Showing 1–50 of 67 results for author: Droz, M

Renormalization group theory for finite-size scaling in extreme statistics

Authors: G. Gyorgyi, N. R. Moloney, K. Ozogany, Z. Racz, M. Droz

Abstract: We present a renormalization group (RG) approach to explain universal features of extreme statistics, applied here to independent, identically distributed variables. The outlines of the theory have been described in a previous Letter, the main result being that finite-size shape corrections to the limit distribution can be obtained from a linearization of the RG transformation near a fixed point,… ▽ More We present a renormalization group (RG) approach to explain universal features of extreme statistics, applied here to independent, identically distributed variables. The outlines of the theory have been described in a previous Letter, the main result being that finite-size shape corrections to the limit distribution can be obtained from a linearization of the RG transformation near a fixed point, leading to the computation of stable perturbations as eigenfunctions. Here we show details of the RG theory which exhibit remarkable similarities to the RG known in statistical physics. Besides the fixed points explaining universality, and the least stable eigendirections accounting for convergence rates and shape corrections, the similarities include marginally stable perturbations which turn out to be generic for the Fisher-Tippett-Gumbel class. Distribution functions containing unstable perturbations are also considered. We find that, after a transitory divergence, they return to the universal fixed line at the same or at a different point depending on the type of perturbation. △ Less

Submitted 5 April, 2010; originally announced April 2010.

Comments: 15 pages, 8 figures, to appear in Phys. Rev. E

Entropy-based characterizations of the observable-dependence of the fluctuation-dissipation temperature

Authors: Kirsten Martens, Eric Bertin, Michel Droz

Abstract: The definition of a nonequilibrium temperature through generalized fluctuation-dissipation relations relies on the independence of the fluctuation-dissipation temperature from the observable considered. We argue that this observable independence is deeply related to the uniformity of the phase-space probability distribution on the hypersurfaces of constant energy. This property is shown explicitly… ▽ More The definition of a nonequilibrium temperature through generalized fluctuation-dissipation relations relies on the independence of the fluctuation-dissipation temperature from the observable considered. We argue that this observable independence is deeply related to the uniformity of the phase-space probability distribution on the hypersurfaces of constant energy. This property is shown explicitly on three different stochastic models, where observable-dependence of the fluctuation-dissipation temperature arises only when the uniformity of the phase-space distribution is broken. The first model is an energy transport model on a ring, with biased local transfer rules. In the second model, defined on a fully connected geometry, energy is exchanged with two heat baths at different temperatures, breaking the uniformity of the phase-space distribution. Finally, in the last model, the system is connected to a zero temperature reservoir, and preserves the uniformity of the phase-space distribution in the relaxation regime, leading to an observable-independent temperature. △ Less

Submitted 12 March, 2010; originally announced March 2010.

Comments: 15 pages, 7 figures

Journal ref: Phys. Rev. E 81, 061107 (2010)

Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis

Authors: Eric Bertin, Michel Droz, Guillaume Grégoire

Abstract: Considering a gas of self-propelled particles with binary interactions, we derive the hydrodynamic equations governing the density and velocity fields from the microscopic dynamics, in the framework of the associated Boltzmann equation. Explicit expressions for the transport coefficients are given, as a function of the microscopic parameters of the model. We show that the homogeneous state with… ▽ More Considering a gas of self-propelled particles with binary interactions, we derive the hydrodynamic equations governing the density and velocity fields from the microscopic dynamics, in the framework of the associated Boltzmann equation. Explicit expressions for the transport coefficients are given, as a function of the microscopic parameters of the model. We show that the homogeneous state with zero hydrodynamic velocity is unstable above a critical density (which depends on the microscopic parameters), signaling the onset of a collective motion. Comparison with numerical simulations on a standard model of self-propelled particles shows that the phase diagram we obtain is robust, in the sense that it depends only slightly on the precise definition of the model. While the homogeneous flow is found to be stable far from the transition line, it becomes unstable with respect to finite-wavelength perturbations close to the transition, implying a non trivial spatio-temporal structure for the resulting flow. We find solitary wave solutions of the hydrodynamic equations, quite similar to the stripes reported in direct numerical simulations of self-propelled particles. △ Less

Submitted 27 July, 2009; originally announced July 2009.

Comments: 33 pages, 11 figures, submitted to J. Phys. A

Journal ref: J. Phys A: Math. Theor. 42, 445001 (2009)

Dependence of the fluctuation-dissipation temperature on the choice of observable

Authors: Kirsten Martens, Eric Bertin, Michel Droz

Abstract: On general grounds, a nonequilibrium temperature can be consistently defined from generalized fluctuation-dissipation relations only if it is independent of the observable considered. We argue that the dependence on the choice of observable generically occurs when the phase-space probability distribution is non-uniform on constant energy shells. We relate quantitatively this observable dependenc… ▽ More On general grounds, a nonequilibrium temperature can be consistently defined from generalized fluctuation-dissipation relations only if it is independent of the observable considered. We argue that the dependence on the choice of observable generically occurs when the phase-space probability distribution is non-uniform on constant energy shells. We relate quantitatively this observable dependence to a fundamental characteristics of nonequilibrium systems, namely the Shannon entropy difference with respect to the equilibrium state with the same energy. This relation is illustrated on a mean-field model in contact with two heat baths at different temperatures. △ Less

Submitted 4 January, 2010; v1 submitted 18 June, 2009; originally announced June 2009.

Comments: 4 pages, 2 figures, final version

Journal ref: Phys.Rev.Lett.103:260602,2009

Width of reaction zones in A + B -> C type reaction-diffusion processes: Effects of an electric current

Authors: Kirsten Martens, Michel Droz, Zoltan Rácz

Abstract: We investigate the effects of an electric current on the width of a stationary reaction zone in an irreversible A^- + B^+ -> C reaction-diffusion process. The ion dynamics of the electrolytes A = (A^+, A^-) and B = (B^+, B^-) is described by reaction-diffusion equations obeying local electroneutrality, and the stationary state is obtained by employing reservoirs of fixed electrolyte concentratio… ▽ More We investigate the effects of an electric current on the width of a stationary reaction zone in an irreversible A^- + B^+ -> C reaction-diffusion process. The ion dynamics of the electrolytes A = (A^+, A^-) and B = (B^+, B^-) is described by reaction-diffusion equations obeying local electroneutrality, and the stationary state is obtained by employing reservoirs of fixed electrolyte concentrations at the opposite ends of a finite domain. We find that the width of the reaction zone decreases when the current drives the reacting ions towards the reaction zone while it increases in the opposite case. The linear response of the width to the current is estimated by developing a phenomenological theory based on conservation laws, and on electroneutrality. The theory is found to reproduce numerical solutions to a good accuracy. △ Less

Submitted 13 July, 2009; v1 submitted 16 March, 2009; originally announced March 2009.

Comments: 8 pages, 4 figures

Journal ref: J. Chem. Phys. 130, 234506 (2009)

Encoding information into precipitation structures

Authors: Kirsten Martens, Ioana Bena, Michel Droz, Zoltan Racz

Abstract: 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 precipitatio… ▽ More 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. △ Less

Submitted 28 October, 2008; originally announced October 2008.

