Practical Implementation: Continuity and The Minimum Image Convention
To implement periodic boundary conditions in practice, at least two steps are needed.
The first is to make an object which leaves the simulation cell on one side enter back on the other. This is of course a simple operation, and could in code be e.g. (for the x dimension, assuming an orthogonal unit cell centered on the origin):
if (periodicx) then if (x < -xsize*0.5) x=x+xsize if (x >= xsize*0.5) x=x-xsize endifThe second is to make sure that every distance between atoms, or other vector calculated from one atom to another, has a length and direction which corresponds to the minimum image criterion. This can be achieved as follows to calculate e.g. the x direction distance component from atom i to atom j:
if (periodicx) then dx = x(j) - x(i) if (abs(dx) > xsize*0.5) dx = dx - sign(xsize,dx) endifNaturally both operations should be repeated in all 3 dimensions.
These operations can be written in much more compact form for orthorhombic cells if the origin is shifted to a corner of the box. Then we have, in one dimension, for positions and distances respectively:
! After x(i) update without regard to PBC: x(i)=x(i)-floor(x(i)/xsize)*xsize !For a box with the origin at the lower left vertex ! Works for xs lying in any image. dx=x(j)-x(i) dx=dx-nint(dx/(0.5*xsize))*xsizeFor non-orthorhombic cells the situation can be considerably more complicated.
In simulations of ionic systems considerably more complicated operations may be needed to handle the long-range Coulomb interactions.
Read more about this topic: Periodic Boundary Conditions
Famous quotes containing the words practical, continuity, minimum, image and/or convention:
“Juggling produces both practical and psychological benefits.... A woman’s involvement in one role can enhance her functioning in another. Being a wife can make it easier to work outside the home. Being a mother can facilitate the activities and foster the skills of the efficient wife or of the effective worker. And employment outside the home can contribute in substantial, practical ways to how one works within the home, as a spouse and as a parent.”
—Faye J. Crosby (20th century)
“There is never a beginning, there is never an end, to the inexplicable continuity of this web of God, but always circular power returning into itself.”
—Ralph Waldo Emerson (1803–1882)
“After decades of unappreciated drudgery, American women just don’t do housework any more—that is, beyond the minimum that is required in order to clear a path from the bedroom to the front door so they can get off to work in the mourning.”
—Barbara Ehrenreich (20th century)
“The image cannot be dispossessed of a primordial freshness, which idea can never claim. An idea is derivative and tamed. The image is in the natural or wild state, and it has to be discovered there, not put there, obeying its own law and none of ours. We think we can lay hold of image and take it captive, but the docile captive is not the real image but only the idea, which is the image with its character beaten out of it.”
—John Crowe Ransom (1888–1974)
“The metaphor of the king as the shepherd of his people goes back to ancient Egypt. Perhaps the use of this particular convention is due to the fact that, being stupid, affectionate, gregarious, and easily stampeded, the societies formed by sheep are most like human ones.”
—Northrop Frye (b. 1912)