supercomputer TaihuLight, whose computational rate is
93 PFlop/s.
12.2. For the network shown in Figure 12.16, determine the degree distribution, the average degree, the average clustering coefficient, and the shortest path between nodes A and D.
12.3. At https://ccl.northwestern.edu/netlogo/index.shtml webpage, download NetLogo software, free of charge (Wilensky 1999). In Models Library, open the NetLogo “Small Worlds”
model (Wilensky 2005) or the “SmallWorldNetworks” model (available at Complexity
Explorer project, http://complexityexplorer.org). Build a regular network and calculate its mean path length sp and its average clustering coefficient C. Then, randomize the links of
the network by increasing the rewiring probability p monotonically. Determine how the
ratios sp ( p) / sp (0) and C ( p) C
/ (0) changes for p = 0 0
. 1;0 05
. ;0 1
. ;0 1
. 5 …
;
1
; .
12.4. Determine the degree distribution and the average clustering coefficient of the modular
network shown in Figure 12.17.
12.5. Figure 12.18 shows an example of a hierarchical network and its iterative construction (Ravasz and Barabási 2003). The starting subgraph is a fully connected cluster of five
nodes with links also between diagonal nodes. The 1° iteration, required to build the net-
work, consists in creating four identical replicas and connecting the peripheral nodes of
G
F
A
H
E
I
B
D
C
FIGURE 12.16 Structure of an undirected network.
448
Untangling Complex Systems
FIGURE 12.17 Example of modular network.
(Starting motif)
(2° iteration)
(1° iteration)
FIGURE 12.18 Construction of a hierarchical network.
each new cluster to the central node of the original subgraph. The output is a network with
25 nodes. In the 2° iteration, four copies of the output of the 1° iteration are created and,
again, the peripheral nodes are connected to the central node of the original motif, obtain-
ing a network with 125 nodes. Determine the clustering coefficient of the nodes at the cen-
ter of (I) the numerous five-node motifs, (II) the 25-node modules, and (III) the 125-node
network.
Complex Systems
449
12.6. Calculate the intensity of the thermal radiation emitted by bodies at 5900, 5500, 5000,
and 4500 K, in the range 10
10010
÷
nm and Wm−2nm−1. Define the function relat-
ing the wavelength that corresponds to the maximum intensity emitted at the various
temperatures versus the reciprocal of temperature. Moreover, confirm the Stephan-
Boltzmann law.
12.7. Calculate the spectral emissivity of our body that has an average temperature of 36°C. In
which spectral region does our body emit?
12.8. Knowing that the solar irradiance at the Earth’s atmosphere is 1366 W/m2, how large is the
pressure of the solar radiation exerted on the terrestrial atmosphere, considering a perpen-
dicular incidence?
12.9. Try to explain how the Crookes radiometer works (see Figure 12.19). A set of vanes, black on one side, white or silver on the other, are enclosed within a sealed glass bulb, evacuated
at about 1 Pa. The vanes can rotate on a low-friction spindle. When exposed to light or heat
source, the dark sides rotate away from the source. On the other hand, if a cold body (for
instance, a block of ice) is placed nearby, the vanes rotate in the opposite direction, i.e., the
black sides turn towards the cold body. This radiometer was invented by the chemist Sir
William Crookes, in 1873, after noticing that sunlight shining on his balance was disturb-
ing his weighing within a partially evacuated chamber. The explanation of the working
mechanism of the Crookes radiometer was a source of scientific controversy for over half
a century.
FIGURE 12.19 Schematic structure of a Crookes radiometer.
450
Untangling Complex Systems
12.10. Knowing that the average temperature of the Earth is 288 K, how much is its total radiant
power, and which is the wavelength of the peak power?
12.11. Open NetLogo software (Wilensky, 1999). Go to the Models Library; select Biology and
open the “Flocking” Model (Wilensky, 1998). I Part: Play with the model by changing
the vision slider. Run simulations with a vision of 1.0 patch, then 3.0 patches, 5.0 patches,
7.0 patches, each time letting the simulation run for at least 300 ticks (Click on “Go”
again to stop, then “Setup” before starting a new run). As you increase the vision range,
does the group flocking behavior become stronger (converging more quickly on larger
flocks) or weaker (converging more slowly, with smaller flocks)?
