1638:. However, this approach was shown to not be consistent: it fails to explain the absence of conservation of angular momentum of a single particle moving in a periodic cell. Lagrangian of the particle is constant and therefore rotationally invariant, while angular momentum of the particle is not conserved. This contradiction is caused by the fact that Noether's theorem is usually formulated for closed systems. The periodic cell exchanges mass momentum, angular momentum, and energy with the neighboring cells.
28:
20:
943:"head" interacting with its own "tail". This produces highly unphysical dynamics in most macromolecules, although the magnitude of the consequences and thus the appropriate box size relative to the size of the macromolecules depends on the intended length of the simulation, the desired accuracy, and the anticipated dynamics. For example, simulations of
939:, and by extension the energies of charged particles like electrons, are not automatically aligned to experimental energy scales. Mathematically, this energy level ambiguity corresponds to the sum of the electrostatic energy being dependent on a surface term that needs to be set by the user of the method.
541:
335:
774:
66:
of some video games; the geometry of the unit cell satisfies perfect two-dimensional tiling, and when an object passes through one side of the unit cell, it re-appears on the opposite side with the same velocity. In topological terms, the space made by two-dimensional PBCs can be thought of as being
971:
An object which has passed through one face of the simulation box should re-enter through the opposite face—or its image should do it. Evidently, a strategic decision must be made: Do we (A) “fold back” particles into the simulation box when they leave it, or do we (B) let them go on (but compute
942:
The size of the simulation box must also be large enough to prevent periodic artifacts from occurring due to the unphysical topology of the simulation. In a box that is too small, a macromolecule may interact with its own image in a neighboring box, which is functionally equivalent to a molecule's
867:
Three-dimensional PBCs are useful for approximating the behavior of macro-scale systems of gases, liquids, and solids. Three-dimensional PBCs can also be used to simulate planar surfaces, in which case two-dimensional PBCs are often more suitable. Two-dimensional PBCs for planar surfaces are also
922:
of the solution in which the molecules naturally appear. Maintenance of the minimum-image convention also generally requires that a spherical cutoff radius for nonbonded forces be at most half the length of one side of a cubic box. Even in electrostatically neutral systems, a net
1645:(constant particle number, volume, and energy, abbreviated NVE), using PBC rather than reflecting walls slightly alters the sampling of the simulation due to the conservation of total linear momentum and the position of the center of mass; this ensemble has been termed the "
1673:
particles. These artifacts have quantifiable consequences for small toy systems containing only perfectly hard particles; they have not been studied in depth for standard biomolecular simulations, but given the size of such systems, the effects will be largely negligible.
983:
Restricting the coordinates is a simple operation which can be described with the following code, where x_size is the length of the box in one direction (assuming an orthogonal unit cell centered on the origin) and x is the position of the particle in the same direction:
1390:
Assuming an orthorhombic simulation box with the origin at the lower left forward corner, the minimum image convention for the calculation of effective particle distances can be calculated with the “nearest integer” function as shown above, here as C/C++ code:
935:. Another consequence of applying PBCs to a simulated system such as a liquid or a solid is that this hypothetical system has no contact with its “surroundings”, due to it being infinite in all directions. Therefore, long-range energy contributions such as the
1090:
Distance and vector between objects should obey the minimum image criterion. This can be implemented according to the following code (in the case of a one-dimensional system where dx is the distance direction vector from object i to object j):
1813:
Cheatham, T. E.; Miller, J. H.; Fox, T.; Darden, P. A.; Kollman, P. A. (1995). "Molecular
Dynamics Simulations on Solvated Biomolecular Systems: The Particle Mesh Ewald Method Leads to Stable Trajectories of DNA, RNA, and Proteins".
341:
135:
577:
1761:
Mai, W.; Li, P.; Bao, H.; Li, X.; Jiang, L.; Hu, J.; Werner, D. H. (April 2019). "Prism-Based DGTD With a
Simplified Periodic Boundary Condition to Analyze FSS With D2n Symmetry in a Rectangular Array Under Normal Incidence".
