- Email: [email protected]
Fragmentations, mergings and order: aspects of entropy
Physica A 305 (2002) 32 – 40
www.elsevier.com/locate/physa
Fragmentations, mergings and order: aspects of entropy P.T. Landsberg ∗ Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, UK
Abstract Eight types of thermodynamics and inequalities are discussed with reference to a generalised gas and black holes. Choices for the statistical entropy are considered along with systems whose heat capacity can be negative. Further, a formal relation between entropy and order is recalled. c 2002 Elsevier Science B.V. All rights reserved. PACS: 65.40; 04.70.D; 82.60.N Keywords: Entropy; Blackholes; Nucleation
1. Introduction It is not possible to prove mathematical theorems from physics! However, there exist a thought-provoking class of arguments in which a truth of pure mathematics is directly accessible from established principles of science. Here is one [1,2] which is related to inequalities (which will occupy us to some extent in this paper). Consider n identical heat reservoirs each of constant heat capacity C and at di
C(Ti − Tf ) = 0 ;
i=1
∗
Tel.: +44-23-8059-3681; fax: +44-23-8059-5147. E-mail address: [email protected] (P.T. Landsberg).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 6 3 6 - 7
(1)
P.T. Landsberg / Physica A 305 (2002) 32 – 40
33
whence n
Tf =
1 Ti : n
(2)
i=1
The second law requires a calculation of the entropy gain Tf dT Tf CS = C ¿0 : = Cn ln (T1 T2 : : : Tn )1=n T Ti i Hence, by Eqs. (2) and (3), n 1 Ti ¿ (T1 T2 : : : Tn )1=n : n
(3)
(4)
1
Further, only if there was equilibrium in the initial state (T1 = T2 = · · · = Tn ) can the equality be attained, since then CS = 0. Thus we have the famous inequality between the arithmetic and the geometric mean, but have obtained it from physics! 2. Types of thermodynamics Using now equations and inequalities, we show the possibility of classifying all entropy functions, and hence the associated types of thermodynamics, into eight categories. These are generated by three characteristics, each of which may hold (S; H; C) E HE ; C) E (see Table 1). Note that in Table 1, x and y can stand for or may not hold (S; a set of variables or for just one variable. The consideration of these categories of thermodynamics goes back to the 1960s [3]. We now proceed to some examples. 3. Types of systems considered The 3rst example is inspired by the ideal gas and is speci=ed by the entropy S = k ln(Uvg =N h ) ;
(7)
where k; ; g; h are positive constants and v is the volume, U the internal energy and N the number of particles in the gas. It turns out that this example can generate three types of thermodynamics [4]: h = g + 1 : type 1;
h ¡ g + 1 : type 4;
h ¿ g + 1 : type 6 :
(8)
Of course, type 1 is the normal thermodynamics; type 4 reveals the purpose of the example: For it was given as a simple classical analogue of the thermodynamics of the Schwarzschild black hole, in that it violates C but keeps S. Type 3 achieves the same but is logically forbidden. Second example: For a Schwarzschild black hole of mass M; M0 being a constant, one knows that S = k (M=M0 )2 :
(9)
34
P.T. Landsberg / Physica A 305 (2002) 32 – 40 Table 1 Types of thermodynamics 3. SH CE E CE 7. SH
1. SHC E 5. SHC
2. S HE C 6. SEHE C
Superadditivity Homogeneity Concavity
S: S(x + y) ¿ S(x) + S(y) H : S(x) = S(x) C: S(x + (1 − )y) ¿ S(x) + (1 − )S(y)
4. S HE CE 8. SEHE CE (5) (6)
Fig. 1. Venn diagram for thermodynamics types.
