Calculation of the Isothermal-Isobaric Partition Function using Nested Sampling

Blake Wilson and Dr. Steven Nielsen

The University of Texas at Dallas

DFW Meeting in Miniature 2014

Nested Sampling

● Originally developed by Skilling for Bayesian computation (Skilling 2004-

Nested Sampling for General Bayesian Computation)

● First applied to atomic simulation by Partay et al. in 2010 – computation

of the NVT partition function (J. Phys. Chem. B 2010, 114, 10502–10512)

● Powerful athermal statistical mechanical sampling technique

● Direct computation of the partition function

● Access to thermodynamic quantities such as free energy, heat

capacities, and entropy

● Achieved through simple post-processing of the Nested Sampling

output (at any Temperature)

Why Develop A Constant Pressure Nested Sampling Method?

● The isothermal-isobaric (NPT) ensemble is one that most resembles many

experiments (especially condensed phase)

– Partay et al. recently published a Nested Sampling Method to compute the

NPT partition function in the special case of the hard sphere model

( PHYSICAL REVIEW E 89, 022302 (2014) )

● A generalized method for Constant Pressure Nested Sampling to compute

the NPT partition function would greatly increase the utility of Nested

Sampling for atomic simulation

● Allow a much wider range of physically relevant systems to be simulated

Thermodynamics in the Isothermal-Isobaric (NPT) Ensemble

ΔNPT=1V o

∫0

∞

dV∫ dx e−β HPartition Function:

H is the instantaneous enthalpy:

Some Thermodynamic Quantities:

H=U (x )+PV

H=−∂ ln ( ΔNPT )

∂ βC p=

∂ H∂T

G=−1β

ln(ΔNPT)

Gibbs Free Energy Enthalpy Heat Capacity

Entropy S=k B ln(ΔNPT )+HT

μ=k BT (∂ ln(ΔNPT )

∂N) Chemical Potential

Nested Sampling to Compute the NPT Partition Function

● Convert the partition function into the density of states form:

ΔNPT=1V o

∫0

∞

dV∫ dx e−β H=

1V o

∫Ω(V ,x )e−β H dH

●Define a constant pressure, P●Initially collect samples (coordinates and volumes) uniformly in phase space (T=∞)●Determine enthalpy (H

m) at a fixed fraction (f) of the initial sample enthalpy

distribution, where enthalpy is given by H = U + PV ●Sample coordinates and volumes uniformly under the restriction energy < H

m.

●Determine enthalpy (Hm+1

) at fixed fraction (f) of the sample enthalpy distribution.●Repeat until global minimum enthalpy is reached ( T=0)

Apply Nested Sampling Algorithm: ΔNPT≈1V o

∑n

wn e−β H n

ΔNPT≈1V o

∑0

m−1

( f m−f m+1)e−β

2(Hm+Hm+1)

Thermodynamics in the Isothermal-Isobaric (NPT) Ensemble

Partition Function:

Some Thermodynamic Quantities:

H=−∂ ln ( ΔNPT )

∂ βC p=

∂ H∂T

G=−1β

ln(ΔNPT)

Gibbs Free Energy Enthalpy Heat Capacity

Entropy S=k B ln(ΔNPT )+HT

ΔNPT≈1V o

∑0

m−1

( f m−f m+1)e−β

2(Hm+Hm+1)

O≈1V o

∑0

m−1

( f m−f m+1)(Om+Om+1)

2e

−β2

(Hm+Hm+1 )

Any Observable

μ=k BT (∂ ln(ΔNPT )

∂N) Chemical Potential

Example System: Lennard-Jones 50 Cluster

● Composed of 50 Lennard-Jones particles

● Spherical root mean squared radius boundary condition

U ij=4πεij [(σ ij

rij)

12

−(σ ij

rij)

6

]

Results: Heat Capacity and Density

Results: Comparison to Monte Carlo

● Results of Energy per particle and Particle density are in good agreement with those Constant Pressure Monte Carlo Simulation

Conclusions

● We have developed a generalized constant pressure Nested

Sampling method (applicable to a wide range of systems)

● We are able to compute the NPT partition function and other

thermodynamic quantities directly

● Results from the test system (LJ 50 cluster) are in good

agreement those from Monte Carlo Simulation

Thank You!

Additional Slides

Results: Comparison to Monte Carlo

LJ 17 Cluster

Computation of the NVT Partition from Nested Sampling

●Developed by Partay et. al in 2010

●Algorithm:

●Initially collect samples uniformly in phase space (T=∞)●Determine potential energy (E

m) at a fixed fraction (f)

of the initial sample energy distribution ●Sample uniformly under the restriction energy < E

m.

●Determine potential energy (Em+1

) at fixed fraction (f) of the sample energy distribution.●Repeat until global minimum energy is reached ( T=0)

Z NVT≈∑ ( f m− f m+1)e− βEn En=12 (Em+Em+1)

Reference