Space-time velocity correlation function for random walks

Space-time correlation functions constitute a useful instrument from the research toolkit of continuous-media and many-body physics. We adopt here this concept for single-particle random walks and demonstrate that the corresponding space-time velocity auto-correlation functions reveal correlations which extend in time much longer than estimated with the commonly employed temporal correlation functions. A generic feature of considered random-walk processes is an effect of velocity echo identified by the existence of time-dependent regions where most of the walkers are moving in the direction opposite to their initial motion. We discuss the relevance of the space-time velocity correlation functions for the experimental studies of cold atom dynamics in an optical potential and charge transport on micro- and nano-scales.Comment: Phys. Rev. Lett., in pres

Chaoticity of the Wet Granular Gas

In this work we derive an analytic expression for the Kolmogorov-Sinai entropy of dilute wet granular matter, valid for any spatial dimension. The grains are modelled as hard spheres and the influence of the wetting liquid is described according to the Capillary Model, in which dissipation is due to the hysteretic cohesion force of capillary bridges. The Kolmogorov-Sinai entropy is expanded in a series with respect to density. We find a rapid increase of the leading term when liquid is added. This demonstrates the sensitivity of the granular dynamics to humidity, and shows that the liquid significantly increases the chaoticity of the granular gas.Comment: 13 pages, 10 figures, Physical Review

L\'evy walks

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The L\'{e}vy walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, bio-physics, and behavioral science demonstrate that this particular type of random walks provides significant insight into complex transport phenomena. This review provides a self-consistent introduction to L\'{e}vy walks, surveys their existing applications, including latest advances, and outlines further perspectives.Comment: 50 page

Levy walks with velocity fluctuations

The standard Levy walk is performed by a particle that moves ballistically between randomly occurring collisions, when the intercollision time is a random variable governed by a power-law distribution. During instantaneous collision events the particle randomly changes the direction of motion but maintains the same constant speed. We generalize the standard model to incorporate velocity fluctuations into the process. Two types of models are considered, namely, (i) with a walker changing the direction and absolute value of its velocity during collisions only, and (ii) with a walker whose velocity continuously fluctuates. We present full analytic evaluation of both models and emphasize the importance of initial conditions. We show that the type of the underlying Levy walk process can be identified by looking at the ballistic regions of the diffusion profiles. Our analytical results are corroborated by numerical simulations

A Riemann hypothesis analogue for invariant rings

AbstractA Riemann hypothesis analogue for coding theory was introduced by I.M. Duursma [A Riemann hypothesis analogue for self-dual codes, in: A. Barg, S. Litsyn (Eds.), Codes and Association Schemes (Piscataway, NJ, 1999), American Mathematical Society, Providence, RI, 2001, pp. 115–124]. In this paper, we extend zeta polynomials for linear codes to ones for invariant rings, and we investigate whether a Riemann hypothesis analogue holds for some concrete invariant rings. Also we shall show that there is some subring of an invariant ring such that the subring is not an invariant ring but extremal polynomials all satisfy the Riemann hypothesis analogue

A Pili-Driven Bacterial Turbine

Work generated by self-propelled bacteria can be harnessed with the help of microdevices. Such nanofabricated microdevices, immersed in a bacterial bath, may exhibit unidirectional rotational or translational motion. Swimming bacteria that propel with the help of actively rotating flagella are a prototypical example of active agents that can power such microdevices. In this work, we propose a computational model of a micron-sized turbine powered by bacteria that rely on active type IV pili appendages for surface-associated motility. We find that the turbine can rotate persistently over a time scale that significantly exceeds the characteristic times of the single cell motility. The persistent rotation is explained by the collective dynamics of multiple pili of groups of cells attaching to and pulling on turbine. Furthermore, we show that the turbine can rotate permanently in the same direction by altering the pili binding to the turbine surface in an asymmetric fashion. We thus can show that by changing the adhesive properties of the turbine while keeping its symmetric geometry, we can still break the symmetry of its rotation. Altogether, this study widely expands the range of bacteria that can be used to power nanofabricated microdevices, and, due to high pili forces generated by pili retraction, promises to push the harnessed work by several orders of magnitude

The hierarchical packing of euchromatin domains can be described as multiplicative cascades

The genome is packed into the cell nucleus in the form of chromatin. Biochemical approaches have revealed that chromatin is packed within domains, which group into larger domains, and so forth. Such hierarchical packing is equally visible in super-resolution microscopy images of large-scale chromatin organization. While previous work has suggested that chromatin is partitioned into distinct domains via microphase separation, it is unclear how these domains organize into this hierarchical packing. A particular challenge is to find an image analysis approach that fully incorporates such hierarchical packing, so that hypothetical governing mechanisms of euchromatin packing can be compared against the results of such an analysis. Here, we obtain 3D STED super-resolution images from pluripotent zebrafish embryos labeled with improved DNA fluorescence stains, and demonstrate how the hierarchical packing of euchromatin in these images can be described as multiplicative cascades. Multiplicative cascades are an established theoretical concept to describe the placement of ever-smaller structures within bigger structures. Importantly, these cascades can generate artificial image data by applying a single rule again and again, and can be fully specified using only four parameters. Here, we show how the typical patterns of euchromatin organization are reflected in the values of these four parameters. Specifically, we can pinpoint the values required to mimic a microphase-separated state of euchromatin. We suggest that the concept of multiplicative cascades can also be applied to images of other types of chromatin. Here, cascade parameters could serve as test quantities to assess whether microphase separation or other theoretical models accurately reproduce the hierarchical packing of chromatin. Author summary DNA is stored inside the cell nucleus in the form of chromatin. Chromatin exhibits a striking three-dimensional organization, where small domains group into larger domains, which again group into larger domains, and so forth. While this hierarchical domain packing is obvious from microscopy images, it is still not entirely clear how it is established, or how it should be properly characterized. Here, we demonstrate that multiplicative cascades-a concept from theoretical physics used to characterize for example cloud patterns, galaxy locations, or soil patterns-are also ideally suited to describe the domain-within-domain organization of chromatin. This description is rather simple, using only four numbers, and can thus facilitate testing of competing theories for the packing of chromatin domains

Mean zero artificial diffusion for stable finite element approximation of convection in cellular aggregate formation

We develop and implement finite element approximations for the coupled problem of cellular aggregate formation. The equation governing evolution of cell density is convective in nature, necessitating a modification of standard approaches to circumvent the instabilities associated with standard finite element approximations. To this end, a novel mean zero artificial diffusion approach is proposed, in which the classical artificial diffusion term is replaced by one that is orthogonal to its projection on to continuous functions. The resulting approach for the convective equation is shown to be well-posed. A range of numerical results illustrate the stability and accuracy of the new approach, and its behaviour in comparison with an alternative approach using Taylor–Hood elements. The results also provide insights into the behaviour of cellular aggregates in the context of the model studied here