Canonical Ensemble NVT

Another way to view MD simulation is as a technique to probe the atomic positions and momenta that are available to a molecular system under certain conditions. In other words, MD is a statistical mechanics method that can be used to obtain a set of configurations distributed according to a certain statistical ensemble. The natural ensemble for MD simulation is the microcanonical ensemble, where the total energy E, volume V, and amount of particles N (NVE) are constant. Modifications of the integration algorithm also allow for the sampling of other ensembles, such as the canonical ensemble (NVT) with constant temperature... 

There are several conditions in which molecular dynamics simulations can take place. These include the microcanonical ensemble (NV ), canonical ensemble (NVT), isothermal-isobaric ensemble (NPT), and grand canonical ensemble (p-VT), where N is the number of particles, V is the volume, E is the energy, P is the pressure, T is the temperature, and x is the chemical potential. In each of these ensembles, the thermodynamic variables held constant are designated by the appropriate letters. 

If there is no time-dependent external force, the dynamics of a molecular system will evolve on a constant-energy surface. Therefore, a natural choice of the statistical ensemble in molecular dynamics simulation is the micro-canonical ensemble (NVE). Other types of ensembles, such as the canonical ensemble (NVT) and the isothermal-isobaric ensemble (NPT), can also be realized by controlling corresponding thermodynamic variables. For the last two ensembles, the temperature of the ensemble needs to be controlled and four different control mechanisms, namely differential control, proportional control, integral control and stochastic control, have been developed in the literature. As an example, a proportional thermostat for the NVT ensemble will be briefly discussed as follows. 

In the case of the hydride transfer catalysed by Escherichia coli DHFR (EcDHFR), one-dimensional PMFs were computed using the antisymmetric combination of distances describing the hydride transfer, = d(CdonorH) — d(CacceptorH)> as the reaction coordinate. The umbrella sampling approach was used, with the system restrained to remain close to the desired value of the reaction coordinate by means of the addition of a harmonic potential. As stated above, the probability distributions obtained from MD simulations within each individual window were combined by means of the WHAM method. Twenty picoseconds of relaxation and 40 ps of production MD, with a time step of 0.5 fs, in the canonical ensemble [NVT (number of molecules, volume, temperature), with a reference temperature of 300 K] and the Langevin-Verlet integrator [85] were used in the simulations. 

The canonical ensemble (NVT), where volume and temperature are kept constant. This ensemble facilitates comparisons with experimental data from structures with fixed dimensions. 

It is instructive to see this in temis of the canonical ensemble probability distribution function for the energy, NVT - Referring to equation B3.3.1 and equation (B3.3.2I. it is relatively easy to see that... 

Monte Carlo Methods. Although several statistical mechanical ensembles may be studied using MC methods (2,12,14), the canonical ensemble has been the most frequently used ensemble for studies of interfacial systems. In the canonical ensemble, the number of molecules (N), cell volume (V) and temperature (T) are fixed. Hence, the canonical ensemble is denoted by the symbols NVT. The choice of ensemble determines which thermodynamic properties can be computed. 

Another parameter that can have a great influence on the results obtained is the type of the simulation performed. Generally, simulations are carried out at constant particle number (N). The volume (V) and energy (E) of the simulated system can be held constant, leading to a so-called NVE, or microcanonical, ensemble. When the volume and temperature are held constant, this yields a canonical or NVT ensemble. In both cases, the size of the simulated system is chosen in such a way as to represent the desired state of the phospholipid, mostly the liquid crystalline La phase. The surface per lipid and the thickness of the bilayer are set based on experimental values and remain unchanged during the simulation. Therefore, the system is not able to adjust its size and thickness. 

While the NVE (microcanonical) ensemble theory is sound and useful, the NVT (canonical) ensemble (which fixes the number of particles, volume, and temperature while allowing the energy to vary) proves more convenient than the NVE for numerous applications. 

In a series of papers,Nose showed that a Hamiltonian mechanics could be written down that would generate the distribution function for the NVT or canonical ensemble. The basic approach involves extending the phase space of the system, in a manner similar to that originally laid out by Andersen and by Parrinello and Rahman. Namely, in addition to the dN coordinates and dN momenta, where d is the number of spatial dimensions, an additional variable s, representing a heat bath, and its conjugate momentum are included. The Hamiltonian for the extended system is given by... 

The energy change, AE, associated with the move is calculated from this the Boltzmann factor B = exp ( — AE/kT) is calculated if AE is positive. We note that MC simulations are most commonly run in the NVT (or canonical) ensemble in which both volume and temperature are fixed. 

Consider simulating a system in the canonical ensemble, close to a first-order phase transition. In one phase, NVt( is essentially a Gaussian centred around a value E, while in the other phase the peak is around Ejj. 

A Monte Carlo simulation traditionally samples from the constant NVT (canonical) ensemble, but the teclmique can also be used to sample from different ensembles. A common alternative is the isothermal-isobaric, or constant NPT, ensemble. To simulate from this ensemble, it is necessary to have a scheme for changing the volume of the simulation cell in order to keep the pressure constant. This is done by combining random displacements of the particles with random changes in the volume of the simulation cell. The size of each volume change is governed by the maximum volume change, V ax-Thus a new volume is generated from the old volume as follows ... 

For the statistical mechanical problem to be well posed, a choice of ensemble is essential. To this point, we have assumed the canonical or NVT ensemble, that is one in which the number of particles, N, the volume, V, and the temperature T are held constant, while the conjugate variables chemical potential, pressure, and energy are allowed to fluctuate. The magnitude of these fluctuations can be related to thermodynamic... 

It is easy to check that detailed balance holds for the Metropolis algorithm described above with p ff) equal to the canonical ensemble distribution in configuration space. In the Metropolis algorithm, IF(F- F ) = a(F F )min(l,/OM(F )//OM(f)), where q (F F ) is the probability of attempting a move from F to F and Pacc(F -> F ) = min, pijj ) PiJr)) is the probability of accepting that move. In the classical Metropolis algorithm, the attempt step is designed such that q (F -> F ) = q (F F). In the NVT ensemble, the acceptance probability is mi (l,exp[—(V —V)/(k5l)]), as described above. One can easily verify that the detailed balance equation (77) is satisfied under these conditions. 

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