Showing 1–50 of 51 results for author: Racz, Z

Finite-size corrections to scaling of the magnetization distribution in the $2d$ $XY$-model at zero temperature

Authors: G. Palma, F. Niedermayer, Z. Rácz, A. Riveros, D. Zambrano

Abstract: The zero-temperature, classical $XY$-model on an $L \times L$ square-lattice is studied by exploring the distribution $Φ_L(y)$ of its centered and normalized magnetization $y$ in the large $L$ limit. An integral representation of the cumulant generating function, known from earlier works, is used for the numerical evaluation of $Φ_L(y)$, and the limit distribution… ▽ More The zero-temperature, classical $XY$-model on an $L \times L$ square-lattice is studied by exploring the distribution $Φ_L(y)$ of its centered and normalized magnetization $y$ in the large $L$ limit. An integral representation of the cumulant generating function, known from earlier works, is used for the numerical evaluation of $Φ_L(y)$, and the limit distribution $Φ_{L \rightarrow \infty} (y) = Φ_0(y)$ is obtained with high precision. The two leading finite-size corrections $Φ_L (y) -Φ_0 (y) \approx a_1(L)\, Φ_1(y) + a_2(L)\,Φ_2(y)$ are also extracted both from numerics and from analytic calculations. We find that the amplitude $a_1(L)$ scales as $\ln(L/L_0) /L^2$ and the shape correction function $Φ_1 (y)$ can be expressed through the low-order derivatives of the limit distribution, $Φ_1 (y) = [\,y\, Φ_0 (y) + Φ'_0 (y)\,]'$. The second finite-size correction has an amplitude $a_2(L)\propto 1/L^2$ and one finds that $a_2\,Φ_2(y) \ll a_1 \,Φ_1(y)$ already for small system size ($L> 10$). We illustrate the feasibility of observing the calculated finite-size corrections by performing simulations of the $XY$-model at low temperatures, including $T = 0$. △ Less

Submitted 4 April, 2016; originally announced April 2016.

Comments: 9 pages, 7 figures, to be submitted to Phys. Rev. E

Journal ref: Phys. Rev. E 94, 022145 (2016)

Helices in the wake of precipitation fronts

Authors: Shibi Thomas, Istvan Lagzi, Ferenc Molnar Jr, Zoltan Racz

Abstract: A theoretical study of the emergence of helices in the wake of precipitation fronts is presented. The precipitation dynamics is described by the Cahn-Hilliard equation and the fronts are obtained by quenching the system into a linearly unstable state. Confining the process onto the surface of a cylinder and using the pulled-front formalism, our analytical calculations show that there are front sol… ▽ More A theoretical study of the emergence of helices in the wake of precipitation fronts is presented. The precipitation dynamics is described by the Cahn-Hilliard equation and the fronts are obtained by quenching the system into a linearly unstable state. Confining the process onto the surface of a cylinder and using the pulled-front formalism, our analytical calculations show that there are front solutions that propagate into the unstable state and leave behind a helical structure. We find that helical patterns emerge only if the radius of the cylinder R is larger than a critical value R>R_c, in agreement with recent experiments. △ Less

Submitted 30 August, 2013; originally announced August 2013.

Comments: 7 pages, 3 figures, to appear in Phys. Rev. E

Probability of the emergence of helical precipitation patterns in the wake of reaction-diffusion fronts

Authors: Shibi Thomas, Istvan Lagzi, Ferenc Molnar Jr, Zoltan Racz

Abstract: Helical and helicoidal precipitation patterns emerging in the wake of reaction-diffusion fronts are studied. In our experiments, these chiral structures arise with well-defined probabilities P_H controlled by conditions such as e.g., the initial concentration of the reagents. We develop a model which describes the observed experimental trends. The results suggest that P_H is determined by a delica… ▽ More Helical and helicoidal precipitation patterns emerging in the wake of reaction-diffusion fronts are studied. In our experiments, these chiral structures arise with well-defined probabilities P_H controlled by conditions such as e.g., the initial concentration of the reagents. We develop a model which describes the observed experimental trends. The results suggest that P_H is determined by a delicate interplay among the time and length scales related to the front and to the unstable precipitation modes and, furthermore, the noise amplitude also plays a quantifiable role. △ Less

Submitted 20 February, 2013; originally announced February 2013.

Comments: 7 pages, 5 composite figures

Journal ref: Phys. Rev. Lett. 110, 078303 (2013)

Full counting statistics in a propagating quantum front and random matrix spectra

Authors: Viktor Eisler, Zoltan Racz

Abstract: One-dimensional free fermions are studied with emphasis on propagating fronts emerging from a step initial condition. The probability distribution of the number of particles at the edge of the front is determined exactly. It is found that the full counting statistics coincides with the eigenvalue statistics of the edge spectrum of matrices from the Gaussian unitary ensemble. The correspondence est… ▽ More One-dimensional free fermions are studied with emphasis on propagating fronts emerging from a step initial condition. The probability distribution of the number of particles at the edge of the front is determined exactly. It is found that the full counting statistics coincides with the eigenvalue statistics of the edge spectrum of matrices from the Gaussian unitary ensemble. The correspondence established between the random matrix eigenvalues and the particle positions yields the order statistics of the right-most particles in the front and, furthermore, it implies their subdiffusive spreading. △ Less

Submitted 19 November, 2012; v1 submitted 10 November, 2012; originally announced November 2012.

