N Molecules Flipping Between Two Compartments

Consider now a constant number N of molecules, identical to the one in the previous section, which switch independently between compartments I and E. Given molecule independence, the probability of finding n/ molecules in compartment I at time t obeys the following master equation 

The mean molecule count in compartment / can be calculated from the corresponding probability distribution as Niit) = niP(rii,t). By differentiating this expression and substituting Eq. (9.7) we obtain that 

From Eqs. (9.10) and (9.11), the average net molecule current from compartment E to compartment 7 (i.e. the average number of molecules crossing from E to I per unit time) results to be 

That is, the stationary state is characterized by equal molecule concentrations on both compartments. 

To give a thermodynamic interpretation to the result in Eq. (9.17), recall the expression for the ehemical potential derived in Chap. 2—Eq. (2.11)  

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