II Part: Set the vision slider back to 3.0 patches. Play with the model by changing the
minimum-separation slider. Run simulations after setting the minimum-separation slider
to 1.00, 2.00, and then 4.00. Which of these settings gives you the most “flock-like”
behavior after about 300 ticks?
12.9 SOLUTIONS TO THE EXERCISES
12.1. The amount of time required to solve the three polynomial problems and the two exponen-
tial problems are listed in Table 12.2. The calculations have been performed for N equal to 10, 20, 100, and 500, and assuming to have the Chinese supercomputer TaihuLight for the
computations.
It is amazing how much the time soars abruptly when we deal with exponential prob-
lems, and we increase N. It is evident that when the size of the instance is large, any expo-
nential problem cannot be solved accurately and in a reasonable time.
12.2. In Table 12.3, there is a list of the degree and clustering coefficient for each node of the network.
The degree distribution is shown in Figure 12.20.
The average degree is d = 3; the average clustering coefficient is C = 0 6
. 2. The shortest
path between node A and D is equal to 2 links.
12.3. First of all, we build a regular network with 200 nodes. The degree is d = 4 for each node,
sp (0) = 25 2
. 5 and C(0) = 0 5
. (obtained by the “SmallWorldNetworks” model). By assign-
ing monotonically growing values to the rewiring probability p, we gain the trends of the
ratios sp ( p)/ sp (0) and C( p) / C(0) that are reported in Figure 12.21.
It is noteworthy that the ratio sp ( p) / sp(0) drops immediately to ≈0.2 when p is just 0.01. By rewiring only 1% of the total number of links, we obtain the onset of the small-world phenomenon. For p > 0 2
. , sp( p) / sp(0) remains almost constant to the value of
≈0.15, even when we have a completely random network. On the other hand, the ratio
C( p) / C(0) decreases less abruptly, but when p > 0.4, it goes to zero, confirming that in a random network the clustering is very low.
TABLE 12.2
Running Times of Polynomial and Exponential Problems
Calculated by the Chinese Supercomputer TaihuLight
Size of the Problem (N)
Running time
10
20
100
500
O(N)
1 10 16
× − s
2 10 16
× − s
1 10 15
× − s
5 4 10 15
. ×
− s
O(N2)
1 10 15
× − s
4 3 10 15
. ×
− s
1 10 13
× − s
2 7 10 12
. ×
− s
O(N3)
1 10 14
× − s
8 6 10 14
. ×
−
s
1 10 11
× − s
1 3 10 9
. ×
− s
O(2N)
1 10 14
× − s
1 10 11
× − s
4 3 105
. ×
years
1 10126
×
years
O(NN)
1 10 7
× − s
36 years
3 4 10175
. ×
years
1 101325
×
years
Complex Systems
451
TABLE 12.3
Degree ( d) and Clustering Coefficient ( C) of the
Nodes Belonging to the Network of Exercise 12.2
Node
d
C
A
4
3/6
B
2
1
C
2
1
D
2
1
E
5
4/10
F
5
4/10
G
1
0
H
3
2/3
0.4
0.3
y
0.2
equencFr
0.1
0.0 0
1
2
3
4
5
d (Degree)
FIGURE 12.20 Degree distribution for the network of this exercise.
1.0
0.8
0.6
C( p)/ C(0)
0.4
0.2
sp( p)/ sp(0)
0.0
0.0
0.2
0.4
0.6
0.8
1.0
P
FIGURE 12.21 Trends of sp( p)/ sp(0) (black squares) and C ( p)/ C(0) (gray circles).
452
Untangling Complex Systems
0.7
0.6
0.5
y 0.4
equenc 0.3
Fr
0.2
0.1
0.0 0
1
2
3
4
d (Degree)
FIGURE 12.22 Degree distribution for the modular network of the exercise 12.4.
12.4. The degree distribution of the modular network is shown in Figure 12.22. The average clustering coefficient is ≈0.51.
12.5. The clustering coefficient of the node at the center of the five-node motifs is 1 because its
degree is 4 and its neighbors are all mutually linked: C = (2*6 4 *3). The node at the cen-
ter of a 25-node module has d = 20 and C = (2*30 20 *19) = 3 19
/ . The node at the center
of the 125-node network has d = 84 and C = (2*126 84 *83) = 3/83. These data confirm
that in a hierarchical network, the clustering coefficient is a power law of the degree of
connectivity, d.