883:
forces in the system. However, PBCs also introduce correlational artifacts that do not respect the translational invariance of the system, and requires constraints on the composition and size of the simulation box.
972:
interactions with the nearest images)? The decision has no effect on the course of the simulation, but if the user is interested in mean displacements, diffusion lengths, etc., the second option is preferable.
127:
1582:
For simulations in 2D and 3D space, cubic periodic boundary condition is most commonly used since it is simplest in coding. In computer simulation of high dimensional systems, however, the
75:). The large systems approximated by PBCs consist of an infinite number of unit cells. In computer simulations, one of these is the original simulation box, and others are copies called
2174:"Periodic boundary condition induced breakdown of the equipartition principle and other kinetic effects of finite sample size in classical hard-sphere molecular dynamics simulation"
918:) in appropriate numbers if the molecules of interest are charged. Sometimes ions are even added to a system in which the molecules of interest are neutral, to approximate the
891:
field arising from any inhomogeneity in the system will be artificially truncated and modified by the periodic boundary. Similarly, the wavelength of sound or shock waves and
1594:. For example, the hypercubic periodic boundary condition corresponds to the hypercubic lattice packing. It is then preferred to choose a unit cell which corresponds to the
83:
is a common form of PBC particle bookkeeping in which each individual particle in the simulation interacts with the closest image of the remaining particles in the system.
1993:
Berthier, Ludovic; Charbonneau, Patrick; Kundu, Joyjit (31 August 2020). "Finite
Dimensional Vestige of Spinodal Criticality above the Dynamical Glass Transition".
831:
804:
571:
1546:
In simulations of ionic systems more complicated operations may be needed to handle the long-range
Coulomb interactions spanning several box images, for instance
902:
of the system must be zero to avoid summing to an infinite charge when PBCs are applied. In some applications it is appropriate to obtain neutrality by adding
536:{\displaystyle {\frac {\partial ^{m}}{\partial x_{2}^{m}}}\phi (x_{1},a_{2},...,x_{n})={\frac {\partial ^{m}}{\partial x_{2}^{m}}}\phi (x_{1},b_{2},...,x_{n}),}
330:{\displaystyle {\frac {\partial ^{m}}{\partial x_{1}^{m}}}\phi (a_{1},x_{2},...,x_{n})={\frac {\partial ^{m}}{\partial x_{1}^{m}}}\phi (b_{1},x_{2},...,x_{n}),}
769:{\displaystyle {\frac {\partial ^{m}}{\partial x_{n}^{m}}}\phi (x_{1},x_{2},...,a_{n})={\frac {\partial ^{m}}{\partial x_{n}^{m}}}\phi (x_{1},x_{2},...,b_{n})}
1558:
PBC requires the unit cell to be a shape that will tile perfectly into a three-dimensional crystal. Thus, a spherical or elliptical droplet cannot be used. A
1586:
periodic boundary condition can be less efficient because corners occupy most part of the space. In general dimension, unit cell can be viewed as the
1693:
844:, PBCs are usually applied to calculate properties of bulk gasses, liquids, crystals or mixtures. A common application uses PBC to simulate solvated
959:
on the observed dynamics – in simulation or in experiment – are not well understood. A common recommendation based on simulations of
1747:
859:
In electromagnetics, PBC can be applied for different mesh types to analyze the electromagnetic properties of periodical structures.
1467:
The fastest way of carrying out this operation depends on the processor architecture. If the sign of dx is not relevant, the method
1816:
1237:
cells if the origin is shifted to a corner of the box. Then we have, in one dimension, for positions and distances respectively:
872:; in this case, PBCs are used for two Cartesian coordinates (e.g., x and y), and the third coordinate (z) extends to infinity.
853:
1723:
89:
1888:"Effect of periodic box size on aqueous molecular dynamics simulation of a DNA dodecamer with particle-mesh Ewald method"
1570:
molecules in the corners, distant from the central macromolecules. A common alternative that requires less volume is the
2173:
2254:
2231:
2153:
2054:
Conway, J.; Sloane, N. (March 1982). "Fast quantizing and decoding and algorithms for lattice quantizers and codes".
