g-fractional diffusion models in bounded domains.

In the recent literature, the g-subdiffusion equation involving Caputo fractional derivatives with respect to another function has been studied in relation to anomalous diffusions with a continuous transition between different subdiffusive regimes. In this paper we study the problem of g-fractional diffusion in a bounded domain with absorbing boundaries. We find the explicit solution for the initial boundary value problem, and we study the first-passage time distribution and the mean first-passage time (MFPT). The main outcome is the proof that with a particular choice of the function g it is possible to obtain a finite MFPT, differently from the anomalous diffusion described by a fractional heat equation involving the classical Caputo derivative.

Run-and-tumble motion in one dimension with space-dependent speed.

We consider a particle performing run-and-tumble dynamics with space-dependent speed. The model has biological relevance as it describes motile bacteria or cells in heterogeneous environments. We give exact expression for the probability density function in the case of free motion in unbounded space. We then analyze the case of a particle moving in a confined interval in the presence of partially absorbing boundaries, reporting the probability density in the Laplace (time) domain and the mean time to absorption. We also discuss the relaxation to the steady state in the case of confinement with reflecting boundaries and drift effects due to direction-dependent tumbling rates, modeling taxis phenomena of cells. The case of diffusive particles with spatially variable diffusivity is obtained as a limiting case.

Self-Sustained Density Oscillations of Swimming Bacteria Confined in Microchambers.

We numerically study the dynamics of run-and-tumble particles confined in two chambers connected by thin channels. Two dominant dynamical behaviors emerge: (i) an oscillatory pumping state, in which particles periodically fill the two vessels, and (ii) a circulating flow state, dynamically maintaining a near constant population level in the containers when connected by two channels. We demonstrate that the oscillatory behavior arises from the combination of a narrow channel, preventing bacteria reorientation, and a density-dependent motility inside the chambers.

Generalized model of blockage in particulate flow limited by channel carrying capacity.

We investigate stochastic models of particles entering a channel with a random time distribution. When the number of particles present in the channel exceeds a critical value N, a blockage occurs and the particle flux is definitively interrupted. By introducing an integral representation of the n-particle survival probabilities, we obtain exact expressions for the survival probability, the distribution of the number of particles that pass before failure, the instantaneous flux of exiting particles, and their time correlation. We generalize previous results for N=2 to an arbitrary distribution of entry times and obtain exact solutions for N=3 for a Poisson distribution and partial results for N≥4.

Run-and-tumble particles in speckle fields.

The random energy landscapes developed by speckle fields can be used to confine and manipulate a large number of micro-particles with a single laser beam. By means of molecular dynamics simulations, we investigate the static and dynamic properties of an active suspension of swimming bacteria embedded into speckle patterns. Looking at the correlation of the density fluctuations and the equilibrium density profiles, we observe a crossover phenomenon when the forces exerted by the speckles are equal to the bacteria's propulsion.

First-passage time of run-and-tumble particles.

We solve the problem of first-passage time for run-and-tumble particles in one dimension. Exact expression is derived for the mean first-passage time in the general case, considering external force fields and chemotactic fields, giving rise to space-dependent swim speed and tumble rate. Agreement between theoretical formulae and numerical simulations is obtained in the analyzed case studies --constant and sinusoidal force fields, constant gradient chemotactic field. Reported findings can be useful to get insights into very different phenomena involving active particles, such as bacterial motion in external fields, intracellular transport, cell migration, animal foraging.

Effective run-and-tumble dynamics of bacteria baths.

E. coli bacteria swim in straight runs interrupted by sudden reorientation events called tumbles. The resulting random walks give rise to density fluctuations that can be derived analytically in the limit of non-interacting particles or equivalently of very low concentrations. However, in situations of practical interest, the concentration of bacteria is always large enough to make interactions an important factor. Using molecular dynamics simulations, we study the dynamic structure factor of a model bacterial bath for increasing values of densities. We show that it is possible to reproduce the dynamics of density fluctuations in the system using a free run-and-tumble model with effective fitting parameters. We discuss the dependence of these parameters, e.g., the tumbling rate, tumbling time and self-propulsion velocity, on the density of the bath.

