13/8 THE SYSTEM WITH LOCAL STABILITIES 



At any given moment, the whole system is at a definite state, and 

 therefore so is each variable ; the state of each variable either is or 

 is not a state of equilibrium for that variable (in the conditions 

 given by the other variables). The number of variables that are 

 at a state of equilibrium will be represented by i. If the whole is 

 of n variables then obviously i must lie in the range of to n. 

 If i equals n, then every variable is at a state of equilibrium 

 in the conditions given by the others, so the whole is at a state of 

 equilibrium (S. 6/8). If i is not equal to n, the other variables, 

 n — i in number, will change value at the next step in time. A 

 new state of the whole will then occur, and i will have a new value. 

 Thus, as the whole moves along a line of behaviour, i will change 

 in value ; and we can get a useful insight into the behaviour of the 

 whole by considering how i will behave as time progresses. 



13/7. The behaviour of i is strictly determinate once the system 

 and its initial state have been given. In a set of systems, how- 

 ever, the behaviour of i is difficult to characterise except at the 

 two extremes, where its behaviour is simple and clear. Com- 

 parison of what happens at the extremes will give us an insight 

 that will be invaluable in the later chapters, for it will go a long 

 way towards answering the fundamental problem of Chapter 11. 

 (By establishing what happens in the two specially simple and 

 clear cases we are following the strategy of S. 2/17.) 



13/8. At one extreme is the polystable system that has been 

 joined very richly, so that almost every variable is joined to almost 

 every other. (Such a system's diagram of immediate effects 

 would show that almost all of the n(n — 1) arrows were present.) 

 Let us consider the case in which, as in S. 12/15, every subsystem 

 has a high probability p of being at a state of equilibrium, and in 

 which the probabilities are all independent. How will * behave ? 

 (Here we want to know what will usually happen; what can 

 happen is of little interest.) 



The probability of each part being at a state of equilibrium is 

 p, and so, if independence (of probability) holds, the probability 

 that the whole (of n variables) is at a state of equilibrium will be 

 p n (by S. 6/8). If p is not very close to 1, and n is large, this 

 quantity will be extremely small (S. 11/4). i will usually have 

 a value not far from np (i.e. about a fraction p of the total will be 



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