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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: 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
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