arXiv subjects

Stefan Grosskinsky

Publications and source records attributed to Stefan Grosskinsky.

At least 37 records · Page 2Linked to original sources

arXiv2016

Monotonicity and Condensation in Homogeneous Stochastic Particle Systems

We study stochastic particle systems that conserve the particle density and exhibit a condensation transition due to particle interactions. We restrict our analysis to spatially homogeneous systems on finite lattices with stationary product measures, which includes previously studied zero-range or misanthrope processes. All known examples of such condensing processes are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space and the canonical measures (with a fixed number of particles) are not monotonically ordered. For our main result we prove that condensing homogeneous particle systems with finite critical density are necessarily non-monotone. On finite lattices condensation can occur even when the critical density is infinite, in this case we give an example of a condensing process that numerical evidence suggests is monotone, and give a partial proof of its monotonicity.

cond-mat.stat-mech↗

arXiv2016

Metastability in a condensing zero-range process in the thermodynamic limit

Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value. We study such a process on a one-dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed, super-critical particle density. We show that the process exhibits metastability with respect to the condensate location, i.e. the suitably accelerated process of the rescaled location converges to a limiting Markov process on the unit torus. This process has stationary, independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice. Our result extends previous work for fixed lattices and diverging density in [J. Beltran, C. Landim, Probab. Theory Related Fields, 152(3-4):781-807, 2012], and we follow the martingale approach developed there and in subsequent publications. Besides additional technical difficulties in estimating error bounds for transition rates, the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells, corresponding to a typical set of configurations with a particular condensate location. The total exit rates from individual wells turn out to diverge in the limit, which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator. Another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells, which is of a general nature and can be adapted to other models.

math.PR↗

arXiv2015

Explosive condensation in symmetric mass transport models

We study the dynamics of condensation in a misanthrope process with nonlinear jump rates and factorized stationary states. For large enough density, it is known that such models have a phase separated state, with a non-zero fraction of the total mass concentrating in a single lattice site. It has been established in [B Waclaw and M R Evans, Phys. Rev. Lett., 108(7):070601, 2012] for asymmetric dynamics that such processes exhibit explosive condensation, where the time to reach the stationary state vanishes with increasing system size. This constitutes a spatially extended version of instantaneous gelation which has previously been studied only in mean-field coagulation models. We show that this phenomenon also occurs for symmetric dynamics in one dimension if the non-linearity is strong enough, and we find a coarsening regime where the time to stationarity diverges with the system size for weak non-linearity. In higher space dimensions explosive condensation is expected to be generic for all parameter values. Our results are based on heuristic mean field arguments which are confirmed by simulation data.

cond-mat.stat-mech↗

arXiv2015

A hydrodynamic limit for chemotaxis in a given heterogeneous environment

In this paper the first equation within a class of well known chemotaxis systems is derived as a hydrodynamic limit from a stochastic interacting many particle system on the lattice. The cells are assumed to interact with attractive chemical molecules on a finite number of lattice sites, but they only directly interact among themselves on the same lattice site. The chemical environment is assumed to be stationary with a slowly varying mean, which results in a non-trivial macroscopic chemotaxis equation for the cells. Methodologically the limiting procedure and its proofs are based on results by Koukkus [18] and Kipnis/Landim [17]. Numerical simulations extend and illustrate the theoretical findings.

math.PR↗

arXiv2015

Collective Relaxation Dynamics of Small-World Networks

Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization, diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the local Jacobian, graph Laplacian or a similar linear operator. The structure of networks with regular, small-world and random connectivities are reasonably well understood, but their collective dynamical properties remain largely unknown. Here we present a two-stage mean-field theory to derive analytic expressions for network spectra. A single formula covers the spectrum from regular via small-world to strongly randomized topologies in Watts-Strogatz networks, explaining the simultaneous dependencies on network size N, average degree k and topological randomness q. We present simplified analytic predictions for the second largest and smallest eigenvalue, and numerical checks confirm our theoretical predictions for zero, small and moderate topological randomness q, including the entire small-world regime. For large q of the order of one, we apply standard random matrix theory thereby overarching the full range from regular to randomized network topologies. These results may contribute to our analytic and mechanistic understanding of collective relaxation phenomena of network dynamical systems.

