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 depends on external variables. An ensemble will always depend on three controlled external 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 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 internal energy E. The internal energy is the sum of the kinetic energy and potential energy of the particle system. The partition function is written as a sum over all states in agreement with these constraints

NVT ensmble

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