Canonical Boltzmann sampling

The most obvious way of gathering information about the set of macrostates is to use canonical Boltzmann sampling to explore them one at a time, with a sampling distribution set to... 

To study protein folding theoretically, simulation methods have proved indispensable. The folding transition is ultimately governed by statistical thermodynamics and hence it is paramount to use sampling methods that are able to reproduce the canonical Boltzmann distribution. Common sampling techniques are molecular dynamics (MD), Langevin or Brownian dynamics (BD) and Monte Carlo (MC). 

When sampling from a (canonical) Boltzmann distribution, it follows from Eq. (1.4) that the limiting probability, pm, is given by... 

Gaussian-like distribution of energy around the energy average. Other ensembles with non-Boltzmann distributions can enhance the sampling considerably for example, in the multi-canonical approach [97, 98], all the conformations are equiprobable in energy in Tsallis statistics [99], the distribution function includes Boltzmann, Lorentzian, and Levy distributions. 

Using multicanonical simulation techniques we can sample all configurations in the pertinent interval of density (order parameter) and determine their free energy. It is important to note that the simulation samples all states at a fixed order parameter with the Boltzmann weight of the canonical ensemble. What are the typical configurations that a finite system of volume V adopts inside the miscibility gap in the canonical ensemble [48-55] ... 

Several variations of ordinary MD have been developed. Andersen has proposed "molecular dynamics at constant temperature," in which an MD system is made to represent a canonical system, by altering the momentum of random particles at sequential random Instants of time. The new momentum is picked from a Boltzmann distribution, with a given parameter p. Since the motion of the system is no longer Hamiltonian, this procedure is a statistical sampling method. A combination technique was used by Wood and Erpenbeck, who ran a set of independent MD calculations, with the initial phase of each calculatlon glcked from a canonical, or mlcrocanonical, distribution. Andersen also described molecular dynamics at constant pressure," in which the pressure is a parameter of the Lagranglan, and the system volume fluctuates. 

The MC technique is a stochastic simulation method designed to generate a long sequence, or Markov chain of configurations that asymptotically sample the probability density of an equilibrium ensemble of statistical mechanics [105, 116]. For example, a MC simulation in the canonical (NVT) ensemble, carried out under the macroscopic constraints of a prescribed number of molecules N, total volume V and temperature T, samples configurations rp with probability proportional to, with, k being the Boltzmann constant and T the... 

---