text: canonical ensemble. The free energy functional is ﬁrst deﬁned on the original KMC lattice. How- 
  
  ever, after re-writing the interaction terms employing gradient operators [78] one ﬁnally obtains 
  
  the free energy functional for a continuous system 
  
  (cid:90) 
  (cid:104) 
  dr 
  (cid:105) 
  
  εll 
  2  εnn 
  2 
  (∇ρl)2 +  (∇ρn)2 + εnl(∇ρn) · (∇ρl) − µρl  F [ρl, ρn] =  f (ρl, ρn) + 
  ,  (4) 
  
  where 
  
  f (ρl, ρn) = kT [ρl ln ρl + (1 − ρl) ln(1 − ρl)] 
  
  + kT [ρn ln ρn + (1 − ρn) ln(1 − ρn)] 
  
  − 2εllρ2 
  l − 2εnnρ2  n − 4εnlρnρl. 
  (5) 
  
  Since the liquid may evaporate from the surface into the vapour above the surface, µ is the (true) 
  
  chemical potential of this reservoir and determines the rate of evaporation [condensation] from 
  
  [to] the surface. Note that normally a free energy of the form in Eq. (4) is obtained by making a 