arXiv subjects

H. K. Janssen

Publications and source records attributed to H. K. Janssen.

18 recordsLinked to original sources

arXiv2007

Universal properties of population dynamics with fluctuating resources

Starting from the well-known field theory for directed percolation, we describe an evolving population, near extinction, in an environment with its own nontrivial spatio-temporal dynamics. Here, we consider the special case where the environment follows a simple relaxational (Model A) dynamics. Two new operators emerge, with upper critical dimension of four, which couple the two theories in a nontrivial way. While the Wilson-Fisher fixed point remains completely unaffected, a mismatch of time scales destabilizes the usual DP fixed point, suggesting a crossover to a first order transition from the active (surviving) to the inactive (extinct) state.

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arXiv2006

Novel surface universality classes with strong anisotropy

Using renormalized field theory, we examine the dynamics of a growing surface, driven by an obliquely incident particle beam. Its projection on the reference (substrate) plane selects a ``parallel'' direction, so that the evolution equation for the surface height becomes anisotropic. The phase diagram of the model is controlled by the properties of an effective anisotropic surface tension. Our renormalization group analysis suggests the existence of a line of continuous transitions and a line of (potentially) first-order transitions, which meet at a multicritical point. The full scaling behavior for the continuous line and the multicritical point is discussed in detail. Two novel universality classes for scale-invariant surface fluctuations are found.

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arXiv2001

Master Operators Govern Multifractality in Percolation

Using renormalization group methods we study multifractality in percolation at the instance of noisy random resistor networks. We introduce the concept of master operators. The multifractal moments of the current distribution (which are proportional to the noise cumulants $C_R^{(l)} (x, x^\prime)$ of the resistance between two sites x and $x^\prime$ located on the same cluster) are related to such master operators. The scaling behavior of the multifractal moments is governed exclusively by the master operators, even though a myriad of servant operators is involved in the renormalization procedure. We calculate the family of multifractal exponents ${ψ_l}$ for the scaling behavior of the noise cumulants, $C_R^{(l)} (x, x^\prime) \sim | x - x^\prime |^{ψ_l /ν}$, where $ν$ is the correlation length exponent for percolation, to two-loop order.

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arXiv1999

Viability of competing field theories for the driven lattice gas

It has recently been suggested that the driven lattice gas should be described by a novel field theory in the limit of infinite drive. We review the original and the new field theory, invoking several well-documented key features of the microscopics. Since the new field theory fails to reproduce these characteristics, we argue that it cannot serve as a viable description of the driven lattice gas. Recent results, for the critical exponents associated with this theory, are re-analyzed and shown to be incorrect.

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arXiv1999

Diluted Networks of Nonlinear Resistors and Fractal Dimensions of Percolation Clusters

We study random networks of nonlinear resistors, which obey a generalized Ohm's law, $V\sim I^r$. Our renormalized field theory, which thrives on an interpretation of the involved Feynman Diagrams as being resistor networks themselves, is presented in detail. By considering distinct values of the nonlinearity r, we calculate several fractal dimensions characterizing percolation clusters. For the dimension associated with the red bonds we show that $d_{\scriptsize red} = 1/ν$ at least to order ${\sl O} (ε^4)$, with $ν$ being the correlation length exponent, and $ε= 6-d$, where d denotes the spatial dimension. This result agrees with a rigorous one by Coniglio. Our result for the chemical distance, $d_{\scriptsize min} = 2 - ε/6 - [ 937/588 + 45/49 (\ln 2 -9/10 \ln 3)] (ε/6)^2 + {\sl O} (ε^3)$ verifies a previous calculation by one of us. For the backbone dimension we find $D_B = 2 + ε/21 - 172 ε^2 /9261 + 2 (- 74639 + 22680 ζ(3))ε^3 /4084101 + {\sl O} (ε^4)$, where $ζ(3) = 1.202057...$, in agreement to second order in $ε$ with a two-loop calculation by Harris and Lubensky.