Comments: 13 pages, 7 figures, presented at SigmaPhi 2008

Journal ref: J. Stat. Mech. (2008) P12003

Designer Patterns: Flexible Control of Precipitation through Electric Currents

Authors: Ioana Bena, Michel Droz, Istvan Lagzi, Kirsten Martens, Zoltan Racz, Andras Volford

Abstract: 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… ▽ More 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. △ Less

Submitted 19 February, 2008; originally announced February 2008.

Comments: 4 pages, 6 figures

Journal ref: Phys. Rev. Lett. 101, 075701 (2008)

How bad is to be slow-reacting ? On the effect of the delay in response to a changing environment on a population's survival

Authors: Ioana Bena, Michel Droz, Janusz Szwabinski, Andrzej Pekalski

Abstract: 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 "slo… ▽ More 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. △ Less

Submitted 17 January, 2008; originally announced January 2008.

Comments: 7 pages, 4 figures

Extinction risk and structure of a food web model

Authors: A. Pȩkalski, J. Szwabiński, I. Bena, M. Droz

Abstract: We investigate in detail the model of a trophic web proposed by Amaral and Meyer [Phys. Rev. Lett. 82, 652 (1999)]. We focused on small-size systems that are relevant for real biological food webs and for which the fluctuations are playing an important role. We show, using Monte Carlo simulations, that such webs can be non-viable, leading to extinction of all species in small and/or weakly coupl… ▽ More We investigate in detail the model of a trophic web proposed by Amaral and Meyer [Phys. Rev. Lett. 82, 652 (1999)]. We focused on small-size systems that are relevant for real biological food webs and for which the fluctuations are playing an important role. We show, using Monte Carlo simulations, that such webs can be non-viable, leading to extinction of all species in small and/or weakly coupled systems. Estimations of the extinction times and survival chances are also given. We show that before the extinction the fraction of highly-connected species ("omnivores") is increasing. Viable food webs exhibit a pyramidal structure, where the density of occupied niches is higher at lower trophic levels, and moreover the occupations of adjacent levels are closely correlated. We also demonstrate that the distribution of the lengths of food chains has an exponential character and changes weakly with the parameters of the model. On the contrary, the distribution of avalanche sizes of the extinct species depends strongly on the connectedness of the web. For rather loosely connected systems we recover the power-law type of behavior with the same exponent as found in earlier studies, while for densely-connected webs the distribution is not of a power-law type. △ Less

Submitted 17 December, 2007; originally announced December 2007.

Comments: 9 pages, 15 figures

Guiding-fields for phase-separation: Controlling Liesegang patterns

Authors: Tibor Antal, Ioana Bena, Michel Droz, Kirsten Martens, Zoltan Racz

Abstract: 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 governe… ▽ More 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. △ Less

Submitted 12 August, 2007; v1 submitted 5 June, 2007; originally announced June 2007.

Comments: Minor changes, to be published in Phys. Rev. E. 10 pages, 8 figures

Journal ref: Published in Phys. Rev. E 76, 046203 (2007).

Complex population dynamics as a competition between multiple time-scale phenomena

Authors: Ioana Bena, Michel Droz, Janusz Szwabinski, Andrzej Pekalski

Abstract: 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 syste… ▽ More 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. △ Less

Submitted 7 June, 2007; v1 submitted 14 March, 2007; originally announced March 2007.

Comments: 15 pages, 12 figures. Accepted for publication in Phys. Rev. E

Journal ref: Physical Review E 76, 011908 (2007).

Intensive thermodynamic parameters in nonequilibrium systems

Authors: Eric Bertin, Kirsten Martens, Olivier Dauchot, Michel Droz

Abstract: Considering a broad class of steady-state nonequilibrium systems for which some additive quantities are conserved by the dynamics, we introduce from a statistical approach intensive thermodynamic parameters (ITPs) conjugated to the conserved quantities. This definition does not require any detailed balance relation to be fulfilled. Rather, the system has to satisfy a general additivity property,… ▽ More Considering a broad class of steady-state nonequilibrium systems for which some additive quantities are conserved by the dynamics, we introduce from a statistical approach intensive thermodynamic parameters (ITPs) conjugated to the conserved quantities. This definition does not require any detailed balance relation to be fulfilled. Rather, the system has to satisfy a general additivity property, which holds in most of the models usually considered in the literature, including those described by a matrix product ansatz with finite matrices. The main property of these ITPs is to take equal values in two subsystems, making them a powerful tool to describe nonequilibrium phase coexistence, as illustrated on different models. We finally discuss the issue of the equalization of ITPs when two different systems are put into contact. This issue is closely related to the possibility of measuring the ITPs using a small auxiliary system, in the same way as temperature is measured with a thermometer, and points at one of the major difficulties of nonequilibrium statistical mechanics. In addition, an efficient alternative determination, based on the measure of fluctuations, is also proposed and illustrated. △ Less

Submitted 10 April, 2007; v1 submitted 27 November, 2006; originally announced November 2006.

Comments: 17 pages, 5 figures; final version, with minor changes

Journal ref: Phys. Rev. E 75, 031120 (2007)

Reaction-diffusion fronts with inhomogeneous initial conditions

Authors: Ioana Bena, Michel Droz, Kirsten Martens, Zoltan Racz

Abstract: 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 ge… ▽ More 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. △ Less

Submitted 11 July, 2006; originally announced July 2006.

Comments: 11 pages, 3 figures, submitted to J. Phys. C

Journal ref: Journal of Physics C 19, 065103 (2007).

Boltzmann and hydrodynamic description for self-propelled particles

Authors: Eric Bertin, Michel Droz, Guillaume Gregoire

Abstract: We study analytically the emergence of spontaneous collective motion within large bidimensional groups of self-propelled particles with noisy local interactions, a schematic model for assemblies of biological organisms. As a central result, we derive from the individual dynamics the hydrodynamic equations for the density and velocity fields, thus giving a microscopic foundation to the phenomenol… ▽ More We study analytically the emergence of spontaneous collective motion within large bidimensional groups of self-propelled particles with noisy local interactions, a schematic model for assemblies of biological organisms. As a central result, we derive from the individual dynamics the hydrodynamic equations for the density and velocity fields, thus giving a microscopic foundation to the phenomenological equations used in previous approaches. A homogeneous spontaneous motion emerges below a transition line in the noise-density plane. Yet, this state is shown to be unstable against spatial perturbations, suggesting that more complicated structures should eventually appear. △ Less

Submitted 3 August, 2006; v1 submitted 3 January, 2006; originally announced January 2006.

Comments: 4 pages, 3 figures, final version

Journal ref: Phys. Rev. E 74, 022101 (2006)

Definition and relevance of nonequilibrium intensive thermodynamic parameters

Authors: Eric Bertin, Olivier Dauchot, Michel Droz

Abstract: We show that intensive thermodynamic parameters associated to additive conserved quantities can be naturally defined from a statistical approach in far-from-equilibrium steady-state systems, under few assumptions, and without any detailed balance requirement. It may apply, e.g., to dissipative systems like granular gases where volume or mass is still conserved, or to systems with periodic bounda… ▽ More We show that intensive thermodynamic parameters associated to additive conserved quantities can be naturally defined from a statistical approach in far-from-equilibrium steady-state systems, under few assumptions, and without any detailed balance requirement. It may apply, e.g., to dissipative systems like granular gases where volume or mass is still conserved, or to systems with periodic boundary conditions where fluxes of conserved quantities are present. We emphasize the usefulness of this concept to characterize the coexistence of different nonequilibrium phases, and discuss the influence of the contact between two different systems, in relation with measurement issues. △ Less

Submitted 3 April, 2006; v1 submitted 6 December, 2005; originally announced December 2005.

Comments: 4 pages, final version

Journal ref: Phys. Rev. Lett. 96, 120601 (2006)

Statistical Mechanics of Equilibrium and Nonequilibrium Phase Transitions: The Yang-Lee Formalism

Authors: Ioana Bena, Michel Droz, Adam Lipowski

Abstract: 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 brea… ▽ More 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. △ Less

Submitted 11 October, 2005; originally announced October 2005.

Comments: 62 pages, 1 figure, review article submitted to the International Journal of Modern Physics B

Journal ref: International Journal of Modern physics B, vol. 19, pp. 4269-4329 (2005)

Stationary state of a heated granular gas: fate of the usual H-functional

Authors: Ioana Bena, Francois Coppex, Michel Droz, Paolo Visco, Emmanuel Trizac, Frederic van Wijland

Abstract: 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 prod… ▽ More 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. △ Less

Submitted 13 March, 2006; v1 submitted 6 October, 2005; originally announced October 2005.

Comments: 17 pages, 2 figures, to be published in Physica A

Journal ref: Physica A 370, 179-189 (2006).

Maxwell and very hard particle models for probabilistic ballistic annihilation: hydrodynamic description

Authors: Francois Coppex, Michel Droz, Emmanuel Trizac

Abstract: The hydrodynamic description of probabilistic ballistic annihilation, for which no conservation laws hold, is an intricate problem with hard sphere-like dynamics for which no exact solution exists. We consequently focus on simplified approaches, the Maxwell and very hard particles (VHP) models, which allows us to compute analytically upper and lower bounds for several quantities. The purpose is… ▽ More The hydrodynamic description of probabilistic ballistic annihilation, for which no conservation laws hold, is an intricate problem with hard sphere-like dynamics for which no exact solution exists. We consequently focus on simplified approaches, the Maxwell and very hard particles (VHP) models, which allows us to compute analytically upper and lower bounds for several quantities. The purpose is to test the possibility of describing such a far from equilibrium dynamics with simplified kinetic models. The motivation is also in turn to assess the relevance of some singular features appearing within the original model and the approximations invoked to study it. The scaling exponents are first obtained from the (simplified) Boltzmann equation, and are confronted against Monte Carlo simulation (DSMC technique). Then, the Chapman-Enskog method is used to obtain constitutive relations and transport coefficients. The corresponding Navier-Stokes equations for the hydrodynamic fields are derived for both Maxwell and VHP models. We finally perform a linear stability analysis around the homogeneous solution, which illustrates the importance of dissipation in the possible development of spatial inhomogeneities. △ Less

Submitted 4 March, 2005; originally announced March 2005.

Comments: 24 pages, 10 eps figures included

Journal ref: Phys. Rev. E 72, 021105 (2005)

Formation of Liesegang patterns in the presence of an electric field

Authors: I. Bena, M. Droz, Z. Racz

Abstract: The effects of an external electric field on the formation of Liesegang patterns are investigated. The patterns are assumed to emerge from a phase separation process in the wake of a diffusive reaction front. The dynamics is described by a Cahn-Hilliard equation with a moving source term representing the reaction zone, and the electric field enters through its effects on the properties of the re… ▽ More The effects of an external electric field on the formation of Liesegang patterns are investigated. The patterns are assumed to emerge from a phase separation process in the wake of a diffusive reaction front. The dynamics is described by a Cahn-Hilliard equation with a moving source term representing the reaction zone, and the electric field enters through its effects on the properties of the reaction zone. We employ our previous results [I. Bena, F. Coppex, M. Droz, and Z. Rácz, J. Chem. Phys. {\bf 122}, 024512 (2005)] on how the electric field changes both the motion of the front, as well as the amount of reaction product left behind the front, and our main conclusion is that the number of precipitation bands becomes finite in a finite electric field. The reason for the finiteness in case when the electric field drives the reagents towards the reaction zone is that the width of consecutive bands increases so that, beyond a distance $\ell_+$, the precipitation is continuous (plug is formed). In case of an electric field of opposite polarity, the bands emerge in a finite interval $\ell_-$, since the reaction product decreases with time and the conditions for phase separation cease to exist. We give estimates of $\ell_{\pm}$ in terms of measurable quantities and thus present an experimentally verifiable prediction of the "Cahn-Hilliard equation with a moving source" description of Liesegang phenomena. △ Less

Submitted 3 February, 2005; originally announced February 2005.

Comments: 11 pages, 10 figures, submitted to J. Chem. Phys

Journal ref: Journal of Chemical Physics 122, 204502 (2005).

Non-equilibrium temperatures in steady-state systems with conserved energy

Authors: Eric Bertin, Olivier Dauchot, Michel Droz

Abstract: We study a class of non-equilibrium lattice models describing local redistributions of a globally conserved quantity, which is interpreted as an energy. A particular subclass can be solved exactly, allowing to define a statistical temperature T_{th} along the same lines as in the equilibrium microcanonical ensemble. We compute the response function and find that when the fluctuation-dissipation… ▽ More We study a class of non-equilibrium lattice models describing local redistributions of a globally conserved quantity, which is interpreted as an energy. A particular subclass can be solved exactly, allowing to define a statistical temperature T_{th} along the same lines as in the equilibrium microcanonical ensemble. We compute the response function and find that when the fluctuation-dissipation relation is linear, the slope T_{FD}^{-1} of this relation differs from the inverse temperature T_{th}^{-1}. We argue that T_{th} is physically more relevant than T_{FD}, since in the steady-state regime, it takes equal values in two subsystems of a large isolated system. Finally, a numerical renormalization group procedure suggests that all models within the class behave similarly at a coarse-grained level, leading to a new parameter which describes the deviation from equilibrium. Quantitative predictions concerning this parameter are obtained within a mean-field framework. △ Less

Submitted 3 December, 2004; originally announced December 2004.

Comments: 16 pages, 2 figures, submitted to Phys. Rev. E

Journal ref: Phys. Rev. E 71, 046140 (2005)

Front motion in an $A+B\to C$ type reaction-diffusion process: Effects of an electric field

Authors: I. Bena, F. Coppex, M. Droz, Z. Racz

Abstract: We study the effects of an external electric field on both the motion of the reaction zone and the spatial distribution of the reaction product, $C$, in an irreversible $A^- +B^+ \to C$ reaction-diffusion process. The electrolytes $A\equiv (A^+,A^-)$ and $B\equiv (B^+,B^-)$ are initially separated in space and the ion-dynamics is described by reaction-diffusion equations obeying local electroneu… ▽ More We study the effects of an external electric field on both the motion of the reaction zone and the spatial distribution of the reaction product, $C$, in an irreversible $A^- +B^+ \to C$ reaction-diffusion process. The electrolytes $A\equiv (A^+,A^-)$ and $B\equiv (B^+,B^-)$ are initially separated in space and the ion-dynamics is described by reaction-diffusion equations obeying local electroneutrality. Without an electric field, the reaction zone moves diffusively leaving behind a constant concentration of $C$-s. In the presence of an electric field which drives the reagents towards the reaction zone, we find that the reaction zone still moves diffusively but with a diffusion coefficient which slightly decreases with increasing field. The important electric field effect is that the concentration of $C$-s is no longer constant but increases linearly in the direction of the motion of the front. The case of an electric field of reversed polarity is also discussed and it is found that the motion of the front has a diffusive, as well as a drift component. The concentration of $C$-s decreases in the direction of the motion of the front, up to the complete extinction of the reaction. Possible applications of the above results to the understanding of the formation of Liesegang patterns in an electric field is briefly outlined. △ Less

Submitted 6 September, 2004; originally announced September 2004.

Comments: 13 pages, 13 figures, submitted to J. Chem. Phys

Journal ref: Journal of Chemical Physics 122, 024512 (2005).

Hydrodynamics of probabilistic ballistic annihilation

Authors: Francois Coppex, Michel Droz, Emmanuel Trizac

Abstract: We consider a dilute gas of hard spheres in dimension $d \geq 2$ that upon collision either annihilate with probability $p$ or undergo an elastic scattering with probability $1-p$. For such a system neither mass, momentum, nor kinetic energy are conserved quantities. We establish the hydrodynamic equations from the Boltzmann equation description. Within the Chapman-Enskog scheme, we determine th… ▽ More We consider a dilute gas of hard spheres in dimension $d \geq 2$ that upon collision either annihilate with probability $p$ or undergo an elastic scattering with probability $1-p$. For such a system neither mass, momentum, nor kinetic energy are conserved quantities. We establish the hydrodynamic equations from the Boltzmann equation description. Within the Chapman-Enskog scheme, we determine the transport coefficients up to Navier-Stokes order, and give the closed set of equations for the hydrodynamic fields chosen for the above coarse grained description (density, momentum and kinetic temperature). Linear stability analysis is performed, and the conditions of stability for the local fields are discussed. △ Less

Submitted 9 December, 2004; v1 submitted 30 July, 2004; originally announced July 2004.

Comments: 19 pages, 3 eps figures included

Journal ref: Phys. Rev. E 70, 061102 (2004)

Temperature in nonequilibrium systems with conserved energy

Authors: Eric Bertin, Olivier Dauchot, Michel Droz

Abstract: We study a class of nonequilibrium lattice models which describe local redistributions of a globally conserved energy. A particular subclass can be solved analytically, allowing to define a temperature T_{th} along the same lines as in the equilibrium microcanonical ensemble. The fluctuation-dissipation relation is explicitely found to be linear, but its slope differs from the inverse temperatur… ▽ More We study a class of nonequilibrium lattice models which describe local redistributions of a globally conserved energy. A particular subclass can be solved analytically, allowing to define a temperature T_{th} along the same lines as in the equilibrium microcanonical ensemble. The fluctuation-dissipation relation is explicitely found to be linear, but its slope differs from the inverse temperature T_{th}^{-1}. A numerical renormalization group procedure suggests that, at a coarse-grained level, all models behave similarly, leading to a two-parameter description of their macroscopic properties. △ Less

Submitted 22 December, 2004; v1 submitted 29 June, 2004; originally announced June 2004.

Comments: 4 pages, 1 figure, final version

Journal ref: Phys. Rev. Lett. 93, 230601 (2004)

Extinction dynamics of Lotka-Volterra ecosystems on evolving networks

Authors: Francois Coppex, Michel Droz, Adam Lipowski

Abstract: We study a model of a multi-species ecosystem described by Lotka-Volterra-like equations. Interactions among species form a network whose evolution is determined by the dynamics of the model. Numerical simulations show power-law distribution of intervals between extinctions, but only for ecosystems with sufficient variability of species and with networks of connectivity above certain threshold t… ▽ More We study a model of a multi-species ecosystem described by Lotka-Volterra-like equations. Interactions among species form a network whose evolution is determined by the dynamics of the model. Numerical simulations show power-law distribution of intervals between extinctions, but only for ecosystems with sufficient variability of species and with networks of connectivity above certain threshold that is very close to the percolation threshold of the network. Effect of slow environmental changes on extinction dynamics, degree distribution of the network of interspecies interactions, and some emergent properties of our model are also examined. △ Less

Submitted 2 June, 2004; v1 submitted 19 December, 2003; originally announced December 2003.

Comments: 8 pages, 6 eps figures included

Journal ref: Phys. Rev. E 69, 061901 (2004)

Synchronization and partial synchronization of linear maps

Authors: Adam Lipowski, Michel Droz

Abstract: We study synchronization of low-dimensional ($d=2,3,4$) chaotic piecewise linear maps. For Bernoulli maps we find Lyapunov exponents and locate the synchronization transition, that numerically is found to be discontinuous (despite continuously vanishing Lyapunov exponent(s)). For tent maps, a limit of stability of the synchronized state is used to locate the synchronization transition that numer… ▽ More We study synchronization of low-dimensional ($d=2,3,4$) chaotic piecewise linear maps. For Bernoulli maps we find Lyapunov exponents and locate the synchronization transition, that numerically is found to be discontinuous (despite continuously vanishing Lyapunov exponent(s)). For tent maps, a limit of stability of the synchronized state is used to locate the synchronization transition that numerically is found to be continuous. For nonidentical tent maps at the partial synchronization transition, the probability distribution of the synchronization error is shown to develop highly singular behavior. We suggest that for nonidentical Bernoulli maps (and perhaps some other discontinuous maps) partial synchronization is merely a smooth crossover rather than a well defined transition. More subtle analysis in the $d=4$ case locates the point where the synchronized state becomes stable. In some cases, however, a riddled basin attractor appears, and synchronized and chaotic behaviors coexist. We also suggest that similar riddling of a basin of attractor might take place in some extended systems where it is known as stable chaos. △ Less

Submitted 2 December, 2003; originally announced December 2003.

Comments: 7 pages, eps figures included

Journal ref: Physica A 347 (2005) 38-50

Probabilistic ballistic annihilation with continuous velocity distributions

Authors: Francois Coppex, Michel Droz, Emmanuel Trizac

Abstract: We investigate the problem of ballistically controlled reactions where particles either annihilate upon collision with probability $p$, or undergo an elastic shock with probability $1-p$. Restricting to homogeneous systems, we provide in the scaling regime that emerges in the long time limit, analytical expressions for the exponents describing the time decay of the density and the root-mean-squa… ▽ More We investigate the problem of ballistically controlled reactions where particles either annihilate upon collision with probability $p$, or undergo an elastic shock with probability $1-p$. Restricting to homogeneous systems, we provide in the scaling regime that emerges in the long time limit, analytical expressions for the exponents describing the time decay of the density and the root-mean-square velocity, as continuous functions of the probability $p$ and of a parameter related to the dissipation of energy. We work at the level of molecular chaos (non-linear Boltzmann equation), and using a systematic Sonine polynomials expansion of the velocity distribution, we obtain in arbitrary dimension the first non-Gaussian correction and the corresponding expressions for the decay exponents. We implement Monte-Carlo simulations in two dimensions, that are in excellent agreement with our analytical predictions. For $p<1$, numerical simulations lead to conjecture that unlike for pure annihilation ($p=1$), the velocity distribution becomes universal, i.e. does not depend on the initial conditions. △ Less

Submitted 1 November, 2003; v1 submitted 4 September, 2003; originally announced September 2003.

Comments: 10 pages, 9 eps figures included

Journal ref: Phys. Rev. E 69, 011303 (2004)

Probability distribution of magnetization in the one-dimensional Ising model: Effects of boundary conditions

Authors: T. Antal, M. Droz, Z. Racz

Abstract: Finite-size scaling functions are investigated both for the mean-square magnetization fluctuations and for the probability distribution of the magnetization in the one-dimensional Ising model. The scaling functions are evaluated in the limit of the temperature going to zero (T -> 0), the size of the system going to infinity (N -> oo) while N[1-tanh(J/k_BT)] is kept finite (J being the nearest ne… ▽ More Finite-size scaling functions are investigated both for the mean-square magnetization fluctuations and for the probability distribution of the magnetization in the one-dimensional Ising model. The scaling functions are evaluated in the limit of the temperature going to zero (T -> 0), the size of the system going to infinity (N -> oo) while N[1-tanh(J/k_BT)] is kept finite (J being the nearest neighbor coupling). Exact calculations using various boundary conditions (periodic, antiperiodic, free, block) demonstrate explicitly how the scaling functions depend on the boundary conditions. We also show that the block (small part of a large system) magnetization distribution results are identical to those obtained for free boundary conditions. △ Less

Submitted 21 August, 2003; originally announced August 2003.

Comments: 8 pages, 5 figures

Journal ref: J.Phys.A: Math.Gen. 37, 1465-78 (2004)

Some recent developments in models with absorbing states

Authors: Michel Droz, Adam Lipowski

Abstract: We describe some of the recent results obtained for models with absorbing states. First, we present the nonequilibrium absorbing-state Potts model and discuss some of the factors that might affect the critical behaviour of such models. In particular we show that in two dimensions the further neighbour interactions might split the voter critical point into two critical points. We also describe so… ▽ More We describe some of the recent results obtained for models with absorbing states. First, we present the nonequilibrium absorbing-state Potts model and discuss some of the factors that might affect the critical behaviour of such models. In particular we show that in two dimensions the further neighbour interactions might split the voter critical point into two critical points. We also describe some of the results obtained in the context of synchronization of chaotic dynamical systems. Moreover, we discuss the relation of the synchronization transition with some interfacial models. △ Less

Submitted 21 August, 2003; originally announced August 2003.

Comments: 8 pages, Brazilian J. of Physics (in press)

Journal ref: Brazilian Journal of Physics 33, 526 (2003)

Yang-Lee zeroes for an urn model for the separation of sand

Authors: Ioana Bena, Francois Coppex, Michel Droz, Adam Lipowski

Abstract: 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… ▽ More 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. △ Less

Submitted 15 October, 2003; v1 submitted 30 June, 2003; originally announced June 2003.

Comments: 4 pages, 3 eps figures included

Journal ref: Phys. Rev. Lett. 91, 160602 (2003)

Synchronization of Random Linear Maps

Authors: Adam Lipowski, Ioana Bena, Michel Droz, Antonio L. Ferreira

Abstract: 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 t… ▽ More 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. △ Less

Submitted 27 March, 2003; originally announced March 2003.

Comments: 5 pages, 6 figures

Journal ref: Physica A 339, 237 (2004).

On the first Sonine correction for granular gases

Authors: Francois Coppex, Michel Droz, Jaroslaw Piasecki, Emmanuel Trizac

Abstract: We consider the velocity distribution for a granular gas of inelastic hard spheres described by the Boltzmann equation. We investigate both the free of forcing case and a system heated by a stochastic force. We propose a new method to compute the first correction to Gaussian behavior in a Sonine polynomial expansion quantified by the fourth cumulant $a_2$. Our expressions are compared to previou… ▽ More We consider the velocity distribution for a granular gas of inelastic hard spheres described by the Boltzmann equation. We investigate both the free of forcing case and a system heated by a stochastic force. We propose a new method to compute the first correction to Gaussian behavior in a Sonine polynomial expansion quantified by the fourth cumulant $a_2$. Our expressions are compared to previous results and to those obtained through the numerical solution of the Boltzmann equation. It is numerically shown that our method yields very accurate results for small velocities of the rescaled distribution. We finally discuss the ambiguities inherent to a linear approximation method in $a_2$. △ Less

Submitted 22 May, 2003; v1 submitted 7 February, 2003; originally announced February 2003.

Comments: 9 pages, 8 eps figures included

Journal ref: Physica A 329/1-2, 114-226 (2003)

Splitting the voter criticality

Authors: Michel Droz, Antonio L. Ferreira, Adam Lipowski

Abstract: Recently some two-dimensional models with double symmetric absorbing states were shown to share the same critical behaviour that was called the voter universality class. We show, that for an absorbing-states Potts model with finite but further than nearest neighbour range of interactions the critical point is splitted into two critical points: one of the Ising type, and the other of the directed… ▽ More Recently some two-dimensional models with double symmetric absorbing states were shown to share the same critical behaviour that was called the voter universality class. We show, that for an absorbing-states Potts model with finite but further than nearest neighbour range of interactions the critical point is splitted into two critical points: one of the Ising type, and the other of the directed percolation universality class. Similar splitting takes place in the three-dimensional nearest-neighbour model. △ Less

Submitted 16 January, 2003; originally announced January 2003.

Comments: 4 pages, eps figures included

Journal ref: Phys.Rev.E 67, 056108 (2003)

Synchronization in coupled map lattices as an interface depinning

Authors: Adam Lipowski, Michel Droz

Abstract: We study an SOS model whose dynamics is inspired by recent studies of the synchronization transition in coupled map lattices (CML). The synchronization of CML is thus related with a depinning of interface from a binding wall. Critical behaviour of our SOS model depends on a specific form of binding (i.e., transition rates of the dynamics). For an exponentially decaying binding the depinning belo… ▽ More We study an SOS model whose dynamics is inspired by recent studies of the synchronization transition in coupled map lattices (CML). The synchronization of CML is thus related with a depinning of interface from a binding wall. Critical behaviour of our SOS model depends on a specific form of binding (i.e., transition rates of the dynamics). For an exponentially decaying binding the depinning belongs to the directed percolation universality class. Other types of depinning, including the one with a line of critical points, are observed for a power-law binding. △ Less

Submitted 21 August, 2003; v1 submitted 6 December, 2002; originally announced December 2002.

