Growing interfaces: A brief review on the tilt method

The tilt method applied to models of growing interfaces is a useful tool to characterize the nonlinearities of their associated equation. Growing interfaces with average slope $m$, in models and equations belonging to Kardar-Parisi-Zhang (KPZ) universality class, have average saturation velocity $\mathcal{V}_\mathrm{sat}=\Upsilon+\frac{1}{2}\Lambda\,m^2$ when $|m|\ll 1$. This property is sufficient to ensure that there is a nonlinearity type square height-gradient. Usually, the constant $\Lambda$ is considered equal to the nonlinear coefficient $\lambda$ of the KPZ equation. In this paper, we show that the mean square height-gradient $\langle |\nabla h|^2\rangle=a+b \,m^2$, where $b=1$ for the continuous KPZ equation and $b\neq 1$ otherwise, e.g. ballistic deposition (BD) and restricted-solid-on-solid (RSOS) models. In order to find the nonlinear coefficient $\lambda$ associated to each system, we establish the relationship $\Lambda=b\,\lambda$ and we test it through the discrete integration of the KPZ equation. We conclude that height-gradient fluctuations as function of $m^2$ are constant for continuous KPZ equation and increasing or decreasing in other systems, such as BD or RSOS models, respectively.Comment: 11 pages, 4 figure

Revisiting random deposition with surface relaxation: approaches from growth rules to Edwards-Wilkinson equation

We present several approaches for deriving the coarse-grained continuous Langevin equation (or Edwards-Wilkinson equation) from a random deposition with surface relaxation (RDSR) model. First we introduce a novel procedure to divide the first transition moment into the three fundamental processes involved: deposition, diffusion and volume conservation. We show how the diffusion process is related to antisymmetric contribution and the volume conservation process is related to symmetric contribution, which renormalizes to zero in the coarse-grained limit. In another approach, we find the coefficients of the continuous Langevin equation, by regularizing the discrete Langevin equation. Finally, in a third approach, we derive these coefficients from the set of test functions supported by the stationary probability density function (SPDF) of the discrete model. The applicability of the used approaches to other discrete random deposition models with instantaneous relaxation to a neighboring site is discussed.Comment: 12 pages, 4 figure

Comprehensive study of phase transitions in relaxational systems with field-dependent coefficients

We present a comprehensive study of phase transitions in single-field systems that relax to a non-equilibrium global steady state. The mechanism we focus on is not the so-called Stratonovich drift combined with collective effects, but is instead similar to the one associated with noise-induced transitions a la Horsthemke-Lefever in zero-dimensional systems. As a consequence, the noise interpretation (e.g., Ito vs Stratonvich) merely shifts the phase boundaries. With the help of a mean-field approximation, we present a broad qualitative picture of the various phase diagrams that can be found in these systems. To complement the theoretical analysis we present numerical simulations that confirm the findings of the mean-field theory

Stochastic model for the CheY-P molarity in the neighbourhood of E. coli flagella motors

E.coli serves as prototype for the study of peritrichous enteric bacteria that perform runs and tumbles alternately. Bacteria run forward as a result of the counterclockwise (CCW) rotation of their flagella bundle and perform tumbles when at least one of their flagella rotates clockwise (CW), moving away from the bundle. The flagella are hooked to molecular rotary motors of nanometric diameter able to make transitions between CCW and CW rotations that last up to one hundredth of a second. At the same time, flagella move or rotate the bacteria's body microscopically during lapses that range between a tenth and ten seconds. We assume that the transitions between CCW and CW rotations occur solely by fluctuations of CheY-P molarity in the presence of two threshold values, and that a veto rule selects the run or tumble motions. We present Langevin eqs for the CheY-P molarity in the vicinity of each molecular motor. This model allows to obtain the run- or tumble-time distribution as a linear combination of decreasing exponentials that is a function of the steady molarity of CheY-P in the neighbourhood of the molecular motor, which fits experimental data. In turn, if the internal signaling system is unstimulated, we show that the runtime distributions reach power-law behaviour, a characteristic of self-organized systems, in some time range and, afterwards, exponential cutoff. In addition, our model explains without any fitting parameters the ultrasensitivity of the flagella motors as a function of the steady state of CheY-P molarity. In addition, we show that the tumble bias for peritrichous bacterium has a similar sigmoid-shape to the CW bias, although shifted to lower concentrations when the flagella number increases. Thus, the increment in the flagella number allows lower operational values for each motor increasing amplification and robustness of the chemotatic pathway.Comment: 13 pages, 7 figure