12.6. The spectral emissivity, as a function of the wavelength λ, can be obtained from the
Planck’s formula u ν
( , T ) dν = ( π
8 hν 3 c 3 )(1 ( ehν / kBT − )
1 ) dν . First, we substitute ν with
c/λ and dν with c λ2
(
) dλ. Second, we multiply the energy density by c/4 (according to
equation [12.7]). The final equation looks like
2 hc 2
1
E ( T,λ
π
) =
5
hc
λ
ekBTλ −1
By using SI units, we obtain a spectral emissivity in Wm−3. Dividing by 109, we achieve the
units required by the exercise, which are Wm−2nm−1 (see Figure 12.23).
From
Figure 12.23, we observe that the hotter the body, the more intense the emissivity
of the thermal radiation. If we calculate the integrals of these functions, and we plot the
log( Areaof E( T,λ)) versus log(T), we can fit the data by a straight line with slope equal
to 4, confirming the Stephan-Boltzmann law (see Figure 12.24).
If we plot the wavelength that corresponds to the maximum of E ( T,λ ) versus 1/ T, we
obtain a linear trend, like that shown in Figure 12.25.
The
relation
λ(
)
nm = A + ( ,
2 9×106/ T ( K)) is known as Wien’s law. The hotter the body,
the shorter the wavelength that corresponds to the maximum spectral emissivity.
Complex Systems
453
Visible
region
9.00E +007
5900 K
5500 K
5000 K
4500 K
6.00E +007
Wm−2nm−1
3.00E +007
0.00E +000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
λ (nm)
FIGURE 12.23 Spectral emissivity of the black body at four distinct temperatures.
10.9
Equation
y = a + b∗x
Adj. R-Square
1 Value Standard error
10.8
C
Intercept −4.25799
8.62089E-4
C
Slope
4.00298
2.31987E-4
10.7
( T, λ)) 10.6
of EeaAr
log( 10.5
10.4
10.3
3.64
3.66
3.68
3.70
3.72
3.74
3.76
3.78
log( T)
FIGURE 12.24 Linear relation between log ( Area of E ( T ,λ)) versus log( T).
12.7. The spectral emissivity of our body at 309 K, calculated by using the formula presented in
the previous exercise, is plotted in Figure 12.26. It is included between 2.5 and 50 μm, i.e.,
within the IR region. Snakes that have IR sensors can perceive the thermal IR radiation
that we emit.
454
Untangling Complex Systems
660 Equation
y = a + b∗x
640 Adj. R-Square
0.99794
Value
Standard error
B
Intercept
14.2766
14.37227
620 B
Slope
2.82198E6
73930.83351
600
580
560
λ (nm) 540
520
500
480
0.00017
0.00018
0.00019
0.00020
0.00021
0.00022
0.00023
1/T (K−1)
FIGURE 12.25 Linear relation between the wavelength (λ) that corresponds to the maximum of E T
( ,λ)
and 1/ T.
45
Visible region
40
35
30
−1
25
−2 nm 20
Wm 15
10
5
0 0
10000
20000
30000
40000
50000
λ (nm)
FIGURE 12.26 Spectral emissivity of the human body.
12.8. By using equation [12.7], we transform the intensity 1366 W/m2 in energy density:
• What is life?
• How does life exploit solar radiation as a source of energy?
• How does life exploit solar radiation as a source of information?
• Where does the complexity of human vision come from?
• What is an emergent property and how does it originate?
• How many kinds of emergent properties do you know?
12.6 KEY WORDS
Natural Complex Systems; Computational Complexity; P, NP, and NP-complete Problems; Regular
Networks; Small-World Networks; Random Networks; Scale-free Networks; Modular Networks;
Hierarchical Networks; Thermal radiation; Biomolecular Information System; Neural Information
System; Immune Information System; Social Information System; Fuzzy set; Bayesian probability;
Multiplicity; Interconnection; Integration; Emergent properties.
Complex Systems
447
12.7 HINTS FOR FURTHER READING
• The subject of Computational Complexity can be studied more deeply by reading the book
by Garey and Johnson (1979) and that by Papadimitriou (1994). The description of some of
the principal NP problems can also be found in (Lewis and Papadimitriou, 1978).