Comments: 4 pages, Phys.Rev.E (in press)

Journal ref: Physical Review E 68, 056119 (2003)

Dynamical properties of the synchronization transition

Authors: Michel Droz, Adam Lipowski

Abstract: We study the dynamics of the synchronization transition (ST) of one-dimensional coupled map lattices. For the Bernoulli map it was recently found by Ahlers and Pikovsky (Phys. Rev. Lett. {\bf 88}, 254101 (2002)) that the ST belongs to the directed percolation (DP) universality class. Spreading dynamics confirms such an identification, only for a certain class of synchronized configurations. For… ▽ More We study the dynamics of the synchronization transition (ST) of one-dimensional coupled map lattices. For the Bernoulli map it was recently found by Ahlers and Pikovsky (Phys. Rev. Lett. {\bf 88}, 254101 (2002)) that the ST belongs to the directed percolation (DP) universality class. Spreading dynamics confirms such an identification, only for a certain class of synchronized configurations. For homogeneous configurations spreading exponents $η$ and $δ$ are different than DP exponents but their sum equals to the corresponding sum of DP exponents. Such a relation is typical to some models with infinitely many absorbing states. Moreover, we calculate the spreading exponents for the tent map for which the ST belongs to the bounded Kardar-Parisi-Zheng (BKPZ) universality class. Our estimation of spreading exponents are consistent with the hyperscaling relation. Finally, we examine the asymmetric tent map. For small asymmetry the ST remains of the BKPZ type. However, for large asymmetry a different critical behaviour appears with exponents being relatively close to the ones of DP. △ Less

Submitted 7 November, 2002; originally announced November 2002.

Comments: 15 pages, 14 eps figures included

Journal ref: Phys.Rev.E 67, 056204 (2003)

Some exact results for Boltzmann's annihilation dynamics

Authors: Francois Coppex, Michel Droz, Jaroslaw Piasecki, Emmanuel Trizac, Peter Wittwer

Abstract: The problem of ballistic annihilation for a spatially homogeneous system is revisited within Boltzmann's kinetic theory in two and three dimensions. Exact analytical results are derived for the time evolution of the particle density for some isotropic discrete bimodal velocity modulus distributions. According to the allowed values of the velocity modulus, different behaviors are obtained: power… ▽ More The problem of ballistic annihilation for a spatially homogeneous system is revisited within Boltzmann's kinetic theory in two and three dimensions. Exact analytical results are derived for the time evolution of the particle density for some isotropic discrete bimodal velocity modulus distributions. According to the allowed values of the velocity modulus, different behaviors are obtained: power law decay with non-universal exponents depending continuously upon the ratio of the two velocities, or exponential decay. When one of the two velocities is equal to zero, the model describes the problem of ballistic annihilation in presence of static traps. The analytical predictions are shown to be in excellent agreement with the results of two-dimensional molecular dynamics simulations. △ Less

Submitted 13 February, 2003; v1 submitted 17 October, 2002; originally announced October 2002.

Comments: 10 pages, 5 figures

Journal ref: Phys. Rev. E 67, 021103 (2003)

Dynamics of ballistic annihilation

Authors: Jaroslaw Piasecki, Emmanuel Trizac, Michel Droz

Abstract: The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the non-linear Boltzmann equation for… ▽ More The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the non-linear Boltzmann equation for dimensions larger than one and provides expressions for the exponents describing the decay of the particle density n(t) ~ t^{-ξ} and the root mean-square velocity ${\bar v} ~ t^{-γ}$ in term of a parameter related to the dissipation of kinetic energy. The Boltzmann equation is then solved perturbatively within a systematic expansion in Sonine polynomials. Analytical expressions for the exponents $ξ$ and $γ$ are obtained in arbitrary dimension as a function of the parameter $μ$ characterizing the small velocity behavior of the initial velocity distribution. Moreover, the leading non-Gaussian corrections to the scaled velocity distribution are computed. These expressions for the scaling exponents are in good agreement with the values reported in the literature for continuous velocity distributions in $d=1$. For the two dimensional case, we implement Monte-Carlo and molecular dynamics simulations that turn out to be in excellent agreement with the analytical predictions. △ Less

Submitted 3 October, 2002; originally announced October 2002.

Comments: LaTeX, 19 pages, 10 figures

Journal ref: Phys. Rev. E 66, 066111 (2002)

Exponential velocity tails in a driven inelastic Maxwell model

Authors: Tibor Antal, Michel Droz, Adam Lipowski

Abstract: The problem of the steady-state velocity distribution in a driven inelastic Maxwell model of shaken granular material is revisited. Numerical solution of the master equation and analytical arguments show that the model has bilateral exponential velocity tails ($P(v)\sim e^{-|v|/\sqrt D}$), where $D$ is the amplitude of the noise. Previous study of this model predicted Gaussian tails (… ▽ More The problem of the steady-state velocity distribution in a driven inelastic Maxwell model of shaken granular material is revisited. Numerical solution of the master equation and analytical arguments show that the model has bilateral exponential velocity tails ($P(v)\sim e^{-|v|/\sqrt D}$), where $D$ is the amplitude of the noise. Previous study of this model predicted Gaussian tails ($P(v)\sim e^{-av^2}$). △ Less

Submitted 3 July, 2002; originally announced July 2002.

Comments: 4 pages, 2eps figures included

Journal ref: Phys.Rev. E 66, 062301 (2002)

On the Role of External Constraints in a Spatially Extended Evolutionary Prisoner's Dilemma Game

Authors: Gyorgy Szabo, Tibor Antal, Peter Szabo, Michel Droz

Abstract: We study the emergency of mutual cooperation in evolutionary prisoner's dilemma games when the players are located on a square lattice. The players can choose one of the three strategies: cooperation (C), defection (D) or "tit for tat" (T), and their total payoffs come from games with the nearest neighbors. During the random sequential updates the players adopt one of their neighboring strategie… ▽ More We study the emergency of mutual cooperation in evolutionary prisoner's dilemma games when the players are located on a square lattice. The players can choose one of the three strategies: cooperation (C), defection (D) or "tit for tat" (T), and their total payoffs come from games with the nearest neighbors. During the random sequential updates the players adopt one of their neighboring strategies if the chosen neighbor has higher payoff. We compare the effect of two types of external constraints added to the Darwinian evolutionary processes. In both cases the strategy of a randomly chosen player is replaced with probability P by another strategy. In the first case, the strategy is replaced by a randomly chosen one among the two others, while in the second case the new strategy is always C. Using generalized mean-field approximations and Monte Carlo simulations the strategy concentrations are evaluated in the stationary state for different strength of external constraints characterized by the probability P. △ Less

Submitted 28 May, 2002; originally announced May 2002.

Comments: 19 pages, 10 figures

Journal ref: in Modeling Complex Systems, AIP Conference Proceedings, Vol 574, Eds. P. L. Garrido and J. Marro, (AIP, Melville, 2001) pp. 38-55

Dynamics of the breakdown of granular clusters

Authors: Francois Coppex, Michel Droz, Adam Lipowski

Abstract: Recently van der Meer et al. studied the breakdown of a granular cluster (Phys. Rev. Lett. {\bf 88}, 174302 (2002)). We reexamine this problem using an urn model, which takes into account fluctuations and finite-size effects. General arguments are given for the absence of a continuous transition when the number of urns (compartments) is greater than two. Monte Carlo simulations show that the lif… ▽ More Recently van der Meer et al. studied the breakdown of a granular cluster (Phys. Rev. Lett. {\bf 88}, 174302 (2002)). We reexamine this problem using an urn model, which takes into account fluctuations and finite-size effects. General arguments are given for the absence of a continuous transition when the number of urns (compartments) is greater than two. Monte Carlo simulations show that the lifetime of a cluster $τ$ diverges at the limits of stability as $τ\sim N^{1/3}$, where $N$ is the number of balls. After the breakdown, depending on the dynamical rules of our urn model, either normal or anomalous diffusion of the cluster takes place. △ Less

Submitted 3 May, 2002; originally announced May 2002.

