Biol. Cybernetics 17, 207--210 (1975) 9 by Springer-Verlag 1975
An Information-Theoretical Approach to a System of Interacting Elements M. Takatsuji Central Research Laboratory of Hitachi, Ltd., Kokubunji, Tokyo, Japan Received: May 12, 1974
Abstract The maximum-entropy principle in information theory is generalized to include the interaction between elements of the system. A complex relation between the probabilities of the events is derived using a familiar technique in statistical mechanics. The relation is explicitly discussed for the case of bilinear interaction and only two events. Quite noteworthy is the existense in the system of a kind of "phase transition" similar to ferromagnetism. The result is applied to the mass behaviour. It is shown that the cooperative mass behaviour such as boom and fashion may be interpreted as a phase transition which would occur below certain "informational temperature".
I. Introduction Information theory has been used extensively in many fields of science and engineering. Not only communication theory but also such different fields as linguistics, molecular biology and social science have enjoyed this theory. Physics is not an exception: Brillouin (1956) discussed a close connection between information-theory entropy and thermodynamic entropy. Particularly important is the work of Jaynes (1957) who reconstructed statistical mechanics based on the maximum-entropy principle. This principle states that our ignorance must be represented in the most unbiased form, that is, the probability distribution must be so generated as to have maximum uncertainty subject to a given information. It is powerful enough to be applied to other fields than physics. Recently Cozzolino and Zahner (1973) used this principle for an econometric problem of future stock price. Although the maximum-entropy principle has large potential, its application has been limited to systems of independent elements. Many real systems are composed of mutually interacting elements and the interaction often plays a decisive role: In a thermodynamic system such as a ferromagnet, the interaction between constituent atoms is so strong as to produce a phase transition (Brout, 1965). In a biological association, the predator-prey interaction between species yields variations and distributions of the populations (Goel et al., 1971). The social interaction between people is very important in some psychological problems such as group dynamics (Rapopart, 1963).
The purposes of this paper are twofold. Firstly, Jaynes' theory of the maximum-entropy principle is generalized to include the interaction between elements of the system. Secondly, the result is shown to be applicable to certain type of the mass behaviour by a careful analysis of the latter. n . The Maximum-Entropy Principle for an Interacting System Let pi(i = t, 2 .... , n) represents the probability of the ith event: E Pi = 1.
(1)
i
It is assumed that the system under consideration allows the average value of a quantity fi being constant: f = E P,fi. (2) i
Here f~ is in general a function of p/s in order that the interaction between elements is taken into account, and written in an expansion: fl = Hi + 89E JiiPj + 89E KijkPJPk + ' " " j
j,k
(31
The maximum-entropy principle states that the most unbiased representation of our ignorance of the state of the system is provided by the probability distribution which maximizes the entropy S = - ~ p~lnp~
(4)
i
under the given Conditions (1) and (2). It is easy to carry out this program. As in the usual way well-known particularly in statistical mechanics (see for example Davidson, 1962), Eqs. (1) and (2) are multiplied by Lagrange multipliers 2 - 1 and fl, subtracted from Eq. (4) and the extreme value is taken for the resultant quantity regarding pi's independent. Interchanging the dummy indices properly in the variational equation we have Pl = Z - l e-flE~
(5)
where Z is called the partition function and given by Z = e - ~ = ~ e -pE' i
(6)
208
and E i is the statistical-mechanical analogue o f energy:
e, = Hi + 89Z (J,j + &) pj J
(7) + 89~'. (K~jk + Kjik + Kkji) PjPk + " ' ' j,k fi- 1 is the analogue of temperature as will be seen from the following consideration. The system under consideration is supposed to be in weak contact with another system. Since there is an interaction between the two systems which is negligibly small compared with f and g, g being the similar constant of another system when separated, only the sum and not each of f and g is constant. Lagrange's method shows that fl is common to the two systems. When the second system is considered to be non-interacting, the probability distribution and the partition function are given by qm = Y - l e - P ~
(8)
Y= ~ e-~"
(9)
probability distributions of these systems. However such problems are not discussed here. In many situations in physics and perhaps in some problems in social science, the coefficients are symmetric:
Jij = Jji,
Kijk = Kjik,
etc.
