Chapter 5: Periodic Boundary Conditions

Question 1

Periodic Boundary Conditions

(Motivation)

Answer 1

Using a finite box makes surface effects extremely non-negligible with respect to bulk effects

Question 2

Periodic Boundary Conditions

(Notes)

Answer 2

• Use minimum image convention for short-range interactions with x_ij |< ^L/₂
• PBC forbids long wavelength fluctuations
  
  • _​_max correlation length l_c = L
• ​PBC is problematic for long-range interaction
  
  • use something like Ewald sum or P3M

Question 3

Periodic Boundary Conditions

(Efficiency)

Answer 3

• Force calculations very expensive N²
• Can increase efficiency using cell division and verlet lists

Question 4

Periodic Boundary Conditions

(Cell Division)

Answer 4

• divide central box into subboxes with length L_c >| r_c
  
  • ​particle only interact with those in same, or neighboring, box
• reduces force calcuation to N

Question 5

Periodic Boundary Conditions

(Verlet Lists)

Answer 5

• Let r_s > r_c be skin radius and list l_i of all particles within sphere of radius r_s around particle i
  
  • choose r_s such that some particles can remain r_c < r < r_s for 10-20 timesteps
• particle i can only interacts with particles j in l_i
• update when max{∆r_ij} > ¹/₂(r_s-r_c)
  
  • ∆r_ij = r_i - r_j

Question 6

MD Observables

Answer 6

• thermodyanmic properties T, P, C_V, etc. simplest to measure
  
  • relevant equations dependent on ensemble
• virial pressure (see below)
• thermodyanmic potentials S, F, G cannot be measure in simulation