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| {{About|a topic in physics|the automobile platform|GM N platform}}
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| {{cleanup|date=October 2009}}
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| {{DISPLAYTITLE:''N''-body simulation}}
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| [[File:AstroMSseqF 063aL (18135101).jpg|thumb|300px|right|The [[Millennium Run]] simulates the universe until the present state, where structures are abundant, manifesting themselves as stars, galaxies and clusters]] | |
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| An '''''N''-body simulation''' is a simulation of a [[dynamical system]] of particles, usually under the influence of physical forces, such as gravity (see [[n-body problem]]). In cosmology, they are used to study processes of non-linear [[structure formation]] such as the process of forming [[galaxy filament]]s and [[galaxy halo]]s from [[dark matter]] in [[physical cosmology]]. Direct ''N''-body simulations are used to study the dynamical evolution of star clusters.
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| ==Nature of the particles== | |
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| The 'particles' treated by the simulation may or may not correspond to physical objects which are particulate in nature. For example, an N-body simulation of a star cluster might have a particle per star, so each particle has some physical significance. On the other hand a simulation of a [[interstellar cloud|gas cloud]] cannot afford to have a particle for each atom or molecule of gas as this would require billions of particles for each gram of material (see [[Avogadro constant]]), so a single 'particle' would represent some much larger quantity of gas (often implemented using [[Smoothed Particle Hydrodynamics]]). This quantity need not have any physical significance, but must be chosen as a compromise between accuracy and manageable computer requirements.
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| ==Direct gravitational ''N''-body simulations==
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| [[File:N-body Simulation.ogv|thumb|N-body simulation of 400 objects with parameters close to those of [[Solar System]] planets.]] | |
| In direct gravitational ''N''-body simulations, the equations of motion of a system of ''N'' particles under the influence of their mutual gravitational forces are integrated numerically without any simplifying approximations. These calculations are used in situations where interactions between individual objects, such as stars or planets, are important to the evolution of the system. The first direct ''N''-body simulations were carried out by [[Sebastian von Hoerner]] at the [[Astronomical Calculation Institute (University of Heidelberg)|Astronomisches Rechen-Institut]] in [[Heidelberg]], [[Germany]]. [[Sverre Aarseth]] at the [[University of Cambridge]] (UK) has dedicated his entire scientific life to the development of a series of highly efficient ''N''-body codes for astrophysical applications which use adaptive (hierarchical) time steps, an Ahmad-Cohen neighbour scheme and regularization of close encounters. Regularization is a mathematical trick to remove the singularity in the Newtonian law of gravitation for two particles which approach each other arbitrarily close. Sverre Aarseth's codes are used to study the dynamics of star clusters, planetary systems and galactic nuclei.
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| ==General relativity simulations==
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| Many simulations are large enough that the effects of [[general relativity]] in establishing a [[Friedmann-Lemaitre-Robertson-Walker cosmology]] are significant. This is incorporated in the simulation as an evolving measure of distance (or [[scale factor (Universe)|scale factor]]) in a [[comoving coordinates|comoving coordinate]] system, which causes the particles to slow in comoving coordinates (as well as due to the [[redshift]]ing of their physical energy). However, the contributions of general relativity and the finite [[speed of gravity]] can otherwise be ignored, as typical dynamical timescales are long compared to the light crossing time for the simulation, and the space-time curvature induced by the particles and the particle velocities are small. The boundary conditions of these cosmological simulations are usually periodic (or toroidal), so that one edge of the simulation volume matches up with the opposite edge.
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| ==Calculation optimizations==
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| ''N''-body simulations are simple in principle, because they merely involve integrating the 6''N'' [[ordinary differential equation]]s defining the particle motions in [[Newtonian gravity]]. In practice, the number ''N'' of particles involved is usually very large (typical simulations include many millions, the [[Millennium simulation]] included ten billion) and the number of particle-particle interactions needing to be computed increases as ''N''<sup>2</sup>, and so direct integration of the differential equations can be prohibitively computationally expensive. Therefore, a number of refinements are commonly used.
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| One of the simplest refinements is that each particle carries with it its own timestep variable, so that particles with widely different dynamical times don't all have to be evolved forward at the rate of that with the shortest time.
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| There are two basic approximation schemes to decrease the computational time for such simulations. These can reduce the [[computational complexity]] to O(N log N) or better.
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| ===Tree methods===
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| In '''tree methods''', such as a [[Barnes–Hut simulation]], an [[octree]] is usually used to divide the volume into cubic cells in, so that only particles from nearby cells need to be treated individually, and particles in distant cells can be treated as a single large particle centered at the cell's center of mass (or as a low-order [[multipole]] expansion). This can dramatically reduce the number of particle pair interactions that must be computed. To prevent the simulation from becoming swamped by computing particle-particle interactions, the cells must be refined to smaller cells in denser parts of the simulation which contain many particles per cell. For simulations where particles are not evenly distributed, the well-separated pair decomposition methods of Callahan and [[S._Rao_Kosaraju|Kosaraju]] yield optimal O(''n'' log ''n'') time per iteration with fixed dimension.