• For those who want to deepen their understanding of networks, I suggest the book by
Caldarelli and Catanzaro (2012) and the recent book by Barabási (2016), which is freely
available at the link http://barabasi.com/networksciencebook/. Recent exciting investigations on the phenomena of synchronization in large networks are presented in a special
issue of the journal Chaos, edited by Duane et al. (2017).
• Whoever is interested particularly about Complexity in ecology, I suggest the paper by
Montoya et al. (2006) and that by Sugihara et al. (2012).
• The books by Haken (2006) and Walker et al. (2017) are relevant for whom wants to learn
more about the role of information in living beings and Complex Systems.
• A program for large network analysis is available at http://mrvar.fdv.uni-lj.si/pajek/
• A list of databases for biochemical networks is available in (Subramanian et al. 2015).
12.8 EXERCISES
12.1. Compare the time required to solve three polynomial problems of the type O(N1), O(N2),
O(N3), and two exponential problems of the type O(2N) and O(NN), for N = 10, 20, 100,
and 500. Image using the Chinese supercomputer TaihuLight, whose computational rate is
93 PFlop/s.
12.2. For the network shown in Figure 12.16, determine the degree distribution, the average degree, the average clustering coefficient, and the shortest path between nodes A and D.
12.3. At https://ccl.northwestern.edu/netlogo/index.shtml webpage, download NetLogo software, free of charge (Wilensky 1999). In Models Library, open the NetLogo “Small Worlds”
model (Wilensky 2005) or the “SmallWorldNetworks” model (available at Complexity
Explorer project, http://complexityexplorer.org). Build a regular network and calculate its mean path length sp and its average clustering coefficient C. Then, randomize the links of
the network by increasing the rewiring probability p monotonically. Determine how the
ratios sp ( p) / sp (0) and C ( p) C
/ (0) changes for p = 0 0
. 1;0 05
. ;0 1
. ;0 1
. 5 …
;
1
; .
12.4. Determine the degree distribution and the average clustering coefficient of the modular
network shown in Figure 12.17.
12.5. Figure 12.18 shows an example of a hierarchical network and its iterative construction (Ravasz and Barabási 2003). The starting subgraph is a fully connected cluster of five
nodes with links also between diagonal nodes. The 1° iteration, required to build the net-
work, consists in creating four identical replicas and connecting the peripheral nodes of
G
F
A
H
E
I
B
D
C
FIGURE 12.16 Structure of an undirected network.
448
Untangling Complex Systems
FIGURE 12.17 Example of modular network.
(Starting motif)
(2° iteration)
(1° iteration)
FIGURE 12.18 Construction of a hierarchical network.
each new cluster to the central node of the original subgraph. The output is a network with
25 nodes. In the 2° iteration, four copies of the output of the 1° iteration are created and,
again, the peripheral nodes are connected to the central node of the original motif, obtain-
ing a network with 125 nodes. Determine the clustering coefficient of the nodes at the cen-
ter of (I) the numerous five-node motifs, (II) the 25-node modules, and (III) the 125-node
network.
Complex Systems
449
12.6. Calculate the intensity of the thermal radiation emitted by bodies at 5900, 5500, 5000,
and 4500 K, in the range 10
10010
÷
nm and Wm−2nm−1. Define the function relat-
ing the wavelength that corresponds to the maximum intensity emitted at the various
temperatures versus the reciprocal of temperature. Moreover, confirm the Stephan-
Boltzmann law.
12.7. Calculate the spectral emissivity of our body that has an average temperature of 36°C. In
which spectral region does our body emit?
12.8. Knowing that the solar irradiance at the Earth’s atmosphere is 1366 W/m2, how large is the
pressure of the solar radiation exerted on the terrestrial atmosphere, considering a perpen-
dicular incidence?
12.9. Try to explain how the Crookes radiometer works (see Figure 12.19). A set of vanes, black on one side, white or silver on the other, are enclosed within a sealed glass bulb, evacuated
at about 1 Pa. The vanes can rotate on a low-friction spindle. When exposed to light or heat
source, the dark sides rotate away from the source. On the other hand, if a cold body (for
instance, a block of ice) is placed nearby, the vanes rotate in the opposite direction, i.e., the
black sides turn towards the cold body. This radiometer was invented by the chemist Sir
William Crookes, in 1873, after noticing that sunlight shining on his balance was disturb-
ing his weighing within a partially evacuated chamber. The explanation of the working
mechanism of the Crookes radiometer was a source of scientific controversy for over half
a century.