79:. During the simulation, only the properties of the original simulation box need to be recorded and propagated. The
951:
may undergo smaller fluctuations, and therefore may not require as large a box, as simulations that begin from a
1649:
ensemble" or the NVEPG ensemble. These additional conserved quantities introduce minor artifacts related to the
1595:
841:
72:
1566:
is the most intuitive and common choice, but can be computationally expensive due to unnecessary amounts of
2279:
2091:
Kuzkin, V. A. (2015). "On angular momentum balance in particle systems with periodic boundary conditions".
1606:
in 8-dimension. The implementation of these high dimensional periodic boundary conditions is equivalent to
1981:
2274:
1683:
888:
2068:
46:
which are often chosen for approximating a large (infinite) system by using a small part called a
1642:
963:
is to require at least 1 nm of solvent around the molecules of interest in every dimension.
936:
2063:
1658:
86:
One example of periodic boundary conditions can be defined according to smooth real functions
1650:
1607:
2185:
2110:
2012:
1899:
1852:
1771:
1635:
1634:, which states that conservation of angular momentum follows from rotational invariance of
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1571:
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809:
782:
547:
8:
51:
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2016:
1903:
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2100:
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2002:
1964:
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1795:
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1646:
1611:
837:
55:
43:
1911:
1661:, and violations of equipartition for systems containing particles with heterogeneous
2250:
2227:
2201:
2149:
2040:
2028:
1968:
1925:
1868:
1799:
1787:
1729:
1719:
1599:
1587:
1563:
2130:
1230:
For three-dimensional PBCs, both operations should be repeated in all 3 dimensions.
2193:
2118:
2073:
2024:
2020:
1956:
1915:
1907:
1860:
1825:
1779:
1627:
932:
924:
880:
1591:
1547:
956:
944:
928:
876:
1843:
Kleinman, Leonard (1981). "Comment on the average potential of a Wigner solid".
919:
2148:. Modern Theoretical Chemistry. Vol. 6. New York: Plenum. pp. 1–40.
1783:
2268:
2242:
2077:
1872:
1864:
1791:
1733:
927:
of the unit cell can introduce a spurious bulk-surface energy, equivalent to
845:
27:
2249:. Interdisciplinary Applied Mathematics. Vol. 21. New York: Springer.
2205:
2122:
2032:
1960:
1234:
948:
1929:
1713:
1669:
particles will behave, in the molecular dynamics ensemble, as a system of
975:
1715:
Understanding molecular simulation : from algorithms to applications
1654:
952:
849:
1829:
1603:
915:
2197:
1583:
967:
Practical implementation: continuity and the minimum image convention
19:
2007:
1944:
1623:
911:
59:
2105:
1567:
1233:
These operations can be written in a much more compact form for
898:
In simulations containing ionic (Coulomb) interactions, the net
1385:
907:
892:
1543:
For non-orthorhombic cells the situation is more complicated.
31:
Unit cell with water molecules, used to simulate flowing water
2247:
Molecular
Modeling and Simulation: An Interdisciplinary Guide
980:
To implement a PBC algorithm, at least two steps are needed.
68:
879:
methods (e.g., the particle mesh Ewald method) to calculate
1662:
1559:
2144:
Erpenbeck, J. J.; Wood, W. W. (1977). Berne, B. J. (ed.).
1630:
is not. Conventional explanation of this fact is based on
960:
903:
1992:
2146:
Statistical
Mechanics, Part B: Time-dependent Processes
1812:
976:(A) Restrict particle coordinates to the simulation box
856:
are periodic boundary conditions for a special system.
1982:
Minimum image convention in non-cubic simulation cells
1540:
was found to be fastest on x86-64 processors in 2013.
122:{\displaystyle \phi :\mathbb {R} ^{n}\to \mathbb {R} }
2047:
1657:, the departure of the velocity distributions from a
812:
785:
580:
550:
344:
138:
92:
1665:. The simplest of these effects is that a system of
1308:! For a box with the origin at the lower left vertex
2172:Shirts, R. B.; Burt, S. R.; Johnson, A. M. (2006).