physics.soc-ph↗

arXiv2015

Large deviation analysis of a simple information engine

Information thermodynamics provides a framework for studying the effect of feedback loops on entropy production. It has enabled the understanding of novel thermodynamic systems such as the information engine which can be seen as a modern version of `Maxwell's Daemon', whereby the feedback controller is acting as a Daemon, processing information gained about the system in order to do work. Here, we analyse a simple model of such an engine and provide a detailed analysis of its fluctuation properties, including the large deviations of information. We find an exact expression of the large deviation rate function for a two-site version of our model, and provide an approximate analysis for larger systems which is corroborated by simulation data.

cond-mat.stat-mech↗

arXiv2015

Dynamics of non-Markovian exclusion processes

Driven diffusive systems are often used as simple discrete models of collective transport phenomena in physics, biology or social sciences. Restricting attention to one-dimensional geometries, the asymmetric simple exclusion process (ASEP) plays a paradigmatic role to describe noise-activated driven motion of entities subject to an excluded volume interaction and many variants have been studied in recent years. While in the standard ASEP the noise is Poissonian and the process is therefore Markovian, in many applications the statistics of the activating noise has non-standard distribution with possible memory effects resulting from internal degrees of freedom or external sources. This leads to temporal correlations and can significantly affect the shape of the current-density relation as has been studied recently for a number of scenarios. In this paper we report a general framework to derive the fundamental diagram of ASEPs driven by non-Poissonian noise by using effectively only two simple quantities, viz., the mean residual lifetime of the jump distribution and a suitably defined temporal correlation length. We corroborate our results by detailed numerical studies for various noise statistics under periodic boundary conditions and discuss how our approach can be applied to more general driven diffusive systems.

cond-mat.stat-mech↗

arXiv2014

A dynamical transition and metastability in a size-dependent zero-range process

We study a zero-range process with system-size dependent jump rates, which is known to exhibit a discontinuous condensation transition. Metastable homogeneous phases and condensed phases coexist in extended phase regions around the transition, which have been fully characterized in the context of the equivalence and non-equivalence of ensembles. In this communication we report rigorous results on the large deviation properties and the free energy landscape which determine the metastable dynamics of the system. Within the condensed phase region we identify a new dynamic transition line which separates two distinct mechanism of motion of the condensate, and provide a complete discussion of all relevant timescales. Our results are directly related to recent interest in metastable dynamics of condensing particle systems. Our approach applies to more general condensing particle systems, which exhibit the dynamical transition as a finite size effect.

cond-mat.stat-mech↗

arXiv2014

Condensation in stochastic particle systems with stationary product measures

We study stochastic particle systems with stationary product measures that exhibit a condensation transition due to particle interactions or spatial inhomogeneities. We review previous work on the stationary behaviour and put it in the context of the equivalence of ensembles, providing a general characterization of the condensation transition for homogeneous and inhomogeneous systems in the thermodynamic limit. This leads to strengthened results on weak convergence for subcritical systems, and establishes the equivalence of ensembles for spatially inhomogeneous systems under very general conditions, extending previous results which were focused on attractive and finite systems. We use relative entropy techniques which provide simple proofs, making use of general versions of local limit theorems for independent random variables.