Probability distributions for the run-and-tumble bacterial dynamics: an analogy to the Lorentz model.

In this paper, we exploit an analogy of the run-and-tumble process for bacterial motility with the Lorentz model of electron conduction in order to obtain analytical results for the intermediate scattering function. This allows to obtain an analytical result for the van Hove function in real space for two-dimensional systems. We furthermore consider the 2D circling motion of bacteria close to solid boundaries with tumbling, and show that the analogy to electron conduction in a magnetic field allows to predict the effective diffusion coefficient of the bacteria. The latter is shown to be reduced by the circling motion of the bacteria.

Transport of self-propelling bacteria in micro-channel flow.

Understanding the collective motion of self-propelling organisms in confined geometries, such as that of narrow channels, is of great theoretical and practical importance. By means of numerical simulations we study the motion of model bacteria in 2D channels under different flow conditions: fluid at rest, steady and unsteady flow. We find aggregation of bacteria near channel walls and, in the presence of external flow, also upstream swimming, which turns out to be a very robust result. Detailed analysis of bacterial velocity and orientation fields allows us to quantify the phenomenon by varying cell density, channel width and fluid velocity. The tumbling mechanism turns out to have strong influence on velocity profiles and particle flow, resulting in a net upstream flow in the case of non-tumbling organisms. Finally we demonstrate that upstream flow can be enhanced by a suitable choice of an unsteady flow pattern.

Effective interactions between colloidal particles suspended in a bath of swimming cells.

The dynamics of passive colloidal tracers in a bath of self-propelled particles is receiving a lot of attention in the context of nonequilibrium statistical mechanics. Here we demonstrate that active baths are also capable of mediating effective interactions between suspended bodies. In particular we observe that a bath of swimming bacteria gives rise to a short range attraction similar to depletion forces in equilibrium colloidal suspensions. Using numerical simulations and experiments we show how the features of this interaction arise from the combination of nonequilibrium dynamics (peculiar of bacterial baths) and excluded volume effects.

Swimming with an image.

The hydrodynamic interactions of a swimming bacterium with a neighboring surface can cause it to swim in circles. For example, when E. coli is above a solid surface it had been observed to swim in a clockwise direction. By contrast we observe that, when swimming near a liquid-air interface, the sense of rotation is reversed. We quantitatively account for this through the hydrodynamic interaction of the bacterium with its own mirror image swimming on the opposite side of a perfect-slip boundary. The strength of the coupling is reduced for longer cells, where the torque is spread over a larger length, resulting in longer bacteria swimming in larger circles. We confirm this through precise video measurements of bacterial trajectories and orientations.

Bacterial ratchet motors.

Self-propelling bacteria are a nanotechnology dream. These unicellular organisms are not just capable of living and reproducing, but they can swim very efficiently, sense the environment, and look for food, all packaged in a body measuring a few microns. Before such perfect machines can be artificially assembled, researchers are beginning to explore new ways to harness bacteria as propelling units for microdevices. Proposed strategies require the careful task of aligning and binding bacterial cells on synthetic surfaces in order to have them work cooperatively. Here we show that asymmetric environments can produce a spontaneous and unidirectional rotation of nanofabricated objects immersed in an active bacterial bath. The propulsion mechanism is provided by the self-assembly of motile Escherichia coli cells along the rotor boundaries. Our results highlight the technological implications of active matter's ability to overcome the restrictions imposed by the second law of thermodynamics on equilibrium passive fluids.

Phase diagram and complexity of mode-locked lasers: from order to disorder.

We investigate mode-locking processes in lasers displaying a variable degree of structural randomness. By a spin-glass theoretic approach, we analyze the mean-field Hamiltonian and derive a phase diagram in terms of pumping rate and degree of disorder. Paramagnetic (noisy continuous wave emission), ferromagnetic (standard passive mode locking), and spin-glass phases with an exponentially large number of configurations are identified. The results are also relevant for other physical systems displaying a random Hamiltonian, such as Bose-condensed gases and nonlinear optics.