(12)
In this case the Relation (5) among pi's is very complicated and cannot in general be solved for Pi explicitly. We therefore limit our consideration to the case of only two events, n--2, and the bilinear interaction: E i = H , + ~ J~jpj (i,j= 1, 2). (13) J
Even this simple situation will help us to understand qualitatively and semi-quantitatively the behaviour of the interacting system. Introducing a new variable x = 2pl - 1 = 1 - 2p2
(14)
and combining Eqs. (5), (6), and (13), we have immediaately the relation:
m
where g,, is independent of qm'S. It is then easy to calculate the mean-square fluctuation of gin: 692_Z
2
2
m q~g" -- g = = _ y - ~ ~(g Y)
x = tanh [89 {H2 -- H 1 + 89 + 89
(15)
2 - J22 - J11) x}].
We further introduce two constants
y - 1 t~zY __g2
8f12 g2 _
- J11)
4 Og
(10)
flO = 2 J 1 2 _ J 2 2 _ J 1 t
Xo= 88
,
(16)
2 -H1)+ Ja2-Jlt-I
(17)
and a variable y where ~g/~fl- 1 is the analogue of heat capacity, fl- ~ is directly related to the fluctuation of the second system which plays the role of "heat bath", and therefore regarded as the "temperature" which is the measure to specify the probability distribution of the first system. If the coefficients are antisymmetric Jij = - J j l ,
K i j k § K j i k T Kkji = 0 ,
etc,
(11)
Eq. (5) is simply the usual Boltzmann distribution in statistical mechanics and there is no contribution of the interaction terms to the partition function. The "thermodynamic" properties of the system have no relation to these coefficients which can contribute only in dynamical processes. This situation occurs in many competition problems such as in the ecological system (Goel et al., 1971) and perhaps partly in the central nervous system (Cowan, 1969). Given the constants of motion of the latter two systems, our approach can easily be used to obtain the steady-state
X= ~/~o y--x
o
(18)
to obtain a simple relation x = tanh y.
(19)
Relations (15) or (18) and (19) are quite familiar in ferromagnetism and show a "phase transition" (Brout, 1965) below the critical temperature flo 1 fl- 1 < flo 1
(20)
as seen from Fig. 1 for Xo = 0. The case of x0 > 0 is also shown in Fig. 2 where the phase transition occurs at a lower temperature: f l - 1 =0~flO 1 '
(Z< 1.
(21)
Since coupled Eqs.(18) and (19) are invariant under the transformation x - - ~ - - x , Xo--~--Xo, the solution for Xo < 0 can be shown by simply mirroring Fig. 2 with respect to the horizontal axis. A ferromagnet obeys a similar relation to (15) under the
209
_•
-?~_ '
1
a
P0 Y
; y
b
p
i
x 730
X x
ix1 - ~ n h y
1 x 1 ~
x2 Y ~Xo 1
•
,
' a
,
2y
~
x l- --) -l~'~ii} b
Fig. 1. (a) Three curves of x=y, x=(~o/~)y and x=tanhy. (b) Schematic representation of the relation x=tanh[(fl/~o)X]
Fig. 2. (a) Three curves of x = a y - x o , x = ( ~ o / , 8 ) y - x o and x = tanhy, x 0 > 0. (b) Schematic representation of the relation x = t a n h [(fl//~o) (x + Xo)]
"Bragg-Williams approximation" (13) because the exchange interaction represented by J's is so strong that all spins align below the Curie temperature. Similarly in our case, the symmetric interaction between elements of the system makes only one event to dominate under the condition (20). When x is much smaller than t, Eq. (19) is expanded to give approximately x 1 (22) 89 - Hi) + 88 - -/11) rio* - fl- t
person taking the ith action is written as Pi which satisfies Eq. (1). The most crucial problem here is to find a constant as given by Eq. (2). In statistical mechanics it is the energy of the system that is conserved exactly. It wil]l be hopeless to find an exact constant in social problems, but even an approximate constant will be very informative to us. We anticipate here that the average amount of individual behaviour for the given set of actions is approximately constant. The grounds for this reasoning are twofold: Firstly, for the case of purchasing activity this quantity will indeed be approximately constant since his allowed expense will be given. Secondly, for the case of nonfinancial activity such as the emotional one, for which the reasoning should be inevitably less persuasive, we rely on Freud's famous psychoenergy-conservation law (see for example Brown, 196t), which essentially states that our direction and degree of movement are controlled by a tendency to reduce tension and thus return to a constant state like homeostasis, the unpleasurable tension being the result of the compromise between the pleasure principle based on the instinct desire and the reality principle based on the culture. If this statement is true, the average amount of individual emotional behaviour will be given by his tension determined by the desirous and cultural levels of his group. F r o m the discussion given above, it will suffice for the gross facts to regard the average amount of individual behaviour for the given set of actions as approximately constant, which is denoted by f. The quantity f will be determined by coherent forces from outside such as advertisement, propaganda and opinion leaders and also by the influence of other people in the group both through direct encounters and mass communication. Other enormous noisy sources of information will b e regarded as '"heat bath", or more suitably '"information bath" in which the group under consideration is immersed.
