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Register a new userUndular diffusion in nonlinear sigma models
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We discuss general features of charge transport in non-relativistic classical field theories invariant under non-abelian unitary Lie groups by examining the full structure of two-point dynamical correlation functions in grand-canonical ensembles at finite charge densities (polarized ensembles). Upon explicit breaking of non-abelian symmetry, two distinct transport laws characterized by dynamical exponent $z=2$ arise. While in the unbroken symmetry sector the Cartan fields exhibit normal diffusion, the transversal sectors governed by the nonlinear analogues of Goldstone modes disclose an unconventional law of diffusion characterized by a complex diffusion constant and undulating patterns in the spatiotemporal correlation profiles. In the limit of strong polarization, one retrieves the imaginary-time diffusion for uncoupled linear Goldstone modes, whereas for weak polarizations the imaginary component of the diffusion constant becomes small. In models of higher rank symmetry, we prove absence of dynamical correlations among distinct transversal sectors.
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It has been argued that there is biological and modeling evidence that a non-linear diffusion coefficient of the type D(b) = D_0 b^{k} underlies the formation of a number of growth patterns of bacterial colonies. We study a reaction-diffusion system with a non-linear diffusion coefficient introduced by Ben-Jacob et al. Due to the fact that the bacterial diffusion coefficient vanishes when the bacterial density b -> 0, the standard linear stability analysis for fronts cannot be used. We introduce an extension of the stability analysis which can be applied to such singular fronts, map out the region of stability in the D-k-plane and derive an interfacial approximation in some limits. Our linear stability analysis and sharp interface formulation will also be applicable to other examples of interface formation due to nonlinear diffusion, like in porous media or in the problem of vortex motion in superconductors.
Heat transport in one-dimensional (1D) momentum-conserving lattices is generally assumed to be anomalous, thus yielding a power-law divergence of thermal conductivity with system length. However, whether heat transport in two-dimensional (2D) system is anomalous or not is still on debate because of the difficulties involved in experimental measurements or due to the insufficiently large simulation size. Here, we simulate energy and momentum diffusion in the 2D nonlinear lattices using the method of fluctuation correlation functions. Our simulations confirm that energy diffusion in the 2D momentum-conserving lattices is anomalous and can be well described by the L{e}vy-stable distribution. We also find that the disappear of side peaks of heat mode may suggest a weak coupling between heat mode and sound mode in the 2D nonlinear system. It is also observed that the harmonic interactions in the 2D nonlinear lattices can accelerate the energy diffusion. Contrary to the hypothesis of 1D system, we clarify that anomalous heat transport in the 2D momentum-conserving system cannot be corroborated by the momentum superdiffusion any more. Moreover, as is expected, lattices with a nonlinear on-site potential exhibit normal energy diffusion, independent of the dimension. Our findings offer some valuable insights into the mechanism of thermal transport in 2D system.
We generalize the reaction-diffusion model A + B -> 0 in order to study the impact of an excess of A (or B) at the reaction front. We provide an exact solution of the model, which shows that linear response breaks down: the average displacement of the reaction front grows as the square-root of the imbalance. We argue that this model provides a highly simplified but generic framework to understand the square-root impact of large orders in financial markets.
In the usual statistical model of a dense polymer (a single space-filling loop on a lattice) in two dimensions the loop does not cross itself. We modify this by including intersections in which {em three} lines can cross at the same point, with some statistical weight w per crossing. We show that our model describes a line of critical theories with continuously-varying exponents depending on w, described by a conformally-invariant non-linear sigma model with varying coupling constant g_sigma^2 >0. For the boundary critical behavior, or the model defined in a strip, we propose an exact formula for the ell-leg exponents, h_ell=g_sigma^2 ell(ell-2)/8, which is shown numerically to hold very well.
Dynamical reaction-diffusion processes and meta-population models are standard modeling approaches for a wide variety of phenomena in which local quantities - such as density, potential and particles - diffuse and interact according to the physical laws. Here, we study the behavior of two basic reaction-diffusion processes ($B to A$ and $A+B to 2B$) defined on networks with heterogeneous topology and no limit on the nodes occupation number. We investigate the effect of network topology on the basic properties of the systems phase diagram and find that the network heterogeneity sustains the reaction activity even in the limit of a vanishing density of particles, eventually suppressing the critical point in density driven phase transitions, whereas phase transition and critical points, independent of the particle density, are not altered by topological fluctuations. This work lays out a theoretical and computational microscopic framework for the study of a wide range of realistic meta-populations models and agent-based models that include the complex features of real world networks.
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