Its thermodynamics is of type 4 as well. Indeed, one can show that this holds more generally [5] for S = am + b ;
(10)
where a ¿ 0; ¿ 1 and b are positive constants (b must not be too large; see Ref. [5, line 1 of Table 2]). The thermodynamics of type 8, for which S; C and H all fail, is rather rare. But it is found for special cases of (10), for example if (for a; b; ¿ 0) S = am2 + b; and for certain dilatonic black holes [5]. As a third example, consider a (1 + 1)-dimensional black hole, as discussed in Ref. [6], for which S = k ln (M=M0 ) :
(11)
In this case, one =nds type 2 if M=M0 ¡ 4 and type 6 if M=M0 ¿ 4. In the second and third examples, there has in e
P.T. Landsberg / Physica A 305 (2002) 32 – 40
35
It is seen from Fig. 1 that no example has been given of a thermodynamics of types 3, 5 and 7. Types 3 and 5 are actually logically impossible because S + H → C and C + H → S. For functions which depend on one independent variable only, we know also [5] that H → S; C which rules out thermodynamics of type 7. (We still need an example of this type. It must involve several independent variables.) Broadly speaking, if S holds, then two identical systems will tend to “clump” together if they are given a chance, for in this way the entropy goes up. This holds for a thermodynamics of types 1, 2 and 4. However, if S fails, as it does in thermodynamics of types 6, 7 and 8, then a system increases its entropy by splitting up into separate systems. This is “fragmentation” [6]. An explicit example of these phenomena is given in the next section. 4. Example of fragmentation and merging For (1 + 1)-dimensional black holes, the dimensionless entropy 2 M 0 ≡ S = ln ˝k M0
(12)
satis=es S HE C for M=M0 ¡ 4 and SEHE C for M=M0 ¿ 4, thus adding an example for each of two relatively unexplored types of thermodynamics. Note that since entropies must be positive the physical applicability of (12) must cease just before the broken portions of the curves of Fig. 2 are reached. Consider next a (1 + 1)-dimensional system consisting of a black hole of mass M which has spawned n identical black holes of mass m. The dimensionless entropy is n M − nm m n = ln ; (13) M0 M0 provided subsystems (12) are suMciently separated to be approximately independent. Eq. (13) becomes (12) for n = 0: It is easily con=rmed, for a given n; M and M0 , that n has its largest value with respect to m if m = M=(n + 1), whence [6] M max n = (n + 1) ln : (n + 1)M0 Fig. 2 shows a plot of 1 as a function of m=M0 and also gives an indication of 0 for each M=M0 -value. The changes indicated by arrows in Fig. 2 must occur because the second law states, that the entropy increases, or possibly stays constant, in spontaneous processes. For M=M0 =3 the separated black holes merge, and the system is of the type S HE C; for M=M0 = 5 the black hole tends to split into equal parts and the system is of the type SEHE C. The concave nature of the logarithmic function ensures the validity of C in both cases. A split into unequal parts has not been studied yet. Equating 0 and 1 yields a curve of ‘indi
36
P.T. Landsberg / Physica A 305 (2002) 32 – 40
Fig. 2. The dimensionless entropy of a combination of two separated (1 + 1)-dimensional black holes of masses m and M − m is shown for m=M0 = 3; 4 and 5. Arrows show transitions which will increase the entropy.
5. Entropy choices The non-extensive nature of the entropy can be encapsulated in a function Y which vanishes in the case of an extensive entropy: S(A + B) − S(A) − S(B) = YS(A)S(B) ; where A and B represent the part systems under consideration [7]. Let us add super=xes R; S; T; U for Renyi, Shannon, Tsallis and a new entropy. Let Q ≡ piq , where the pi are the probabilities of the various states of the system involved, and let q be a real constant. Then we can construct a table of matching entropies and Y -functions [8]. k ln Q; Y (R) = 0 ; 1−q S (S) (A) = −k pi ln pi ; Y (S) = 0 ;
S (R) (A) =
(14) (15)
i
k 1−q (Q − 1); Y (T ) = ; 1−q k 1 k q−1 (U ) 1− ; Y (U ) = : S (A) = 1−q Q k
S (T ) (A) =
For each S, one has limq→1 S (i) (A) = S (S) (A).