which is the analogue of the Curie-Weiss law. In the next section we apply the result of this section to a problem in social science, namely the mass behaviour. IIL Application to the M a s s Behavioar We suppose a group of N people whose lifestyles and views of things are more or less the same and therefore who act in a similar way for a given stimulus. Such a homogenity is required in order that the people can be regarded as "molecules" although the required degree of homogenity depends upon the kind and scale of the behaviour they take. Namely, for the investigation of the mass behaviour we seek for a law of behaviour which is most unbiased subject to a given information instead of adopting a very difficult and quite unsettled method of starting from the elucidation of individual behaviour. This approach is somewhat similar to the situation in statistical mechanics where one completely gives up to solve enormous equations of motion obeyed by individual molecules. However the situation is even worse in our case since the definite law of individual behaviour is not established and thus our result should be only an approximation, although we believe it is important as one step towards a new principle for the mass behaviour. Each of N people is supposedly allowed freedom of choice of n ways of action. The probability of a
210
Furthermore the mutual interaction between people in the mass behaviour will be such that one tends to follow others, and thus will be symmetric. This "law of imitation" is the consequence of the egodefence mechanism in psychoanalysis: One will feel uneasy if he does not adopt the action which most of the people in the group take. This situation is frequently encountered in many cultural, social, political and economic phenomena, namely more or less in all kinds of booms and fashions. It seems therefore natural to represent f in the form (2) where f is given by Eq. (3) up to the second term with symmetric coefficients. Hi means here the coherent force from outside on people who have chosen the ith action and Jzj represents the mutual influence between people of the ith and jth actions. Given the limited information (2), the most uncertain state of the system with regard to the missing information is given by the maximum-entropy principle. General results (5), (6), and (13) follow immediately by the procedure of the preceeding section. The group under consideration is always receiving noisy informations from the outside world, largely from mass communication, in addition to coherent informations and therefore member's decision towards certain behaviour is disturbed by them. Thus according to relation (10),/3-1 may be identified as "informational temperature" which is proportional to the fluctuation of information in mass communication. For qualitative and semi-quantitative discussions for the mass behaviour to be possible, we hereafter limit our consideration to the case where only two ways of action are allowed. Fig. 1b for x0 = 0 shows that x =0, i.e. Pl =P2 results when the temperature /3-1 is high enough or the mutual influence /3o 1 given by Eq. (16) is small, namely no prefered behaviour occurs in the system. People behave as if there were no interaction between them: Diversification takes place. On the other hand when either the temperature is low enough or the mutual influence is strong enough, perhaps because a specific behaviour is very appealing, to satisfy the condition (20), one of the two behaviours dominates spontaneously, namely x=~0, which means that a cooperative behaviour occurs within the group. All kinds of booms and fashions have in common more or less such a drastic property. Next we discuss the case of x o > 0. From Fig. 2b there are three situations: (i) The behaviour which is more strongly subjected to the external force always dominates and becomes increasingly preferred as the temperature is lowered. This situation is the usual case.
