Journal articles: 'Discrete-to-continuum limit'

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Müller, Stefan, and Anja Schlömerkemper. "Discrete-to-continuum limit of magnetic forces." Comptes Rendus Mathematique 335, no. 4 (January 2002): 393–98. http://dx.doi.org/10.1016/s1631-073x(02)02494-9.

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Cesana, Pierluigi, and Patrick van Meurs. "Discrete-to-continuum limits of planar disclinations." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 23. http://dx.doi.org/10.1051/cocv/2021025.

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In materials science, wedge disclinations are defects caused by angular mismatches in the crystallographic lattice. To describe such disclinations, we introduce an atomistic model in planar domains. This model is given by a nearest-neighbor-type energy for the atomic bonds with an additional term to penalize change in volume. We enforce the appearance of disclinations by means of a special boundary condition. Our main result is the discrete-to-continuum limit of this energy as the lattice size tends to zero. Our proof relies on energy relaxation methods. The main mathematical novelty of our proof is a density theorem for the special boundary condition. In addition to our limit theorem, we construct examples of planar disclinations as solutions to numerical minimization of the model and show that classical results for wedge disclinations are recovered by our analysis.

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LING, YI. "DISCRETE GRAVITY AND ITS CONTINUUM LIMIT." Modern Physics Letters A 20, no. 03 (January 30, 2005): 213–25. http://dx.doi.org/10.1142/s0217732305015793.

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Recently Gambini and Pullin proposed a new consistent discrete approach to quantum gravity and applied it to cosmological models. One remarkable result of this approach is that the cosmological singularity can be avoided in a general fashion. However, whether the continuum limit of such discretized theories exists is model dependent. In the case of massless scalar field coupled to gravity with Λ=0, the continuum limit can only be achieved by fine tuning the recurrence constant. We regard this failure as the implication that cosmological constant should vary with time. For this reason we replace the massless scalar field by Chaplygin gas which may contribute an effective cosmological constant term with the evolution of the universe. It turns out that the continuum limit can indeed be reached in this case.

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Español, Malena I., Dmitry Golovaty, and J. Patrick Wilber. "Euler elastica as a Γ-limit of discrete bending energies of one-dimensional chains of atoms." Mathematics and Mechanics of Solids 23, no. 7 (May 26, 2017): 1104–16. http://dx.doi.org/10.1177/1081286517707997.

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In the 1920s, Hencky proposed a discrete elastica model describing a chain of identical rigid bars connected by torsional springs. Hencky observed that this discrete elastica model converges to Euler’s elastica as the number of bars increases while their lengths decrease, and Hencky’s bar-chain model has been used primarily as an approximation of Euler’s elastica. A Hencky-type bar-chain model can also be incorporated into a Frenkel–Kontorova-type discrete atomistic model, where the joints and bars represent the atoms and interatomic bonds, respectively, while the entire chain of atoms interacts with either a substrate or other chains. The energy of a continuum system corresponding to this Frenkel–Kontorova-type model can then be recovered by taking an appropriate discrete-to-continuum limit. Developing a correct limiting procedure for the discrete elastica establishes the bending component of this continuum energy. In this paper we use Γ-convergence to rigorously show that as the bar length in the discrete elastica model we consider goes to 0, the bending energies of the chain Γ-converge to the continuum bending energy associated with Euler’s elastica.

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REQUARDT, MANFRED. "THE CONTINUUM LIMIT OF DISCRETE GEOMETRIES." International Journal of Geometric Methods in Modern Physics 03, no. 02 (March 2006): 285–313. http://dx.doi.org/10.1142/s0219887806001156.

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In various areas of modern physics and in particular in quantum gravity or foundational space–time physics, it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be constructed from a more primordial and basically discrete underlying substratum, which may behave in a quite erratic and irregular way. We develop such a framework within the category of general metric spaces by combining recent work of our own and ingeneous ideas of Gromov et al. developed in pure mathematics. A central role is played by two core concepts. For one, the notion of intrinsic scaling dimension of a (discrete) space or, in mathematical terms, the growth degree of a metric space at infinity, on the other hand, the concept of a metrical distance between general metric spaces and an appropriate scaling limit (called by us a geometric renormalization group) performed in this metric space of spaces. In doing this, we prove a variety of physically interesting results about the nature of this limit process, properties of the limit space, e.g., what preconditions qualify it as a smooth classical space–time and, in particular, its dimension.

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Kevrekidis, P. G., and D. E. Pelinovsky. "Discrete vector on-site vortices." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2073 (April 4, 2006): 2671–94. http://dx.doi.org/10.1098/rspa.2006.1693.

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We study discrete vortices in coupled discrete nonlinear Schrödinger equations. We focus on the vortex cross configuration that has been experimentally observed in photorefractive crystals. Stability of the single-component vortex cross in the anti-continuum limit of small coupling between lattice nodes is proved. In the vector case, we consider two coupled configurations of vortex crosses, namely the charge-one vortex in one component coupled in the other component to either the charge-one vortex (forming a double-charge vortex) or the charge-negative-one vortex (forming a, so-called, hidden-charge vortex). We show that both vortex configurations are stable in the anti-continuum limit, if the parameter for the inter-component coupling is small and both of them are unstable when the coupling parameter is large. In the marginal case of the discrete two-dimensional Manakov system, the double-charge vortex is stable while the hidden-charge vortex is linearly unstable. Analytical predictions are corroborated with numerical observations that show good agreement near the anti-continuum limit, but gradually deviate for larger couplings between the lattice nodes.

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ANGUIGE, K. "A one-dimensional model for the interaction between cell-to-cell adhesion and chemotactic signalling." European Journal of Applied Mathematics 22, no. 4 (February 10, 2011): 291–316. http://dx.doi.org/10.1017/s0956792511000040.