Comments: 5 pages, 3 figures

Journal ref: Phys. Rev. Lett. 110, 060602 (2013)

Order statistics of 1/f^α signals

Authors: N. R. Moloney, K. Ozogany, Z. Racz

Abstract: Order statistics of periodic, Gaussian noise with 1/f^α power spectrum is investigated. Using simulations and phenomenological arguments, we find three scaling regimes for the average gap d_k=<x_k-x_{k+1}> between the k-th and (k+1)-st largest values of the signal. The result d_k ~ 1/k known for independent, identically distributed variables remains valid for 0<α<1. Nontrivial, α-dependent scaling… ▽ More Order statistics of periodic, Gaussian noise with 1/f^α power spectrum is investigated. Using simulations and phenomenological arguments, we find three scaling regimes for the average gap d_k=<x_k-x_{k+1}> between the k-th and (k+1)-st largest values of the signal. The result d_k ~ 1/k known for independent, identically distributed variables remains valid for 0<α<1. Nontrivial, α-dependent scaling exponents d_k ~ k^{(α-3)/2} emerge for 1<α<5 and, finally, α-independent scaling, d_k ~ k is obtained for α>5. The spectra of average ordered values ε_k=<x_1-x_k> ~ k^β is also examined. The exponent β is derived from the gap scaling as well as by relating ε_k to the density of near extreme states. Known results for the density of near extreme states combined with scaling suggest that β(α=2)=1/2, β(4)=3/2, and beta(infinity)=2 are exact values. We also show that parallels can be drawn between ε_k and the quantum mechanical spectra of a particle in power-law potentials. △ Less

Submitted 25 September, 2011; originally announced September 2011.

Comments: 8 pages, 5 figures

Casimir effect in the nonequilibrium steady-state of a quantum spin chain

Authors: Diego Luis Gonzalez Cabrera, Zoltan Racz, Frederic van Wijland

Abstract: We present a fully microscopics-based calculation of the Casimir effect in a nonequilibrium system, namely an energy flux driven quantum XX chain. The force between the walls (transverse-field impurities) is calculated in a nonequilibrium steady state which is prepared by letting the system evolve from an initial state with the two halves of the chain prepared at equilibrium at different temperatu… ▽ More We present a fully microscopics-based calculation of the Casimir effect in a nonequilibrium system, namely an energy flux driven quantum XX chain. The force between the walls (transverse-field impurities) is calculated in a nonequilibrium steady state which is prepared by letting the system evolve from an initial state with the two halves of the chain prepared at equilibrium at different temperatures. The steady state emerging in the large-time limit is homogeneous but carries an energy flux. The Casimir force in this nonequilibrium state is calculated analytically in the limit when the transverse fields are small. We find that the the Casimir force range is reduced compared to the equilibrium case, and suggest that the reason for this is the reduction of fluctuations in the flux carrying steady state. △ Less

Submitted 22 April, 2010; originally announced April 2010.

Comments: 11 pages

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

Coarsening of Precipitation Patterns in a Moving Reaction-Diffusion Front

Authors: A. Volford, I. Lagzi, F. Molnar Jr, Z. Racz

Abstract: Precipitation patterns emerging in a 2D moving front are investigated on the example of NaOH diffusing into a gel containing AlCl_3. The time evolution of the precipitate Al(OH)_3 can be observed since the precipitate redissolves in the excess outer electrolyte NaOH and thus it exists only in a narrow, optically accessible region of the reaction front. The patterns display self-similar coarsenin… ▽ More Precipitation patterns emerging in a 2D moving front are investigated on the example of NaOH diffusing into a gel containing AlCl_3. The time evolution of the precipitate Al(OH)_3 can be observed since the precipitate redissolves in the excess outer electrolyte NaOH and thus it exists only in a narrow, optically accessible region of the reaction front. The patterns display self-similar coarsening with a characteristic length, xi(t), increasing with time t as xi(t) ~ t^(1/2). A theory based on Cahn-Hilliard phase-separation dynamics including dissolution is shown to yield agreement with the experiments. △ Less

Submitted 10 November, 2009; originally announced November 2009.

Comments: 4 pages, 7 figures, to appear in Phys. Rev. E

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)

Finite-size scaling in extreme statistics

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

Abstract: We study the convergence and shape correction to the limit distributions of extreme values due to the finite size (FS) of data sets. A renormalization method is introduced for the case of independent, identically distributed (iid) variables, showing that the iid universality classes are subdivided according to the exponent of the FS convergence, which determines the leading order FS shape correc… ▽ More We study the convergence and shape correction to the limit distributions of extreme values due to the finite size (FS) of data sets. A renormalization method is introduced for the case of independent, identically distributed (iid) variables, showing that the iid universality classes are subdivided according to the exponent of the FS convergence, which determines the leading order FS shape correction function as well. We find that, for the correlated systems of subcritical percolation and 1/f^alpha stationary (alpha<1) noise, the iid shape correction compares favorably to simulations. Furthermore, for the strongly correlated regime (alpha>1) of 1/f^alpha noise, the shape correction is obtained in terms of the limit distribution itself. △ Less

Submitted 24 December, 2007; originally announced December 2007.

Comments: 4 pages, 3 figures

Extreme statistics for time series: Distribution of the maximum relative to the initial value

Authors: T. W. Burkhardt, G. Gyorgyi, N. R. Moloney, Z. Racz

Abstract: The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1/f^alpha power spectr… ▽ More The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1/f^alpha power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRH_I). The exact MRH_I distribution is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random acceleration), and alpha=infinity (single sinusoidal mode). For other, intermediate values of alpha, the distribution is determined from simulations. We find that the MRH_I distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all alpha. The two main distinguishing features of the MRH_I distribution are the much larger weight for small relative heights and the divergence at zero height for alpha>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some non-periodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRH_I distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions. △ Less

Submitted 18 July, 2007; originally announced July 2007.

Comments: 29 pages, 4 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 motion of precipitation bands

Authors: Istvan Lagzi, Peter Papai, Zoltan Racz

Abstract: Formation and dynamics of an Al(OH)_3 precipitation ring is studied by diffusing NaOH into a gel containing AlCl_3. Limited feeding of the outer electrolyte (NaOH) is found to yield an intricate ring-dynamics which involves stopping and reversal of the direction of motion of the precipitation ring, and evolution into stationary multi-ring structures. A model of the ring-dynamics is developed by… ▽ More Formation and dynamics of an Al(OH)_3 precipitation ring is studied by diffusing NaOH into a gel containing AlCl_3. Limited feeding of the outer electrolyte (NaOH) is found to yield an intricate ring-dynamics which involves stopping and reversal of the direction of motion of the precipitation ring, and evolution into stationary multi-ring structures. A model of the ring-dynamics is developed by combining a phase separation scenario for the precipitation with the redissolution (complex formation) of the precipitate in the excess of the outer electrolyte. △ Less

Submitted 1 December, 2006; originally announced December 2006.