cond-mat.stat-mech↗

arXiv2013

Dynamics of condensation in the totally asymmetric inclusion process

We study the dynamics of condensation of the inclusion process on a one-dimensional periodic lattice in the thermodynamic limit, generalising recent results on finite lattices for symmetric dynamics. Our main focus is on totally asymmetric dynamics which have not been studied before, and which we also compare to exact solutions for symmetric systems. We identify all relevant dynamical regimes and corresponding time scales as a function of the system size, including a coarsening regime where clusters move on the lattice and exchange particles, leading to a growing average cluster size. Suitable observables exhibit a power law scaling in this regime before they saturate to stationarity following an exponential decay depending on the system size. Our results are based on heuristic derivations and exact computations for symmetric systems, and are supported by detailed simulation data.

cond-mat.stat-mech↗

arXiv2013

Interacting Particle Systems in Time-Dependent Geometries

Many complex structures and stochastic patterns emerge from simple kinetic rules and local interactions, and are governed by scale invariance properties in combination with effects of the global geometry. We consider systems that can be described effectively by space-time trajectories of interacting particles, such as domain boundaries in two-dimensional growth or river networks. We study trajectories embedded in time-dependent geometries, and the main focus is on uniformly expanding or decreasing domains for which we obtain an exact mapping to simple fixed domain systems while preserving the local scale invariance properties. This approach was recently introduced in [A. Ali et al., Phys. Rev. E 87, 020102(R) (2013)] and here we provide a detailed discussion on its applicability for self-affince Markovian models, and how it can be adapted to self-affine models with memory or explicit time dependence. The mapping corresponds to a non-linear time transformation which convergences to a finite value for a large class of trajectories, enabling an exact analysis of asymptotic properties in expanding domains. We further provide a detailed discussion of different particle interactions and generalized geometries. All our findings are based on exact computations and are illustrated numerically for various examples, including Lévy processes and fractional Brownian motion.

cond-mat.stat-mech↗

arXiv2013

Dynamics of condensation in the symmetric inclusion process

The inclusion process is a stochastic lattice gas, which is a natural bosonic counterpart of the well-studied exclusion process and has strong connections to models of heat conduction and applications in population genetics. Like the zero-range process, due to attractive interaction between the particles, the inclusion process can exhibit a condensation transition. In this paper we present first rigorous results on the dynamics of the condensate formation for this class of models. We study the symmetric inclusion process on a finite set $S$ with total number of particles $N$ in the regime of strong interaction, i.e. with independent diffusion rate $m=m_N \to 0$. For the case $N m_N\to\infty$ we show that on the time scale $1/m_N$ condensates emerge from general homogeneous initial conditions, and we precisely characterize their limiting dynamics. In the simplest case of two sites or a fully connected underlying random walk kernel, there is a single condensate hopping over $S$ as a continuous-time random walk. In the non fully connected case several condensates can coexist and exchange mass via intermediate sites in an interesting coarsening process, which consists of a mixture of a diffusive motion and a jump process, until a single condensate is formed. Our result is based on a general two-scale form of the generator, with a fast-scale neutral Wright-Fisher diffusion and a slow-scale deterministic motion. The motion of the condensates is described in terms of the generator of the deterministic motion and the harmonic projection corresponding to the absorbing states of the Wright-Fisher diffusion.

math-ph↗

arXiv2012

Scale invariant growth processes in expanding space

Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting particles, and their large scale behaviour depends on the overall growth geometry. We establish an exact relation between statistical properties of structures in uniformly expanding and fixed geometries, which preserves the local scale invariance and is independent of other properties such as the dimensionality. This relation generalizes standard conformal transformations as the natural symmetry of self-affine growth processes. We illustrate our main result numerically for various structures of coalescing Lévy flights and fractional Brownian motions, including also branching and finite particle sizes. One of the main benefits of this new approach is a full, explicit description of the asymptotic statistics in expanding domains, which are often non-trivial and random due to amplification of initial fluctuations.