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We develop and analyse a discrete, one-dimensional model of cell motility which incorporates the effects of volume filling, cell-to-cell adhesion and chemotaxis. The formal continuum limit of the model is a non-linear generalisation of the parabolic-elliptic Keller–Segel equations, with a diffusivity which can become negative if the adhesion coefficient is large. The consequent ill-posedness results in the appearance of spatial oscillations and the development of plateaus in numerical solutions of the underlying discrete model. A global-existence result is obtained for the continuum equations in the case of favourable parameter values and data, and a steady-state analysis, which, amongst other things, accounts for high-adhesion plateaus, is carried out. For ill-posed cases, a singular Stefan-problem formulation of the continuum limit is written down and solved numerically, and the numerical solutions are compared with those of the original discrete model.

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MIRONOV, A., and S. PAKULIAK. "ON THE CONTINUUM LIMIT OF THE CONFORMAL MATRIX MODELS." International Journal of Modern Physics A 08, no. 18 (July 20, 1993): 3107–37. http://dx.doi.org/10.1142/s0217751x93001247.

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The double scaling limit of a new class of the multi-matrix models proposed in Ref. 1, which possess the W-symmetry at the discrete level, is investigated in detail. These models are demonstrated to fall into the same universality class as the standard multi-matrix models. In particular, the transformation of the W-algebra at the discrete level into the continuum one of the papers2 is proposed and the corresponding partition functions compared. All calculations are demonstrated in full in the first nontrivial case of W(3)-constraints.

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BAKER, GEORGE A., and JAMES P. HAGUE. "RISE OF THE CENTRIST: FROM BINARY TO CONTINUOUS OPINION DYNAMICS." International Journal of Modern Physics C 19, no. 09 (September 2008): 1459–75. http://dx.doi.org/10.1142/s0129183108013023.

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We propose a model that extends the binary "united we stand, divided we fall" opinion dynamics of Sznajd-Weron to handle continuous and multi-state discrete opinions on a linear chain. Disagreement dynamics are often ignored in continuous extensions of the binary rules, so we make the most symmetric continuum extension of the binary model that can treat the consequences of agreement (debate) and disagreement (confrontation) within a population of agents. We use the continuum extension as an opportunity to develop rules for persistence of opinion (memory). Rules governing the propagation of centrist views are also examined. Monte Carlo simulations are carried out. We find that both memory effects and the type of centrist significantly modify the variance of average opinions in the large timescale limits of the models. Finally, we describe the limit of applicability for Sznajd-Weron's model of binary opinions as the continuum limit is approached. By comparing Monte Carlo results and long time-step limits, we find that the opinion dynamics of binary models are significantly different to those where agents are permitted more than 3 opinions.

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Schlömerkemper, Anja, and Bernd Schmidt. "Discrete-to-Continuum Limit of Magnetic Forces: Dependence on the Distance Between Bodies." Archive for Rational Mechanics and Analysis 192, no. 3 (July 25, 2008): 589–611. http://dx.doi.org/10.1007/s00205-008-0134-4.

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Pickering, Andrew, Hai-qiong Zhao, and Zuo-nong Zhu. "On the continuum limit for a semidiscrete Hirota equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2195 (November 2016): 20160628. http://dx.doi.org/10.1098/rspa.2016.0628.

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In this paper, we propose a new semidiscrete Hirota equation which yields the Hirota equation in the continuum limit. We focus on the topic of how the discrete space step δ affects the simulation for the soliton solution to the Hirota equation. The Darboux transformation and explicit solution for the semidiscrete Hirota equation are constructed. We show that the continuum limit for the semidiscrete Hirota equation, including the Lax pair, the Darboux transformation and the explicit solution, yields the corresponding results for the Hirota equation as δ → 0 .

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Gavrilov, Serge N. "Discrete and continuum fundamental solutions describing heat conduction in a 1D harmonic crystal: Discrete-to-continuum limit and slow-and-fast motions decoupling." International Journal of Heat and Mass Transfer 194 (September 2022): 123019. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2022.123019.

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DARBHA, SWAROOP, and K. R. RAJAGOPAL. "LIMIT OF A COLLECTION OF DYNAMICAL SYSTEMS: AN APPLICATION TO MODELING THE FLOW OF TRAFFIC." Mathematical Models and Methods in Applied Sciences 12, no. 10 (October 2002): 1381–99. http://dx.doi.org/10.1142/s0218202502002161.

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The flow of traffic is usually described using a continuum approach as that of a compressible fluid, a statistical approach via the kinetic theory of gases or cellular automata models. These approaches are not suitable for modeling dynamical systems such as traffic. While such systems are large collections, they are not large enough to be treated as a continuum. We provide a rationale for why they cannot be appropriately described using a continuum model, the kinetic theory of gases, or by appealing to cellular automata models. As an alternative, we develop a discrete dynamical systems approach that is particularly well suited to describe the dynamics of large systems such as traffic.

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AVAN, J. "GENERALIZED TODA AND VOLTERRA MODELS." International Journal of Modern Physics A 07, no. 20 (August 10, 1992): 4855–69. http://dx.doi.org/10.1142/s0217751x92002192.

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The mean procedure of Faddeev-Reshetikhin with non-Abelian automorphism groups is applied to construct generalizations of the open Toda sl(n, C) chain. These models admit a consistent reduction to integrable generalized Volterra models. An example of such models is analyzed: it leads in the continuum limit to the [Formula: see text] Hirota differential system, associated with two-matrix models of discrete gravity. The continuum limit of the general Volterra models and their relation with discretized versions of Wn-algebra are analyzed.

