The Kardar-Parisi-Zhang equation in the weak noise limit: Pattern formation and upper critical dimension

We extend the previously developed weak noise scheme, applied to the noisy Burgers equation in 1D, to the Kardar-Parisi-Zhang equation for a growing interface in arbitrary dimensions. By means of the Cole-Hopf transformation we show that the growth morphology can be interpreted in terms of dynamically evolving textures of localized growth modes with superimposed diffusive modes. In the Cole-Hopf representation the growth modes are static solutions to the diffusion equation and the nonlinear Schroedinger equation, subsequently boosted to finite velocity by a Galilei transformation. We discuss the dynamics of the pattern formation and, briefly, the superimposed linear modes. Implementing the stochastic interpretation we discuss kinetic transitions and in particular the properties in the pair mode or dipole sector. We find the Hurst exponent H=(3-d)/(4-d) for the random walk of growth modes in the dipole sector. Finally, applying Derrick's theorem based on constrained minimization we show that the upper critical dimension is d=4 in the sense that growth modes cease to exist above this dimension.Comment: 27 pages, 19 eps figs, revte

A primer on noise-induced transitions in applied dynamical systems

Noise plays a fundamental role in a wide variety of physical and biological dynamical systems. It can arise from an external forcing or due to random dynamics internal to the system. It is well established that even weak noise can result in large behavioral changes such as transitions between or escapes from quasi-stable states. These transitions can correspond to critical events such as failures or extinctions that make them essential phenomena to understand and quantify, despite the fact that their occurrence is rare. This article will provide an overview of the theory underlying the dynamics of rare events for stochastic models along with some example applications

High-fidelity rendering on shared computational resources

The generation of high-fidelity imagery is a computationally expensive process and parallel computing has been traditionally employed to alleviate this cost. However, traditional parallel rendering has been restricted to expensive shared memory or dedicated distributed processors. In contrast, parallel computing on shared resources such as a computational or a desktop grid, offers a low cost alternative. But, the prevalent rendering systems are currently incapable of seamlessly handling such shared resources as they suffer from high latencies, restricted bandwidth and volatility. A conventional approach of rescheduling failed jobs in a volatile environment inhibits performance by using redundant computations. Instead, clever task subdivision along with image reconstruction techniques provides an unrestrictive fault-tolerance mechanism, which is highly suitable for high-fidelity rendering. This thesis presents novel fault-tolerant parallel rendering algorithms for effectively tapping the enormous inexpensive computational power provided by shared resources. A first of its kind system for fully dynamic high-fidelity interactive rendering on idle resources is presented which is key for providing an immediate feedback to the changes made by a user. The system achieves interactivity by monitoring and adapting computations according to run-time variations in the computational power and employs a spatio-temporal image reconstruction technique for enhancing the visual fidelity. Furthermore, algorithms described for time-constrained offline rendering of still images and animation sequences, make it possible to deliver the results in a user-defined limit. These novel methods enable the employment of variable resources in deadline-driven environments

Role of particle conservation in self-propelled particle systems

Actively propelled particles undergoing dissipative collisions are known to develop a state of spatially distributed coherently moving clusters. For densities larger than a characteristic value, clusters grow in time and form a stationary well-ordered state of coherent macroscopic motion. In this work we address two questions. (i) What is the role of the particles’ aspect ratio in the context of cluster formation, and does the particle shape affect the system’s behavior on hydrodynamic scales? (ii) To what extent does particle conservation influence pattern formation? To answer these questions we suggest a simple kinetic model permitting us to depict some of the interaction properties between freely moving particles and particles integrated in clusters. To this end, we introduce two particle species: single and cluster particles. Specifically, we account for coalescence of clusters from single particles, assembly of single particles on existing clusters, collisions between clusters and cluster disassembly. Coarse graining our kinetic model, (i) we demonstrate that particle shape (i.e. aspect ratio) shifts the scale of the transition density, but does not impact the instabilities at the ordering threshold and (ii) we show that the validity of particle conservation determines the existence of a longitudinal instability, which tends to amplify density heterogeneities locally, and in turn triggers a wave pattern with wave vectors parallel to the axis of macroscopic order. If the system is in contact with a particle reservoir, this instability vanishes due to a compensation of density heterogeneities

Conditioning the logistic branching process on non-extinction

We consider a birth and death process in which death is due to both `natural death' and to competition between individuals, modelled as a quadratic function of population size. The resulting `logistic branching process' has been proposed as a model for numbers of individuals in populations competing for some resource, or for numbers of species. However, because of the quadratic death rate, even if the intrinsic growth rate is positive, the population will, with probability one, die out in finite time. There is considerable interest in understanding the process conditioned on non-extinction. In this paper, we exploit a connection with the ancestral selection graph of population genetics to find expressions for the transition rates in the logistic branching process conditioned on survival until some fixed time $T$, in terms of the distribution of a certain one-dimensional diffusion process at time $T$. We also find the probability generating function of the Yaglom distribution of the process and rather explicit expressions for the transition rates for the so-called Q-process, that is the logistic branching process conditioned to stay alive into the indefinite future. For this process, one can write down the joint generator of the (time-reversed) total population size and what in population genetics would be called the `genealogy' and in phylogenetics would be called the `reconstructed tree' of a sample from the population. We explore some ramifications of these calculations numerically

Out of equilibrium dynamics of classical and quantum complex systems

Equilibrium is a rather ideal situation, the exception rather than the rule in Nature. Whenever the external or internal parameters of a physical system are varied its subsequent relaxation to equilibrium may be either impossible or take very long times. From the point of view of fundamental physics no generic principle such as the ones of thermodynamics allows us to fully understand their behaviour. The alternative is to treat each case separately. It is illusionary to attempt to give, at least at this stage, a complete description of all non-equilibrium situations. Still, one can try to identify and characterise some concrete but still general features of a class of out of equilibrium problems - yet to be identified - and search for a unified description of these. In this report I briefly describe the behaviour and theory of a set of non-equilibrium systems and I try to highlight common features and some general laws that have emerged in recent years.Comment: 36 pages, to be published in Compte Rendus de l'Academie de Sciences, T. Giamarchi e