Ibrahim Fatkullin

Interests

Research

Applied and computational mathematics, mathematical physics, mathematical models related to liquid crystals and polymers.

Courses

Scholarly Contributions

Journals/Publications

• Xue, J., Sethuraman, S., & Fatkullin, I. (2017). Hydrodynamics for a class of dynamic Young tableau. Have submitted to Elec. J. Probab..
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  Different hydrodynamic limits for the shape of a class of random dynamic Young tableau are found in different scales depending on the type of interactions.
• Xue, J., Xue, J., Sethuraman, S., Sethuraman, S., Fatkullin, I., & Fatkullin, I. (2019). On Hydrodynamic Limits of Young Diagrams. Elec. J. Probab. (accepted subject to minor revision), 43pgs.
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  We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. ‘Static’ scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding ‘dynamical’ limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types parabolic PDEs, depending on the energy structure.
• Fatkullin, I., & Xue, J. (2021). Limit shapes for Gibbs partitions of sets. Journal of Statistical Physics, 183(22), 23.
• Fatkullin, I., & Xue, J. (2020). Limit Shapes for Gibbs Partitions of Sets. Journal of Statistical Physics. doi:10.1007/s10955-021-02756-8
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  This study extends a prior investigation of limit shapes for partitions of integers, which was based on analysis of sums of geometric random variables. Here we compute limit shapes for grand canonical Gibbs ensembles of partitions of sets, which lead to the sums of Poisson random variables. Under mild monotonicity assumptions, we study all possible scenarios arising from different asymptotic behaviors of the energy, and also compute local limit shape profiles for cases in which the limit shape is a step function.
• Xue, J., Sethuraman, S., & Fatkullin, I. (2020). On Hydrodynamic Limits of Young Diagrams. Elec. J. Probab., 25, Paper no. 58, 44pgs. doi:doi:10.1214/20-EJP455
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  We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. ‘Static’ scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding ‘dynamical’ limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types parabolic PDEs, depending on the energy structure.
• Fatkullin, I., & Slastikov, V. (2017). Limit shapes for Gibbs ensembles of partitions. Journal of Statistical Physics, 172(6), 1545.
• Fatkullin, I., Sethuraman, S., & Xue, J. (2018). On Hydrodynamic Limits of Young Diagrams. Electronic Journal of Probability, 25(58), 1-44.
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  Different hydrodynamic limits for the shape of a class of random dynamic Young tableau are found in different scales depending on the type of interactions.
• Slastikov, V., & Fatkullin, I. (2018). Limit shapes for Gibbs ensembles of partitions. Journal of Statistical Physics, 172(6), 1545-1563. doi:10.1007/s10955-018-2117-7
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  We explicitly compute limit shapes for several grand canonical Gibbs ensembles of partitions of integers. These ensembles appear in models of aggregation and are also related to invariant measures of zero range and coagulation-fragmentation processes. We show, that all possible limit shapes for these ensembles fall into several distinct classes determined by the asymptotics of the internal energies of aggregates.
• Zhelezov, G., & Fatkullin, I. (2018). Coalescing particle systems and applications to nonlinear Fokker–Planck equations. Communications in Mathematical Sciences, 16(2), 463-490. doi:10.4310/cms.2018.v16.n2.a8
• Fatkullin, I., & Slastikov, V. (2015). DIFFUSIVE TRANSPORT IN TWO-DIMENSIONAL NEMATICS. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S, 8(2), 323-340.
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  We discuss a dynamical theory for nematic liquid crystals describing the stage of evolution in which the hydrodynamic fluid motion has already equilibrated and the subsequent evolution proceeds via diffusive motion of the orientational degrees of freedom. This diffusion induces a slow motion of singularities of the order parameter field. Using asymptotic methods for gradient flows, we establish a relation between the Doi-Smoluchowski kinetic equation and vortex dynamics in two-dimensional systems. We also discuss moment closures for the kinetic equation and Landau-de Gennes-type free energy dissipation.
• Slastikov, V., & Fatkullin, I. (2014). Diffusive transport in two-dimensional nematics. Discrete and Continuous Dynamical Systems - Series S, 8(2), 323-340. doi:10.3934/dcdss.2015.8.323
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  We discuss a dynamical theory for nematic liquid crystals describing the stage of evolution in which the hydrodynamic fluid motion has already equilibrated and the subsequent evolution proceeds via diffusive motion of the orientational degrees of freedom. This diffusion induces a slow motion of singularities of the order parameter field. Using asymptotic methods for gradient flows, we establish a relation between the Doi-Smoluchowski kinetic equation and vortex dynamics in two-dimensional systems. We also discuss moment closures for the kinetic equation and Landau-de Gennes-type free energy dissipation.
• Fatkullin, I. (2013). A study of blow-ups in the Keller-Segel model of chemotaxis. Nonlinearity, 26(1), 81-94.
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  Abstract: We study the Keller-Segel model of chemotaxis and develop a composite particle-grid numerical method with adaptive time stepping which allows us to resolve and propagate singular solutions. We compare the numerical findings (in two dimensions) with analytical predictions regarding formation and interaction of singularities obtained through analysis of the stochastic differential equations associated with the model. © 2013 IOP Publishing Ltd & London Mathematical Society.
• Fatkullin, I., & Slastikov, V. (2015). Diffusive transport in two-dimensional nematics. Discrete and Continuous Dynamical Systems S, 8(2), 323-340.
• Capponi, A., Shi, L., Fatkullin, I., & Capponi, A. (2011). Stochastic Filtering for Diffusion Processes With Level Crossings. IEEE Transactions on Automatic Control, 56(9), 2201-2206. doi:10.1109/tac.2011.2157404
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  We provide a general framework for computing the state density of a noisy system given the sequence of hitting times of predefined thresholds. Our method relies on eigenfunction expansion corresponding to the Fokker-Planck operator of the diffusion process. For illustration, we present a particular example in which the state and the noise are one-dimensional Gaussian processes and observations are generated when the magnitude of the observed signal is a multiple of some threshold value. We present numerical simulations confirming the convergence and the accuracy of the recovered density estimator. Applications of the filtering methodology will be illustrated.
• Fatkullin, I., Kovačič, G., & Eijnden, E. (2010). Reduced dynamics of stochastically perturbed gradient flows. Communications in Mathematical Sciences, 8(2), 439-461.
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  Abstract: We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds. We derive explicit equations for this limiting dynamics and illustrate them on a few finite-dimensional examples. Finally, we formally extrapolate the reduction technique to several infinite-dimensional examples and derive equations of the stochastic kink motion in Allen-Cahn-type systems. © 2010 International Press.
• Fatkullin, I., & Slastikov, V. (2009). Vortices in two-dimensional nematics. Communications in Mathematical Sciences, 7(4), 917-938.
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  Abstract: We study a two-dimensional model describing spatial variations of orientational ordering in nematic liquid crystals. In particular, we show that the spatially extended Onsager-Maier-Saupe free energy may be decomposed into Landau-de Gennes-type and relative entropy-type contributions. We then prove that in the high concentration limit the states of the system display characteristic vortex-like patterns and derive an asymptotic expansion for the free energy of the system. © 2009 International Press.
• Slastikov, V., & Fatkullin, I. (2009). VORTICES IN TWO-DIMENSIONAL NEMATICS ∗. Communications in Mathematical Sciences, 7(4), 917-938. doi:10.4310/cms.2009.v7.n4.a6
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  We study a two-dimensional model describing spatial variations of orientational ordering in nematic liquid crystals. In particular, we show that the spatially extended Onsager- Maier-Saupe free energy may be decomposed into Landau-de Gennes-type and relative entropy-type contributions. We then prove that in the high concentration limit the states of the system display characteristic vortex-like patterns and derive an asymptotic expansion for the free energy of the system.
• Fatkullin, I., & Slastikov, V. (2008). On spatial variations of nematic ordering. Physica D: Nonlinear Phenomena, 237(20), 2577-2586.
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  Abstract: We present a theory of orientational order in nematic liquid crystals which interpolates between several distinct approaches based on the director field (Oseen and Frank), order parameter tensor (Landau and de Gennes), and orientation probability density function (Onsager). As in density-functional theories, the suggested free energy is a functional of spatially-dependent orientation distribution, however, the nonlocal effects are taken into account via phenomenological elastic terms rather than by means of a direct pair-correlation function. In illustration of this approach we consider a simplified model with orientation parameter on a circle and reveal its relation to the complex Ginzburg-Landau theory. © 2008 Elsevier B.V. All rights reserved.
• Fatkullin, I., & Slastikov, V. (2005). Critical points of the Onsager functional on a sphere. Nonlinearity, 18(6), 2565-2580.
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  Abstract: We study Onsager's model of isotropic-nematic phase transitions with orientation parameter on a sphere. We consider two interaction potentials: the antisymmetric (with respect to orientation inversion) dipolar potential and symmetric Maier-Saupe potential. We prove the axial symmetry and derive explicit formulae for all critical points, thus obtaining their complete classification. Finally, we investigate their stability and construct the corresponding bifurcation diagrams. © 2005 IOP Publishing Ltd and London Mathematical Society.
• Slastikov, V., & Fatkullin, I. (2005). A Note on the Onsager Model of Nematic Phase Transitions. Communications in Mathematical Sciences, 3(1), 21-26. doi:10.4310/cms.2005.v3.n1.a2
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  We study Onsager's free energy functional for nematic liquid crystals with an orientation parameter on a unit circle. For a class of interaction potentials we obtain explicit expressions for all critical points, analyze their stability, and construct the corresponding bifurcation diagram. We also derive asymptotic expansions of the equilibrium density of orientations near the critical and zero temperatures.
• Fatkullin, I., & Vanden-Eijnden, E. (2004). A computational strategy for multiscale systems with applications to Lorenz 96 model. Journal of Computational Physics, 200(2), 605-638.
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  Abstract: Numerical schemes for systems with multiple spatio-temporal scales are investigated. The multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales. The multiscale schemes are analyzed in general, then illustrated on a specific example of a moderately large deterministic system displaying chaotic behavior due to Lorenz. Issues like consistency, accuracy, and efficiency are discussed in detail. The role of possible hidden slow variables as well as additional effects arising on the diffusive time-scale are also investigated. As a byproduct we obtain a rather complete characterization of the effective dynamics in Lorenz model. © 2004 Elsevier Inc. All rights reserved.
• Fatkullin, I., & Vanden-Eijnden, E. (2004). A computational strategy for multiscale systems with applications to Lorenz 96 model. Journal of Computational Physics. doi:10.1016/j.jcp.2004.04.013
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  Numerical schemes for systems with multiple spatio-temporal scales are investigated. The multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales. The multiscale schemes are analyzed in general, then illustrated on a specific example of a moderately large deterministic system displaying chaotic behavior due to Lorenz. Issues like consistency, accuracy, and efficiency are discussed in detail. The role of possible hidden slow variables as well as additional effects arising on the diffusive time-scale are also investigated. As a byproduct we obtain a rather complete characterization of the effective dynamics in Lorenz model.
• Fatkullin, I., & Vanden-Eijnden, E. (2003). Statistical Description of Contact-Interacting Brownian Walkers on the Line. Journal of Statistical Physics, 112(1-2), 155-163.
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  Abstract: The distribution of interval lengths between Brownian walkers on the line is investigated. The walkers are independent until collision; at collision, the left walker disappears, and the right walker survives with probability p. This problem arises in the context of diffusion-limited reactions and also in the scaling limit of the voter model. A systematic expansion in correlation between neighbor intervals gives a series of approximations of increasing accuracy for the probability density functions of interval lengths. The first approximation beyond mere statistical independence between successive intervals already gives excellent results, as established by comparison with direct numerical simulations.
• Fatkullin, I., & Hesthaven, J. S. (2001). Adaptive High-Order Finite-Difference Method for Nonlinear Wave Problems. Journal of Scientific Computing, 16(1), 47-67.
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  Abstract: We discuss a scheme for the numerical solution of one-dimensional initial value problems exhibiting strongly localized solutions or finite-time singularities. To accurately and efficiently model such phenomena we present a full space-time adaptive scheme, based on a variable order spatial finite-difference scheme and a 4th order temporal integration with adaptively chosen time step. A wavelet analysis is utilized at regular intervals to adaptively select the order and the grid in accordance with the local behavior of the solution. Through several examples, taken from gasdynamics and nonlinear optics, we illustrate the performance of the scheme, the use of which results in several orders of magnitude reduction in the required degrees of freedom to solve a problem to a particular fidelity.
• Fatkullin, I., Kladko, K., Mitkov, I., & Bishop, A. R. (2001). Anomalous relaxation and self-organization in nonequilibrium processes. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 63(6 II), 067102/1-067102/4.
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  Abstract: A model of dynamic self-organization is introduced. It is conjectured that the model can also be applied to stretched-exponential relaxation in many biological and physical systems. For demonstration purposes, the properties of self-organization in a simple model are studied.

Presentations

• Fatkullin, I. (2021, May). Gibbs measures on partitions: limit shapes and hydrodynamic limits. SIAM Dynamical Systems. online.
• Fatkullin, I. (2021, October). Gibbs measures on partitions and their evolutions. AMS Fall Western Virtual Sectional Meeting. online.
• Fatkullin, I. (2019, April). Limit shapes for Gibbs ensembles of partitions. Materials Research Society Spring Meeting. Phoenix, AZ: MRS.
• Fatkullin, I. (2019, December). Limit shapes for Gibbs ensembles of partitions. New Trends in the variational modeling and simulation of liquid crystals. Erwing Schroedinger Institute, Vienna, Austria: ESI.
• Fatkullin, I. (2019, May). Gibbs Ensembles of Partitions: from limit shapes to hydrodynamic limits. Applied and Computational Mathematics seminar. University of Edinburgh, UK: University of Edinburgh, INI.
• Fatkullin, I. (2019, Spring). Gibbs ensembles of partitions: from limit shapes to thermodynamic limits. Mathematical Design of New Materials. Isaac Newton Institute, Cambridge, UK: INI.
• Fatkullin, I. (2016, June). A model of aggregation and limit shapes of Young diagrams. Frontiers of Applied and Comp. Math.. NJIT.
• Fatkullin, I., & Slastikov, V. (2014, April). An aggregation model and discotic liquid crystals. AMS Western Spring Sectional Meeting. Albuquerque, NM: AMS.
• Fatkullin, I., & Slastikov, V. (2014, April). Diffusive transport in two-dimensional nematics. AMS Western Spring Sectional Meeting. Albuquerque, NM: AMS.