(ii) The behaviour which is less strongly subjected to the external force suddenly dominates at a lower temperature (21). The discontinuity is the result of the reaction to the strengthened behaviour. This case is however unstable and corresponds perhaps to certain type of temporal and local fad. (iii) This case is similar to (ii) but stable. An important aspect from the point of view of propaganda is given by Eq.(22). When J z z = J l l , the left side represents the efficiency of propaganda which is very high near the transition temperature. Namely, propaganda should be performed timely at the beginnings of booms and fashions.
IV. Conclusions
We have developed the maximum-entropy principle of information theory to include the interaction between elements of the system. The result is general enough to be applied other system than the physical one. A particular case can explain a ferromagnet if desired. Our approach is shown to be capable of treating certain mass behaviour which by some analyses may be regarded as a kind of phase transition. Several problems in social and life sciences where the interaction between elements plays a major role will be treated by the present method. The results for the mass behaviour have some implications for the fashion forecast and marketing which will be discussed elsewhere. References Brillouin, L.: Science and information theory. New York: Academic Press 1956 Brout, R.: Phase transitions. New York: Benjamin 1965 Brown, J.A.: Freud and the Post-Freudians. Middlesex: Penguin Books 1961 Cowan, J.D.: A statistical mechanics of nervous activity. Technical Report JDC 69-1 : The University of Chicago 1969 Cozzolino, J.M., Zahner, M.J.: The maximum-entropy distribution of the future market price of a stock. Op. Res. 21,1200~12t 1 (1973) Davidson, N.: Statistical mechanics. New York: McGraw-Hill 1962 Goel, N.S., Maitra, S.C., Montroll, E.W.: On the volterra and other nonlinear models of interacting populations. Rev. Mod. Phys. 43, 231--276 (1971) Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620--630 (1957) Rapopart, A.: Mathematical models of social interaction. In: Luce, R.D., Bush, R., Galanter, E. (Eds.): Handbook of mathematical psychology, Vol. II. New York: John Wiley & Sons 1963 Dr. M. Takatsuji Central Res. Laboratory of Hitachi, Ltd. 1-280 Higashi Koigakubo Kokubunji, Tokyo 185, Japan
An Information-Theoretical Approach to a System of Interacting Elements M. Takatsuji Central Research Laboratory of Hitachi, Ltd., Kokubunji, Tokyo, Japan Received: May 12, 1974
Abstract The maximum-entropy principle in information theory is generalized to include the interaction between elements of the system. A complex relation between the probabilities of the events is derived using a familiar technique in statistical mechanics. The relation is explicitly discussed for the case of bilinear interaction and only two events. Quite noteworthy is the existense in the system of a kind of "phase transition" similar to ferromagnetism. The result is applied to the mass behaviour. It is shown that the cooperative mass behaviour such as boom and fashion may be interpreted as a phase transition which would occur below certain "informational temperature".
I. Introduction Information theory has been used extensively in many fields of science and engineering. Not only communication theory but also such different fields as linguistics, molecular biology and social science have enjoyed this theory. Physics is not an exception: Brillouin (1956) discussed a close connection between information-theory entropy and thermodynamic entropy. Particularly important is the work of Jaynes (1957) who reconstructed statistical mechanics based on the maximum-entropy principle. This principle states that our ignorance must be represented in the most unbiased form, that is, the probability distribution must be so generated as to have maximum uncertainty subject to a given information. It is powerful enough to be applied to other fields than physics. Recently Cozzolino and Zahner (1973) used this principle for an econometric problem of future stock price. Although the maximum-entropy principle has large potential, its application has been limited to systems of independent elements. Many real systems are composed of mutually interacting elements and the interaction often plays a decisive role: In a thermodynamic system such as a ferromagnet, the interaction between constituent atoms is so strong as to produce a phase transition (Brout, 1965). In a biological association, the predator-prey interaction between species yields variations and distributions of the populations (Goel et al., 1971). The social interaction between people is very important in some psychological problems such as group dynamics (Rapopart, 1963).
