Computational Techniques in Theoretical Physics
Section 18
Molecular dynamics in canonical and grand canonical ensembles
 Key Point
 
 
Canonical Ensemble:
Closed systems:
A closed system can exchange heat with its surroundings.
 

Chacateristics:
 

Physical description:
 
  • Total number of particles fixed.
  • Volume of system is fixed.
  • Total energy fluctuating.
  • Average energy (temperature) fixed at equilibrium.
 

Mathematical description:
 

sum P(x) = 1

< E > = sum E(x) P(x)
          = sum H(x) P(x)

where H(x) is the Hamiltonian of the system.
 

 

Canonical ensemble is a statistical description of a closed system in equilibrium:
 
 
Entropy is maximum with the condition that the number of particles and average energy is constant:
 
d  int dx [ a P(x) + b H(x) P(x) - kB P(x) ln cN P(x) ] = 0
 
 
From this equation, we can obtain:
 
 
P(x) = exp{-H(x)/kBT} / cN Z
 

Z = int dx exp{ -H(x)/kBT }  / cN

 
Z is the partition function. All physical quantities can be derived from it:
 
 
Free energy:
 
F = - kBT ln Z
Entropy:
 
S = - dF / dT  (with V and N fixed).
Internal energy:
 
< E > = F + S T
Pressure:
 
P = dF / dV  (with T and N fixed)
Heat Capacity:
 
CV = d< E > / dT = T dS / dT
                                 (with V and N fixed)
 
 
 
 
 
 Grand canonical ensemble:
 
 
Open systems:            Mathematical description:
 

               < N > = sum N P(x

               < E > = sum E(x) P(x)
                         = sum H(x) P(x)

               where H(x) is the Hamiltonian of the system.
 

 
Grand canonical ensemble is a statistical description of a closed system in equilibrium:

 

        From this equation, we can obtain:
 
 

           P(x) = exp{-[H(x) - mu N]/kBT} / cNZ'

           Z' = int dx exp{ -[H(x) - mu N]/kBT }  / cN
 
Z' is the grand partition function, and mu is the chemical potential. All physical quantities can be derived from Z':
 
 
Free energy:
 
G = - kBT ln Z'
Entropy:
 
S = - dG / dT  (with P and mu fixed).
Internal energy:
 
< E > = F + S T
Volume:
 
P =  -dG / dP (with T and N fixed)
Heat Capacity:
 
Cp =  - T d2G /dT2
                                 (with P and N fixed)
Nose-Hoover Molecular Dynamics for canonical ensembles
General principles: Algorithm for isothermal molecular dynamics (canonical ensemble)
Grand canonical ensemble
(Isothermal-isobaric)  molecular dynamics
An isothermal-isobaric ensemble is a one with fixed number of particles N, temperature T, and pressure P0.

Most experimental conditions are of this type.

In an isothermal-isobaric ensemble volume is a variable. The canonical distribution is replaced by:
 

where H=K+V is total energy, P0 is the constant pressure. In this ensemble, the logarithm of the (isothermal-isobaric) grand partition function is -G/kB T, G is Gibbs free energy.

The generalization of the Newton's equations of motion is

where xi = ri/L is the coordinate vector normalized by the length of the box; the volume of the box is V = Ld; aV and aT are two parameters; P and K are the instantaneous pressure and kinetic energy, defined by
  Homework