Lattice gas simulations of dynamical geometry in two dimensions

Klales, Anna; Cianci, Donato; Needell, Zachary; Meyer, David; Love, Peter J.

doi:10.1103/physreve.82.046705

Cited by 19 publications

(27 citation statements)

References 44 publications

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“…There are many, more specific-purpose models which a posteriori can be viewed as instances of the notion of causal graph dynamics developed in this paper [20,29,26,28]. We now show via two examples that our model subsumes, but is not restricted to, cellular automata.…”

Section: Examples Of Localisable Functionsmentioning

confidence: 92%

“…Sometimes, when a set of rule is non-conflicting, this evaluation strategy happens to coincide with full synchronism. This pragmatic approach to extending cellular automata to time-varying graphs is advocated in [41,24,20], and has led to some advanced algorithmic constructions [40,42]. This paper aims at proposing simple formalisations of the notions of translation-invariance and causality when referring to functions from labelled graphs to labelled graphs.…”

Section: Introductionmentioning

confidence: 98%

“…physical systems [20], computer processes [29], biochemical agents [28], economical agents [23], users of social networks, etc.) interact with their neighbours, leading to a global dynamics.…”

Section: Introductionmentioning

confidence: 99%

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Causal graph dynamics

Arrighi

Dowek

2013

Information and Computation

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International audienceWe extend the theory of cellular automata to arbitrary, time-varying graphs. In other words we formalise, and prove theorems about, the intuitive idea of a labelled graph which evolves in time -- but under the natural constraint that information can only ever be transmitted at a bounded speed, with respect to the distance given by the graph. The notion of translation-invariance is also generalised. The definition we provide for these 'causal graph dynamics' is simple and axiomatic. The theorems we provide also show that it is robust. For instance, causal graph dynamics are stable under composition and under restriction to radius one. In the finite case some fundamental facts of cellular automata theory carry through: causal graph dynamics admit a characterisation as continuous functions, and they are stable under inversion. The provided examples suggest a wide range of applications of this mathematical object, from complex systems science to theoretical physics

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Section: Examples Of Localisable Functionsmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 98%

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Causal graph dynamics

Arrighi

Dowek

2013

Information and Computation

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“…Bouzidi et al 24 proposed a di®erent form of the transformation matrix for the multiple relaxation time (MRT) model. Mendoza et al 25,27 solved discrete Boltzmann equation using the covariant and contravariant transformation, while Klales et al 26 presented a hydrodynamic lattice gas model for two-dimensional°ows on curved surfaces with dynamical geometry. The most recent attempt of Budinski 28,29 to establish LBM for solving°ows in arbitrary domains utilizes a complete transformation of°ow equations in the curvilinear coordinate system.…”

Section: Introductionmentioning

confidence: 99%

Modeling groundwater flow by lattice Boltzmann method in curvilinear coordinates

Budinski

Fabian

Stipić

2015

Int. J. Mod. Phys. C

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In order to promote the use of the lattice Boltzmann method (LBM) for the simulation of isotropic groundwater°ow in a con¯ned aquifer with arbitrary geometry, Poisson's equation was transformed into a curvilinear coordinate system. With the metric function between the physical and the computational domain established, Poisson's equation written in Cartesian coordinates was transformed in curvilinear coordinates. Following, the appropriate equilibrium function for the D2Q9 square lattice has been de¯ned. The resulting curvilinear formulation of the LBM for groundwater°ow is capable of modeling°ow in domains of complex geometry with the opportunity of local re¯ning/coarsening of the computational mesh corresponding to the complexity of the°ow pattern and the required accuracy. Since the proposed form of the LBM uses the transformed equation of°ow implemented in the equilibrium function,¯nding a solution does not require supplementary procedures along the curvilinear boundaries, nor in the zones requiring mesh density adjustments. Thus, the basic concept of the LBM is completely maintained. The improvement of the proposed LBM over the previously published classical methods is completely veri¯ed by three examples with analytical solutions. The results demonstrate the advantages of the proposed curvilinear LBM in modeling groundwater°ow in complex°ow domains.

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“…• Through concrete instances advocating the concept of CA extended to timevarying graphs as in [31,20,19], some of which are advanced algorithmic constructions [30,29].…”

Section: Introductionmentioning

confidence: 99%

Cellular automata over generalized Cayley graphs

Arrighi¹,

Martiel²,

Nesme³

2017

Math. Struct. Comp. Sci.

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Abstract. Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their relations; to name all vertices relative to a point; and the fact that they have a well-defined notion of translation. We propose a notion of graph associated to a language, which conserves or generalizes these features. Whereas Cayley graphs are very regular; associated graphs are arbitrary, although of a bounded degree. Moreover, it is well-known that cellular automata can be characterized as the set of translation-invariant continuous functions for a distance on the set of configurations that makes it a compact metric space; this point of view makes it easy to extend their definition from grids to Cayley graphs. Similarly, we extend their definition to these arbitrary, bounded degree, time-varying graphs. The obtained notion of Cellular Automata over generalized Cayley graphs is stable under composition and under inversion.

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Lattice gas simulations of dynamical geometry in two dimensions

Cited by 19 publications

References 44 publications

Causal graph dynamics

Causal graph dynamics

Modeling groundwater flow by lattice Boltzmann method in curvilinear coordinates

Cellular automata over generalized Cayley graphs

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