Ultrashort pulse propagation and the Anderson localization.

We investigate the dynamics of a 10 fs light pulse propagating in a random medium by the direct solution of the three-dimensional Maxwell equations. Our approach employs molecular dynamics to generate a distribution of spherical scatterers and a parallel finite-difference time-domain code for the vectorial wave propagation. We calculate the disorder-averaged energy velocity and the decay time of the transmitted pulse versus the localization length for an increasing refractive index.

Condensation in disordered lasers: theory, 3D+1 simulations, and experiments.

The complex processes underlying the generation of a coherent emission from the multiple scattering of photons and wave localization in the presence of structural disorder are still mostly unexplored. Here we show that a single nonlinear Schrödinger equation, playing the role of the Schwalow-Townes law for standard lasers, quantitatively reproduces experimental results and three-dimensional time-domain parallel simulations of a colloidal laser system.

Role of saddles in topologically driven phase transitions: the case of the d -dimensional spherical model.

Analyzing the d -dimensional spherical model, we show that underlying saddles, defined through a map in the configuration space, play a central role in driving the phase transition. At the phase transition point the underlying saddle energy reaches its lowest value, corresponding to the trivial boundary topological singularity.

Phase transitions and topology in 2+k XY mean-field models.

The thermodynamics and topology of mean-field models with 2+k body interaction terms (generalizing XY model) are derived. Focusing on two particular cases (2+4 and 2+6 body interaction terms), a comparison between thermodynamic (phase transition energy, thermodynamically forbidden energy regions) and topological (singularity and curvature of saddle entropy) properties is performed. We find that (i) a topological change is present at the phase transition energy; however, (ii) only one topological change occurs, also for those models exhibiting two phase transitions; (iii) the order of a phase transition is not completely signaled by the curvature of topological quantities.

Glassy behavior of light.

We study the nonlinear dynamics of a multimode random laser using the methods of statistical physics of disordered systems. A replica-symmetry breaking phase transition is predicted as a function of the pump intensity. We thus show that light propagating in a random nonlinear medium displays glassy behavior; i.e., the photon gas has a multitude of metastable states and a nonvanishing complexity, corresponding to mode-locking processes in random lasers. The present work reveals the existence of new physical phenomena, and demonstrates how nonlinear optics and random lasers can be a benchmark for the modern theory of complex systems and glasses.

Relationship between phase transitions and topological changes in one-dimensional models.

We address the question of the quantitative relationship between thermodynamic phase transitions and topological changes in the potential energy manifold analyzing two classes of one dimensional models, the Burkhardt solid-on-solid model and the Peyrard-Bishop model for DNA thermal denaturation, both in the confining and nonconfining version. These models, apparently, do not fit [M. Kastner, Phys. Rev. Lett. 93, 150601 (2004)] in the general idea that the phase transition is signaled by a topological discontinuity. We show that in both models the phase transition energy v(c) is actually noncoincident with, and always higher than, the energy v(theta) at which a topological change appears. However, applying a procedure already successfully employed in other cases as the mean field phi4 model, i.e., introducing a map M:v-->v(s) from levels of the energy hypersurface V to the level of the stationary points "visited" at temperature T, we find that M (v(c))=v(theta). This result enhances the relevance of the underlying stationary points in determining the thermodynamics of a system, and extends the validity of the topological approach to the study of phase transition to the elusive one-dimensional systems considered here.

Generalized fluctuation relation and effective temperatures in a driven fluid.

By numerical simulation of a Lennard-Jones-like liquid driven by a velocity gradient gamma we test the fluctuation relation (FR) below the (numerical) glass transition temperature T(g) . We show that, in this region, the FR deserves to be generalized introducing a numerical factor X (T, gamma) <1 that defines an "effective temperature" T(FR) =T/X . On the same system we also measure the effective temperature T(eff) , as defined from the generalized fluctuation-dissipation relation, and find a qualitative agreement between the two different nonequilibrium temperatures.