(16) (17)
P.T. Landsberg / Physica A 305 (2002) 32 – 40
37
A great deal of work has been done using the Tsallis expression S (T ) . The object was to =nd various =ts to experimental data and limits on the amount by which q is likely to di
S = 4 GkM 2 =˝c :
Here, r is the radius, S is the entropy and G is Newton’s gravitational constant. One =nds a negative heat capacity C = dU=dT = −˝c5 =8 GkT 2 : Now an isolated system whose temperature T is de=ned in terms of its internal variables cannot sustain equilibrium with a heat reservoir at that temperature if its heat capacity is negative. This may be seen from the expression for the energy Quctuation of a system of a given number of particles (N ); T and V; which is E 2 = kT 2 Cv : (E − E) This canonical ensemble expression shows that a system of negative Cv cannot be in equilibrium with a heat reservoir at temperature T: The Hawking radiation, leading to the evaporation of a black hole, is therefore thermodynamically not surprising,
38
P.T. Landsberg / Physica A 305 (2002) 32 – 40
since there must have been an absence of equilibrium between the black hole and its surroundings. Conditions for stability of equilibrium between two parts (labelled 1 and 2) of an isolated system yield the heat capacity inequality 1 1 + ¿0 : C1 C2
(18)
If, therefore, black-body radiation (system 1) and a black-hole (system 2) are in equilibrium, one has C1 ¿ 0 ¿ C2 and one requires for stability by (18) 1 1 ¿ 0; − C1 |C2 |
i:e:;
|C2 | ¿ C1 :
(19)
The combined heat capacity C1 + C2 is therefore negative. With a2 U1 = 34 a1 T 4 ; U2 = ; U ≡ U1 + U2 ; T this implies a2 ¿ 3a1 T 3 ; T2
i:e:; U2 ¿ 4U1 = 4(U − U2 );
i:e:; U2 ¿ 45 U :
(20)
For stable equilibrium, at least 80% of the energy of such a system must therefore reside in the black hole. One can understand result (19) intuitively, noting that the energy given up by one system is absorbed by the other. If initially T2 ¿ T1 ; both systems become hotter on equilibration, and to reach a common temperature the initially hotter system must have the smaller rise in temperature. Hence, (19) results again. Statistical mechanics can supply details which go beyond (20) by noting that energy and entropy of a combined system of a Schwarzschild black hole and black-body radiation are 8 5 k 4 ; b≡ E = Mc2 + bvT 4 ; S = f(mc2 )2 + 43 bvT 3 ; 15h3 c3 where f ≡ 4 Gk=˝c5 . From this, one can determine conditions for entropy maxima [12]. It is normally possible to produce a negative heat capacity by changing to a set of new variables. Table 2 explains this process for the case of black-body radiation. It is worth noting here that a self-gravitating sphere of normal black-body radiation can also exhibit a negative heat capacity [13], even without a change to unnatural variables. 7. Order and entropy One can de=ne “disorder” (in a technical sense) as [14] D≡
entropy of the system s ≡ : max:entropy it can have S
P.T. Landsberg / Physica A 305 (2002) 32 – 40
39
Table 2 Black-body radiation with negative heat capacity Normal relations p=
a 3
T4
Relations in which the variable p has been replaced by X 2 X = vp2 = v a3 T 8
U = avT 4 S=
4 3
avT 3
@U @T v
= 3avT 3
9 a2
XT −8
U = (aT 4 ) a92 XT −8 = )v=
4 9 a a2 XT −8 T 3 = 3
@U @T X
12 a
9 a
XT −4
XT −5
= − 36 XT −5 a
Then an estimate of the “D” of a model universe can be made by assuming its 1080 baryons to be coalesced into 1023 solar masses [12]. The greatest entropy is believed to occur if they form a single massive black hole. Its entropy would be very large: S = 10123 k : The comparison entropy of the current model universe can be estimated as due only to the 2.7 K background radiation, assumed to pervade all of it. Neglecting all other e
40
P.T. Landsberg / Physica A 305 (2002) 32 – 40
[10] R.J.V. dos Santos, Generalization of Shannon’s theorem for Tsallis entropy, J. Math. Phys. 38 (1997) 4104. [11] A.R.R. Papa, On one-parameter-dependent generalizations of Boltzmann–Gibbs statistical mechanics, J. Phys. A 31 (1998) 5271. [12] P.T. Landsberg, Thermodynamics and black holes, in: V. De Sabbata, Z. Zhang (Eds.), Black Hole Physics, 12th Course NATO Advanced Study Institute, Erice, 1991, pp. 99 –145. [13] D. Pavon, P.T. Landsberg, Heat capacity of a self-gravitating radiation sphere, Gen. Relat. Gravitation 20 (1988) 457. [14] J.S. Shiner, M. Davison, P.T. Landsberg, Simple measure of complexity, Phys. Rev. E 59 (1999) 1459.