FIGURE 12.19 Schematic structure of a Crookes radiometer.
450
Untangling Complex Systems
12.10. Knowing that the average temperature of the Earth is 288 K, how much is its total radiant
power, and which is the wavelength of the peak power?
12.11. Open NetLogo software (Wilensky, 1999). Go to the Models Library; select Biology and
open the “Flocking” Model (Wilensky, 1998). I Part: Play with the model by changing
the vision slider. Run simulations with a vision of 1.0 patch, then 3.0 patches, 5.0 patches,
7.0 patches, each time letting the simulation run for at least 300 ticks (Click on “Go”
again to stop, then “Setup” before starting a new run). As you increase the vision range,
does the group flocking behavior become stronger (converging more quickly on larger
flocks) or weaker (converging more slowly, with smaller flocks)?
II Part: Set the vision slider back to 3.0 patches. Play with the model by changing the
minimum-separation slider. Run simulations after setting the minimum-separation slider
to 1.00, 2.00, and then 4.00. Which of these settings gives you the most “flock-like”
behavior after about 300 ticks?
12.9 SOLUTIONS TO THE EXERCISES
12.1. The amount of time required to solve the three polynomial problems and the two exponen-
tial problems are listed in Table 12.2. The calculations have been performed for N equal to 10, 20, 100, and 500, and assuming to have the Chinese supercomputer TaihuLight for the
computations.
It is amazing how much the time soars abruptly when we deal with exponential prob-
lems, and we increase N. It is evident that when the size of the instance is large, any expo-
nential problem cannot be solved accurately and in a reasonable time.
12.2. In Table 12.3, there is a list of the degree and clustering coefficient for each node of the network.
The degree distribution is shown in Figure 12.20.
The average degree is d = 3; the average clustering coefficient is C = 0 6
. 2. The shortest
path between node A and D is equal to 2 links.
12.3. First of all, we build a regular network with 200 nodes. The degree is d = 4 for each node,
sp (0) = 25 2
. 5 and C(0) = 0 5
. (obtained by the “SmallWorldNetworks” model). By assign-
ing monotonically growing values to the rewiring probability p, we gain the trends of the
ratios sp ( p)/ sp (0) and C( p) / C(0) that are reported in Figure 12.21.
It is noteworthy that the ratio sp ( p) / sp(0) drops immediately to ≈0.2 when p is just 0.01. By rewiring only 1% of the total number of links, we obtain the onset of the small-world phenomenon. For p > 0 2
. , sp( p) / sp(0) remains almost constant to the value of
≈0.15, even when we have a completely random network. On the other hand, the ratio
C( p) / C(0) decreases less abruptly, but when p > 0.4, it goes to zero, confirming that in a random network the clustering is very low.
TABLE 12.2
Running Times of Polynomial and Exponential Problems
Calculated by the Chinese Supercomputer TaihuLight
Size of the Problem (N)
Running time
10
20
100
500
O(N)
1 10 16
× − s
2 10 16
× − s
1 10 15
× − s
5 4 10 15
. ×
− s
O(N2)
1 10 15
× − s
4 3 10 15
. ×
− s
1 10 13
× − s
2 7 10 12
. ×
− s
O(N3)
1 10 14
× − s
8 6 10 14
. ×
−
s
1 10 11
× − s
1 3 10 9
. ×
− s
O(2N)
1 10 14
× − s
1 10 11
× − s
4 3 105
. ×
years
1 10126
×
years
O(NN)
1 10 7
× − s
36 years
3 4 10175
. ×
years
1 101325
×
years
Complex Systems
451
TABLE 12.3
Degree ( d) and Clustering Coefficient ( C) of the
Nodes Belonging to the Network of Exercise 12.2
Node
d
C
A
4
3/6
B
2
1
C
2
1
D
2
1
E
5
4/10
F
5
4/10
G
1
0
H
3
2/3
0.4
0.3
y
0.2
equencFr
0.1
0.0 0
1
2
3
4
5
d (Degree)
FIGURE 12.20 Degree distribution for the network of this exercise.