Comments: 6 pages, 15 figures

Maximal height statistics for 1/f^alpha signals

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

Abstract: Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one dimensional interfaces) displaying a 1/f^alpha power spectrum. For 0<alpha<1 (regime of decaying correlations), we observe that the mathematically established limiting distribution (Fisher-Tippett-Gumbel distribution) is approached extremely slowly as the sampl… ▽ More Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one dimensional interfaces) displaying a 1/f^alpha power spectrum. For 0<alpha<1 (regime of decaying correlations), we observe that the mathematically established limiting distribution (Fisher-Tippett-Gumbel distribution) is approached extremely slowly as the sample size increases. The convergence is rapid for alpha>1 (regime of strong correlations) and a highly accurate picture gallery of distribution functions can be constructed numerically. Analytical results can be obtained in the limit alpha -> infinity and, for large alpha, by perturbation expansion. Furthermore, using path integral techniques we derive a trace formula for the distribution function, valid for alpha=2n even integer. From the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha > 1 is found to be in agreement with simulations. Comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions. △ Less

Submitted 17 October, 2006; originally announced October 2006.

Comments: 17 pages, 8 figures, RevTex4

Fluctuations in subsystems of the zero temperature XX chain: Emergence of an effective temperature

Authors: V. Eisler, O. Legeza, Z. Racz

Abstract: The zero-temperature XX chain is studied with emphasis on the properties of a block of $L$ spins inside the chain. We investigate the quantum fluctuations resulting from the entanglement of the block with the rest of the chain using analytical as well as numerical (density matrix renormalization group) methods. It is found that the rest of the chain acts as a thermal environment and an effective… ▽ More The zero-temperature XX chain is studied with emphasis on the properties of a block of $L$ spins inside the chain. We investigate the quantum fluctuations resulting from the entanglement of the block with the rest of the chain using analytical as well as numerical (density matrix renormalization group) methods. It is found that the rest of the chain acts as a thermal environment and an effective temperature can be introduced to describe the fluctuations. We show that the effective temperature description is robust in the sense that several independent definitions (through fluctuation dissipation theorem, comparing with a finite temperature system) yield the same functional form in the limit of large block size ($L\to\infty$). The effective temperature can also be shown to satisfy the basic requirements on how it changes when two bodies of equal or unequal temperatures are brought into contact. △ Less

Submitted 24 November, 2006; v1 submitted 27 September, 2006; originally announced September 2006.

Comments: 19 pages, 7 figures

Journal ref: J. Stat. Mech. (2006) P11013

Fisher Waves and Front Roughening in a Two-Species Invasion Model with Preemptive Competition

Authors: L. O'Malley, B. Kozma, G. Korniss, Z. Racz, T. Caraco

Abstract: We study front propagation when an invading species competes with a resident; we assume nearest-neighbor preemptive competition for resources in an individual-based, two-dimensional lattice model. The asymptotic front velocity exhibits power-law dependence on the difference between the two species' clonal propagation rates (key ecological parameters). The mean-field approximation behaves similar… ▽ More We study front propagation when an invading species competes with a resident; we assume nearest-neighbor preemptive competition for resources in an individual-based, two-dimensional lattice model. The asymptotic front velocity exhibits power-law dependence on the difference between the two species' clonal propagation rates (key ecological parameters). The mean-field approximation behaves similarly, but the power law's exponent slightly differs from the individual-based model's result. We also study roughening of the front, using the framework of non-equilibrium interface growth. Our analysis indicates that initially flat, linear invading fronts exhibit Kardar-Parisi-Zhang (KPZ) roughening in one transverse dimension. Further, this finding implies, and is also confirmed by simulations, that the temporal correction to the asymptotic front velocity is of ${\cal O}(t^{-2/3})$. △ Less

Submitted 1 August, 2006; originally announced August 2006.

Comments: 8 pages, 5 figures; Papers on related work can be found at http://www.rpi.edu/~korniss/Research

Journal ref: Phys. Rev. E 74, 041116 (2006) .

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).

Fisher Waves and the Velocity of Front Propagation in a Two-Species Invasion Model with Preemptive Competition

Authors: L. O'Malley, B. Kozma, G. Korniss, Z. Racz, T. Caraco

Abstract: We consider an individual-based two-dimensional spatial model with nearest-neighbor preemptive competition to study front propagation between an invader and a resident species. In particular, we investigate the asymptotic front velocity and compare it with mean-field predictions. We consider an individual-based two-dimensional spatial model with nearest-neighbor preemptive competition to study front propagation between an invader and a resident species. In particular, we investigate the asymptotic front velocity and compare it with mean-field predictions. △ Less

Submitted 13 March, 2006; originally announced March 2006.

Comments: Computer Simulation Studies in Condensed Matter Physics XIX, edited by D.P. Landau, S.P. Lewis, and H.-B. Schuttler, (Springer, Heidelberg, Berlin, in press)

Synchronization Landscapes in Small-World-Connected Computer Networks

Authors: H. Guclu, G. Korniss, M. A. Novotny, Z. Toroczkai, Z. Racz

Abstract: Motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small-world networks, embedded in one and two-dimensions. We find that the synchronization landscape (corresponding to the progress of the individual processors) exhibits Kardar-Parisi-Zhang-like kinetic roughening on regular networks with short-range communication links. Although the… ▽ More Motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small-world networks, embedded in one and two-dimensions. We find that the synchronization landscape (corresponding to the progress of the individual processors) exhibits Kardar-Parisi-Zhang-like kinetic roughening on regular networks with short-range communication links. Although the processors, on average, progress at a nonzero rate, their spread (the width of the synchronization landscape) diverges with the number of nodes (desynchronized state) hindering efficient data management. When random communication links are added on top of the one and two-dimensional regular networks (resulting in a small-world network), large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system-size limit (synchronized state). In the resulting synchronization scheme, the processors make close-to-uniform progress with a nonzero rate without global intervention. We obtain our results by ``simulating the simulations", based on the exact algorithmic rules, supported by coarse-grained arguments. △ Less

Submitted 4 January, 2006; v1 submitted 3 January, 2006; originally announced January 2006.

Comments: 20 pages, 22 figures

Journal ref: Phys. Rev. E 73, 066115 (2006)

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).

Energy flux distribution in a two-temperature Ising model

Authors: Vivien Lecomte, Zoltan Racz, Frederic van Wijland

Abstract: The nonequilibrium steady state of an infinite-range Ising model is studied. The steady state is obtained by dividing the spins into two groups and attaching them to two heat baths generating spin flips at different temperatures. In the thermodynamic limit, the resulting dynamics can be solved exactly, and the probability flow in the phase space can be visualized. We can calculate the steady sta… ▽ More The nonequilibrium steady state of an infinite-range Ising model is studied. The steady state is obtained by dividing the spins into two groups and attaching them to two heat baths generating spin flips at different temperatures. In the thermodynamic limit, the resulting dynamics can be solved exactly, and the probability flow in the phase space can be visualized. We can calculate the steady state fluctuations far from equilibrium and, in particular, we find the exact probability distribution of the energy current in both the high- and low-temperature phase. △ Less

Submitted 20 December, 2004; originally announced December 2004.

Comments: 19 pages, 4 figures

Journal ref: J. Stat. Mech. (2005) P02008

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).

Large deviations in weakly interacting boundary driven lattice gases

Authors: Frederic van Wijland, Zoltan Racz

Abstract: One-dimensional, boundary-driven lattice gases with local interactions are studied in the weakly interacting limit. The density profiles and the correlation functions are calculated to first order in the interaction strength for zero-range and short-range processes differing only in the specifics of the detailed-balance dynamics. Furthermore, the effective free-energy (large-deviation function)… ▽ More One-dimensional, boundary-driven lattice gases with local interactions are studied in the weakly interacting limit. The density profiles and the correlation functions are calculated to first order in the interaction strength for zero-range and short-range processes differing only in the specifics of the detailed-balance dynamics. Furthermore, the effective free-energy (large-deviation function) and the integrated current distribution are also found to this order. From the former, we find that the boundary drive generates long-range correlations only for the short-range dynamics while the latter provides support to an additivity principle recently proposed by Bodineau and Derrida. △ Less

Submitted 15 April, 2004; originally announced April 2004.

Comments: 27 pages, 2 figures

Report number: LPT Orsay 04-29

Dynamic scaling of fronts in the quantum XX chain

Authors: V. Hunyadi, Z. Racz, L. Sasvari

Abstract: The dynamics of the transverse magnetization in the zero-temperature XX chain is studied with emphasis on fronts emerging from steplike initial magnetization profiles. The fronts move with fixed velocity and display a staircase like internal structure whose dynamic scaling is explored both analytically and numerically. The front region is found to spread with time sub-diffusively with the height… ▽ More The dynamics of the transverse magnetization in the zero-temperature XX chain is studied with emphasis on fronts emerging from steplike initial magnetization profiles. The fronts move with fixed velocity and display a staircase like internal structure whose dynamic scaling is explored both analytically and numerically. The front region is found to spread with time sub-diffusively with the height and the width of the staircase steps scaling as t^(-1/3) and t^1/3, respectively. The areas under the steps are independent of time, thus the magnetization relaxes in quantized "steps" of spin-flips. △ Less

Submitted 10 December, 2003; originally announced December 2003.

Comments: 4 pages, 3 eps figures, RevTex

Journal ref: Phys. Rev. E 69, 066103 (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)

Statistics of extremal intensities for Gaussian interfaces

Authors: G. Gyorgyi, P. C. W. Holdsworth, B. Portelli, Z. Racz

Abstract: The extremal Fourier intensities are studied for stationary Edwards-Wilkinson-type, Gaussian, interfaces with power-law dispersion. We calculate the probability distribution of the maximal intensity and find that, generically, it does not coincide with the distribution of the integrated power spectrum (i.e. roughness of the surface), nor does it obey any of the known extreme statistics limit dis… ▽ More The extremal Fourier intensities are studied for stationary Edwards-Wilkinson-type, Gaussian, interfaces with power-law dispersion. We calculate the probability distribution of the maximal intensity and find that, generically, it does not coincide with the distribution of the integrated power spectrum (i.e. roughness of the surface), nor does it obey any of the known extreme statistics limit distributions. The Fisher-Tippett-Gumbel limit distribution is, however, recovered in three cases: (i) in the non-dispersive (white noise) limit, (ii) for high dimensions, and (iii) when only short-wavelength modes are kept. In the last two cases the limit distribution emerges in novel scenarios. △ Less

Submitted 25 July, 2003; originally announced July 2003.

Comments: 15 pages, including 7 ps figures

Scaling functions for nonequilibrium fluctuations: A picture gallery

Authors: Zoltan Racz

Abstract: The emergence of non-gaussian distributions for macroscopic quantities in nonequilibrium steady states is discussed with emphasis on the effective criticality and on the ensuing universality of distribution functions. The following problems are treated in more detail: nonequilibrium interface fluctuations (the problem of upper critical dimension of the Kardar-Parisi-Zhang equation), roughness of… ▽ More The emergence of non-gaussian distributions for macroscopic quantities in nonequilibrium steady states is discussed with emphasis on the effective criticality and on the ensuing universality of distribution functions. The following problems are treated in more detail: nonequilibrium interface fluctuations (the problem of upper critical dimension of the Kardar-Parisi-Zhang equation), roughness of signals displaying Gaussian 1/f power spectra (the relationship to extreme-value statistics), effects of boundary conditions (randomness of the digits of pi). △ Less

Submitted 20 July, 2003; originally announced July 2003.

