Injormation Concept in Ecology 145 



|6] ey = eO'; Tia, Ub, . . .;Xi, X2, . . .). 



If/?, is the probability for a particular small subsystem to be in 

 state I, then, 



[7] Z V, = 1 



i 

 [8] 11V^^^ = <f> 



[y] Y^V^na.i = <>L>, (a,l>,c, . . .) 



and 



[10] *S:= -kY^P^np,, 



where S is the entropy of the system, and < > denotes expected 

 values. The maximum uncertainty of selecting" a subsystem in state 

 I is obtained when the p^s are all ecjual (10). Maximizing [10] 



[11] "T=? ^^''^'^ l)dp, = 0; 



difTerentiating [7], [8] and [9] and introducing the undetermined 

 Lagrangian multipliers ««, «&, • . ■ , 1^, ^a, we obtain 



[12] (Qo- l)T.dp. = 



[13] (3Y.e.dp,=0 



[14] a^X) na.,dpi = 0. {a,h,c, . . .) 



Adding" [11-14] and collecting" terms: 



[15] XI Oil P' + ^^0 -\- ^U + a,, Ha., + abni,, + . . .) fZ/J/ = 0, 



from which 



[16] p, = exp ( — Qo — (Se,- — aa nn.i — ab Hb.i —...). 



Thus, a system's entropy is maximal when the e,'s, na,i's, «6, /s, 

 etc., of all its subsystems are identical, signifying a homogeneous 

 distribution of matter and energy throughout the system. It can 

 be shown (22) that the rate of entropy change is 