1.0
0.8
0.6
C( p)/ C(0)
0.4
0.2
sp( p)/ sp(0)
0.0
0.0
0.2
0.4
0.6
0.8
1.0
P
FIGURE 12.21 Trends of sp( p)/ sp(0) (black squares) and C ( p)/ C(0) (gray circles).
452
Untangling Complex Systems
0.7
0.6
0.5
y 0.4
equenc 0.3
Fr
0.2
0.1
0.0 0
1
2
3
4
d (Degree)
FIGURE 12.22 Degree distribution for the modular network of the exercise 12.4.
12.4. The degree distribution of the modular network is shown in Figure 12.22. The average clustering coefficient is ≈0.51.
12.5. The clustering coefficient of the node at the center of the five-node motifs is 1 because its
degree is 4 and its neighbors are all mutually linked: C = (2*6 4 *3). The node at the cen-
ter of a 25-node module has d = 20 and C = (2*30 20 *19) = 3 19
/ . The node at the center
of the 125-node network has d = 84 and C = (2*126 84 *83) = 3/83. These data confirm
that in a hierarchical network, the clustering coefficient is a power law of the degree of
connectivity, d.
12.6. The spectral emissivity, as a function of the wavelength λ, can be obtained from the
Planck’s formula u ν
( , T ) dν = ( π
8 hν 3 c 3 )(1 ( ehν / kBT − )
1 ) dν . First, we substitute ν with
c/λ and dν with c λ2
(
) dλ. Second, we multiply the energy density by c/4 (according to
equation [12.7]). The final equation looks like
2 hc 2
1
E ( T,λ
π
) =
5
hc
λ
ekBTλ −1
By using SI units, we obtain a spectral emissivity in Wm−3. Dividing by 109, we achieve the
units required by the exercise, which are Wm−2nm−1 (see Figure 12.23).
From
Figure 12.23, we observe that the hotter the body, the more intense the emissivity
of the thermal radiation. If we calculate the integrals of these functions, and we plot the
log( Areaof E( T,λ)) versus log(T), we can fit the data by a straight line with slope equal
to 4, confirming the Stephan-Boltzmann law (see Figure 12.24).
If we plot the wavelength that corresponds to the maximum of E ( T,λ ) versus 1/ T, we
obtain a linear trend, like that shown in Figure 12.25.
The
relation
λ(
)
nm = A + ( ,
2 9×106/ T ( K)) is known as Wien’s law. The hotter the body,
the shorter the wavelength that corresponds to the maximum spectral emissivity.
Complex Systems
453
Visible
region
9.00E +007
5900 K
5500 K
5000 K
4500 K
6.00E +007
Wm−2nm−1
3.00E +007
0.00E +000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
λ (nm)
FIGURE 12.23 Spectral emissivity of the black body at four distinct temperatures.
10.9
Equation
y = a + b∗x
Adj. R-Square
1 Value Standard error
10.8
C
Intercept −4.25799
8.62089E-4
C
Slope
4.00298
2.31987E-4
10.7
( T, λ)) 10.6
of EeaAr
log( 10.5
10.4
10.3
3.64
3.66
3.68
3.70
3.72
3.74
3.76
3.78
log( T)
FIGURE 12.24 Linear relation between log ( Area of E ( T ,λ)) versus log( T).
12.7. The spectral emissivity of our body at 309 K, calculated by using the formula presented in
the previous exercise, is plotted in Figure 12.26. It is included between 2.5 and 50 μm, i.e.,
within the IR region. Snakes that have IR sensors can perceive the thermal IR radiation
that we emit.
454
Untangling Complex Systems
660 Equation
y = a + b∗x
640 Adj. R-Square
0.99794
Value
Standard error
B
Intercept
14.2766
14.37227
620 B
Slope
2.82198E6
73930.83351
600
580
560
λ (nm) 540
520
500
480
0.00017
0.00018
0.00019
0.00020
0.00021
0.00022
0.00023
1/T (K−1)
FIGURE 12.25 Linear relation between the wavelength (λ) that corresponds to the maximum of E T
( ,λ)
and 1/ T.
45
Visible region
40
35
30
−1
25
−2 nm 20
Wm 15
10
5
0 0
10000
20000
30000
40000
50000
λ (nm)
FIGURE 12.26 Spectral emissivity of the human body.
12.8. By using equation [12.7], we transform the intensity 1366 W/m2 in energy density:
