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Material design at submicron scales would be profoundly affected if the formation of precipitation patterns could be easily controlled. It would allow the direct building of bulk structures, in contrast to traditional techniques which consist of removing material in order to create patterns. Here, we discuss an extension of our recent proposal of using electrical currents to control precipitation bands which emerge in the wake of reaction fronts in A^{+} + B^{-} -> C reaction-diffusion processes. Our main result, based on simulating the reaction-diffusion-precipitation equations, is that the dynamics of the charged agents can be guided by an appropriately designed time-dependent electric current so that, in addition to the control of the band spacing, the width of the precipitation bands can also be tuned. This makes straightforward the encoding of information into precipitation patterns and, as an amusing example, we demonstrate the feasibility by showing how to encode a musical rhythm., Comment: 13 pages, 7 figures, presented at SigmaPhi 2008

We consider a Jepsen gas of $N$ hard-point particles undergoing free expansion on a line, starting from random initial positions of the particles having random initial velocities. The particles undergo binary elastic collisions upon contact and move freely in-between collisions. After a certain ordering time $T_{o}$, the system reaches a ``fan'' state where all the velocities are completely ordered from left to right in an increasing fashion and there is no further collision. We compute analytically the distributions of (i) the total number of collisions and (ii) the ordering time $T_{o}$. We show that several features of these distributions are universal., Comment: 18 pages, 5 figures

Precipitation patterns generated by A^{+}+B^{-} --> C type reaction-diffusion processes are studied. It is shown both theoretically and experimentally that the patterns can be controlled by an appropriately designed, time-dependent electric current in the system. We describe examples of current dynamics yielding periodic bands of prescribed wavelength, as well as more complicated structures. The pattern control is demonstrated experimentally on the reaction-diffusion-precipitation process 2AgNO_3+K_2Cr_2O_7 --> Ag_2Cr_2O_7+2KNO_3 taking place in a gel., Comment: 4 pages, 6 figures

We consider a simple-model population, whose individuals react with a certain delay to temporal variations of their habitat. We investigate the impact of such a delayed-answer on the survival chances of the population, both in a periodically changing environment, and in the case of an abrupt change of it. It is found that for population with low degree of mutation-induced variability, being "slow-reacting" decreases the extinction risk face to environmental changes. On the contrary, for populations with high mutation amplitude, the delayed reaction reduces the survival chances., Comment: 7 pages, 4 figures

Liesegang patterns emerge from precipitation processes and may be used to build bulk structures at submicron lengthscales. Thus they have significant potential for technological applications provided adequate methods of control can be devised. Here we describe a simple, physically realizable pattern-control based on the notion of driven precipitation, meaning that the phase-separation is governed by a guiding field such as, for example, a temperature or a pH field. The phase-separation is modeled through a non-autonomous Cahn-Hilliard equation whose spinodal is determined by the evolving guiding field. Control over the dynamics of the spinodal gives control over the velocity of the instability front which separates the stable and unstable regions of the system. Since the wavelength of the pattern is largely determined by this velocity, the distance between successive precipitation bands becomes controllable. We demonstrate the above ideas by numerical studies of a 1D system with diffusive guiding field. We find that the results can be accurately described by employing a linear stability analysis (pulled-front theory) for determining the velocity -- local-wavelength relationship. From the perspective of the Liesegang theory, our results indicate that the so-called revert patterns may be naturally generated by diffusive guiding fields., Comment: Minor changes, to be published in Phys. Rev. E. 10 pages, 8 figures

The role of the selection pressure and mutation amplitude on the behavior of a single-species population evolving on a two-dimensional lattice, in a periodically changing environment, is studied both analytically and numerically. The mean-field level of description allows to highlight the delicate interplay between the different time-scale processes in the resulting complex dynamics of the system. We clarify the influence of the amplitude and period of the environmental changes on the critical value of the selection pressure corresponding to a phase-transition "extinct-alive" of the population. However, the intrinsic stochasticity and the dynamically-built in correlations among the individuals, as well as the role of the mutation-induced variety in population's evolution are not appropriately accounted for. A more refined level of description, which is an individual-based one, has to be considered. The inherent fluctuations do not destroy the phase transition "extinct-alive", and the mutation amplitude is strongly influencing the value of the critical selection pressure. The phase diagram in the plane of the population's parameters -- selection and mutation is discussed as a function of the environmental variation characteristics. The differences between a smooth variation of the environment and an abrupt, catastrophic change are also addressesd., Comment: 15 pages, 12 figures. Accepted for publication in Phys. Rev. E

We study the extremal dynamics emerging in an out-of-equilibrium one-dimensional Jepsen gas of $(N+1)$ hard-point particles. The particles undergo binary elastic collisions, but move ballistically in-between collisions. The gas is initally uniformly distributed in a box $[-L,0]$ with the "leader" (or the rightmost particle) at X=0, and a random positive velocity, independently drawn from a distribution $\phi(V)$, is assigned to each particle. The gas expands freely at subsequent times. We compute analytically the distribution of the leader's velocity at time $t$, and also the mean and the variance of the number of collisions that are undergone by the leader up to time $t$. We show that in the thermodynamic limit and at fixed time $t\gg 1$ (the so-called "growing regime"), when interactions are strongly manifest, the velocity distribution exhibits universal scaling behavior of only three possible varieties, depending on the tail of $\phi(V)$. The associated scaling functions are novel and different from the usual extreme-value distributions of uncorrelated random variables. In this growing regime the mean and the variance of the number of collisions of the leader up to time $t$ increase logarithmically with $t$, with universal prefactors that are computed exactly. The implications of our results in the context of biological evolution modeling are pointed out., Comment: 13 pages, 3 figures. To appear in Phys. Rev. E

Properties of reaction zones resulting from A+B -> C type reaction-diffusion processes are investigated by analytical and numerical methods. The reagents A and B are separated initially and, in addition, there is an initial macroscopic inhomogeneity in the distribution of the B species. For simple two-dimensional geometries, exact analytical results are presented for the time-evolution of the geometric shape of the front. We also show using cellular automata simulations that the fluctuations can be neglected both in the shape and in the width of the front., Comment: 11 pages, 3 figures, submitted to J. Phys. C

Nonequilibrium systems driven by additive or multiplicative dichotomous Markov noise appear in a wide variety of physical and mathematical models. We review here some prototypical examples, with an emphasis on {\em analytically-solvable} situations. In particular, it has escaped attention till recently that the standard results for the long-time properties of such systems cannot be applied when unstable fixed points are crossed in the asymptotic regime. We show how calculations have to be modified to deal with these cases and present a few relevant applications -- the hypersensitive transport, the rocking ratchet, and the stochastic Stokes' drift. These results reinforce the impression that dichotomous noise can be put on a par with Gaussian white noise as far as obtaining analytical results is concerned. They convincingly illustrate the interplay between noise and nonlinearity in generating nontrivial behaviors of nonequilibrium systems and point to various practical applications., Comment: Review article, 85 pages, 24 figures. Some references added. Published in the International Journal of Modern Physics B 20, pp. 2825-2888 (2006). Comments and suggestions from the readers are highly welcome

Showing that the location of the zeros of the partition function can be used to study phase transitions, Yang and Lee initiated an ambitious and very fruitful approach. We give an overview of the results obtained using this approach. After an elementary introduction to the Yang-Lee formalism, we summarize results concerning equilibrium phase transitions. We also describe recent attempts and breakthroughs in extending this theory to nonequilibrium phase transitions., Comment: 62 pages, 1 figure, review article submitted to the International Journal of Modern Physics B

We consider the characterization of the nonequilibrium stationary state of a randomly-driven granular gas in terms of an entropy-production based variational formulation. Enforcing spatial homogeneity, we first consider the temporal stability of the stationary state reached after a transient. In connection, two heuristic albeit physically motivated candidates for the non-equilibrium entropy production are put forward. It turns out that none of them displays an extremum for the stationary velocity distribution selected by the dynamics. Finally, the relevance of the relative Kullbach entropy is discussed., Comment: 17 pages, 2 figures, to be published in Physica A

We study the exact statistical mechanics of Lam\'e solitons using a transfer matrix method. This requires a knowledge of the first forbidden band of the corresponding Schr\"odinger equation with the periodic Lam\'e potential. Since the latter is a quasi-exactly solvable system, an analytical evaluation of the partition function can be done only for a few temperatures. We also study approximately the finite temperature thermodynamics using the ideal kink gas phenomenology. The zero-temperature "thermodynamics" of the soliton lattice solutions is also addressed. Moreover, in appropriate limits our results reduce to that of the sine-Gordon problem., Comment: 29 pages, 5 figures, submitted to Physica Scripta

We obtain the exact nontopological soliton lattice solutions of the Associated Lam\'e equation in different parameter regimes and compute the corresponding energy for each of these solutions. We show that in specific limits these solutions give rise to nontopological (pulse-like) single solitons, as well as to different types of topological (kink-like) single soliton solutions of the Associated Lam\'e equation. Following Manton, we also compute, as an illustration, the asymptotic interaction energy between these soliton solutions in one particular case. Finally, in specific limits, we deduce the soliton lattices, as well as the topological single soliton solutions of the Lam\'e equation, and also the sine-Gordon soliton solution., Comment: 23 pages, 5 figures. Submitted to J. Math. Phys

We apply the Yang-Lee theory of phase transitions to an urn model of separation of sand. The effective partition function of this nonequilibrium system can be expressed as a polynomial of the size-dependent effective fugacity $z$. Numerical calculations show that in the thermodynamic limit, the zeros of the effective partition function are located on the unit circle in the complex $z$-plane. In the complex plane of the actual control parameter certain roots converge to the transition point of the model. Thus the Yang-Lee theory can be applied to a wider class of nonequilibrium systems than those considered previously., Comment: 4 pages, 3 eps figures included

We study synchronization of random one-dimensional linear maps for which the Lyapunov exponent can be calculated exactly. Certain aspects of the dynamics of these maps are explained using their relation with a random walk. We confirm that the Lyapunov exponent changes sign at the complete synchronization transition. We also consider partial synchronization of nonidentical systems. It turns out that the way partial synchronization manifests depends on the type of differences (in Lyapunov exponent or in contraction points) between the systems. The crossover from partial synchronization to complete synchronization is also examined., Comment: 5 pages, 6 figures

We study the scattering of a moving discrete breather (DB) on a junction in a Fermi-Pasta-Ulam (FPU) chain consisting of two segments with different masses of the particles. We consider four distinct cases: (i) a light-heavy (abrupt) junction in which the DB impinges on the junction from the segment with lighter mass, (ii) a heavy-light junction, (iii) an up mass-ramp in which the mass in the heavier segment increases continuously as one moves away from the junction point, and (iv) a down mass-ramp. Depending on the mass difference and DB characteristics (frequency and velocity), the DB can either reflect from, or transmit through, or get trapped at the junction or on the ramp. For the heavy-light junction, the DB can even split at the junction into a reflected and a transmitted DB. The latter is found to subsequently split into two or more DBs. For the down mass-ramp the DB gets accelerated in several stages, with accompanying radiation (phonons). These results are rationalized by calculating the Peierls-Nabarro barrier for the various cases. We also point out implications of our results in realistic situations such as electron-phonon coupled chains., Comment: 17 pages, 16 figures