www.elsevier.com/locate/physa
Fragmentations, mergings and order: aspects of entropy P.T. Landsberg ∗ Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, UK
Abstract Eight types of thermodynamics and inequalities are discussed with reference to a generalised gas and black holes. Choices for the statistical entropy are considered along with systems whose heat capacity can be negative. Further, a formal relation between entropy and order is recalled. c 2002 Elsevier Science B.V. All rights reserved. PACS: 65.40; 04.70.D; 82.60.N Keywords: Entropy; Blackholes; Nucleation
1. Introduction It is not possible to prove mathematical theorems from physics! However, there exist a thought-provoking class of arguments in which a truth of pure mathematics is directly accessible from established principles of science. Here is one [1,2] which is related to inequalities (which will occupy us to some extent in this paper). Consider n identical heat reservoirs each of constant heat capacity C and at di
C(Ti − Tf ) = 0 ;
i=1
∗
Tel.: +44-23-8059-3681; fax: +44-23-8059-5147. E-mail address: [email protected] (P.T. Landsberg).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 1 ) 0 0 6 3 6 - 7
(1)
P.T. Landsberg / Physica A 305 (2002) 32 – 40
33
whence n
Tf =
1 Ti : n
(2)
i=1
The second law requires a calculation of the entropy gain Tf dT Tf CS = C ¿0 : = Cn ln (T1 T2 : : : Tn )1=n T Ti i Hence, by Eqs. (2) and (3), n 1 Ti ¿ (T1 T2 : : : Tn )1=n : n
(3)
(4)
1
Further, only if there was equilibrium in the initial state (T1 = T2 = · · · = Tn ) can the equality be attained, since then CS = 0. Thus we have the famous inequality between the arithmetic and the geometric mean, but have obtained it from physics! 2. Types of thermodynamics Using now equations and inequalities, we show the possibility of classifying all entropy functions, and hence the associated types of thermodynamics, into eight categories. These are generated by three characteristics, each of which may hold (S; H; C) E HE ; C) E (see Table 1). Note that in Table 1, x and y can stand for or may not hold (S; a set of variables or for just one variable. The consideration of these categories of thermodynamics goes back to the 1960s [3]. We now proceed to some examples. 3. Types of systems considered The 3rst example is inspired by the ideal gas and is speci=ed by the entropy S = k ln(Uvg =N h ) ;
(7)
where k; ; g; h are positive constants and v is the volume, U the internal energy and N the number of particles in the gas. It turns out that this example can generate three types of thermodynamics [4]: h = g + 1 : type 1;
h ¡ g + 1 : type 4;
h ¿ g + 1 : type 6 :
(8)
Of course, type 1 is the normal thermodynamics; type 4 reveals the purpose of the example: For it was given as a simple classical analogue of the thermodynamics of the Schwarzschild black hole, in that it violates C but keeps S. Type 3 achieves the same but is logically forbidden. Second example: For a Schwarzschild black hole of mass M; M0 being a constant, one knows that S = k (M=M0 )2 :
(9)
34
P.T. Landsberg / Physica A 305 (2002) 32 – 40 Table 1 Types of thermodynamics 3. SH CE E CE 7. SH
1. SHC E 5. SHC
2. S HE C 6. SEHE C
Superadditivity Homogeneity Concavity
S: S(x + y) ¿ S(x) + S(y) H : S(x) = S(x) C: S(x + (1 − )y) ¿ S(x) + (1 − )S(y)
4. S HE CE 8. SEHE CE (5) (6)
Fig. 1. Venn diagram for thermodynamics types.
