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Browsing by Author "Täuber, Uwe C."

Now showing 1 - 20 of 137
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    Aging phenomena in the two-dimensional complex Ginzburg-Landau equation
    Liu, Weigang; Täuber, Uwe C. (2019-11)
    The complex Ginzburg-Landau equation with additive noise is a stochastic partial differential equation that describes a remarkably wide range of physical systems which include coupled non-linear oscillators subject to external noise near a Hopf bifurcation instability and spontaneous structure formation in non-equilibrium systems, e.g., in cyclically competing populations or oscillatory chemical reactions. We employ a finite-difference method to numerically solve the noisy complex Ginzburg-Landau equation on a two-dimensional domain with the goal to investigate its non-equilibrium dynamics when the system is quenched into the "defocusing spiral quadrant". We observe slow coarsening dynamics as oppositely charged topological defects annihilate each other, and characterize the ensuing aging scaling behavior. We conclude that the physical aging features in this system are governed by non-universal aging scaling exponents. We also investigate systems with control parameters residing in the "focusing quadrant", and identify slow aging kinetics in that regime as well. We provide heuristic criteria for the existence of slow coarsening dynamics and physical aging behavior in the complex Ginzburg-Landau equation.
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    Applications of field-theoretic renormalization group methods to reaction-diffusion problems
    Täuber, Uwe C.; Howard, M.; Vollmayr-Lee, B. P. (IOP, 2005-04-29)
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    Boundary Effects on Population Dynamics in Stochastic Lattice Lotka-Volterra Models
    Heiba, B.; Chen, S.; Täuber, Uwe C. (2017-08)
    We investigate spatially inhomogeneous versions of the stochastic Lotka-Volterra model for predator-prey competition and coexistence by means of Monte Carlo simulations on a two-dimensional lattice with periodic boundary conditions. To study boundary effects for this paradigmatic population dynamics system, we employ a simulation domain split into two patches: Upon setting the predation rates at two distinct values, one half of the system resides in an absorbing state where only the prey survives, while the other half attains a stable coexistence state wherein both species remain active. At the domain boundary, we observe a marked enhancement of the predator population density. The predator correlation length displays a minimum at the boundary, before reaching its asymptotic constant value deep in the active region. The frequency of the population oscillations appears only very weakly affected by the existence of two distinct domains, in contrast to their attenuation rate, which assumes its largest value there. We also observe that boundary effects become less prominent as the system is successively divided into subdomains in a checkerboard pattern, with two different reaction rates assigned to neighboring patches. When the domain size becomes reduced to the scale of the correlation length, the mean population densities attain values that are very similar to those in a disordered system with randomly assigned reaction rates drawn from a bimodal distribution.
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    Coexistence in the two-dimensional May-Leonard model with random rates
    He, Q.; Mobilia, M.; Täuber, Uwe C. (Springer, 2011-07-01)
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    Computing macroscopic reaction rates in reaction-diffusion systems using Monte Carlo simulations
    Swailem, Mohamed; Täuber, Uwe C. (2024-07-17)
    Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale, longtime kinetics in such systems are effective, scale-dependent renormalized parameters that need to be either measured experimentally or computed by means of a microscopic model. In a Monte Carlo simulation of stochastic reaction-diffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. To match the results of any computer simulation to observations or experiments carried out on the macroscale, a mapping is required between the microscopic probabilities that define the Monte Carlo algorithm and the macroscopic reaction rates that are experimentally measured. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities (subject to specific rules of interaction) is a very difficult problem, and there is currently no systematic, accurate analytical way to achieve this goal. Therefore, we introduce a straightforward numerical method of using lattice Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka–Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka–Volterra model as well as its May–Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a more thorough and deeper understanding of coarse-graining in spatially extended stochastic reaction-diffusion systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters, with a focus on ecological systems. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation results to experimental or observational data.
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    Continuous elastic phase transitions and disordered crystals
    Schwabl, Franz; Täuber, Uwe C. (1996-12-15)
    We review the theory of second–order (ferro–)elastic phase transitions, where the order parameter consists of a certain linear combination of strain tensor components, and the accompanying soft mode is an acoustic phonon. In three–dimensional crystals, the softening can occur in one– or two–dimensional soft sectors. The ensuing anisotropy reduces the effect of fluctuations, rendering the critical behaviour of these systems classical for a one–dimensional soft sector, and classical with logarithmic corrections in case of a two–dimensional soft sector. The dynamical critical exponent is z = 2, and as a consequence the sound velocity vanishes as cs ∝ |T − Tc|1/2, while the phonon damping coefficient is essentially temperature–independent. Even if the elastic phase transition is driven by the softening of an optical mode linearly coupled to a transverse acoustic phonon, the critical exponents retain their mean–field values. Disorder may lead to a variety of precursor effects and modified critical behaviour. Defects that locally soften the crystal may induce the phenomenon of local order parameter condensation. When the correlation length of the pure system exceeds the average defect separation nD−1/3, a disorder–induced phase transition to a state with non–zero average order parameter can occur at a temperature Tc(nD) well above the transition temperature T0c of the pure crystal. Near T0c, the order–parameter curve, susceptibility, and specific heat appear rounded. For T < Tc(nD) the spatial inhomogeneity induces a static central peak with finite q width in the scattering cross section, accompanied by a dynamical component that is confined to the very vicinity of the disorder–induced phase transition.
