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430 chapter 162 3 2 3 2 31 4 1 4 1 45552 3 2 3 2 31 4 1 4 1 45552 3 2 3 2 31 4 1 4 1 4555Figure 16.4 The infinite space generated by imposing periodic boundary conditions on theparticles within the simulation volume (shaded box). The two-headed arrows indicate how aparticle interacts with the nearest version of another particle, be that within the simulationvolume or an image. The vertical arrows indicate how the image of particle 4 enters whenparticle 4 exits.from the surface, that is, 49% of the molecules, while for 10 6 particles this fractionfalls to 6%.The imposition of periodic boundary conditions (PBCs) strives to minimize theshortcomings of both the small numbers of particles and of artificial boundaries.Even though we limit our simulation to an L x × L y × L z box, we imagine this boxbeing replicated to infinity in all directions (Figure 16.4). Accordingly, after eachtime-integration step we examine the position of each particle and check if it has leftthe simulation region. If it has, then we bring an image of the particle back throughthe opposite boundary (Figure 16.4):x ⇒{x + Lx , if x ≤ 0,x − L x , if x>L x .(16.13)Consequently, each box looks the same and has continuous properties at the edges.As shown by the one-headed arrows in Figure 16.4, if a particle exits the simulationvolume, its image enters from the other side, and so balance is maintained.In principle a molecule interacts with all others molecules and their images, soeven though there is a finite number of atoms in the interaction volume, thereis an effective infinite number of interactions [Erco]. Nonetheless, because theLennard–Jones potential falls off so rapidly for large r, V (r =3σ) ≃ V (1.13σ)/200,−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 430