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ALVAREZ-GAUMÉ, L., H. ITOYAMA, J. L. MAÑES, and A. ZADRA. "SUPERLOOP EQUATIONS AND TWO-DIMENSIONAL SUPERGRAVITY." International Journal of Modern Physics A 07, no. 21 (August 20, 1992): 5337–67. http://dx.doi.org/10.1142/s0217751x92002441.

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We propose a discrete model whose continuum limit reproduces the string susceptibility and the scaling dimensions of (2, 4m) minimal superconformal models coupled to 2D supergravity. The basic assumption in our presentation is a set of super-Virasoro constraints imposed on the partition function. We recover the Neveu-Schwarz and Ramond sectors of the theory, and we are also able to evaluate all planar loop correlation functions in the continuum limit. We find evidence to identify the integrable hierarchy of nonlinear equations describing the double scaling limit as a supersymmetric generalization of KP studied by Rabin.

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Hill, James M., and Barry D. Hughes. "On the general random walk formulation for diffusion in media with Diffusivities." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 27, no. 1 (July 1985): 73–87. http://dx.doi.org/10.1017/s033427000000477x.

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AbstractA general discrete multi-dimensional and multi-state random walk model is proposed to describe the phenomena of diffusion in media with multiple diffusivities. The model is a generalization of a two-state one-dimensional discrete random walk model (Hill [8]) which gives rise to the partial differential equations of double diffusion. The same partial differential equations are shown to emerge as a special case of the continuous version of the present general model. For two states a particular generalization of the model given in [8] is presented which is not restricted to nearest neighbour transitions. Under appropriate circumstances this two-state model still yields the partial differential equations of double diffusion in the continuum limit, but an example of circumstances leading to a radically different continuum limit is presented.

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Field, Timothy R., and Robert J. A. Tough. "Coupled dynamics of populations supported by discrete sites and their continuum limit." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2121 (April 29, 2010): 2561–86. http://dx.doi.org/10.1098/rspa.2010.0049.

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The illumination of single population behaviour subject to the processes of birth, death and immigration has provided a basis for the discussion of the non-Gaussian statistical and temporal correlation properties of scattered radiation. As a first step towards the modelling of its spatial correlations, we consider the populations supported by an infinite chain of discrete sites, each subject to birth, death and immigration and coupled by migration between adjacent sites. To provide some motivation, and illustrate the techniques we will use, the migration process for a single particle on an infinite chain of sites is introduced and its diffusion dynamics derived. A certain continuum limit is identified and its properties studied via asymptotic analysis. This forms the basis of the multi-particle model of a coupled population subject to single site birth, death and immigration processes, in addition to inter-site migration. A discrete rate equation is formulated and its generating function dynamics derived. This facilitates derivation of the equations of motion for the first- and second-order cumulants, thus generalizing the earlier results of Bailey through the incorporation of immigration at each site. We present a novel matrix formalism operating in the time domain that enables solution of these equations yielding the mean occupancy and inter-site variances in the closed form. The results for the first two moments at a single time are used to derive expressions for the asymptotic time-delayed correlation functions, which relates to Glauber’s analysis of an Ising model. The paper concludes with an analysis of the continuum limit of the birth–death–immigration–migration process in terms of a path integral formalism. The continuum rate equation and evolution equation for the generating function are developed, from which the evolution equation of the mean occupancy is derived, in this limit. Its solution is provided in closed form.

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Hafiene, Yosra, Jalal M. Fadili, Christophe Chesneau, and Abderrahim Elmoataz. "Continuum limit of the nonlocal p-Laplacian evolution problem on random inhomogeneous graphs." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 2 (February 25, 2020): 565–89. http://dx.doi.org/10.1051/m2an/2019066.

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In this paper we study numerical approximations of the evolution problem governed by the nonlocal p-Laplacian operator with a given kernel and homogeneous Neumann boundary conditions. More precisely, we consider discretized versions on inhomogeneous random graph sequences, establish their continuum limits and provide error bounds with nonasymptotic rate of convergence of solutions of the discrete problems to their continuum counterparts as the number of vertices grows. Our bounds reveal the role of the different parameters that come into play, and in particular that of p and of the geometry/regularity of the initial data and the kernel.

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Brzeski, Piotr, and Grzegorz Kondrat. "Percolation of hyperspheres in dimensions 3 to 5: from discrete to continuous." Journal of Statistical Mechanics: Theory and Experiment 2022, no. 5 (May 1, 2022): 053202. http://dx.doi.org/10.1088/1742-5468/ac6519.

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Abstract We study the onset of percolation of overlapping discrete hyperspheres on hypercubic lattices in dimension D = 3, 4, 5. Taking the continuum limit of the thresholds for discrete hyperspheres we obtain the values of percolation thresholds for continuous hyperspheres. In D = 3 we improved the value of the correlation length exponent: ν = 0.8762(7). In D = 4 and 5 we obtained the continuous percolation thresholds of hyperspheres with much better quality than previously known (the uncertainties reduced by the factor of 230 and 10 respectively). We discuss the hypothesis of constant exponent governing the rate of convergence of discrete models to the continuous one for hyperspheres.

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SEMIKHATOV, A. M. "CONTINUUM REDUCTION OF VIRASORO-CONSTRAINED LATTICE HIERARCHIES." Modern Physics Letters A 06, no. 28 (September 14, 1991): 2601–12. http://dx.doi.org/10.1142/s0217732391003043.

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Integrable hierarchies with Virasoro constraints have been observed to describe matrix models. I suggest to define general Virasoro-constrained integrable hierarchies by imposing Virasora-highest-weight conditions on the dressing operators. This simplifies the study of the Virasoro constraints and allows an explicit construction of a scaling which implements the continuum limit of discrete (lattice) hierarchies. Applied to the Toda lattice hierarchy subjected to the Virasoro constraints, this scaling leads to the Virasoro-constrained KP hierarchy. Therefore, in particular, the KP hierarchy is shown to arise as the scaling limit of a matrix model.

