Periodic boundary condition

Periodic boundary conditions are selected in analytical or numerical model calculations in order to avoid a separate treatment of edges or to be able to reduce the area over which the calculation extends. Periodic boundary conditions in dimensions can be understood as compacting the space into a flat torus in dimensions. ${\ displaystyle d}$ ${\ displaystyle d + 1}$

Areas of application are solid-state physics of crystalline materials, molecular dynamics , Monte Carlo simulations and simulations on lattices such as lattice scale theories .

Periodic boundary conditions in two dimensions. The edge of the simulation box is shown in black.

In continuous particle simulations with periodic boundary conditions, particles exit at one edge of the simulation box (the simulation area) and re- enter at the opposite edge.

Mathematical definition

A partial differential equation is an equation in which derivatives of an unknown function

{\ displaystyle u \ colon \ Omega \ to \ mathbb {R}}

occur for an open subset . One speaks of periodic boundary conditions when of the form ${\ displaystyle \ Omega \ subset \ mathbb {R} ^ {d}, d \ geq 2}$ ${\ displaystyle \ Omega}$

{\ displaystyle \ Omega = (a_ {1}, b_ {1}) \ times \ ldots \ times (a_ {d}, b_ {d}) \ subset \ mathbb {R} ^ {d}}

is and one demands that there is a steady continuation with it ${\ displaystyle u \ colon {\ overline {\ Omega}} \ to \ mathbb {R}}$

{\ displaystyle u (x_ {1}, \ ldots, x_ {i-1}, a_ {i}, x_ {i + 1}, \ ldots, x_ {d}) = u (x_ {1}, \ ldots , x_ {i-1}, b_ {i}, x_ {i + 1}, \ ldots, x_ {d}) \ \ forall (x_ {1}, \ ldots, x_ {d}) \ in \ Omega, i = 1, \ ldots, d}

should give.

${\ displaystyle \ Omega}$ does not have to be cuboid, but allow complete periodic coverage of the room (see room filling and parquet flooring ).

Compactification

A rectangular area is merged into a torus; their edges disappear.
The distortion required for the 3D view does not occur in the application.

Periodic boundary conditions in all dimensions correspond to a flat torus in dimensions, i.e. H. with a non-curved surface. The surface can therefore not be described as a subset of the -dimensional space, but as a Cartesian product of circles. ${\ displaystyle d}$ ${\ displaystyle d + 1}$ ${\ displaystyle d + 1}$ ${\ displaystyle d}$

Conservation quantities

Periodic boundary conditions allow energy and momentum conservation , but violate the conservation of angular momentum . Formal: Conservation of angular momentum is a consequence of the invariance of the physical laws towards rotations of the reference system, which is not given here . Clearly: a locally started vortex grows through impulse diffusion up to the length scale of the simulation box and is then systematically destroyed.

Examples

Periodic solution to a periodic problem

In the first example, periodic boundary conditions are combined with various non-periodic boundary conditions. It is about the simulation of the flow around turbine blades in an annular gap. The assumption (or approximation) applies that the flow around all N blades is the same. The problem is thus an N -fold rotational symmetry , which should be exploited by choosing a sector as the simulation area that contains only one of the blades. In the end, the solution found for the sector should be multiplied N times and put together to form a ring. The computational effort is thus limited to a fraction of the total area, and the total solution automatically has the expected symmetry, i.e. H. it is periodic.

In order for the solution to be physically meaningful, however, it must fit together as smoothly at the seams as is required of the solution inside the simulation area. This requirement defines the periodic boundary conditions . In addition, in this special example, the solution on the solid inner and outer surfaces must meet corresponding boundary conditions (e.g. flow velocity = wall velocity, if the boundary layer is modeled in detail) and match upstream and downstream with the solutions for further turbine stages (es is to find a common solution).

How one arrives at solutions that fulfill the periodic boundary conditions depends on how the physical quantities of the problem (here pressure, temperature and velocity components) should be represented. One possibility would be to use the sum of basis functions that are individually periodic and smooth, e.g. B. sinusoidally depend on the angle around the axis of rotation, see Fourier series . An edge then does not occur. Another class of methods that is more suitable here uses numerous support points (grid points), see e.g. B. Finite Volume Method . Grid points on one side of the simulation area are defined as adjacent to those on the other side in order to satisfy the boundary condition.

Ignoring surface waves

In a cuboid single crystal, phonons are excitations of mechanical standing waves . If one ignores that the atoms in the edge position have different neighborhood relationships, the wave equation is simplified to that for the infinitely extended lattice. The solutions are superpositions of plane waves of the kind

{\ displaystyle u (x, y, t) = u_ {0} \ cdot e ^ {\ mathrm {i} (k_ {x} \ cdot x + k_ {y} \ cdot y- \ omega \ cdot t)} }

Are in it

the components of the displacements of all atoms of the crystallographic unit cell from their rest positions ${\ displaystyle u}$
${\ displaystyle x}$ and location coordinates (for the sake of simplicity in only two dimensions) ${\ displaystyle y}$
${\ displaystyle t}$ the time
${\ displaystyle \ omega}$ the frequency , which depends on the wave vector and the polarization contained in the amplitude factor . ${\ displaystyle k}$ ${\ displaystyle u_ {0}}$

The only surface effect taken into account is the restriction to discrete wave vectors:

{\ displaystyle k_ {x} = {\ frac {\ pi \ cdot n_ {x}} {N_ {x} \ cdot a_ {x}}}}

{\ displaystyle k_ {y} = {\ frac {\ pi \ cdot n_ {y}} {N_ {y} \ cdot a_ {y}}}}

With

natural numbers ${\ displaystyle n}$
the size of the crystal in units of the size of its unit cell. ${\ displaystyle N}$ ${\ displaystyle a}$

With these , the solutions meet the periodicity condition ${\ displaystyle k}$

{\ displaystyle u (x, y, t) = u (x + 2 \ cdot N_ {x} \ cdot a_ {x}, y, t) = u (x, y + 2 \ cdot N_ {y} \ cdot a_ {y}, t)}

.

