Exact dynamics of a class of aggregation models

Spatial correlations, additivity, and fluctuations in conserved-mass transport processes

We exactly calculate two-point spatial correlation functions in steady state in a broad class of conserved-mass transport processes, which are governed by chipping, diffusion, and coalescence of masses. We find that the spatial correlations are in general short-ranged and, consequently, on a large scale, these transport processes possess a remarkable thermodynamic structure in the steady state. That is, the processes have an equilibrium-like additivity property and, consequently, a fluctuation-response relation, which help us to obtain subsystem mass distributions in the limit of subsystem size large.

Additivity property and emergence of power laws in nonequilibrium steady states

We show that an equilibriumlike additivity property can remarkably lead to power-law distributions observed frequently in a wide class of out-of-equilibrium systems. The additivity property can determine the full scaling form of the distribution functions and the associated exponents. The asymptotic behavior of these distributions is solely governed by branch-cut singularity in the variance of subsystem mass. To substantiate these claims, we explicitly calculate, using the additivity property, subsystem mass distributions in a wide class of previously studied mass aggregation models as well as in their variants. These results could help in the thermodynamic characterization of nonequilibrium critical phenomena.

Statistical mechanics of the "Chinese restaurant process": lack of self-averaging, anomalous finite-size effects, and condensation

The Pitman-Yor, or Chinese restaurant process, is a stochastic process that generates distributions following a power law with exponents lower than 2, as found in numerous physical, biological, technological, and social systems. We discuss its rich behavior with the tools and viewpoint of statistical mechanics. We show that this process invariably gives rise to a condensation, i.e., a distribution dominated by a finite number of classes. We also evaluate thoroughly the finite-size effects, finding that the lack of stationary state and self-averaging of the process creates realization-dependent cutoffs and behavior of the distributions with no equivalent in other statistical mechanical models.

Breakdown of Kolmogorov scaling in models of cluster aggregation

We describe a model of cluster aggregation with a source which provides a rare example of an analytically tractable turbulent system. The steady state is characterized by a constant mass flux from small masses to large. Thus it can be studied using a phenomenological theory, inspired by Kolmogorov's 1941 theory, which assumes constant flux and self-similarity. We prove that such self-similarity is violated in dimensions less than or equal to two. We then use dynamical renormalization group techniques to show that the scaling of multipoint correlation functions implies nontrivial multifractality. The analytical results are supported by Monte Carlo simulations.

Nonequilibrium phase transitions in models of adsorption and desorption

The nonequilibrium phase transition in a system of diffusing, coagulating particles in the presence of a steady input and evaporation of particles is studied. The system undergoes a transition from a phase in which the average number of particles is finite to one in which it grows linearly in time. The exponents characterizing the mass distribution near the critical point are calculated in all dimensions.

Dynamics and critical behavior of the q model

The q model, a random walk model rich in behavior and applications, is investigated. We introduce and motivate the q model via its application proposed by Coppersmith et al. to the flow of stress through granular matter at rest. For a special value of its parameters the q model has a critical point that we analyze. To characterize the critical point we imagine that a uniform load has been applied to the top of the granular medium and we study the evolution with depth of fluctuations in the distribution of load. Close to the critical point explicit calculation reveals that the evolution of load exhibits scaling behavior analogous to thermodynamic critical phenomena. The critical behavior is remarkably tractable: the harvest of analytic results includes scaling functions that describe the evolution of the variance of the load distribution close to the critical point and of the entire load distribution right at the critical point, values of the associated critical exponents, and determination of the upper critical dimension. These results are of intrinsic interest as a tractable example of a random critical point. Of the many applications of the q model, the critical behavior is particularly relevant to network models of river basins, as we briefly discuss. Finally we discuss circumstances under which quantum network models that describe the surface electronic states of a quantum Hall multilayer can be mapped onto the classical q model. For mesoscopic multilayers of finite circumference the mapping fails; instead a mapping to a ferromagnetic supersymmetric spin chain has proved fruitful. We discuss aspects of the superspin mapping and give an elementary derivation of it making use of operator rather than functional methods.

Exact calculation of the spatiotemporal correlations in the takayasu model and in the q model of force fluctuations in bead packs

We calculate exactly the two point mass-mass correlations in arbitrary spatial dimensions in the aggregation model of Takayasu. In this model, masses diffuse on a lattice, coalesce upon contact, and adsorb unit mass from outside at a constant rate. Our exact calculation of the variance of mass at a given site proves explicitly, without making any assumption of scaling, that the upper critical dimension of the model is 2. We also extend our method to calculate the spatiotemporal correlations in a generalized class of models with aggregation, fragmentation, and injection, which include, in particular, the q model of force fluctuations in bead packs. We present explicit expressions for the spatiotemporal force-force correlation function in the q model. These can be used to test the applicability of the q model in experiments.

Phase transition in the takayasu model with desorption

We study a lattice model where particles carrying different masses diffuse and coalesce upon contact, and also unit masses adsorb to a site with rate q or desorb from a site with nonzero mass with rate p. In the limit p=0 (without desorption), our model reduces to the well studied Takayasu model where the steady-state single site mass distribution has a power-law tail P(m) approximately m(-tau) for large mass. We show that varying the desorption rate p induces a nonequilibrium phase transition in all dimensions. For fixed q, there is a critical p(c)(q) such that if p<p(c)(q), the steady-state mass distribution, P(m) approximately m(-tau) for large m as in the Takayasu case. For p=p(c)(q), we find P(m) approximately m(-tau(c)) where tau(c) is a new exponent, while for p>p(c)(q), P(m) approximately exp(-m/m(*)) for large m. The model is studied analytically within a mean-field theory and numerically in one dimension.

Numerical renormalization group of vortex aggregation in two-dimensional decaying turbulence: the role of three-body interactions

We introduce a numerical renormalization group procedure which permits long-time simulations of vortex dynamics and coalescence in a two-dimensional turbulent decaying fluid. The number of vortices decreases as N approximately t(-xi), with xi approximately 1 instead of the value xi=4/3 predicted by a naive kinetic theory. For short time, we find an effective exponent xi approximately 0.7 consistent with previous simulations and experiments. We show that the mean square displacement of surviving vortices grows as <x(2)> approximately t(1+xi/2). Introducing effective dynamics for two- and three-body collisions, we justify that only the latter become relevant at a small vortex area coverage. A kinetic theory consistent with this mechanism leads to xi=1. We find that the theoretical relations between kinetic parameters are all in good agreement with experiments.

Topology of the fittest transportation network

The presence or absence of loops in the emergent transportation networks, that are characterized by a minimum overall cost, is shown to depend on the convexity of the cost function for the local transportation of material. Our results are directly applicable to a variety of situations across disciplines.

Analytical results for random walks in the presence of disorder and traps

In this paper, we study the dynamics of a random walker diffusing on a disordered one-dimensional lattice with random trappings. The distribution of escape probabilities is computed exactly for any strength of the disorder. These probabilities do not display any multifractal properties, contrary to previous numerical claims. The explanation for this apparent multifractal behavior is given, and our conclusions are supported by numerical calculations. These exact results are exploited to compute the large time asymptotics of the survival probability (or the density) which is found to decay as exp[-Ct(1/3)ln(2/3)(t)]. An exact lower bound for the density is found to decay in a similar way.