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Bahr, Benjamin, and Klaus Liegener. "Towards exploring features of Hamiltonian renormalisation relevant for quantum gravity." Classical and Quantum Gravity 39, no. 7 (March 7, 2022): 075010. http://dx.doi.org/10.1088/1361-6382/ac5050.

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Abstract We consider the Hamiltonian renormalisation group (RG) flow of discretised one-dimensional physical theories. In particular, we investigate the influence the choice of different embedding maps has on the RG flow and the resulting continuum limit, and show in which sense they are, and in which sense they are not equivalent as physical theories. We are furthermore elucidating on the interplay of the RG flow and the algebras which operators satisfy, both on the discrete and the continuum. Further, we propose preferred renormalisation prescriptions for operator algebras guaranteeing to arrive at preferred algebraic relations in the continuum, if suitable extension properties are assumed. Finally, we introduce a weaker form of distributional equivalence, and show how unitarily inequivalent continuum limits, which arise due to a choice of different embedding maps, can still be weakly equivalent in that sense. We expect these results to have application in defining an RG flow in loop quantum gravity.

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Shapiro, Jacob, and Michael I. Weinstein. "Is the continuum SSH model topological?" Journal of Mathematical Physics 63, no. 11 (November 1, 2022): 111901. http://dx.doi.org/10.1063/5.0064037.

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The discrete Hamiltonian of Su, Schrieffer, and Heeger (SSH) [Phys. Rev. Lett. 42, 1698–1701 (1979)] is a well-known one-dimensional translation-invariant model in condensed matter physics. The model consists of two atoms per unit cell and describes in-cell and out-of-cell electron-hopping between two sub-lattices. It is among the simplest models exhibiting a non-trivial topological phase; to the SSH Hamiltonian, one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on values 0 and 1 labeling the two distinct phases. We display two homotopically equivalent continuum Hamiltonians whose tight binding limits are SSH models with different topological indices. The topological character of the SSH model is, therefore, an emergent rather than fundamental property, associated with emergent chiral or sublattice symmetry in the tight-binding limit. In order to establish that the tight-binding limit of these continuum Hamiltonians is the SSH model, we extend our recent results on the tight-binding approximation [J. Shapiro and M. I. Weinstein, Adv. Math. 403, 108343 (2022)] to lattices, which depend on the tight-binding asymptotic parameter λ.

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Arrighi, Pablo, Giuseppe Di Molfetta, and Stefano Facchini. "Quantum walking in curved spacetime: discrete metric." Quantum 2 (August 22, 2018): 84. http://dx.doi.org/10.22331/q-2018-08-22-84.

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A discrete-time quantum walk (QW) is essentially a unitary operator driving the evolution of a single particle on the lattice. Some QWs have familiar physics PDEs as their continuum limit. Some slight generalization of them (allowing for prior encoding and larger neighbourhoods) even have the curved spacetime Dirac equation, as their continuum limit. In the(1+1)−dimensional massless case, this equation decouples as scalar transport equations with tunable speeds. We characterise and construct all those QWs that lead to scalar transport with tunable speeds. The local coin operator dictates that speed; we provide concrete techniques to tune the speed of propagation, by making use only of a finite number of coin operators-differently from previous models, in which the speed of propagation depends upon a continuous parameter of the quantum coin. The interest of such a discretization is twofold : to allow for easier experimental implementations on the one hand, and to evaluate ways of quantizing the metric field, on the other.

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Eberle, Andreas. "Spectral Gaps on Discretized Loop Spaces." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, no. 02 (June 2003): 265–300. http://dx.doi.org/10.1142/s021902570300116x.

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We study spectral gaps w.r.t. marginals of pinned Wiener measures on spaces of discrete loops (or, more generally, pinned paths) on a compact Riemannian manifold M. The asymptotic behaviour of the spectral gap as the time parameter T of the underlying Brownian bridge goes to 0 is investigated. It turns out that depending on the choice of a Riemannian metric on the base manifold, very different asymptotic behaviours can occur. For example, on discrete loop spaces over sufficiently round ellipsoids the gap grows of order α/T as T ↓ 0. The strictly positive rate α stabilizes as the discretization approaches the continuum limit. On the other extreme, if there exists a closed geodesic γ : S1 → M such that the sectional curvature on γ(S1) is strictly negative, and the loop is pinned close to γ(S1), then the gap decays of order exp (-β/T), and the decay rate β approaches +∞ as the discretization approaches the continuum limit.

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Carpio, A., and G. Duro. "Instability and Collapse in Discrete Wave Equations." Computational Methods in Applied Mathematics 5, no. 3 (2005): 223–41. http://dx.doi.org/10.2478/cmam-2005-0011.

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AbstractUnstable growth phenomena in spatially discrete wave equations are studied. We characterize sets of initial states leading to instability and collapse and obtain analytical predictions for the blow-up time. The theoretical predictions are con- trasted with the numerical solutions computed by a variety of schemes. The behavior of the systems in the continuum limit and the impact of discreteness and friction are discussed.

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Scalari, Giacomo, Shima Rajabali, Elsa Jöchl, Sergej Markmann, Simone de Liberato, Erika Cortese, Mattias Beck, and Jérôme Faist. "Non-locality and single meta-atom spectroscopy in THz Landau polaritons." EPJ Web of Conferences 266 (2022): 08013. http://dx.doi.org/10.1051/epjconf/202226608013.