1945:"Efficient coding of the minimum image convention"
825:
798:
768:
565:
535:
329:
121:
2171:
2266:
1885:
1622:Under periodic boundary conditions, the linear
1414:// compute only when box size is set or changed
1764:IEEE Antennas and Wireless Propagation Letters
2143:
1986:
862:
62:of two-dimensional PBC is equal to that of a
2226:(2nd ed.). Cambridge University Press.
2053:
1386:(B) Do not restrict the particle coordinates
1711:
779:for all m = 0, 1, 2, ... and for constants
1760:
1746:: CS1 maint: location missing publisher (
1242:! After x(i) update without regard to PBC:
895:in the system is limited by the box size.
2167:
2165:
2137:
2104:
2067:
2006:
1919:
1886:de Souza, O. N.; Ornstein, R. L. (1997).
1694:Software for molecular mechanics modeling
115:
101:
2224:The Art of Molecular Dynamics Simulation
2221:
1842:
1817:Journal of the American Chemical Society
26:
18:
2241:
2056:IEEE Transactions on Information Theory
1942:
1879:
1806:
1617:
1553:
1311:! Works for x's lying in any image.
2267:
2162:
2090:
2084:
955:conformation. However, the effects of
887:In simulations of solid systems, the
875:PBCs can be used in conjunction with
1712:Frenkel, Daan; Smit, Berend (2002).
1577:
854:Born–von Karman boundary conditions
13:
688:
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584:
452:
442:
358:
348:
246:
236:
152:
142:
23:Periodic boundary conditions in 2D
14:
2291:
1598:of that dimension. In 4D this is
1626:of the system is conserved, but
2025:10.1103/PhysRevLett.125.108001
1975:
1936:
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842:Monte Carlo molecular modeling
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16:Concept in molecular modelling
1:
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1912:10.1016/s0006-3495(97)78884-2
36:Periodic boundary conditions
7:
1943:Deiters, Ulrich K. (2013).
1718:(2nd ed.). San Diego.
1684:Helical boundary conditions
1677:
10:
2296:
863:Requirements and artifacts
1784:10.1109/LAWP.2019.2902340
50:. PBCs are often used in
2222:Rapaport, D. C. (2004).
2078:10.1109/TIT.1982.1056484
1865:10.1103/PhysRevB.24.7412
1699:
1469:
1393:
1239:
1093:
986:
870:slab boundary conditions
81:minimum-image convention
1995:Physical Review Letters
1643:microcanonical ensemble
937:electrostatic potential
2123:10.1002/zamm.201400045
1961:10.1524/zpch.2013.0311
1659:Boltzmann distribution
1651:statistical mechanical
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123:
32:
24:
1608:error correction code
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826:{\displaystyle b_{i}}
801:
799:{\displaystyle a_{i}}
771:
568:
538:
332:
124:
30:
22:
1641:When applied to the
1618:Conserved properties
1572:truncated octahedron
1554:Unit cell geometries
947:that begin from the
900:electrostatic charge
810:
783:
578:
566:{\displaystyle ...,}
548:
342:
136:
90:
52:computer simulations
2280:Boundary conditions
2190:2006JChPh.125p4102S
2115:2015ZaMM...95.1290K
2017:2020PhRvL.125j8001B
1904:1997BpJ....72.2395D
1857:1981PhRvB..24.7412K
1830:10.1021/ja00119a045
1776:2019IAWPL..18..771M
705:
611:
469:
375:
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56:mathematical models
44:boundary conditions
2275:Molecular dynamics
1689:Molecular modeling
1647:molecular dynamics
1612:information theory
838:molecular dynamics
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2261:See esp. pp272–6.
2238:See esp. pp15–20.
2198:10.1063/1.2359432
2099:(11): 1290–1295.
1851:(12): 7412–7414.
1845:Physical Review B
1824:(14): 4193–4194.
1725:978-0-08-051998-2
1632:Noether's theorem
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1955:(2–3): 345–352.
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920:ionic strength
864:
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916:counterions
2269:Categories
2216:References
2008:1912.11510
1636:Lagrangian
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1600:D4 lattice
1584:hypercubic
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2106:1312.7008
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