1549380323-Statistical Mechanics Theory and Molecular Simulation

208 Canonical ensemble

the box edge. Let these intervals be indexed by an integeri= 0,...,Nr−1 with

radial valuesr 1 ,...,rNr.

2. Generate a histogramhab(i) by counting the number of times the distance between
   two atoms of typeaandblies betweenriandri+∆r. For this histogram, all atoms
   of the desired types in the system can be used and all configurations generated
   in the simulation should be considered. Thus, if we are interested in the oxygen–
   oxygen histogram of water, we would use the oxygens of all watersin the system
   and all configurations generated in the simulation. For each distancercalculated,
   the index into the histogram is given by

i= int(r/∆r). (4.13.25)

3. Once the histogram is generated, the radial distribution function is obtained by

gab(ri) =

hab(i)

4 πρbri^2 ∆rNconfNa

, (4.13.26)

whereNconfis the number of configurations in the simulation,Nais the number

of atoms of typea, andρbis the number density of the atom typeb.

This procedure was employed to produce the plots in Figs. 4.2 and 4.3.

4.14 Problems

4.1. Prove that the microcanonical partition function in the Nos ́e–Poincar ́e Hamil-

tonian of eqn. (4.8.15) is equivalent to a canonical partition functionin the

physical HamiltonianH(r,p). What choice must be made for the parameter

gin eqn. (4.8.15)?

4.2. Consider a one-dimensional system with momentumpand coordinateqcou-

pled to an extended-system thermostat for which the equations of motion

take the form

q ̇=

p

m

p ̇=F(q)−

pη 1

Q 1

p−

pη 2

Q 2

[

(kT)p+

p^3

3 m

]

η ̇ 1 =

pη 1

Q 1

η ̇ 2 =

[

(kT) +

p^2

m

]

pη 2

Q 2