Its thermodynamics is of type 4 as well. Indeed, one can show that this holds more generally [5] for S = am + b ;
(10)
where a ¿ 0; ¿ 1 and b are positive constants (b must not be too large; see Ref. [5, line 1 of Table 2]). The thermodynamics of type 8, for which S; C and H all fail, is rather rare. But it is found for special cases of (10), for example if (for a; b; ¿ 0) S = am2 + b; and for certain dilatonic black holes [5]. As a third example, consider a (1 + 1)-dimensional black hole, as discussed in Ref. [6], for which S = k ln (M=M0 ) :
(11)
In this case, one =nds type 2 if M=M0 ¡ 4 and type 6 if M=M0 ¿ 4. In the second and third examples, there has in e
P.T. Landsberg / Physica A 305 (2002) 32 – 40
35
It is seen from Fig. 1 that no example has been given of a thermodynamics of types 3, 5 and 7. Types 3 and 5 are actually logically impossible because S + H → C and C + H → S. For functions which depend on one independent variable only, we know also [5] that H → S; C which rules out thermodynamics of type 7. (We still need an example of this type. It must involve several independent variables.) Broadly speaking, if S holds, then two identical systems will tend to “clump” together if they are given a chance, for in this way the entropy goes up. This holds for a thermodynamics of types 1, 2 and 4. However, if S fails, as it does in thermodynamics of types 6, 7 and 8, then a system increases its entropy by splitting up into separate systems. This is “fragmentation” [6]. An explicit example of these phenomena is given in the next section. 4. Example of fragmentation and merging For (1 + 1)-dimensional black holes, the dimensionless entropy 2 M 0 ≡ S = ln ˝k M0
(12)
satis=es S HE C for M=M0 ¡ 4 and SEHE C for M=M0 ¿ 4, thus adding an example for each of two relatively unexplored types of thermodynamics. Note that since entropies must be positive the physical applicability of (12) must cease just before the broken portions of the curves of Fig. 2 are reached. Consider next a (1 + 1)-dimensional system consisting of a black hole of mass M which has spawned n identical black holes of mass m. The dimensionless entropy is n M − nm m n = ln ; (13) M0 M0 provided subsystems (12) are suMciently separated to be approximately independent. Eq. (13) becomes (12) for n = 0: It is easily con=rmed, for a given n; M and M0 , that n has its largest value with respect to m if m = M=(n + 1), whence [6] M max n = (n + 1) ln : (n + 1)M0 Fig. 2 shows a plot of 1 as a function of m=M0 and also gives an indication of 0 for each M=M0 -value. The changes indicated by arrows in Fig. 2 must occur because the second law states, that the entropy increases, or possibly stays constant, in spontaneous processes. For M=M0 =3 the separated black holes merge, and the system is of the type S HE C; for M=M0 = 5 the black hole tends to split into equal parts and the system is of the type SEHE C. The concave nature of the logarithmic function ensures the validity of C in both cases. A split into unequal parts has not been studied yet. Equating 0 and 1 yields a curve of ‘indi
36
P.T. Landsberg / Physica A 305 (2002) 32 – 40
Fig. 2. The dimensionless entropy of a combination of two separated (1 + 1)-dimensional black holes of masses m and M − m is shown for m=M0 = 3; 4 and 5. Arrows show transitions which will increase the entropy.
5. Entropy choices The non-extensive nature of the entropy can be encapsulated in a function Y which vanishes in the case of an extensive entropy: S(A + B) − S(A) − S(B) = YS(A)S(B) ; where A and B represent the part systems under consideration [7]. Let us add super=xes R; S; T; U for Renyi, Shannon, Tsallis and a new entropy. Let Q ≡ piq , where the pi are the probabilities of the various states of the system involved, and let q be a real constant. Then we can construct a table of matching entropies and Y -functions [8]. k ln Q; Y (R) = 0 ; 1−q S (S) (A) = −k pi ln pi ; Y (S) = 0 ;
S (R) (A) =
(14) (15)
i
k 1−q (Q − 1); Y (T ) = ; 1−q k 1 k q−1 (U ) 1− ; Y (U ) = : S (A) = 1−q Q k
S (T ) (A) =
For each S, one has limq→1 S (i) (A) = S (S) (A).