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    Coulomb Gap and Correlated Vortex Pinning in Superconductors
    Täuber, Uwe C.; Dai, H.; Nelson, D.; Lieber, C. (1995-06-19)
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    Coupled two-species model for the pair contact process with diffusion
    Deng, S.; Li, W.; Täuber, Uwe C. (American Physical Society, 2020-10-22)
    The contact process with diffusion (PCPD) defined by the binary reactions B+B→B+B+B, B+B→∅ and diffusive particle spreading exhibits an unusual active to absorbing phase transition whose universality class has long been disputed. Multiple studies have indicated that an explicit account of particle pair degrees of freedom may be required to properly capture this system's effective long-time, large-scale behavior. We introduce a two-species representation for the PCPD in which single particles B and particle pairs A are dynamically coupled according to the stochastic reaction processes B+B→A, A→A+B, A→∅, and A→B+B, with each particle type diffusing independently. Mean-field analysis reveals that the phase transition of this model is driven by competition and balance between the two species. We employ Monte Carlo simulations in one, two, and three dimensions to demonstrate that this model consistently captures the pertinent features of the PCPD. In the inactive phase, A particles rapidly go extinct, effectively leaving the B species to undergo pure diffusion-limited pair annihilation kinetics B+B→∅. At criticality, both A and B densities decay with the same exponents (within numerical errors) as the corresponding order parameters of the original PCPD, and display mean-field scaling above the upper critical dimension dc=2. In one dimension, the critical exponents for the B species obtained from seed simulations also agree well with previously reported exponent value ranges. We demonstrate that the scaling properties of consecutive particle pairs in the PCPD are identical with that of the A species in the coupled model. This two-species picture resolves the conceptual difficulty for seed simulations in the original PCPD and naturally introduces multiple length scales and timescales to the system, which are also the origin of strong corrections to scaling. The extracted moment ratios from our simulations indicate that our model displays the same temporal crossover behavior as the PCPD, which further corroborates its full dynamical equivalence with our coupled model.
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    Critical behaviour of driven bilayer systems: a field-theoretic renormalization group study
    Täuber, Uwe C.; Schmittmann, Beate; Zia, Royce K. P. (IOP, 2001-10-26)
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    Critical dynamics at incommensurate phase transitions and NMR relaxation experiments
    Kaufmann, B. A.; Schwabl, Franz; Täuber, Uwe C. (American Physical Society, 1999-05)
    We study the critical dynamics of crystals which undergo a second-order phase transition from a high-temperature normal phase to a structurally incommensurate (IC) modulated phase. We give a comprehensive description of the critical dynamics of such systems, e.g., valid for crystals of the A(2)BX(4) family. Using an extended renormalization scheme, we present a framework in which we analyze the phases above and below the critical temperature T-I. Above T-I, the crossover from the critical behavior to the mean-field regime is studied. Specifically, the resulting width of the critical region is investigated. In the IC modulated phase, we consider explicitly the coupling of the order parameter modes to one-loop order. Here the Goldstone anomalies and their effect on measurable quantities are investigated. We show their relation with the postulated phason gap. While the theory can be applied to a variety of experiments, we concentrate on quadrupole-perturbed nuclear magnetic resonance (NMR) experiments. We find excellent agreement with these dynamical measurements and provide answers for some questions that arose from recent results. [S0163-1829(99)03417-7].
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    Critical dynamics at incommensurate phase transitions and NMR relaxation experiments
    Kaufmann, B. A.; Schwabl, Franz; Täuber, Uwe C. (American Physical Society, 1999-05-01)
    We study the critical dynamics of crystals which undergo a second-order phase transition from a high-temperature normal phase to a structurally incommensurate (IC) modulated phase. We give a comprehensive description of the critical dynamics of such systems, e.g. valid for crystals of the A2BX4 family. Using an extended renormalization scheme, we present a framework in which we analyze the phases above and below the critical temperature TI . Above TI , the crossover from the critical behavior to the mean-field regime is studied. Specifically, the resulting width of the critical region is investigated. In the IC modulated phase, we consider explicitly the coupling of the order parameter modes to one-loop order. Here the Goldstone anomalies and their effect on measurable quantities are investigated. We show their relation with the postulated phason gap. While the theory can be applied to a variety of experiments, we concentrate on quadrupole-perturbed nuclear magnetic resonance (NMR) experiments. We find excellent agreement with these dynamical measurements and provide answers for some questions that arose from recent results.