In the Born-von-Kármán model, periodic boundary conditions are also called Born-von Kármán boundary conditions .

Compression without a vice

The image of the vice represents great external forces. If simulated vice jaws were to act on the atoms of a small simulation box, the results would be useless. Particularly large forces are necessary to examine materials under conditions that cannot be realized in the laboratory, such as those in the Earth's core (e.g. to determine their elasticity tensor ). By using periodic boundary conditions, this problem is avoided by not specifying forces ( mechanical stresses ) and observing deformations , but vice versa. The simulation box contains a single unit cell of the crystal and is arbitrarily deformed (edge lengths and angles). For each geometry of the unit cell, the position of the atoms is varied and the electronic energy is calculated in each case (see Born-Oppenheimer approximation ), using the density functional theory with periodic basis functions. The external mechanical stresses result as a calculation result from the dependence of the electronic energy on the geometry of the unit cell with a relaxed arrangement of the atoms.

Areas of application with non-periodic problems

The advantage of walllessness through periodic boundary conditions can also be used when simulating systems that are actually not periodic. The size of the simulation box is then set arbitrarily, but not arbitrarily: it must be greater than the distances over which correlations occur. This can be a large molecule in a solution , a material deformation or a density fluctuation near a phase transition .

For short-range interactions , cutoff radii can be introduced from which an explicit interaction between particles is no longer calculated; however, additional terms analytically obtained for the cut interactions can still be taken into account. If these cutoff radii are used, you can specify a criterion for a size of the simulation box that is not to be undercut, whereby other criteria must often be observed that force the significantly larger box size: The smallest diameter of the simulation box should be at least twice as large as the largest cutoff radius used in this way, otherwise a particle in the simulation box (central box) sees a copy of itself in an adjacent box. Furthermore, this is the criterion for using the Minimum Image Convention for these interactions. This convention says that you only have to consider interactions with particles in the next neighboring boxes of the central box.

Thermodynamic Limes and Continuum Limes

In order to obtain information on thermodynamic quantities from molecular dynamics simulations , the thermodynamic limit must be determined, i.e. H. Particle number N and volume V must each be increased to infinity ( ), with constant particle number density . For this purpose, the measured variables for different sizes of the simulation box are interpolated to give an infinite number of particles and an infinite volume . ${\ displaystyle N \ to \ infty, V \ to \ infty}$ ${\ displaystyle N / V}$

Note that the application of periodic boundary conditions alone does not lead to the thermodynamic limit. This is particularly clear if, for example, you want to simulate a liquid, but only place one liquid particle in the simulation box. If you look at the simulation box and its copies, you will always find a crystal. In order to simulate the liquid correctly, there must already be enough particles in the main simulation box (N large, goes in the direction of thermodynamic limits) so that the interesting correlation effects already occur here and finite size effects (N small) can be neglected.

Periodic boundary conditions are only used to avoid introducing any boundary effects into the simulation and thus to simulate a system without boundaries, as is the case e.g. B. is present inside (English bulk ) a liquid.

Individual evidence

↑ Hugo Reinhardt: Quantum Mechanics 1: Path integral formulation and operator formalism. ISBN 348671516X , p. 312 ( Google Books ).
↑ Jürgen Jost: Partial differential equations (= Graduate Texts in Mathematics. 214) Third edition, Springer, New York 2013, ISBN 978-1-4614-4808-2 .
↑ Oleg Jardetzky, Michael D. Finucane (Ed.): Dynamics, Structure, and Function of Biological Macromolecules. IOS Press, 2001, ISBN 1586030329 , p. 3 ( Google Books ).
↑ On angular momentum balance for particle systems with periodic boundary conditions, http://arxiv.org/pdf/1312.7008.pdf
↑ Mireille Defranceschi, Claude Le Bris: Mathematical Models and Methods for Ab Initio Quantum Chemistry. (= Lectures in Chemistry. 74), Springer, 2000 ISBN 3540676317 , p. 96 ( Google Books ).
^ MP Allen, DJ Tildesley: Computer Simulation of Liquids . Oxford University Press, 1989, ISBN 0-19-855645-4 , pp. 24 ( limited preview in Google Book search).

[1] Hugo Reinhardt: Quantum Mechanics 1: Path integral formulation and operator formalism. ISBN 348671516X , p. 312 ( Google Books ).

[2] Jürgen Jost: Partial differential equations (= Graduate Texts in Mathematics. 214) Third edition, Springer, New York 2013, ISBN 978-1-4614-4808-2 .

[3] Oleg Jardetzky, Michael D. Finucane (Ed.): Dynamics, Structure, and Function of Biological Macromolecules. IOS Press, 2001, ISBN 1586030329 , p. 3 ( Google Books ).

[4] On angular momentum balance for particle systems with periodic boundary conditions, http://arxiv.org/pdf/1312.7008.pdf

[5] Mireille Defranceschi, Claude Le Bris: Mathematical Models and Methods for Ab Initio Quantum Chemistry. (= Lectures in Chemistry. 74), Springer, 2000 ISBN 3540676317 , p. 96 ( Google Books ).

[6] MP Allen, DJ Tildesley: Computer Simulation of Liquids . Oxford University Press, 1989, ISBN 0-19-855645-4 , pp. 24 ( limited preview in Google Book search).