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We will discuss, theoretically and experimentally, the existence of a limit to the possibility of arbitrarily increasing electromagnetic confinement in polaritonic systems, where strongly sub-wavelength fields can excite a continuum of high-momenta propagative magnetoplasmons. This leads to peculiar nonlocal polaritonic effects, as certain polaritonic features disappear and the system enters in the regime of discrete-to-continuum strong coupling. We will as well present experiments reporting spectroscopy of a single, ultrastrongly coupled, highly subwavelength resonator operating at 300 GHz.

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SMITH, WARREN R., and JONATHAN A. D. WATTIS. "Necessary conditions for breathers on continuous media to approximate breathers on discrete lattices." European Journal of Applied Mathematics 27, no. 1 (June 23, 2015): 23–41. http://dx.doi.org/10.1017/s0956792515000273.

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The sine-Gordon (SG) partial differential equation (PDE) with an arbitrary perturbation is initially considered. Using the method of Kuzmak–Luke, we investigate the conditions, which the perturbation must satisfy, for a breather solution to be a valid leading-order asymptotic approximation to the perturbed problem. We analyse the cases of both stationary and moving breathers. As examples, we consider perturbing terms which include typical linear damping, periodic sinusoidal driving, and dispersion. The motivation for this study is that the mathematical modelling of physical systems often leads to the discrete SG system of ordinary differential equations, which are then approximated in the long wavelength limit by the continuous SG PDE. Such limits typically produce fourth-order spatial derivatives as correction terms. The new results show that the stationary breather solution is a consistent solution of both the quasi-continuum SG equation and the forced/damped SG system. However, the moving breather is only a consistent solution of the quasi-continuum SG equation and not the damped SG system.

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Arrighi, Pablo, and Stefno Facchini. "Quantum walking in curved spacetime: (3+1) dimensions, and beyond." Quantum Information and Computation 17, no. 9&10 (August 2017): 810–24. http://dx.doi.org/10.26421/qic17.9-10-4.

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A discrete-time Quantum Walk (QW) is essentially an operator driving the evolution of a single particle on the lattice, through local unitaries. Some QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac equation). Recently it was discovered that prior grouping and encoding allows for more general continuum limit equations (e.g. the Dirac equation in (1+ 1) curved spacetime). In this paper, we extend these results to arbitrary space dimension and internal degree of freedom. We recover an entire class of PDEs encompassing the massive Dirac equation in (3 + 1) curved spacetime. This means that the metric field can be represented by a field of local unitaries over a lattice.

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Boyd, I. D., and J. P. W. Stark. "Assessment of Impingement Effects in the Isentropic Core of a Small Satellite Control Thruster Plume." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 203, no. 2 (July 1989): 97–103. http://dx.doi.org/10.1243/pime_proc_1989_203_060_01.

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Two computational techniques commonly employed in the calculation of rocket and thruster expansion plumes are assessed. These are the method of characteristics (MOC), which is derived from the continuum Euler equations, and the direct simulation Monte Carlo (DSMC) method, which adopts a discrete particle approach. These techniques vary both in the computational expense and in the accuracy and detail of the solutions that they provide, depending upon the regime of application. The assessment is made with reference to the plume expanding from a small monopropellant hydrazine thruster and concentrates on the isentropic core of the jet for the flow regime lying between the continuum and free molecular limits. It is found that the more numerically intensive DSMC method offers the better correspondence to the available experimental data. In addition, large differences in typical impingement effects such as drag force and heat transfer are found at the free molecular limit of the plume expansion for the two predictive techniques. It is concluded that accurate estimation of impingement potential may only be achieved through application of the discrete particle method.

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Eichhorn, Astrid, Tim Koslowski, and Antonio Pereira. "Status of Background-Independent Coarse Graining in Tensor Models for Quantum Gravity." Universe 5, no. 2 (February 5, 2019): 53. http://dx.doi.org/10.3390/universe5020053.

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A background-independent route towards a universal continuum limit in discrete models of quantum gravity proceeds through a background-independent form of coarse graining. This review provides a pedagogical introduction to the conceptual ideas underlying the use of the number of degrees of freedom as a scale for a Renormalization Group flow. We focus on tensor models, for which we explain how the tensor size serves as the scale for a background-independent coarse-graining flow. This flow provides a new probe of a universal continuum limit in tensor models. We review the development and setup of this tool and summarize results in the two- and three-dimensional case. Moreover, we provide a step-by-step guide to the practical implementation of these ideas and tools by deriving the flow of couplings in a rank-4-tensor model. We discuss the phenomenon of dimensional reduction in these models and find tentative first hints for an interacting fixed point with potential relevance for the continuum limit in four-dimensional quantum gravity.

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Wheeler, S. J., W. K. Sham, and S. D. Thomas. "Gas pressure in unsaturated offshore soils." Canadian Geotechnical Journal 27, no. 1 (February 1, 1990): 79–89. http://dx.doi.org/10.1139/t90-008.

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Direct measurement of gas pressure within unsaturated offshore soils is very difficult because the gas occurs in the form of large, discrete bubbles. However, consideration of the soil structure and analysis of a continuum model for the soil suggest two independent sets of limits for the gas bubble pressure. Surface tension effects limit the difference between gas pressure and pore-water pressure, while cavity expansion and contraction considerations limit the difference between gas pressure and mean total stress. If the gas pressure lies within these limits, it should remain almost unaffected by changes to the total stress or pore-water pressure. These suggestions are supported by results from an oedometer test on a reconstituted soil sample containing large bubbles of methane gas. Key words: bubbles, cavity expansion, gas, oedometer tests, offshore geotechnics, pore pressure, surface tension, unsaturated.

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Esposito, Antonio, Francesco S. Patacchini, André Schlichting, and Dejan Slepčev. "Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit." Archive for Rational Mechanics and Analysis 240, no. 2 (March 15, 2021): 699–760. http://dx.doi.org/10.1007/s00205-021-01631-w.