Comments: 5 pages, 6 eps figures included

Journal ref: Phys. Rev. E 66, 011305 (2002)

Killer Geometries in Competing Species Dynamics

Authors: Serge Galam, Bastien Chopard, Michel Droz

Abstract: We discuss a cellular automata model to study the competition between an emergent better fitted species against an existing majority species. The model implement local fights among small group of individual and a synchronous random walk on a 2D lattice. The faith of the system, i.e. the spreading or disappearance of the species is determined by their initial density and fight frequency. The init… ▽ More We discuss a cellular automata model to study the competition between an emergent better fitted species against an existing majority species. The model implement local fights among small group of individual and a synchronous random walk on a 2D lattice. The faith of the system, i.e. the spreading or disappearance of the species is determined by their initial density and fight frequency. The initial density of the emergent species has to be higher than a critical threshold for total spreading but this value depends in a non-trivial way of the fight frequency. Below the threshold any better adapted species disappears showing that a qualitative advantage is not enough for a minority to win. No strategy is involved but spatial organization turns out to be crucial. For instance at minority densities of zero measure some very rare local geometries which occur by chance are found to be killer geometries. Once set they lead with high probability to the total destruction of the preexisting majority species. The occurrence rate of these killer geometries is function of the system size. This model may apply to a large spectrum of competing groups like smoker-non smoker, opinion forming, diffusion of innovation setting of industrial standards, species evolution, epidemic spreading and cancer growth. △ Less

Submitted 8 April, 2002; originally announced April 2002.

Comments: 11 pages, 8 figures

Binder cumulants of an urn model and Ising model above critical dimension

Authors: A. Lipowski, M. Droz

Abstract: Solving numerically master equation for a recently introduced urn model, we show that the fourth- and sixth-order cumulants remain constant along an exactly located line of critical points. Obtained values are in very good agreement with values predicted by Brezin and Zinn-Justin for the Ising model above the critical dimension. At the tricritical point cumulants acquire values which also agree… ▽ More Solving numerically master equation for a recently introduced urn model, we show that the fourth- and sixth-order cumulants remain constant along an exactly located line of critical points. Obtained values are in very good agreement with values predicted by Brezin and Zinn-Justin for the Ising model above the critical dimension. At the tricritical point cumulants acquire values which also agree with a suitably extended Brezin and Zinn-Justin approach. △ Less

Submitted 25 January, 2002; originally announced January 2002.

Comments: 5 pages, eps figures included

Journal ref: Physical Review E 66, 016118 (2002)

Roughness distributions for 1/f^alpha signals

Authors: T. Antal, M. Droz, G. Gyorgyi, Z. Racz

Abstract: The probability density function (PDF) of the roughness, i.e., of the temporal variance, of 1/f^alpha noise signals is studied. Our starting point is the generalization of the model of Gaussian, time-periodic, 1/f noise, discussed in our recent Letter [T. Antal et al., PRL, vol. 87, 240601 (2001)], to arbitrary power law. We investigate three main scaling regions, distinguished by the scaling of… ▽ More The probability density function (PDF) of the roughness, i.e., of the temporal variance, of 1/f^alpha noise signals is studied. Our starting point is the generalization of the model of Gaussian, time-periodic, 1/f noise, discussed in our recent Letter [T. Antal et al., PRL, vol. 87, 240601 (2001)], to arbitrary power law. We investigate three main scaling regions, distinguished by the scaling of the cumulants in terms of the microscopic scale and the total length of the period. Various analytical representations of the PDF allow for a precise numerical evaluation of the scaling function of the PDF for any alpha. A simulation of the periodic process makes it possible to study also non-periodic signals on short intervals embedded in the full period. We find that for alpha=<1/2 the scaled PDF-s in both the periodic and the non-periodic cases are Gaussian, but for alpha>1/2 they differ from the Gaussian and from each other. Both deviations increase with growing alpha. That conclusion, based on numerics, is reinforced by analytic results for alpha=2 and alpha->infinity. We suggest that our theoretical and numerical results open a new perspective on the data analysis of 1/f^alpha processes. △ Less

Submitted 15 December, 2001; v1 submitted 14 December, 2001; originally announced December 2001.

Comments: 12 pages incl. 6 figures, with RevTex4, for A4 paper, in v2 some references were corrected

Journal ref: Phys. Rev. E, 65, 046140 (2002)

Phase transitions in nonequilibrium d-dimensional models with q absorbing states

Authors: A. Lipowski, M. Droz

Abstract: A nonequilibrium Potts-like model with $q$ absorbing states is studied using Monte Carlo simulations. In two dimensions and $q=3$ the model exhibits a discontinuous transition. For the three-dimensional case and $q=2$ the model exhibits a continuous, transition with $β=1$ (mean-field). Simulations are inconclusive, however, in the two-dimensional case for $q=2$. We suggest that in this case the… ▽ More A nonequilibrium Potts-like model with $q$ absorbing states is studied using Monte Carlo simulations. In two dimensions and $q=3$ the model exhibits a discontinuous transition. For the three-dimensional case and $q=2$ the model exhibits a continuous, transition with $β=1$ (mean-field). Simulations are inconclusive, however, in the two-dimensional case for $q=2$. We suggest that in this case the model is close to or at the crossing point of lines separating three different types of phase transitions. The proposed phase diagram in the $(q,d)$ plane is very similar to that of the equilibrium Potts model. In addition, our simulations confirm field-theory prediction that in two dimensions a branching-annihilating random walk model without parity conservation belongs to the directed percolation universality class. △ Less

Submitted 26 February, 2002; v1 submitted 7 December, 2001; originally announced December 2001.

Comments: 8 pages, figures included, accepted in Phys.Rev.E

Journal ref: Phys. Rev. E 65, 056114 (2002)

Urn model of separation of sand

Authors: Adam Lipowski, Michel Droz

Abstract: We introduce an urn model which describes spatial separation of sand. In this dynamical model, in a certain range of parameters spontaneous symmetry breaking takes place and equipartitioning of sand into two compartments is broken. The steady-state equation for an order parameter, a critical line, and the tricritical point on the phase diagram are found exactly. Master equation and the first-pas… ▽ More We introduce an urn model which describes spatial separation of sand. In this dynamical model, in a certain range of parameters spontaneous symmetry breaking takes place and equipartitioning of sand into two compartments is broken. The steady-state equation for an order parameter, a critical line, and the tricritical point on the phase diagram are found exactly. Master equation and the first-passage problem for the model are solved numerically and the results are used to locate first-order transitions. Exponential divergence of a certain characteristic time shows that the model can also exhibit very strong metastability. In certain cases characteristic time diverges as N^{z}, where N is the number of balls and z=1/2 (critical line), 2/3 (tricritical point), or 1/3 (limits of stability). △ Less

Submitted 26 November, 2001; originally announced November 2001.