(16) (17)
P.T. Landsberg / Physica A 305 (2002) 32 – 40
37
A great deal of work has been done using the Tsallis expression S (T ) . The object was to =nd various =ts to experimental data and limits on the amount by which q is likely to di
S = 4 GkM 2 =˝c :
Here, r is the radius, S is the entropy and G is Newton’s gravitational constant. One =nds a negative heat capacity C = dU=dT = −˝c5 =8 GkT 2 : Now an isolated system whose temperature T is de=ned in terms of its internal variables cannot sustain equilibrium with a heat reservoir at that temperature if its heat capacity is negative. This may be seen from the expression for the energy Quctuation of a system of a given number of particles (N ); T and V; which is E 2 = kT 2 Cv : (E − E) This canonical ensemble expression shows that a system of negative Cv cannot be in equilibrium with a heat reservoir at temperature T: The Hawking radiation, leading to the evaporation of a black hole, is therefore thermodynamically not surprising,
38
P.T. Landsberg / Physica A 305 (2002) 32 – 40
since there must have been an absence of equilibrium between the black hole and its surroundings. Conditions for stability of equilibrium between two parts (labelled 1 and 2) of an isolated system yield the heat capacity inequality 1 1 + ¿0 : C1 C2
(18)
If, therefore, black-body radiation (system 1) and a black-hole (system 2) are in equilibrium, one has C1 ¿ 0 ¿ C2 and one requires for stability by (18) 1 1 ¿ 0; − C1 |C2 |
i:e:;
|C2 | ¿ C1 :
(19)
The combined heat capacity C1 + C2 is therefore negative. With a2 U1 = 34 a1 T 4 ; U2 = ; U ≡ U1 + U2 ; T this implies a2 ¿ 3a1 T 3 ; T2
i:e:; U2 ¿ 4U1 = 4(U − U2 );
i:e:; U2 ¿ 45 U :
(20)
For stable equilibrium, at least 80% of the energy of such a system must therefore reside in the black hole. One can understand result (19) intuitively, noting that the energy given up by one system is absorbed by the other. If initially T2 ¿ T1 ; both systems become hotter on equilibration, and to reach a common temperature the initially hotter system must have the smaller rise in temperature. Hence, (19) results again. Statistical mechanics can supply details which go beyond (20) by noting that energy and entropy of a combined system of a Schwarzschild black hole and black-body radiation are 8 5 k 4 ; b≡ E = Mc2 + bvT 4 ; S = f(mc2 )2 + 43 bvT 3 ; 15h3 c3 where f ≡ 4 Gk=˝c5 . From this, one can determine conditions for entropy maxima [12]. It is normally possible to produce a negative heat capacity by changing to a set of new variables. Table 2 explains this process for the case of black-body radiation. It is worth noting here that a self-gravitating sphere of normal black-body radiation can also exhibit a negative heat capacity [13], even without a change to unnatural variables. 7. Order and entropy One can de=ne “disorder” (in a technical sense) as [14] D≡
entropy of the system s ≡ : max:entropy it can have S
P.T. Landsberg / Physica A 305 (2002) 32 – 40
39
Table 2 Black-body radiation with negative heat capacity Normal relations p=
a 3
T4
Relations in which the variable p has been replaced by X 2 X = vp2 = v a3 T 8
U = avT 4 S=
4 3
avT 3
@U @T v
= 3avT 3
9 a2
XT −8
U = (aT 4 ) a92 XT −8 = )v=
4 9 a a2 XT −8 T 3 = 3
@U @T X
12 a
9 a
XT −4
XT −5
= − 36 XT −5 a
Then an estimate of the “D” of a model universe can be made by assuming its 1080 baryons to be coalesced into 1023 solar masses [12]. The greatest entropy is believed to occur if they form a single massive black hole. Its entropy would be very large: S = 10123 k : The comparison entropy of the current model universe can be estimated as due only to the 2.7 K background radiation, assumed to pervade all of it. Neglecting all other e
40
P.T. Landsberg / Physica A 305 (2002) 32 – 40
[10] R.J.V. dos Santos, Generalization of Shannon’s theorem for Tsallis entropy, J. Math. Phys. 38 (1997) 4104. [11] A.R.R. Papa, On one-parameter-dependent generalizations of Boltzmann–Gibbs statistical mechanics, J. Phys. A 31 (1998) 5271. [12] P.T. Landsberg, Thermodynamics and black holes, in: V. De Sabbata, Z. Zhang (Eds.), Black Hole Physics, 12th Course NATO Advanced Study Institute, Erice, 1991, pp. 99 –145. [13] D. Pavon, P.T. Landsberg, Heat capacity of a self-gravitating radiation sphere, Gen. Relat. Gravitation 20 (1988) 457. [14] J.S. Shiner, M. Davison, P.T. Landsberg, Simple measure of complexity, Phys. Rev. E 59 (1999) 1459.