Comments: Invited contribution at SPIE conference on Fluctuations and Noise, Santa Fe, 2003; 9 pages, 6 figures

Journal ref: SPIE Proceedings, Vol. 5112, pp. 248-258 (2003)

Magnetization distribution in the transverse Ising chain with energy flux

Authors: V. Eisler, Z. Racz, F. van Wijland

Abstract: The zero-temperature transverse Ising chain carrying an energy flux j_E is studied with the aim of determining the nonequilibrium distribution functions, P(M_z) and P(M_x), of its transverse and longitudinal magnetizations, respectively. An exact calculation reveals that P(M_z) is a Gaussian both at j_E=0 and j_E not equal 0, and the width of the distribution decreases with increasing energy flu… ▽ More The zero-temperature transverse Ising chain carrying an energy flux j_E is studied with the aim of determining the nonequilibrium distribution functions, P(M_z) and P(M_x), of its transverse and longitudinal magnetizations, respectively. An exact calculation reveals that P(M_z) is a Gaussian both at j_E=0 and j_E not equal 0, and the width of the distribution decreases with increasing energy flux. The distribution of the order-parameter fluctuations, P(M_x), is evaluated numerically for spin-chains of up to 20 spins. For the equilibrium case (j_E=0), we find the expected Gaussian fluctuations away from the critical point while the critical order-parameter fluctuations are shown to be non-gaussian with a scaling function Phi(x)=Phi(M_x/<M_x>)=<M_x>P(M_x) strongly dependent on the boundary conditions. When j_E not equal 0, the system displays long-range, oscillating correlations but P(M_x) is a Gaussian nevertheless, and the width of the Gaussian decreases with increasing j_E. In particular, we find that, at critical transverse field, the width has a j_E^(-3/8) asymptotic in the j_E -> 0 limit. △ Less

Submitted 24 January, 2003; originally announced January 2003.

Comments: 8 pages, 5 ps figures

Journal ref: Phys. Rev. E 67, 056129 (2003)

Nonequilibrium Phase Transitions

Authors: Zoltan Racz

Abstract: Nonequilibrium phase transitions are discussed with emphasis on general features such as the role of detailed balance violation in generating effective (long-range) interactions, the importance of dynamical anisotropies, the connection between various mechanisms generating power-law correlations, and the emergence of universal distribution functions for macroscopic quantities. Quantum spin chain… ▽ More Nonequilibrium phase transitions are discussed with emphasis on general features such as the role of detailed balance violation in generating effective (long-range) interactions, the importance of dynamical anisotropies, the connection between various mechanisms generating power-law correlations, and the emergence of universal distribution functions for macroscopic quantities. Quantum spin chains are also discussed in order to demonstrate how to construct steady-states carrying fluxes in quantum systems, and to explain how the fluxes may generate power-law correlations. △ Less

Submitted 20 October, 2002; originally announced October 2002.

Comments: Lecture Notes, Les Houches, July 2002, 43 pages, 11 figures

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)

On the velocity distributions of the one-dimensional inelastic gas

Authors: A. Barrat, T. Biben, Z. Racz, E. Trizac, F. van Wijland

Abstract: We consider the single-particle velocity distribution of a one-dimensional fluid of inelastic particles. Both the freely evolving (cooling) system and the non-equilibrium stationary state obtained in the presence of random forcing are investigated, and special emphasis is paid to the small inelasticity limit. The results are obtained from analytical arguments applied to the Boltzmann equation al… ▽ More We consider the single-particle velocity distribution of a one-dimensional fluid of inelastic particles. Both the freely evolving (cooling) system and the non-equilibrium stationary state obtained in the presence of random forcing are investigated, and special emphasis is paid to the small inelasticity limit. The results are obtained from analytical arguments applied to the Boltzmann equation along with three complementary numerical techniques (Molecular Dynamics, Direct Monte Carlo Simulation Methods and iterative solutions of integro-differential kinetic equations). For the freely cooling fluid, we investigate in detail the scaling properties of the bimodal velocity distribution emerging close to elasticity and calculate the scaling function associated with the distribution function. In the heated steady state, we find that, depending on the inelasticity, the distribution function may display two different stretched exponential tails at large velocities. The inelasticity dependence of the crossover velocity is determined and it is found that the extremely high velocity tail may not be observable at ``experimentally relevant'' inelasticities. △ Less

Submitted 17 October, 2001; originally announced October 2001.

Comments: Latex, 14 pages, 12 eps figures

Journal ref: J. Phys. A 35, 463 (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)

Width Distributions and the Upper Critical Dimension of KPZ Interfaces

Authors: E. Marinari, A. Pagnani, G. Parisi, Z. Racz

Abstract: Simulations of restricted solid-on-solid growth models are used to build the width-distributions of d=2-5 dimensional KPZ interfaces. We find that the universal scaling function associated with the steady-state width-distribution changes smoothly as d is increased, thus strongly suggesting that d=4 is not an upper critical dimension for the KPZ equation. The dimensional trends observed in the sc… ▽ More Simulations of restricted solid-on-solid growth models are used to build the width-distributions of d=2-5 dimensional KPZ interfaces. We find that the universal scaling function associated with the steady-state width-distribution changes smoothly as d is increased, thus strongly suggesting that d=4 is not an upper critical dimension for the KPZ equation. The dimensional trends observed in the scaling functions indicate that the upper critical dimension is at infinity. △ Less

Submitted 8 May, 2001; originally announced May 2001.

Comments: 4 pages, 4 postscript figures, RevTex

Journal ref: Phys. Rev. E 65, 026136 (2002)

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)

Presence of energy flux in quantum spin chains: An experimental signature

Authors: Z. Racz

Abstract: Using the XXZ model for the description of one-dimensional magnetic materials we show that an energy flux, j_E, produces a shift, dk (j_E)^0.5, in the characteristic wavenumber of the spin-spin correlations. We estimate dk for a realistic experimental setup and find that it is measurable in inelastic neutron scattering experiments. Using the XXZ model for the description of one-dimensional magnetic materials we show that an energy flux, j_E, produces a shift, dk (j_E)^0.5, in the characteristic wavenumber of the spin-spin correlations. We estimate dk for a realistic experimental setup and find that it is measurable in inelastic neutron scattering experiments. △ Less

Submitted 25 May, 2000; originally announced May 2000.