The purposes of this paper are twofold. Firstly, Jaynes' theory of the maximum-entropy principle is generalized to include the interaction between elements of the system. Secondly, the result is shown to be applicable to certain type of the mass behaviour by a careful analysis of the latter. n . The Maximum-Entropy Principle for an Interacting System Let pi(i = t, 2 .... , n) represents the probability of the ith event: E Pi = 1.
(1)
i
It is assumed that the system under consideration allows the average value of a quantity fi being constant: f = E P,fi. (2) i
Here f~ is in general a function of p/s in order that the interaction between elements is taken into account, and written in an expansion: fl = Hi + 89E JiiPj + 89E KijkPJPk + ' " " j
j,k
(31
The maximum-entropy principle states that the most unbiased representation of our ignorance of the state of the system is provided by the probability distribution which maximizes the entropy S = - ~ p~lnp~
(4)
i
under the given Conditions (1) and (2). It is easy to carry out this program. As in the usual way well-known particularly in statistical mechanics (see for example Davidson, 1962), Eqs. (1) and (2) are multiplied by Lagrange multipliers 2 - 1 and fl, subtracted from Eq. (4) and the extreme value is taken for the resultant quantity regarding pi's independent. Interchanging the dummy indices properly in the variational equation we have Pl = Z - l e-flE~
(5)
where Z is called the partition function and given by Z = e - ~ = ~ e -pE' i
(6)
208
and E i is the statistical-mechanical analogue o f energy:
e, = Hi + 89Z (J,j + &) pj J
(7) + 89~'. (K~jk + Kjik + Kkji) PjPk + " ' ' j,k fi- 1 is the analogue of temperature as will be seen from the following consideration. The system under consideration is supposed to be in weak contact with another system. Since there is an interaction between the two systems which is negligibly small compared with f and g, g being the similar constant of another system when separated, only the sum and not each of f and g is constant. Lagrange's method shows that fl is common to the two systems. When the second system is considered to be non-interacting, the probability distribution and the partition function are given by qm = Y - l e - P ~
(8)
Y= ~ e-~"
(9)
probability distributions of these systems. However such problems are not discussed here. In many situations in physics and perhaps in some problems in social science, the coefficients are symmetric:
Jij = Jji,
Kijk = Kjik,
etc.
(12)
In this case the Relation (5) among pi's is very complicated and cannot in general be solved for Pi explicitly. We therefore limit our consideration to the case of only two events, n--2, and the bilinear interaction: E i = H , + ~ J~jpj (i,j= 1, 2). (13) J
Even this simple situation will help us to understand qualitatively and semi-quantitatively the behaviour of the interacting system. Introducing a new variable x = 2pl - 1 = 1 - 2p2
(14)
and combining Eqs. (5), (6), and (13), we have immediaately the relation:
m
where g,, is independent of qm'S. It is then easy to calculate the mean-square fluctuation of gin: 692_Z
2
2
m q~g" -- g = = _ y - ~ ~(g Y)
x = tanh [89 {H2 -- H 1 + 89 + 89
(15)
2 - J22 - J11) x}].
We further introduce two constants
y - 1 t~zY __g2
8f12 g2 _
- J11)
4 Og
(10)
flO = 2 J 1 2 _ J 2 2 _ J 1 t
Xo= 88
,
(16)
2 -H1)+ Ja2-Jlt-I
(17)
and a variable y where ~g/~fl- 1 is the analogue of heat capacity, fl- ~ is directly related to the fluctuation of the second system which plays the role of "heat bath", and therefore regarded as the "temperature" which is the measure to specify the probability distribution of the first system. If the coefficients are antisymmetric Jij = - J j l ,
K i j k § K j i k T Kkji = 0 ,
etc,
(11)
Eq. (5) is simply the usual Boltzmann distribution in statistical mechanics and there is no contribution of the interaction terms to the partition function. The "thermodynamic" properties of the system have no relation to these coefficients which can contribute only in dynamical processes. This situation occurs in many competition problems such as in the ecological system (Goel et al., 1971) and perhaps partly in the central nervous system (Cowan, 1969). Given the constants of motion of the latter two systems, our approach can easily be used to obtain the steady-state
X= ~/~o y--x
o
(18)
to obtain a simple relation x = tanh y.