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AbstractWe consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

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BARTELT, M. C., and V. PRIVMAN. "KINETICS OF IRREVERSIBLE MONOLAYER AND MULTILAYER ADSORPTION." International Journal of Modern Physics B 05, no. 18 (November 10, 1991): 2883–907. http://dx.doi.org/10.1142/s0217979291001127.

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We review recent theoretical developments of microscopic lattice and continuum models of the kinetics of irreversible monolayer and multilayer surface adsorption. Such models have been used to describe adhesion and reaction processes of colloidal particles and proteins at solid surfaces. Theoretical results surveyed here include the void-filling rate equation approach, exact results for low dimensionalities, the mean-field theory, and large-time kinetics arguments. Numerical simulations serving to test and supplement the analytical theories, are reviewed as well. We also elucidate the crossover from the discrete to continuum behavior, analyzed via scaling arguments in the large-time limit of the deposition kinetics.

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Swinburne, Thomas D., and Danny Perez. "Reaction–drift–diffusion models from master equations: application to material defects." Modelling and Simulation in Materials Science and Engineering 30, no. 3 (March 1, 2022): 034004. http://dx.doi.org/10.1088/1361-651x/ac54c5.

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Abstract We present a general method to produce well-conditioned continuum reaction–drift–diffusion equations directly from master equations on a discrete, periodic state space. We assume the underlying data to be kinetic Monte Carlo models (i.e. continuous-time Markov chains) produced from atomic sampling of point defects in locally periodic environments, such as perfect lattices, ordered surface structures or dislocation cores, possibly under the influence of a slowly varying external field. Our approach also applies to any discrete, periodic Markov chain. The analysis identifies a previously omitted non-equilibrium drift term, present even in the absence of external forces, which can compete in magnitude with the reaction rates, thus being essential to correctly capture the kinetics. To remove fast modes which hinder time integration, we use a generalized Bloch relation to efficiently calculate the eigenspectrum of the master equation. A well conditioned continuum equation then emerges by searching for spectral gaps in the long wavelength limit, using an established kinetic clustering algorithm to define a proper reduced, Markovian state space.

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Ha, Seung-Yeal, Gyuyoung Hwang, and Dohyun Kim. "Two-point correlation function and its applications to the Schrödinger-Lohe type models." Quarterly of Applied Mathematics 80, no. 4 (May 10, 2022): 669–99. http://dx.doi.org/10.1090/qam/1623.

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We study the asymptotic emergent dynamics and the continuum limit for the Schrödinger-Lohe (SL) model and semi-discrete SL model. For the SL model, emergent dynamics has been mostly studied for systems with identical potentials in literature. In this paper, we further extend emergent dynamics and stability estimate for the SL model with nonidentical potentials. To achieve this, we use two-point correlation functions defined as an inner product between wave functions. For the semi-discrete SL model, we provide a global unique solvability and a sufficient framework for the smooth transition from the semi-discrete SL model to the SL model in any finite-time interval, as the mesh size tends to zero. Our convergence estimate depends on the uniform-in- h h Strichartz estimate and the uniform-stability of the SL models with respect to initial data.

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Hayter, J. B., and H. A. Mook. "Discrete thin-film multilayer design for X-ray and neutron supermirrors." Journal of Applied Crystallography 22, no. 1 (February 1, 1989): 35–41. http://dx.doi.org/10.1107/s0021889888010003.

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Multilayer structures, analogous to broadband optical filters, may be used to reflect X-rays or neutrons at angles larger than the total external reflection angle intrinsic to the bulk mirror material. These so-called supermirrors have generally been designed using continuum theories, which predict that the thickness of the jth layer in the multilayer should vary smoothly as j −1/4. A new approach is proposed, based on considering the contribution of each bilayer to the extinction in a given stack of bilayers, and the discrete equations governing the choice of layer thicknesses are derived and solved. In the limit of zero layer thickness, the continuum result is recovered. The design produces essentially perfect reflectivity over the entire supermirror range. An optimal technique for trading reflectivity to gain angular range also emerges naturally from the physics of the design.

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FILIPPOV, A. T., and A. P. ISAEV. "GAUGE MODELS OF “DISCRETE STRINGS”." Modern Physics Letters A 04, no. 22 (October 30, 1989): 2167–76. http://dx.doi.org/10.1142/s0217732389002434.

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A new class of constrained hamiltonian systems with a finite number of degrees of freedom is proposed in which excitations can be divided into two groups analogous to the left and right movers of string theories. Some of these models can be regarded as discrete analogs of the bosonic string, and in the continuum limit with the infinite dimensional constraint algebra Vect (S1)⊗ Vect (S1) one can obtain the classical theory of closed bosonic strings. We also discuss the problem of quantizing these models and constructing the propagator by using path integral methods. A possibility of a supersymmetric extension of our models is also pointed out.

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GIOVANNINI, L., F. NIZZOLI, and A. M. MARVIN. "THEORY OF BRILLOUIN SCATTERING FROM A PERIODICALLY CORRUGATED SURFACE OF AN OPAQUE SOLID." Modern Physics Letters B 07, no. 05 (February 28, 1993): 291–97. http://dx.doi.org/10.1142/s0217984993000308.

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We calculate, in the small corrugation limit, the surface-acoustic phonon normal modes and the Brillouin scattering ripple cross-section for a grating on the Si(001) surface. Both the discrete and continuous spectra of acoustic modes have been studied within the elasticity theory. In the continuum the Rayleigh wave becomes a resonance and hybridizes with the longitudinal pseudo-mode of the flat surface, giving rise to a gap. The theory explains quantitatively recent experimental results.

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Quinn, D. Dane, and Daniel J. Segalman. "Using Series-Series Iwan-Type Models for Understanding Joint Dynamics." Journal of Applied Mechanics 72, no. 5 (August 10, 2004): 666–73. http://dx.doi.org/10.1115/1.1978918.

