Category:Ensembles

Introduction

A central concept of statistical mechanics is the ensemble. An ensemble consists of a large number of virtual copies of the system of interest. An ensemble will always depend on three thermodynamic state variables, as for example particle number N, temperature T and pressure p. These three variables determine the type of ensemble that is studied. Depending on these three variables there is a thermodynamic potential associated with the ensemble, which would be the Helmholtz free energy in the case of N,T and p. Therefore, the concept of the ensemble gives access to any thermodynamic quantity. The configurations of your system building up the ensemble can be obtained from molecular dynamics simulations. The molecular-dynamics approach generates the configurations for the ensemble by integrating Newton's equations of motion.

Theory

In this section various ensembles will be introduced. To describe an ensemble mathematically the partition function will be used. The partition function is the central mathematical entity in statistical mechanics. As the wave function in quantum mechanics it contains all the information about a statistical system. The partition function depends on three thermodynamic state variables such as N,T and volume V.

Microcanonical ensemble (N,V,E)

To start, the three controlled external parameters have to be defined. In the case of the microcanonical ensemble these are the particle number, the volume and the total energy E of the system. The total energy is the sum of the kinetic energy and potential energy of the particle system. Therefore the total energy depends on the momenta and the positions of the system. Furthermore, the energy is an extensive variable depending on the particle number and the volume of the system. The partition function is written as a sum over all microstates in agreement with the NVE constraints

${\displaystyle \Omega(N,V,E) = \sum_{E-\delta E < E(N,V,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\}) < E + \delta E } 1. }$

In this equation ${\displaystyle \delta E }$ denotes a infinitesimal energy. ${\displaystyle \{\mathbf{r}_{i}\} }$ denotes the set of particle positions and ${\displaystyle \{\mathbf{p}_{i}\}}$ are the conjugate momenta. The sum is over all particle positions and momenta giving an energy in agreement with the energy constraint. Therefore, ${\displaystyle \Omega(N,V,E) }$ is the number of microstates under the given conditions. From the microcanonical partition function it is possible to obtain the entropy by

${\displaystyle S(N,V,E) = k_{B}log\Omega(N,V,E), }$

where ${\displaystyle k_{B}}$ is the Boltzmann constant. The probabilities for the different micro states (r) are given by

${\displaystyle P_{r}(N,V,E) = \begin{cases} \frac{1}{\Omega(N,V,E)}\ for\ E-\delta E < E(N,V) < E + \delta E \\ 0\ else \end{cases} }$.

The microcanonical ensemble is a theoretical realization of a N particle system which is placed in a box with fixed volume and fully isolated from its surroundings. Therefore, no energy exchange with the rest of the universe is possible. In VASP this ensemble can be approximated by sampling the configurations in a molecular-dynamics run.

Canonical ensemble ensemble (N,V,T)

In the canonical ensemble the controlled thermodynamic state variables are the particle number, the volume and the temperature. The ensemble consists of all configurations accessible to the system at the given (N,V,T) conditions. The canonical partition function can be derived to be

${\displaystyle Z(N,V,T) = \sum_{r}e^{-\beta E(N,V,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\})} }$,

where ${\displaystyle \beta }$ is the inverse temperature ${\displaystyle \frac{1}{k_{B}T}}$. The probability for a single microstate r is

Failed to parse (Conversion error. Server ("cli") reported: "[INVALID]"): {\displaystyle P_{r}(N,V,T) = \frac{1}{Z(N,V,T)}e^{-\beta E(N,V,\{\mathbf{r}_{i}\},\{\mathbf{p}_{i}\}) }

NpT ensmble

NpH ensmble

How To

The following table gives an overview of the possible combination of ensembles and thermostats in VASP:

	Thermostat
Ensemble	Andersen	Nose-Hoover	Langevin	NHC	CSVR	Multiple Andersen
Microcanonical (NVE)	MDALGO=1, ANDERSEN_PROB=0.0
Canonical (NVT)	MDALGO=1	MDALGO=2	MDALGO=3	MDALGO=4	MDALGO=5	MDALGO=13
	ISIF=2	ISIF=2	ISIF=2	ISIF=2	ISIF=2	ISIF=2
Isobaric-isothermal (NpT)	not available	not available	MDALGO=3	not available	not available	not available
			ISIF=3
Isoenthalpic-isobaric (NpH)	MDALGO=3, ISIF=3, LANGEVIN_GAMMA=LANGEVIN_GAMMA_L=0.0