# Browsing by Subject "Physics, Mathematical"

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Item Open Access A stochastic-Lagrangian particle system for the Navier-Stokes equations(Nonlinearity, 2008-11-01) Iyer, Gautam; Mattingly, JonathanThis paper is based on a formulation of the Navier-Stokes equations developed by Constantin and the first author (Commun. Pure Appl. Math. at press, arXiv:math.PR/0511067), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take N copies of the above process (each based on independent Wiener processes), and replace the expected value with 1/N times the sum over these N copies. (We note that our formulation requires one to keep track of N stochastic flows of diffeomorphisms, and not just the motion of N particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space C1,α which consists of differentiable functions whose first derivative is α Hölder continuous (see section 3 for the precise definition). Further, we show that as N → ∞ the system converges to the solution of Navier-Stokes equations on any finite interval [0, T]. However for fixed N, we prove that this system retains roughly O(1/N) times its original energy as t → ∞. Hence the limit N → ∞ and T → ∞ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as t → ∞ explicitly. © 2008 IOP Publishing Ltd and London Mathematical Society.Item Open Access An energy stable C^{0}finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density(Journal of Computational Physics, 2020-03-15) Shen, L; Huang, H; Lin, P; Song, Z; Xu, SIn this paper, we focus on modeling and simulation of two-phase flow problems with moving contact lines and variable density. A thermodynamically consistent phase-field model with general Navier boundary condition is developed based on the concept of quasi-incompressibility and the energy variational method. A mass conserving C0 finite element scheme is proposed to solve the PDE system. Energy stability is achieved at the fully discrete level. Various numerical results confirm that the proposed scheme for both P1 element and P2 element are energy stable.Item Open Access Anomalous dissipation in a stochastically forced infinite-dimensional system of coupled oscillators(Journal of Statistical Physics, 2007-09-01) Mattingly, JC; Suidan, TM; Vanden-Eijnden, EWe study a system of stochastically forced infinite-dimensional coupled harmonic oscillators. Although this system formally conserves energy and is not explicitly dissipative, we show that it has a nontrivial invariant probability measure. This phenomenon, which has no finite dimensional equivalent, is due to the appearance of some anomalous dissipation mechanism which transports energy to infinity. This prevents the energy from building up locally and allows the system to converge to the invariant measure. The invariant measure is constructed explicitly and some of its properties are analyzed. © 2007 Springer Science+Business Media, LLC.Item Open Access Elusive Worldsheet Instantons in Heterotic String Compactifications(STRING-MATH 2011, 2012-01-01) Aspinwall, Paul S; Plesser, M RonenWe compute the spectrum of massless gauge singlets in some heterotic string compactifications using Landau-Ginzburg, orbifold and non-linear sigma-model methods. This probes the worldsheet instanton corrections to the quadratic terms in the spacetime superpotential. Previous results predict that some of these states remain massless when instanton effects are included. We find vanishing masses in many cases not covered by these predictions. However, we discover that in the case of the Z-manifold the corrections do not vanish. Despite this, in all the examples studied, we find that the massless spectrum in the orbifold limit agrees with the nonlinear sigma-model computation.Item Open Access Reduced-order deep learning for flow dynamics. The interplay between deep learning and model reduction(Journal of Computational Physics, 2020-01) Wang, Min; Cheung, Siu Wun; Leung, Wing Tat; Chung, Eric T; Efendiev, Yalchin; Wheeler, MaryItem Open Access Reinitialization of the Level-Set Function in 3d Simulation of Moving Contact Lines(Communications in Computational Physics, 2016-11-01) Xu, S; Ren, WThe level set method is one of the most successful methods for the simulation of multi-phase flows. To keep the level set function close the signed distance function, the level set function is constantly reinitialized by solving a Hamilton-Jacobi type of equation during the simulation. When the fluid interface intersects with a solid wall, a moving contact line forms and the reinitialization of the level set function requires a boundary condition in certain regions on the wall. In this work, we propose to use the dynamic contact angle, which is extended from the contact line, as the boundary condition for the reinitialization of the level set function. The reinitialization equation and the equation for the normal extension of the dynamic contact angle form a coupled system and are solved simultaneously. The extension equation is solved on the wall and it provides the boundary condition for the reinitialization equation; the level set function provides the directions along which the contact angle is extended from the contact line. The coupled system is solved using the 3rd order TVD Runge-Kutta method and the Godunov scheme. The Godunov scheme automatically identifies the regions where the angle condition needs to be imposed. The numerical method is illustrated by examples in three dimensions.Item Open Access Simple systems with anomalous dissipation and energy cascade(Communications in Mathematical Physics, 2007-11-01) Mattingly, JC; Suidan, T; Vanden-Eijnden, EWe analyze a class of dynamical systems of the type ȧn(t) = cn-1 an-1(t) - cn an+1(t) + f n(t), n ∈ ℕ, a 0=0, where f n (t) is a forcing term with fn(t) ≠ = 0 only for ≤n n* < ∞ and the coupling coefficients c n satisfy a condition ensuring the formal conservation of energy 1/2 Σn |a n(t)|2. Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f n (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients c n . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely E |an|2 scales as n-α as n→∞. Here the exponents α depend on the coupling coefficients c n and E denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable. © 2007 Springer-Verlag.Item Open Access Stationary state volume fluctuations in a granular medium.(Physical review. E, Statistical, nonlinear, and soft matter physics, 2005-03-30) Schröter, Matthias; Goldman, Daniel I; Swinney, Harry LA statistical description of static granular material requires ergodic sampling of the phase space spanned by the different configurations of the particles. We periodically fluidize a column of glass beads and find that the sequence of volume fractions phi of postfluidized states is history independent and Gaussian distributed about a stationary state. The standard deviation of phi exhibits, as a function of phi, a minimum corresponding to a maximum in the number of statistically independent regions. Measurements of the fluctuations enable us to determine the compactivity X , a temperaturelike state variable introduced in the statistical theory of Edwards and Oakeshott [Physica A 157, 1080 (1989)].Item Open Access Topological strings, D-model, and knot contact homology(Advances in Theoretical and Mathematical Physics, 2014) Aganagic, M; Ekholm, T; Ng, L; Vafa, C© 2014 International Press. We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov- Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the Q-deformed A-polynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the "D-model". In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large N limit of the colored HOMFLY, which we conjecture to be related to a Qdeformation of the extension of A-polynomials associated with the link complement.