Some Membrane Phenomena from the Point of View of Information Theory 199 



where A^" is the rate of entropy production. The J's are 'fluxes' and the A"s 

 are 'forces', u is energy, in matter, and // chemical potential. The J's are related 

 to the A"s through 



Assuming an isothermal system, the A"s and y's are 



X, = -^m^T J,-^ 



dt 



dt 



Whereupon, equation (4) becomes 



AS = -J„, ^fl|T - J^ AmIT. (5) 



Substituting into equation (5), 



/* ^ //q + i^rin a 

 where a is the activity, we get 



A^ = (R In ajao) ^ + 7? Am d/dt (In ajoo). (6) 



dt 



The a's refer to the activities in the two different regions. 



Equation (6) is the basic relation replacing both equations (2) and (3). If we 

 assume that the activities may be replaced by the concentrations and that we 

 are concerned with the passage of A^ particles from a lower to a higher concen- 

 tration, Cq > Q, then equation (6) becomes 



^^Ii=k-^ In Co/C, + Nk didt (In CJC,). 



If the outside is taken to be very large with no change in concentration by the 

 addition of the particles from within, 



H=A:^lnC./C,--^. (7) 



Equation (7) gives the rate of decrease of entropy or the rate of production 

 of negentropy or information by a membrane which is transferring material 

 from a lower to a higher concentration where the particles leave at the rate 

 dNjdt and the concentration within changes at the rate dCjjdt. Thus one may 

 look upon equation (7) as the dynamic equation describing real transport. On 

 the other hand equation (2) describes a situation where there is no macroscopically 

 discernible change in concentration within or without. But inasmuch as the 

 membrane maintains a concentration difference, it is producing information 

 at the rate given to continue the imbalance. 



