Extended ensemble formalism

Upon expressing from the equilibrium condition the complex concentration M12 through the concentrations of monomers, and substituting the expression found into relationship (21) we obtain, invoking the formalism of the Markov chains, final formulas enabling us to calculate instantaneous statistical characteristics of the ensemble of macromolecules with colored units. A subsequent color erasing procedure is carried out in the manner described above. For example, when calculating instantaneous copolymer composition, this procedure corresponds to the summation of the appropriate components of the stationary vector jt of the extended Markov chain ... 

The Car-Parrinello method is similar in spirit to the extended system methods [37] for constant temperature [38, 39] or constant pressure dynamics [40], Extensions of the original scheme to the canonical NVT-ensemble, the NPT-ensemble, or to variable cell constant-pressure dynamics [41] are hence in principle straightforward [42, 43]. The treatment of quantum effects on the ionic motion is also easily included in the framework of a path-integral formalism [44-47]. 

Realizations of this formalism go under the names adaptive umbrella sampling [26] and the multicanonical ensemble introduced by Berg and Neuhaus [27]. It seems right to attribute the recent revival in interest in extended sampling to the latter work. 

Fokker-Planck equation The Langevin equation describes the Brownian motion of a single particle which experiences a random force (due to collisions with the solvent particles) causing the velocity to behave in a stochastic way. The Fokker-Planck equation (also known as Kolmogorov forward equation) extends the Langevin equation to an ensemble of identical Brownian particles by finding the probability distribution P v, t) of N particles in the ensemble having velocities in the interval v, V -t- St) at time t. The Fokker-Planck equation can be formally expressed as... 

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