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In mechanical assemblies, the energy loss induced by joints and interfaces can account for a significant portion of the overall structural dissipation. This work considers the dynamical behavior of an elastic rod on a frictional foundation as a model for the dissipation introduced by micro-slip in mechanical joints. In a quasi-static loading limit, the deformation of the rod and hence the frictional dissipation can be solved in closed form. The resulting model is a continuum model of series arrangements of parallel Jenkins elements. For a general class of normal load distributions, the resulting energy loss per forcing cycle follows a power-law and is qualitatively similar to observed experimental findings. Finally, these results are compared with those obtained from a discrete formulation of the rod including inertial effects. For loading conditions that are consistent with mechanical joints, the numerical results from the discrete model are consistent with the closed form predictions obtained in the quasistatic limit.

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Petti, Richard James. "Derivation of Einstein–Cartan theory from general relativity." International Journal of Geometric Methods in Modern Physics 18, no. 06 (April 19, 2021): 2150083. http://dx.doi.org/10.1142/s0219887821500833.

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This paper derives the elements of classical Einstein–Cartan theory (EC) from classical general relativity (GR) in two ways. (I) Derive discrete versions of torsion (translational holonomy) and the spin-torsion field equation of EC from one Kerr solution in GR. (II) Derive the field equations of EC as the continuum limit of a distribution of many Kerr masses in classical GR. The convergence computations employ “epsilon-delta” arguments, and are not as rigorous as convergence in Sobolev norm. Inequality constraints needed for convergence restrict the limits from continuing to an infinitesimal length scale. EC enables modeling exchange of intrinsic and orbital angular momentum, which GR cannot do. Derivation of EC from GR strengthens the case for EC and for new physics derived from EC.

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Boghosian, Bruce M., and Washington Taylor. "Quantum Lattice-Gas Models for the Many-Body Schrödinger Equation." International Journal of Modern Physics C 08, no. 04 (August 1997): 705–16. http://dx.doi.org/10.1142/s0129183197000606.

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A general class of discrete unitary models are described whose behavior in the continuum limit corresponds to a many-body Schrödinger equation. On a quantum computer, these models could be used to simulate quantum many-body systems with an exponential speedup over analogous simulations on classical computers. On a classical computer, these models give an explicitly unitary and local prescription for discretizing the Schrödinger equation. It is shown that models of this type can be constructed for an arbitrary number of particles moving in an arbitrary number of dimensions with an arbitrary interparticle interaction.

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Bossa, Guilherme Volpe, and Sylvio May. "Debye-Hückel Free Energy of an Electric Double Layer with Discrete Charges Located at a Dielectric Interface." Membranes 11, no. 2 (February 14, 2021): 129. http://dx.doi.org/10.3390/membranes11020129.

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Poisson–Boltzmann theory provides an established framework to calculate properties and free energies of an electric double layer, especially for simple geometries and interfaces that carry continuous charge densities. At sufficiently small length scales, however, the discreteness of the surface charges cannot be neglected. We consider a planar dielectric interface that separates a salt-containing aqueous phase from a medium of low dielectric constant and carries discrete surface charges of fixed density. Within the linear Debye-Hückel limit of Poisson–Boltzmann theory, we calculate the surface potential inside a Wigner–Seitz cell that is produced by all surface charges outside the cell using a Fourier-Bessel series and a Hankel transformation. From the surface potential, we obtain the Debye-Hückel free energy of the electric double layer, which we compare with the corresponding expression in the continuum limit. Differences arise for sufficiently small charge densities, where we show that the dominating interaction is dipolar, arising from the dipoles formed by the surface charges and associated counterions. This interaction propagates through the medium of a low dielectric constant and alters the continuum power of two dependence of the free energy on the surface charge density to a power of 2.5 law.

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BOULATOV, D. V., and A. KRZYWICKI. "ON THE PHASE DIAGRAM OF THREE-DIMENSIONAL SIMPLICIAL QUANTUM GRAVITY." Modern Physics Letters A 06, no. 32 (October 20, 1991): 3005–14. http://dx.doi.org/10.1142/s0217732391003511.

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The phase diagram of three-dimensional (3d) simplicial quantum gravity is studied using the Monte–Carlo simulation method. It is found that the behavior of the entropy of manifolds, unambiguously defined in this discrete formulation, insures the existence of a vacuum, in spite of the Einstein action being bottomless. A critical line is found as well as some evidence for the existence of a tricritical point. Only to the right of this point, where the bare coupling of the Einstein term in the action is strong enough, a sensible continuum limit can possibly exist.

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Palumbo, S., A. R. Carotenuto, A. Cutolo, D. R. Owen, L. Deseri, and M. Fraldi. "Bulky auxeticity, tensile buckling and deck-of-cards kinematics emerging from structured continua." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2246 (February 2021): 20200729. http://dx.doi.org/10.1098/rspa.2020.0729.

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Complex mechanical behaviours are generally met in macroscopically homogeneous media as effects of inelastic responses or as results of unconventional material properties, which are postulated or due to structural systems at the meso/micro-scale. Examples are strain localization due to plasticity or damage and metamaterials exhibiting negative Poisson’s ratios resulting from special porous, eventually buckling, sub-structures. In this work, through ad hoc conceived mechanical paradigms, we show that several non-standard behaviours can be obtained simultaneously by accounting for kinematical discontinuities, without invoking inelastic laws or initial voids. By allowing mutual sliding among rigid tesserae connected by pre-stressed hyperelastic links, we find several unusual kinematics such as localized shear modes and tensile buckling-induced instabilities, leading to deck-of-cards deformations—uncapturable with classical continuum models—and unprecedented ‘bulky’ auxeticity emerging from a densely packed, geometrically symmetrical ensemble of discrete units that deform in a chiral way. Finally, after providing some analytical solutions and inequalities of mechanical interest, we pass to the limit of an infinite number of tesserae of infinitesimal size, thus transiting from discrete to continuum, without the need to introduce characteristic lengths. In the light of the theory of structured deformations, this result demonstrates that the proposed architectured material is nothing else than the first biaxial paradigm of structured continuum —a body that projects, at the macroscopic scale, geometrical changes and disarrangements occurring at the level of its sub-macroscopic elements.