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arXiv1999

Field Theory of Critical Behaviour in Driven Diffusive Systems with Quenched Disorder

We present a field theoretic renormalization group study for the critical behaviour of a uniformly driven diffusive system with quenched disorder, which is modelled by different kinds of potential barriers between sites. Due to their symmetry properties, these different realizations of the random potential barriers lead to three different models for the phase transition to transverse order and to one model for the phase transition to longitudinal order all belonging to distinct universality classes. In these four models that have different upper critical dimensions d_{c} we find the critical scaling behaviour of the vertex functions in spatial dimensions d < d_{c} . Its deviation from purely diffusive behaviour is characterized by the anomaly-exponent ηthat we calculate at first and second order, respectively in $ε= d_{c} -d$. In each model ηturns out to be positive which means superdiffusive spread of density fluctuations in the driving force direction.

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arXiv1999

The Resistance of Feynman Diagrams and the Percolation Backbone Dimension

We present a new view of Feynman diagrams for the field theory of transport on percolation clusters. The diagrams for random resistor networks are interpreted as being resistor networks themselves. This simplifies the field theory considerably as we demonstrate by calculating the fractal dimension $D_B$ of the percolation backbone to three loop order. Using renormalization group methods we obtain $D_B = 2 + ε/21 - 172ε^2 /9261 + 2 ε^3 (- 74639 + 22680 ζ(3))/4084101$, where $ε= 6-d$ with $d$ being the spatial dimension and $ζ(3) = 1.202057..$.

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arXiv1999

On Coupled Directed Percolation Processes: A Unifying View

It is shown that the universal critical properties of two recently introduced coupled directed percolation processes can be described by a single rapidity reversal invariant stochastic reaction-diffusion model. It is demonstrated that all renormalizations needed for the calculation of the universal scaling behavior near the multicritical point can be gained from the Gribov process (Reggeon field theory). Consequently the crossover exponent describing the scaling of the linear coupling parameter is given by Phi = 1 to all orders of the perturbation expansion.

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arXiv1998

Critical Exponents for Diluted Resistor Networks

An approach by Stephen is used to investigate the critical properties of randomly diluted resistor networks near the percolation threshold by means of renormalized field theory. We reformulate an existing field theory by Harris and Lubensky. By a decomposition of the principal Feynman diagrams we obtain a type of diagrams which again can be interpreted as resistor networks. This new interpretation provides for an alternative way of evaluating the Feynman diagrams for random resistor networks. We calculate the resistance crossover exponent $ϕ$ up to second order in $ε=6-d$, where $d$ is the spatial dimension. Our result $ϕ=1+ε/42 +4ε^2 /3087$ verifies a previous calculation by Lubensky and Wang, which itself was based on the Potts--model formulation of the random resistor network.

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arXiv1998

Equation of state for directed percolation

Using field-theoretic renormalization group methods we calculate the equation of state for non-equilibrium systems belonging to the universality class of directed percolation (Gribov process) to second order in epsilon = 4-d. By introducing a parametric representation the result can be written to this order in a very simple form. We use our result to obtain a universal amplitude ratio to second order in epsilon.

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arXiv1998

Biased Diffusion with Correlated Noise

The diffusion of hard-core particles subject to a global bias is described by a nonlinear, anisotropic generalization of the diffusion equation with conserved, local noise. Using renormalization group techniques, we analyze the effect of an additional noise term, with spatially long-ranged correlations, on the long-time, long-wavelength behavior of this model. Above an upper critical dimension $d_{LR}$, the long-ranged noise is always relevant. In contrast, for $d<d_{LR}$, we find a ``weak noise'' regime dominated by short-range noise. As the range of the noise correlations increases, an intricate sequence of stability exchanges between different fixed points of the renormalization group occurs. Both smooth and discontinuous crossovers between the associated universality classes are observed, reflected in the scaling exponents. We discuss the necessary techniques in some detail since they are applicable to a much wider range of problems.