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    Critical dynamics of anisotropic antiferromagnets in an external field
    Nandi, Riya; Täuber, Uwe C. (American Physical Society, 2020-03-03)
    We numerically investigate the non-equilibrium critical dynamics in three-dimensional anisotropic antiferromagnets in the presence of an external magnetic field. The phase diagram of this system exhibits two critical lines that meet at a bicritical point. The non-conserved components of the staggered magnetization order parameter couple dynamically to the conserved component of the magnetization density along the direction of the external field. Employing a hybrid computational algorithm that combines reversible spin precession with relaxational Monte Carlo updates, we study the aging scaling dynamics for the model C critical line, identifying the critical initial slip, autocorrelation, and aging exponents for both the order parameter and conserved field, thus also verifying the dynamic critical exponent. We further probe the model F critical line by investigating the system size dependence of the characteristic spin wave frequencies near criticality, and measure the dynamic critical exponents for the order parameter including its aging scaling at the bicritical point.
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    Critical dynamics of the antiferromagnetic O(3) nonlinear sigma model with conserved magnetization
    Yao, Louie Hong; Täuber, Uwe C. (American Physical Society, 2022-06-01)
    We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equations of motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d=2+ɛ expansion about its lower critical dimension dlc=2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension dc=4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ϵ=4-d expansion near dc. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent zΓ=d-2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values zc=d/2 in an intermediate crossover region.
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    Critical initial-slip scaling for the noisy complex Ginzburg-Landau equation
    Liu, W.; Täuber, Uwe C. (IOP, 2016-10-28)
    We employ the perturbative field-theoretic renormalization group method to investigate the universal critical behavior near the continuous non-equilibrium phase transition in the complex Ginzburg-Landau equation with additive white noise. This stochastic partial differential describes a remarkably wide range of physical systems: coupled non-linear oscillators subject to external noise near a Hopf bifurcation instability; spontaneous structure formation in non-equilibrium systems, e.g., in cyclically competing populations; and driven-dissipative Bose--Einstein condensation, realized in open systems on the interface of quantum optics and many-body physics, such as cold atomic gases and exciton-polaritons in pumped semiconductor quantum wells in optical cavities. Our starting point is a noisy, dissipative Gross-Pitaevski or non-linear Schr\"odinger equation, or equivalently purely relaxational kinetics originating from a complex-valued Landau-Ginzburg functional, which generalizes the standard equilibrium model A critical dynamics of a non-conserved complex order parameter field. We study the universal critical behavior of this system in the early stages of its relaxation from a Gaussian-weighted fully randomized initial state. In this critical aging regime, time translation invariance is broken, and the dynamics is characterized by the stationary static and dynamic critical exponents, as well as an independent `initial-slip' exponent. We show that to first order in the dimensional expansion about the upper critical dimension, this initial-slip exponent in the complex Ginzburg-Landau equation is identical to its equilibrium model A counterpart. We furthermore employ the renormalization group flow equations as well as construct a suitable complex spherical model extension to argue that this conclusion likely remains true to all orders in the perturbation expansion.
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    Critical Scaling and Aging near the Flux Line Depinning Transition
    Chaturvedi, Harshwardhan; Dobramysl, Ulrich; Pleimling, Michel J.; Täuber, Uwe C. (2019-12-03)
    We utilize Langevin molecular dynamics simulations to study dynamical critical behavior of magnetic flux lines near the depinning transition in type-II superconductors subject to randomly distributed attractive point defects. We employ a coarse-grained elastic line Hamiltonian for the mutually repulsive vortices and purely relaxational kinetics. In order to infer the stationary-state critical exponents for the continuous non-equilibrium depinning transition at zero temperature T = 0 and at the critical driving current density j_c, we explore two-parameter scaling laws for the flux lines' gyration radius and mean velocity as functions of the two relevant scaling fields T and j - j_c. We also investigate critical aging scaling for the two-time height auto-correlation function in the early-time non-equilibrium relaxation regime to independently measure critical exponents. We provide numerical exponent values for the distinct universality classes of non-interacting and repulsive vortices.
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    Crossover from Isotropic to Directed Percolation
    Frey, E.; Täuber, Uwe C.; Schwabl, Franz (1994-06)
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    Crossover From Self-Similar to Self-Affine Structures in Percolation
    Frey, E.; Täuber, Uwe C.; Schwabl, Franz (Editions Physique, 1994-05-20)
    We study the crossover from self-similar scaling behavior to asymptotically self-affine (anisotropic) structures. As an example, we consider bond percolation with one preferred direction. Our theory is based on a field-theoretical representation, and takes advantage of a renormalization group approach designed for crossover phenomena. We calculate effective exponents for the connectivity describing the entire crossover region from isotropic to directed percolation, and predict at which scale of the anisotropy the crossover should occur. We emphasize the broad range of applicability of our method.