Comments: 10 pages, eps figures included

Journal ref: Phys.Rev.E 65, 031307 (2002)

Dynamics-dependent criticality in models with q absorbing states

Authors: A. Lipowski, M. Droz

Abstract: We study a one-dimensional, nonequilibrium Potts-like model which has $q$ symmetric absorbing states. For $q=2$, as expected, the model belongs to the parity conserving universality class. For $q=3$ the critical behaviour depends on the dynamics of the model. Under a certain dynamics it remains generically in the active phase, which is also the feature of some other models with three absorbing s… ▽ More We study a one-dimensional, nonequilibrium Potts-like model which has $q$ symmetric absorbing states. For $q=2$, as expected, the model belongs to the parity conserving universality class. For $q=3$ the critical behaviour depends on the dynamics of the model. Under a certain dynamics it remains generically in the active phase, which is also the feature of some other models with three absorbing states. However, a modified dynamics induces a parity conserving phase transition. Relations with branching-annihilating random walk models are discussed in order to explain such a behaviour. △ Less

Submitted 25 May, 2002; v1 submitted 19 October, 2001; originally announced October 2001.

Comments: 5 pages, 5 eps figures included, Phys.Rev.E (accepted)

Journal ref: Physical Review E 66, 016106 (2002)

1/f Noise and Extreme Value Statistics

Authors: T. Antal, M. Droz, G. Gyorgyi, Z. Racz

Abstract: We study the finite-size scaling of the roughness of signals in systems displaying Gaussian 1/f power spectra. It is found that one of the extreme value distributions (Gumbel distribution) emerges as the scaling function when the boundary conditions are periodic. We provide a realistic example of periodic 1/f noise, and demonstrate by simulations that the Gumbel distribution is a good approximat… ▽ More We study the finite-size scaling of the roughness of signals in systems displaying Gaussian 1/f power spectra. It is found that one of the extreme value distributions (Gumbel distribution) emerges as the scaling function when the boundary conditions are periodic. We provide a realistic example of periodic 1/f noise, and demonstrate by simulations that the Gumbel distribution is a good approximation for the case of nonperiodic boundary conditions as well. Experiments on voltage fluctuations in GaAs films are analyzed and excellent agreement is found with the theory. △ Less

Submitted 30 May, 2001; originally announced May 2001.

Comments: 4 pages, 4 postscript figures, RevTex

Journal ref: Phys.Rev.Lett. 87, 240601 (2001)

On the critical behavior of a lattice prey-predator model

Authors: Tibor Antal, Michel Droz, Adam Lipowski, Geza Odor

Abstract: The critical properties of a simple prey-predator model are revisited. For some values of the control parameters, the model exhibits a line of directed percolation like transitions to a single absorbing state. For other values of the control parameters one finds a second line of continuous transitions toward infinite number of absorbing states, and the corresponding steady-state exponents are me… ▽ More The critical properties of a simple prey-predator model are revisited. For some values of the control parameters, the model exhibits a line of directed percolation like transitions to a single absorbing state. For other values of the control parameters one finds a second line of continuous transitions toward infinite number of absorbing states, and the corresponding steady-state exponents are mean-field like. The critical behavior of the special point T (bicritical point), where the two transition lines meet, belongs to a different universality class. The use of dynamical Monte-Carlo method shows that a particular strategy for preparing the initial state should be devised to correctly describe the physics of the system near the second transition line. Relationships with a forest fire model with immunization are also discussed. △ Less

Submitted 11 April, 2001; originally announced April 2001.

Comments: 6 RevTex pages, 7 ps figures

Journal ref: Phys. Rev. E 64, 036118 (2001)

Criticality of natural absorbing states

Authors: A. Lipowski, M. Droz

Abstract: We study a recently introduced ladder model which undergoes a transition between an active and an infinitely degenerate absorbing phase. In some cases the critical behaviour of the model is the same as that of the branching annihilating random walk with $N\geq 2$ species both with and without hard-core interaction. We show that certain static characteristics of the so-called natural absorbing st… ▽ More We study a recently introduced ladder model which undergoes a transition between an active and an infinitely degenerate absorbing phase. In some cases the critical behaviour of the model is the same as that of the branching annihilating random walk with $N\geq 2$ species both with and without hard-core interaction. We show that certain static characteristics of the so-called natural absorbing states develop power law singularities which signal the approach of the critical point. These results are also explained using random walk arguments. In addition to that we show that when dynamics of our model is considered as a minimum finding procedure, it has the best efficiency very close to the critical point. △ Less

Submitted 18 May, 2001; v1 submitted 9 February, 2001; originally announced February 2001.

Comments: 6 pages

Journal ref: Phys.Rev.E 64, 031107 (2001)

Formation of Liesegang patterns: Simulations using a kinetic Ising model

Authors: T. Antal, M. Droz, J. Magnin, A. Pekalski, Z. Racz

Abstract: A kinetic Ising model description of Liesegang phenomena is studied using Monte Carlo simulations. The model takes into account thermal fluctuations, contains noise in the chemical reactions, and its control parameters are experimentally accessible. We find that noisy, irregular precipitation takes place in dimension d=2 while, depending on the values of the control parameters, either irregular… ▽ More A kinetic Ising model description of Liesegang phenomena is studied using Monte Carlo simulations. The model takes into account thermal fluctuations, contains noise in the chemical reactions, and its control parameters are experimentally accessible. We find that noisy, irregular precipitation takes place in dimension d=2 while, depending on the values of the control parameters, either irregular patterns or precipitation bands satisfying the regular spacing law emerge in d=3. △ Less

Submitted 29 September, 2000; originally announced September 2000.

Comments: 7 pages, 8 ps figures, RevTex

Journal ref: J. Chem. Phys. 114, 3770 (2001)

Phase Transitions and Oscillations in a Lattice Prey-Predator Model

Authors: Tibor Antal, Michel Droz

Abstract: A coarse grained description of a two-dimensional prey-predator system is given in terms of a 3-state lattice model containing two control parameters: the spreading rates of preys and predators. The properties of the model are investigated by dynamical mean-field approximations and extensive numerical simulations. It is shown that the stationary state phase diagram is divided into two phases: a… ▽ More A coarse grained description of a two-dimensional prey-predator system is given in terms of a 3-state lattice model containing two control parameters: the spreading rates of preys and predators. The properties of the model are investigated by dynamical mean-field approximations and extensive numerical simulations. It is shown that the stationary state phase diagram is divided into two phases: a pure prey phase and a coexistence phase of preys and predators in which temporal and spatial oscillations can be present. The different type of phase transitions occuring at the boundary of the prey absorbing phase, as well as the crossover phenomena occuring between the oscillatory and non-oscillatory domains of the coexistence phase are studied. The importance of finite size effects are discussed and scaling relations between different quantities are established. Finally, physical arguments, based on the spatial structure of the model, are given to explain the underlying mechanism leading to oscillations. △ Less

Submitted 28 September, 2000; originally announced September 2000.

Comments: 11 pages, 13 figures

Journal ref: Phys. Rev. E 63, 056119 (2001)