(19)
Relations (15) or (18) and (19) are quite familiar in ferromagnetism and show a "phase transition" (Brout, 1965) below the critical temperature flo 1 fl- 1 < flo 1
(20)
as seen from Fig. 1 for Xo = 0. The case of x0 > 0 is also shown in Fig. 2 where the phase transition occurs at a lower temperature: f l - 1 =0~flO 1 '
(Z< 1.
(21)
Since coupled Eqs.(18) and (19) are invariant under the transformation x - - ~ - - x , Xo--~--Xo, the solution for Xo < 0 can be shown by simply mirroring Fig. 2 with respect to the horizontal axis. A ferromagnet obeys a similar relation to (15) under the
209
_•
-?~_ '
1
a
P0 Y
; y
b
p
i
x 730
X x
ix1 - ~ n h y
1 x 1 ~
x2 Y ~Xo 1
•
,
' a
,
2y
~
x l- --) -l~'~ii} b
Fig. 1. (a) Three curves of x=y, x=(~o/~)y and x=tanhy. (b) Schematic representation of the relation x=tanh[(fl/~o)X]
Fig. 2. (a) Three curves of x = a y - x o , x = ( ~ o / , 8 ) y - x o and x = tanhy, x 0 > 0. (b) Schematic representation of the relation x = t a n h [(fl//~o) (x + Xo)]
"Bragg-Williams approximation" (13) because the exchange interaction represented by J's is so strong that all spins align below the Curie temperature. Similarly in our case, the symmetric interaction between elements of the system makes only one event to dominate under the condition (20). When x is much smaller than t, Eq. (19) is expanded to give approximately x 1 (22) 89 - Hi) + 88 - -/11) rio* - fl- t
person taking the ith action is written as Pi which satisfies Eq. (1). The most crucial problem here is to find a constant as given by Eq. (2). In statistical mechanics it is the energy of the system that is conserved exactly. It wil]l be hopeless to find an exact constant in social problems, but even an approximate constant will be very informative to us. We anticipate here that the average amount of individual behaviour for the given set of actions is approximately constant. The grounds for this reasoning are twofold: Firstly, for the case of purchasing activity this quantity will indeed be approximately constant since his allowed expense will be given. Secondly, for the case of nonfinancial activity such as the emotional one, for which the reasoning should be inevitably less persuasive, we rely on Freud's famous psychoenergy-conservation law (see for example Brown, 196t), which essentially states that our direction and degree of movement are controlled by a tendency to reduce tension and thus return to a constant state like homeostasis, the unpleasurable tension being the result of the compromise between the pleasure principle based on the instinct desire and the reality principle based on the culture. If this statement is true, the average amount of individual emotional behaviour will be given by his tension determined by the desirous and cultural levels of his group. F r o m the discussion given above, it will suffice for the gross facts to regard the average amount of individual behaviour for the given set of actions as approximately constant, which is denoted by f. The quantity f will be determined by coherent forces from outside such as advertisement, propaganda and opinion leaders and also by the influence of other people in the group both through direct encounters and mass communication. Other enormous noisy sources of information will b e regarded as '"heat bath", or more suitably '"information bath" in which the group under consideration is immersed.