Comments: Talk presented at NATO Advanced Workshop on Nonlinear Science and Stochasticity, (Brussels, July 1999); to appear in J.Stat.Phys., 5 pages, 4 ps figures

Journal ref: J. Stat. Phys. 101, 273 (2000)

Concentration Profiles and Reaction Fronts in A + B -> C Type Processes: Effect of Background Ions

Authors: T. Unger, Z. Racz

Abstract: Diffusion and reaction of initially separated ions A- and B+ in the presence of counter ions A'+ and B'- is studied. The dynamics is described in terms of reaction-diffusion equations obeying local electroneutrality, and the time-evolution of ion-concentrations is determined. We find that, in the absence of reactions, unequal mobility of ions generate nontrivial features in the macroscopically o… ▽ More Diffusion and reaction of initially separated ions A- and B+ in the presence of counter ions A'+ and B'- is studied. The dynamics is described in terms of reaction-diffusion equations obeying local electroneutrality, and the time-evolution of ion-concentrations is determined. We find that, in the absence of reactions, unequal mobility of ions generate nontrivial features in the macroscopically observable concentration profiles. Switching on the reaction (A-) + (B+) -> C leads to the formation of a localized, diffusive reaction front and one finds that the properties of the front (e.g. effective diffusion constant) are affected by the background ions. The consequences of this effect on the formation of Liesegang patterns is discussed. △ Less

Submitted 9 November, 1999; originally announced November 1999.

Comments: 7 pages, 8 ps figures

Journal ref: Phys. Rev. E 61, 3583 (2000)

Formation of Liesegang Patterns

Authors: Zoltan Racz

Abstract: It has been recently shown that precipitation bands characteristic of Liesegang patterns emerge from spinodal decomposition of reaction products in the wake of moving reaction fronts. This mechanism explains the geometric sequence of band positions x_n ~ Q(1+p)^n and, furthermore, it yields a spacing coefficient, p, that is in agreement with the experimentally observed Matalon-Packter law. Here… ▽ More It has been recently shown that precipitation bands characteristic of Liesegang patterns emerge from spinodal decomposition of reaction products in the wake of moving reaction fronts. This mechanism explains the geometric sequence of band positions x_n ~ Q(1+p)^n and, furthermore, it yields a spacing coefficient, p, that is in agreement with the experimentally observed Matalon-Packter law. Here I examine the assumptions underlying this theory and discuss the choice of input parameters that leads to experimentally observable patterns. I also show that the so called width law relating the position and the width of the bands w_n ~ x_n follows naturally from this theory. △ Less

Submitted 4 October, 1999; originally announced October 1999.

Comments: Talk presented at NATO Advanced Workshop on Statistical Physics Applied to Practical Problems (Budapest, May 1999); to appear in Physica A. 6 pages, 1 jpeg and 3 ps figures

Journal ref: Physica A 274, 50 (1999)

Formation of Liesegang patterns: A spinodal decomposition scenario

Authors: T. Antal, M. Droz, J. Magnin, Z. Rácz

Abstract: Spinodal decomposition in the presence of a moving particle source is proposed as a mechanism for the formation of Liesegang bands. This mechanism yields a sequence of band positions x_n that obeys the spacing law x_n~Q(1+p)^n. The dependence of the parameters p and Q on the initial concentration of the reagents is determined and we find that the functional form of p is in agreement with the exp… ▽ More Spinodal decomposition in the presence of a moving particle source is proposed as a mechanism for the formation of Liesegang bands. This mechanism yields a sequence of band positions x_n that obeys the spacing law x_n~Q(1+p)^n. The dependence of the parameters p and Q on the initial concentration of the reagents is determined and we find that the functional form of p is in agreement with the experimentally observed Matalon-Packter law. △ Less

Submitted 29 March, 1999; originally announced March 1999.

Comments: RevTex, 4 pages, 4 eps figures

Journal ref: Phys. Rev. Lett. 83, 2880 (1999).

Transport in the XX chain at zero temperature: Emergence of flat magnetization profiles

Authors: T. Antal, Z. Rácz, A. Rákos, G. M. Schütz

Abstract: We study the connection between magnetization transport and magnetization profiles in zero-temperature XX chains. The time evolution of the transverse magnetization, m(x,t), is calculated using an inhomogeneous initial state that is the ground state at fixed magnetization but with m reversed from -m_0 for x<0 to m_0 for x>0. In the long-time limit, the magnetization evolves into a scaling form m… ▽ More We study the connection between magnetization transport and magnetization profiles in zero-temperature XX chains. The time evolution of the transverse magnetization, m(x,t), is calculated using an inhomogeneous initial state that is the ground state at fixed magnetization but with m reversed from -m_0 for x<0 to m_0 for x>0. In the long-time limit, the magnetization evolves into a scaling form m(x,t)=P(x/t) and the profile develops a flat part (m=P=0) in the |x/t|< c(m_0) region. The flat region shrinks to zero if m_0->1/2 while it expands with the maximum velocity, c_0=1, for m_0->0. The states emerging in the scaling limit are compared to those of a homogeneous system where the same magnetization current is driven by a bulk field, and we find that the expectation values of various quantities (energy, occupation number in the fermionic representation) agree in the two systems. △ Less

Submitted 14 December, 1998; originally announced December 1998.

Comments: RevTex, 8 pages, 3 ps figures

Journal ref: Phys. Rev. E59, 4912 (1999)

Liesegang patterns: Effect of dissociation of the invading electrolyte

Authors: B. Chopard, M. Droz, J. Magnin, Z. Rácz, M. Zrinyi

Abstract: The effect of dissociation of the invading electrolyte on the formation of Liesegang bands is investigated. We find, using organic compounds with known dissociation constants, that the spacing coefficient, 1+p, that characterizes the position of the n-th band as x_n ~ (1+p)^n, decreases with increasing dissociation constant, K_d. Theoretical arguments are developed to explain these experimental… ▽ More The effect of dissociation of the invading electrolyte on the formation of Liesegang bands is investigated. We find, using organic compounds with known dissociation constants, that the spacing coefficient, 1+p, that characterizes the position of the n-th band as x_n ~ (1+p)^n, decreases with increasing dissociation constant, K_d. Theoretical arguments are developed to explain these experimental findings and to calculate explicitly the K_d dependence of 1+p. △ Less

Submitted 29 September, 1998; originally announced September 1998.