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Chakraborty, Ishita, Atanu K. Mohanty, and Anindya Chatterjee. "Localized waves along a line of masses on a plate: propagation and sub-exponential attenuation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2097 (April 15, 2008): 2229–46. http://dx.doi.org/10.1098/rspa.2008.0051.

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We have studied waves propagating in an infinite plate with a line of equally spaced point masses on it. In particular, we consider waves that are laterally decaying, i.e. waves that exist due to the point masses, not in spite of them. We begin with a simple continuum limit and then study the discrete case using a suitable Green function and a superposition method used earlier for other structures by Mead. The superposition involves a slowly decaying series that we transform to a rapidly converging one using the Poisson summation formula. The system has two non-dimensional parameters, and on this parameter plane we find a propagation zone and its boundary. The boundary involves successive point masses vibrating π radians out of phase, a situation far from the continuum limit. Outside the propagation zone, waves get attenuated; however, unlike common examples of linear periodic structures, and contrary to the usual assumption made in studies thereof, here the waves show sub-exponential attenuation. In particular, the eventual decay of the wave amplitude is like the 3/2 power of distance from the point of excitation. We conclude with a theoretical explanation of the 3/2 power in the decay.

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Artina, Marco, Filippo Cagnetti, Massimo Fornasier, and Francesco Solombrino. "Linearly constrained evolutions of critical points and an application to cohesive fractures." Mathematical Models and Methods in Applied Sciences 27, no. 02 (February 2017): 231–90. http://dx.doi.org/10.1142/s0218202517500014.

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We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite-dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite-dimensional. Nevertheless, in the infinite-dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite-dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one- and two-dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.

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Bisi, M., and G. Spiga. "Hydrodynamic limits of kinetic equations for polyatomic and reactive gases." Communications in Applied and Industrial Mathematics 8, no. 1 (March 28, 2017): 23–42. http://dx.doi.org/10.1515/caim-2017-0002.

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Abstract Starting from a kinetic BGK-model for a rarefied polyatomic gas, based on a molecular structure of discrete internal energy levels, an asymptotic Chapman-Enskog procedure is developed in the asymptotic continuum limit in order to derive consistent fluid-dynamic equations for macroscopic fields at Navier-Stokes level. In this way, the model allows to treat the gas as a mixture of mono-atomic species. Explicit expressions are given not only for dynamical pressure, but also for shear stress, diffusion velocities, and heat flux. The analysis is shown to deal properly also with a mixture of reactive gases, endowed for simplicity with translational degrees of freedom only, in which frame analogous results can be achieved.

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Elze, Hans-Thomas. "Multipartite cellular automata and the superposition principle." International Journal of Quantum Information 14, no. 04 (June 2016): 1640001. http://dx.doi.org/10.1142/s0219749916400013.

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Cellular automata (CA) can show well known features of quantum mechanics (QM), such as a linear updating rule that resembles a discretized form of the Schrödinger equation together with its conservation laws. Surprisingly, a whole class of “natural” Hamiltonian CA, which are based entirely on integer-valued variables and couplings and derived from an action principle, can be mapped reversibly to continuum models with the help of sampling theory. This results in “deformed” quantum mechanical models with a finite discreteness scale l, which for [Formula: see text] reproduce the familiar continuum limit. Presently, we show, in particular, how such automata can form “multipartite” systems consistently with the tensor product structures of non-relativistic many-body QM, while maintaining the linearity of dynamics. Consequently, the superposition principle is fully operative already on the level of these primordial discrete deterministic automata, including the essential quantum effects of interference and entanglement.

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Quan, Guo Zheng, Yi Xin Wang, Jie Zhou, and Bin Chen. "A Study on Al-6061-T6 Tube Drawing Limit Based on Critical Damage Value." Advanced Materials Research 102-104 (March 2010): 69–73. http://dx.doi.org/10.4028/www.scientific.net/amr.102-104.69.

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he Al-6061-T6 tube’s drawing limit and the drawing process’s elasto-plastic behavior were investigated based on the foundational theories of larger deformation and continuum damage mechanics. A mathematical computation equation about the maximum Cockcroft-Latham damage value was converted to an appropriate discrete expression which is easy for FE code to programme, thus the corresponding finite element numerical algorithm for damage computation was developed. By an approach that physical experiments and numerical simulation provide mutual support for the critical damage value, the crack criterion of Al-6061-T6 was evaluated as 1.34. The crack criterion obtained was introduced as important design considerations of tube drawing process. An 3D graph which reflects maximum damage variation according with the diameters at different tube thickness was achieved, according to which the drawing process’s safe and unsafe areas of Al-6061-T6 tube with diameter 10mm at temperature 20°C and drawing velocity 100mm/s was ploted.

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Godio, Michele, Ioannis Stefanou, Karam Sab, Jean Sulem, and Seddik Sakji. "A limit analysis approach based on Cosserat continuum for the evaluation of the in-plane strength of discrete media: Application to masonry." European Journal of Mechanics - A/Solids 66 (November 2017): 168–92. http://dx.doi.org/10.1016/j.euromechsol.2017.06.011.

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Journal articles on the topic 'Discrete-to-continuum limit'

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