## Wednesday, May 20, 2015

### introduction to NVE, NVT, NpT ensembles

The first term with which to get acquainted is “ensemble”. In the context of statistical mechanics, an ensemble is a collection of configurations of a system. These configurations will have a set of common constraints such as temperature, volume and pressure, but will differ in the positions and velocities of the individual components (atom/molecule/fragment) across the different configurations. So, even though these configurations differ microscopically, they possess the same common macroscopic properties (based on the ensemble they belong to). This means that ensemble averages for these properties can be computed. The simplest such ensemble is the microcanonical ensemble, in which the constraints are number of particles, volume and total energy, i.e., these quantities are constrained to be the same across all the configurations belonging to the microcanonical ensemble, and hence this ensemble is also known as the “NVE” ensemble. The common nomenclature of the ensemble provides information about the constraints in that ensemble. Hence, the NVE ensemble contains all possible combinations of positions and velocities for the particles which satisfy an energy constraint, provided all these configurations have the same particle density. The total number of configurations for an ensemble is given by its “partition function”, and for the microcanonical ensemble, the microcanonical partition function (also called density of states) is represented as Ω(N,V,E).
The fundamental importance of the density of states is based on the relationship
S(N,V,E) = kB ln Ω(N,V,E) (2.2)
where S is the entropy and kB is Boltzmann’s constant. Eq. 2.2 shows how a macroscopic property like entropy is directly related to the density of states for a system at constant density. A more convenient and commonly used ensemble is the canonical (NVT) ensemble, where the density and temperature are constrained across the configuration.
Similar to the microcanonical ensemble, a thermodynamic quantity can be related to the partition function. In this case, it is the Helmholtz free energy A = −kBT ln Q(N,V,T) (2.6) A commonly used ensemble in MC is the grand canonical ensemble, in which the chemical potential, volume and temperature are constrained. This means that the number of particles is allowed to fluctuate to satisfy the chemical potential constraint. The NpT ensemble, also called the isothermal-isobaric ensemble is also a commonly used ensemble. Here, the number of particles, temperature and pressure are constrained.
The fundamental relationship for the NpT ensemble is the expression relating the Gibbs free energy to the partition function: G = −kBT ln ∆(N,P,T) (2.8) where G is the Gibbs free energy, a thermodynamic quantity of fundamental importance to the problem of phase equilibrium. In all the ensembles listed above, the partition function integrals grow rapidly with increasing complexity and system sizes. Hence, trying to solve these integrals analytically is impractical. Molecular simulations aid in sampling the configuration space for these ensembles subject to the probability distribution for a given ensemble, and ensemble averages can be obtained using expressions similar to Eq. 2.5.

Ref: http://etd.nd.edu/ETD-db/theses/available/etd-08182009-110710/unrestricted/JayaramanS082009D.pdf

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