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| ===Particle mesh method===
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| Another possibility is the '''[[Particle Mesh|particle mesh method]]''' in which space is discretised on a mesh and, for the purposes of computing the [[gravitational potential]], particles are assumed to be divided between the nearby vertices of the mesh. Finding the potential energy Φ is easy, because the [[Poisson equation]]
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| :<math>\nabla^2\Phi=4\pi G{\rho},\,</math>
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| where ''G'' is [[Newton's constant]] and <math>{\rho}</math> is the density (number of particles at the mesh points), is trivial to solve by using the [[fast Fourier transform]] to go to the [[frequency domain]] where the Poisson equation has the simple form
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| :<math>\hat{\Phi}= -4\pi G\frac{\hat{\rho}}{k^2},</math> | |
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| where <math>\vec{k}</math> is the comoving wavenumber and the hats denote Fourier transforms. The gravitational field can now be found by multiplying by <math>\vec{k}</math> and computing the inverse Fourier transform (or computing the inverse transform and then using some other method). Since this method is limited by the mesh size, in practice a smaller mesh or some other technique (such as combining with a tree or simple particle-particle algorithm) is used to compute the small-scale forces. Sometimes an adaptive mesh is used, in which the mesh cells are much smaller in the denser regions of the simulation.
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| ==Two-particle systems==
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| Although there are millions or billions of particles in typical simulations, they typically correspond to a real particle with a very large mass, typically 10<sup>9</sup> [[solar mass]]es. This can introduce problems with short-range interactions between the particles such as the formation of two-particle [[binary (astronomy)|binary]] systems. As the particles are meant to represent large numbers of dark matter particles or groups of stars, these binaries are unphysical. To prevent this, a softened Newtonian force law is used, which does not diverge as the inverse-square radius at short distances. Most simulations implement this quite naturally by running the simulations on cells of finite size. It is important to implement the discretization procedure in such a way that particles always exert a vanishing force on themselves.
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| ==Incorporating baryons, leptons and photons into simulations== | |
| Many simulations simulate only [[cold dark matter]], and thus include only the gravitational force. Incorporating [[baryon]]s, [[lepton]]s and [[photon]]s into the simulations dramatically increases their complexity and often radical simplifications of the underlying physics must be made. However, this is an extremely important area and many modern simulations are now trying to understand processes that occur during [[galaxy formation]] which could account for [[galaxy bias]].
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| ==See also==
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| *[[Millennium Run]]
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| *[[Structure formation]]
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| *[[Large-scale structure of the cosmos]]
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| *[[GADGET]]
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| *[[Galaxy formation and evolution]]
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| *[[n-body problem|''n''-body problem]]
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| *[[natural units]]
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| *[[Virgo Consortium]]
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| *[[Bolshoi Cosmological Simulation]]
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| ==References==
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| * {{cite journal|author = Sebastian von Hoerner|title=Die numerische Integration des n-Körper-Problemes für Sternhaufen. I|journal=Zeitschrift für Astrophysik|volume=50|pages=184|bibcode=1960ZA.....50..184V|year=1960}}
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| * {{cite journal|author = Sebastian von Hoerner|title=Die numerische Integration des ''n''-Körper-Problemes für Sternhaufen. II|journal=Zeitschrift für Astrophysik|volume=57|pages=47|bibcode=1963ZA.....57...47V|year=1963}}
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| * {{cite book|title=Gravitational ''N''-body Simulations: Tools and Algorithms|author=Sverre J. Aarseth|year=2003|publisher=Cambridge University Press|isbn=0-521-12153-1}}
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| * {{cite journal|author = Edmund Bertschinger|title=Simulations of structure formation in the universe|journal=Annual Review of Astronomy and Astrophysics|volume=36|issue = 1|pages=599–654|url=http://arjournals.annualreviews.org/doi/abs/10.1146%2Fannurev.astro.36.1.599|year=1998|doi=10.1146/annurev.astro.36.1.599|bibcode=1998ARA&A..36..599B}}
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| * {{cite book|title=Galactic Dynamics|author=James Binney and Scott Tremaine|year=1988|publisher=Princeton University Press|isbn=0-691-08445-9}}
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| *{{Cite book
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| | last1 = Callahan | first1 = Paul B.
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| | last2 = Kosaraju | first2 = Sambasiva Rao
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| | contribution = A decomposition of multidimensional point sets with applications to ''k''-nearest-neighbors and n-body potential fields (preliminary version)
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| | publisher = ACM
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| | title = STOC '92: Proc. ACM Symp. Theory of Computing
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| | year = 1992}}.
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| ==External links==
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| * [http://www.scholarpedia.org/article/N-body_simulations N-body Simulations] on [http://www.scholarpedia.org Scholarpedia]
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| [[Category:Physical cosmology]]
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| [[Category:Gravitation]]
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| [[Category:Simulation]]
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| [[Category:Cosmological Simulation]]
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