Comments: RevTex, 8 pages, 3 eps figures

Journal ref: J. Phys. Chem. A 103, 1432 (1999)

Derivation of the Matalon-Packter law for Liesegang patterns

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

Abstract: Theoretical models of the Liesegang phenomena are studied and simple expressions for the spacing coefficients characterizing the patterns are derived. The emphasis is on displaying the explicit dependences on the concentrations of the inner- and the outer-electrolytes. Competing theories (ion-product supersaturation, nucleation and droplet growth, induced sol- coagulation) are treated with the a… ▽ More Theoretical models of the Liesegang phenomena are studied and simple expressions for the spacing coefficients characterizing the patterns are derived. The emphasis is on displaying the explicit dependences on the concentrations of the inner- and the outer-electrolytes. Competing theories (ion-product supersaturation, nucleation and droplet growth, induced sol- coagulation) are treated with the aim of finding the distinguishing features of the theories. The predictions are compared with experiments and the results suggest that the induced sol-coagulation theory is the best candidate for describing the experimental observations embodied in the Matalon-Packter law. △ Less

Submitted 17 July, 1998; originally announced July 1998.

Comments: 9 pages, 7 figures, RevTex

Report number: UGVA DPT 1998/07-1012

Journal ref: J. Chem. Phys. 109, 9479 (1998)

Non-equilibrium critical behavior of O(n)-symmetric systems: Effect of reversible mode-coupling terms and dynamical anisotropy

Authors: Uwe C. Täuber, Jaime E. Santos, Zoltán Rácz

Abstract: Phase transitions in non-equilibrium steady states of O(n)-symmetric models with reversible mode couplings are studied using dynamic field theory and the renormalization group. The systems are driven out of equilibrium by dynamical anisotropy in the noise for the conserved quantities, i.e., by constraining their diffusive dynamics to be at different temperatures T^\par and T^\perp in d_\par- and… ▽ More Phase transitions in non-equilibrium steady states of O(n)-symmetric models with reversible mode couplings are studied using dynamic field theory and the renormalization group. The systems are driven out of equilibrium by dynamical anisotropy in the noise for the conserved quantities, i.e., by constraining their diffusive dynamics to be at different temperatures T^\par and T^\perp in d_\par- and d_\perp-dimensional subspaces, respectively. In the case of the Sasv'ari-Schwabl-Sz'epfalusy (SSS) model for planar ferro- and isotropic antiferromagnets, we assume a dynamical anisotropy in the noise for the non-critical conserved quantities that are dynamically coupled to the non-conserved order parameter. We find the equilibrium fixed point (with isotropic noise) to be stable with respect to these non-equilibrium perturbations, and the familiar equilibrium exponents therefore describe the asymptotic static and dynamic critical behavior. Novel critical features are only found in extreme limits, where the ratio of the effective noise temperatures T^\par/T^\perp is either zero or infinite. On the other hand, for model J for isotropic ferromagnets with a conserved order parameter, the dynamical noise anisotropy induces effective long-range elastic forces, which lead to a softening only of the d_\perp-dimensional sector in wavevector space with lower noise temperature T^\perp < T^\par. The ensuing static and dynamic critical behavior is described by power laws of a hitherto unidentified universality class, which, however, is not accessible by perturbational means for d_\par \geq 1. We obtain formal expressions for the novel critical exponents in a double expansion about the static and dynamic upper critical dimensions and d_\par, i.e., about the equilibrium theory. △ Less

Submitted 28 September, 1998; v1 submitted 14 July, 1998; originally announced July 1998.

Comments: EPJ style LaTex2e (style files included), 22 pages, eps file for one figure included; final version (misprints corrected), to appear in EPJB

Journal ref: Eur. Phys. J. B 7 (1999) 309; e: Eur. Phys. J. B 9 (1999) 567

Isotropic Transverse XY Chain with Energy- and Magnetization Currents

Authors: T. Antal, Z. Racz, A. Rakos, G. M. Schutz

Abstract: The ground-state correlations are investigated for an isotropic transverse XY chain which is constrained to carry either a current of magnetization J_M or a current of energy J_E. We find that the effect of nonzero J_M on the large-distance decay of correlations is twofold: i) oscillations are introduced and ii) the amplitude of the power law decay increases with increasing current. The effect o… ▽ More The ground-state correlations are investigated for an isotropic transverse XY chain which is constrained to carry either a current of magnetization J_M or a current of energy J_E. We find that the effect of nonzero J_M on the large-distance decay of correlations is twofold: i) oscillations are introduced and ii) the amplitude of the power law decay increases with increasing current. The effect of energy current is more complex. Generically, correlations in current carrying states are found to decay faster than in the J_E=0 states, contrary to expectations that correlations are increased by the presence of currents. However, increasing the current, one reaches a special line where the correlations become comparable to those of the J_E=0 states. On this line, the symmetry of the ground state is enhanced and the transverse magnetization vanishes. Further increase of the current destroys the extra symmetry but the transverse magnetization remains at the high-symmetry, zero value. △ Less

Submitted 12 November, 1997; originally announced November 1997.

Comments: 7 pages, RevTex, 4 PostScript figures

Journal ref: Phys. Rev. E 57, 5184 (1998)

Localization-delocalization transition of a reaction-diffusion front near a semipermeable wall

Authors: B. Chopard, M. Droz, J. Magnin, Z. Racz

Abstract: The A+B --> C reaction-diffusion process is studied in a system where the reagents are separated by a semipermeable wall. We use reaction-diffusion equations to describe the process and to derive a scaling description for the long-time behavior of the reaction front. Furthermore, we show that a critical localization-delocalization transition takes place as a control parameter which depends on th… ▽ More The A+B --> C reaction-diffusion process is studied in a system where the reagents are separated by a semipermeable wall. We use reaction-diffusion equations to describe the process and to derive a scaling description for the long-time behavior of the reaction front. Furthermore, we show that a critical localization-delocalization transition takes place as a control parameter which depends on the initial densities and on the diffusion constants is varied. The transition is between a reaction front of finite width that is localized at the wall and a front which is detached and moves away from the wall. At the critical point, the reaction front remains at the wall but its width diverges with time [as t^(1/6) in mean-field approximation]. △ Less

Submitted 30 June, 1997; originally announced June 1997.