which is the analogue of the Curie-Weiss law. In the next section we apply the result of this section to a problem in social science, namely the mass behaviour. IIL Application to the M a s s Behavioar We suppose a group of N people whose lifestyles and views of things are more or less the same and therefore who act in a similar way for a given stimulus. Such a homogenity is required in order that the people can be regarded as "molecules" although the required degree of homogenity depends upon the kind and scale of the behaviour they take. Namely, for the investigation of the mass behaviour we seek for a law of behaviour which is most unbiased subject to a given information instead of adopting a very difficult and quite unsettled method of starting from the elucidation of individual behaviour. This approach is somewhat similar to the situation in statistical mechanics where one completely gives up to solve enormous equations of motion obeyed by individual molecules. However the situation is even worse in our case since the definite law of individual behaviour is not established and thus our result should be only an approximation, although we believe it is important as one step towards a new principle for the mass behaviour. Each of N people is supposedly allowed freedom of choice of n ways of action. The probability of a
210
Furthermore the mutual interaction between people in the mass behaviour will be such that one tends to follow others, and thus will be symmetric. This "law of imitation" is the consequence of the egodefence mechanism in psychoanalysis: One will feel uneasy if he does not adopt the action which most of the people in the group take. This situation is frequently encountered in many cultural, social, political and economic phenomena, namely more or less in all kinds of booms and fashions. It seems therefore natural to represent f in the form (2) where f is given by Eq. (3) up to the second term with symmetric coefficients. Hi means here the coherent force from outside on people who have chosen the ith action and Jzj represents the mutual influence between people of the ith and jth actions. Given the limited information (2), the most uncertain state of the system with regard to the missing information is given by the maximum-entropy principle. General results (5), (6), and (13) follow immediately by the procedure of the preceeding section. The group under consideration is always receiving noisy informations from the outside world, largely from mass communication, in addition to coherent informations and therefore member's decision towards certain behaviour is disturbed by them. Thus according to relation (10),/3-1 may be identified as "informational temperature" which is proportional to the fluctuation of information in mass communication. For qualitative and semi-quantitative discussions for the mass behaviour to be possible, we hereafter limit our consideration to the case where only two ways of action are allowed. Fig. 1b for x0 = 0 shows that x =0, i.e. Pl =P2 results when the temperature /3-1 is high enough or the mutual influence /3o 1 given by Eq. (16) is small, namely no prefered behaviour occurs in the system. People behave as if there were no interaction between them: Diversification takes place. On the other hand when either the temperature is low enough or the mutual influence is strong enough, perhaps because a specific behaviour is very appealing, to satisfy the condition (20), one of the two behaviours dominates spontaneously, namely x=~0, which means that a cooperative behaviour occurs within the group. All kinds of booms and fashions have in common more or less such a drastic property. Next we discuss the case of x o > 0. From Fig. 2b there are three situations: (i) The behaviour which is more strongly subjected to the external force always dominates and becomes increasingly preferred as the temperature is lowered. This situation is the usual case.
(ii) The behaviour which is less strongly subjected to the external force suddenly dominates at a lower temperature (21). The discontinuity is the result of the reaction to the strengthened behaviour. This case is however unstable and corresponds perhaps to certain type of temporal and local fad. (iii) This case is similar to (ii) but stable. An important aspect from the point of view of propaganda is given by Eq.(22). When J z z = J l l , the left side represents the efficiency of propaganda which is very high near the transition temperature. Namely, propaganda should be performed timely at the beginnings of booms and fashions.
IV. Conclusions
We have developed the maximum-entropy principle of information theory to include the interaction between elements of the system. The result is general enough to be applied other system than the physical one. A particular case can explain a ferromagnet if desired. Our approach is shown to be capable of treating certain mass behaviour which by some analyses may be regarded as a kind of phase transition. Several problems in social and life sciences where the interaction between elements plays a major role will be treated by the present method. The results for the mass behaviour have some implications for the fashion forecast and marketing which will be discussed elsewhere. References Brillouin, L.: Science and information theory. New York: Academic Press 1956 Brout, R.: Phase transitions. New York: Benjamin 1965 Brown, J.A.: Freud and the Post-Freudians. Middlesex: Penguin Books 1961 Cowan, J.D.: A statistical mechanics of nervous activity. Technical Report JDC 69-1 : The University of Chicago 1969 Cozzolino, J.M., Zahner, M.J.: The maximum-entropy distribution of the future market price of a stock. Op. Res. 21,1200~12t 1 (1973) Davidson, N.: Statistical mechanics. New York: McGraw-Hill 1962 Goel, N.S., Maitra, S.C., Montroll, E.W.: On the volterra and other nonlinear models of interacting populations. Rev. Mod. Phys. 43, 231--276 (1971) Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620--630 (1957) Rapopart, A.: Mathematical models of social interaction. In: Luce, R.D., Bush, R., Galanter, E. (Eds.): Handbook of mathematical psychology, Vol. II. New York: John Wiley & Sons 1963 Dr. M. Takatsuji Central Res. Laboratory of Hitachi, Ltd. 1-280 Higashi Koigakubo Kokubunji, Tokyo 185, Japan