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arXiv1998

Exact results for the Kardar--Parisi--Zhang equation with spatially correlated noise

We investigate the Kardar--Parisi--Zhang (KPZ) equation in $d$ spatial dimensions with Gaussian spatially long--range correlated noise --- characterized by its second moment $R(\vec{x}-\vec{x}') \propto |\vec{x}-\vec{x}'|^{2ρ-d}$ --- by means of dynamic field theory and the renormalization group. Using a stochastic Cole--Hopf transformation we derive {\em exact} exponents and scaling functions for the roughening transition and the smooth phase above the lower critical dimension $d_c = 2 (1+ρ)$. Below the lower critical dimension, there is a line $ρ_*(d)$ marking the stability boundary between the short-range and long-range noise fixed points. For $ρ\geq ρ_*(d)$, the general structure of the renormalization-group equations fixes the values of the dynamic and roughness exponents exactly, whereas above $ρ_*(d)$, one has to rely on some perturbational techniques. We discuss the location of this stability boundary $ρ_* (d)$ in light of the exact results derived in this paper, and from results known in the literature. In particular, we conjecture that there might be two qualitatively different strong-coupling phases above and below the lower critical dimension, respectively.

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arXiv1998

Levy-flight spreading of epidemic processes leading to percolating clusters

We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of short-range infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability distribution decaying in d dimensions with the distance as 1/R^{d+σ}. By means of Wilson's momentum shell renormalization-group recursion relations, the critical exponents characterizing the growing fractal clusters are calculated to first order in an ε-expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection σ=σ_c>2.

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arXiv1998

Influence of Long-range Interactions on the Critical Behavior of Systems with negative Fisher-Exponent

The influence of long-range interactions decaying in d dimensions as 1/R^{d+σ} on the critical behavior of systems with Fisher's correlation-function exponent for short-range interactions η_{SR}<0, is re-examined. Such systems, typically described by Φ^{3}-field theories, are e.g. the Potts-model in the percolation-limit, the Edwards-Anderson spin-glass, and the Yang-Lee edge singularity. In contrast to preceding studies, it is shown by means of Wilson's momentum-shell renormalization-group recursion relations that the long-range interactions dominate as long as σ<2-η_{SR}. Exponents change continuously to their short-range values at the boundary of this region.

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arXiv1998

Surface critical behavior of driven diffusive systems with open boundaries

Using field theoretic renormalization group methods we study the critical behavior of a driven diffusive system near a boundary perpendicular to the driving force. The boundary acts as a particle reservoir which is necessary to maintain the critical particle density in the bulk. The scaling behavior of correlation and response functions is governed by a new exponent eta_1 which is related to the anomalous scaling dimension of the chemical potential of the boundary. The new exponent and a universal amplitude ratio for the density profile are calculated at first order in epsilon = 5-d. Some of our results are checked by computer simulations.

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arXiv1996

On Critical Exponents and the Renormalization of the Coupling Constant in Growth Models with Surface Diffusion

It is shown by the method of renormalized field theory that in contrast to a statement based on a mathematically ill-defined invariance transformation and found in most of the recent publications on growth models with surface diffusion, the coupling constant of these models renormalizes nontrivially. This implies that the widely accepted supposedly exact scaling exponents are to be corrected. A two-loop calculation shows that the corrections are small and these exponents seem to be very good approximations.

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arXiv1996

Spontaneous Symmetry Breaking in Directed Percolation with Many Colors: Differentiation of Species in the Gribov Process

A general field theoretic model of directed percolation with many colors that is equivalent to a population model (Gribov process) with many species near their extinction thresholds is presented. It is shown that the multicritical behavior is always described by the well known exponents of Reggeon field theory. In addition this universal model shows an instability that leads in general to a total asymmetry between each pair of species of a cooperative society.

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arXiv1996

Renormalized field theory and particle density profile in driven diffusive systems with open boundaries

We investigate the density profile in a driven diffusive system caused by a plane particle source perpendicular to the driving force. Focussing on the case of critical bulk density $\bar{c}$ we use a field theoretic renormalization group approach to calculate the density $c(z)$ as a function of the distance from the particle source at first order in $ε=2-d$ ($d$: spatial dimension). For $d=1$ we find reasonable agreement with the exact solution recently obtained for the asymmetric exclusion model. Logarithmic corrections to the mean field profile are computed for $d=2$ with the result $c(z)-\bar{c} \sim z^{-1} (\ln(z))^{2/3}$ for $z \rightarrow \infty$.

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