Comments: 7 pages, PS file

Nonequilibrium Steady State in a Quantum System: One-dimensional Transverse Ising Model with Energy Current

Authors: T. Antal, Z. Racz, L. Sasvari

Abstract: We study the nonequilibrium steady states of an Ising chain in a transverse field, h, by investigating the effect of a field, lambda, which drives the current of energy. The zero-temperature, h-lambda phase diagram is determined exactly and it is found that the energy current appears continuously above a threshold, lambda > lambda_c(h). The long-range magnetic order (present for h<h_c, lambda <… ▽ More We study the nonequilibrium steady states of an Ising chain in a transverse field, h, by investigating the effect of a field, lambda, which drives the current of energy. The zero-temperature, h-lambda phase diagram is determined exactly and it is found that the energy current appears continuously above a threshold, lambda > lambda_c(h). The long-range magnetic order (present for h<h_c, lambda < lambda_c) is destroyed in the current-carrying state where the correlations are characterized by power-law, oscillatory decay. The mechanism which generates the power-law correlations in the current-carrying state is discussed. △ Less

Submitted 4 December, 1996; originally announced December 1996.

Comments: 4 pages, RevTeX, 2 PostScript figures, to appear in Phys.Rev.Lett

Journal ref: Phys. Rev. Lett. 78, 167 (1997)

Critical Behavior of O(n)-symmetric Systems With Reversible Mode-coupling Terms: Stability Against Detailed-balance Violation

Authors: Uwe C. Täuber, Zoltán Rácz

Abstract: We investigate nonequilibrium critical properties of $O(n)$-symmetric models with reversible mode-coupling terms. Specifically, a variant of the model of Sasvári, Schwabl, and Szépfalusy is studied, where violation of detailed balance is incorporated by allowing the order parameter and the dynamically coupled conserved quantities to be governed by heat baths of different temperatures $T_S$ and… ▽ More We investigate nonequilibrium critical properties of $O(n)$-symmetric models with reversible mode-coupling terms. Specifically, a variant of the model of Sasvári, Schwabl, and Szépfalusy is studied, where violation of detailed balance is incorporated by allowing the order parameter and the dynamically coupled conserved quantities to be governed by heat baths of different temperatures $T_S$ and $T_M$, respectively. Dynamic perturbation theory and the field-theoretic renormalization group are applied to one-loop order, and yield two new fixed points in addition to the equilibrium ones. The first one corresponds to $Θ= T_S / T_M = \infty$ and leads to model A critical behavior for the order parameter and to anomalous noise correlations for the generalized angular momenta; the second one is at $Θ= 0$ and is characterized by mean-field behavior of the conserved quantities, by a dynamic exponent $z = d / 2$ equal to that of the equilibrium SSS model, and by modified static critical exponents. However, both these new fixed points are unstable, and upon approaching the critical point detailed balance is restored, and the equilibrium static and dynamic critical properties are recovered. △ Less

Submitted 22 October, 1996; originally announced October 1996.

Comments: 18 pages, RevTeX, 1 figure included as eps-file; submitted to Phys. Rev. E

Report number: OUTP-96-35-S

Journal ref: Phys. Rev. E 55 (1997) 4120-4136

Dynamic Scaling of Width Distribution in Edwards--Wilkinson Type Models of Interface Dynamics

Authors: T. Antal, Z. Racz

Abstract: Edwards--Wilkinson type models are studied in 1+1 dimensions and the time-dependent distribution, P_L(w^2,t), of the square of the width of an interface, w^2, is calculated for systems of size L. We find that, using a flat interface as an initial condition, P_L(w^2,t) can be calculated exactly and it obeys scaling in the form <w^2>_\infty P_L(w^2,t) = Phi(w^2 / <w^2>_\infty, t/L^2) where <w^2>_\… ▽ More Edwards--Wilkinson type models are studied in 1+1 dimensions and the time-dependent distribution, P_L(w^2,t), of the square of the width of an interface, w^2, is calculated for systems of size L. We find that, using a flat interface as an initial condition, P_L(w^2,t) can be calculated exactly and it obeys scaling in the form <w^2>_\infty P_L(w^2,t) = Phi(w^2 / <w^2>_\infty, t/L^2) where <w^2>_\infty is the stationary value of w^2. For more complicated initial states, scaling is observed only in the large- time limit and the scaling function depends on the initial amplitude of the longest wavelength mode. The short-time limit is also interesting since P_L(w^2,t) is found to closely approximate the log-normal distribution. These results are confirmed by Monte Carlo simulations on a `roof-top' model of surface evolution. △ Less

Submitted 31 October, 1995; originally announced October 1995.

Comments: 5 pages, latex, 3 ps figures in a separate files, submitted to Phys.Rev.E

Journal ref: Phys. Rev. E 54, 2256 (1996)

Correlations in Ising chains with non-integrable interactions

Authors: B. Bergersen, Z. Racz, H. -J. Xu

Abstract: Two-spin correlations generated by interactions which decay with distance r as r^{-1-sigma} with -1 <sigma <0 are calculated for periodic Ising chains of length L. Mean-field theory indicates that the correlations, C(r,L), diminish in the thermodynamic limit L -> \infty, but they contain a singular structure for r/L -> 0 which can be observed by introducing magnified correlations, LC(r,L)-\sum_r… ▽ More Two-spin correlations generated by interactions which decay with distance r as r^{-1-sigma} with -1 <sigma <0 are calculated for periodic Ising chains of length L. Mean-field theory indicates that the correlations, C(r,L), diminish in the thermodynamic limit L -> \infty, but they contain a singular structure for r/L -> 0 which can be observed by introducing magnified correlations, LC(r,L)-\sum_r C(r,L). The magnified correlations are shown to have a scaling form F(r/L) and the singular structure of F(x) for x->0 is found to be the same at all temperatures including the critical point. These conclusions are supported by the results of Monte Carlo simulations for systems with sigma =-0.50 and -0.25 both at the critical temperature T=Tc and at T=2Tc. △ Less

Submitted 7 September, 1995; originally announced September 1995.

Comments: 13 pages, latex, 5 eps figures in a separate uuencoded file, to appear in Phys.Rev.E