cond-mat.stat-mech↗

arXiv2012

Frequency-dependent fitness induces multistability in coevolutionary dynamics

Evolution is simultaneously driven by a number of processes such as mutation, competition and random sampling. Understanding which of these processes is dominating the collective evolutionary dynamics in dependence on system properties is a fundamental aim of theoretical research. Recent works quantitatively studied coevolutionary dynamics of competing species with a focus on linearly frequency-dependent interactions, derived from a game-theoretic viewpoint. However, several aspects of evolutionary dynamics, e.g. limited resources, may induce effectively nonlinear frequency dependencies. Here we study the impact of nonlinear frequency dependence on evolutionary dynamics in a model class that covers linear frequency dependence as a special case. We focus on the simplest non-trivial setting of two genotypes and analyze the co-action of nonlinear frequency dependence with asymmetric mutation rates. We find that their co-action may induce novel metastable states as well as stochastic switching dynamics between them. Our results reveal how the different mechanisms of mutation, selection and genetic drift contribute to the dynamics and the emergence of metastable states, suggesting that multistability is a generic feature in systems with frequency-dependent fitness.

q-bio.PE↗

arXiv2012

Condensation in randomly perturbed zero-range processes

The zero-range process is a stochastic interacting particle system that exhibits a condensation transition under certain conditions on the dynamics. It has recently been found that a small perturbation of a generic class of jump rates leads to a drastic change of the phase diagram and prevents condensation in an extended parameter range. We complement this study with rigorous results on a finite critical density and quenched free energy in the thermodynamic limit, as well as quantitative heuristic results for small and large noise which are supported by detailed simulation data. While our new results support the initial findings, they also shed new light on the actual (limited) relevance in large finite systems, which we discuss via fundamental diagrams obtained from exact numerics for finite systems.

cond-mat.stat-mech↗

arXiv2012

Network representations of non-equilibrium steady states: Cycle decompositions, symmetries and dominant paths

Non-equilibrium steady states (NESS) of Markov processes give rise to non-trivial cyclic probability fluxes. Cycle decompositions of the steady state offer an effective description of such fluxes. Here, we present an iterative cycle decomposition exhibiting a natural dynamics on the space of cycles that satisfies detailed balance. Expectation values of observables can be expressed as cycle "averages", resembling the cycle representation of expectation values in dynamical systems. We illustrate our approach in terms of an analogy to a simple model of mass transit dynamics. Symmetries are reflected in our approach by a reduction of the minimal number of cycles needed in the decomposition. These features are demonstrated by discussing a variant of an asymmetric exclusion process (TASEP). Intriguingly, a continuous change of dominant flow paths in the network results in a change of the structure of cycles as well as in discontinuous jumps in cycle weights.

math-ph↗

arXiv2012

Reproduction time statistics and segregation patterns in growing populations

Pattern formation in microbial colonies of competing strains under purely space-limited population growth has recently attracted considerable research interest. We show that the reproduction time statistics of individuals has a significant impact on the sectoring patterns. Generalizing the standard Eden growth model, we introduce a simple one-parameter family of reproduction time distributions indexed by the variation coefficient δ \in [0, 1], which includes deterministic (δ = 0) and memoryless exponential distribution (δ = 1) as extreme cases. We present convincing numerical evidence and heuristic arguments that the generalized model is still in the KPZ universality class, and the changes in patterns are due to changing prefactors in the scaling relations, which we are able to predict quantitatively. At the example of Saccharomyces cerevisiae, we show that our approach using the variation coefficient also works for more realistic reproduction time distributions.

q-bio.PE↗

arXiv2012

Lattice permutations and Poisson-Dirichlet distribution of cycle lengths

We study random spatial permutations on Z^3 where each jump x -> π(x) is penalized by a factor exp(-T ||x-π(x)||^2). The system is known to exhibit a phase transition for low enough T where macroscopic cycles appear. We observe that the lengths of such cycles are distributed according to Poisson-Dirichlet. This can be explained heuristically using a stochastic coagulation-fragmentation process for long cycles, which is supported by numerical data.

cond-mat.stat-mech↗