arXiv subjects

Brian Berkowitz

Publications and source records attributed to Brian Berkowitz.

6 recordsLinked to original sources

arXiv2022

Controls that expedite first passage times in disordered systems

First passage time statistics in disordered systems exhibiting scale invariance are studied widely. In particular, long trapping times in energy or entropic traps are fat-tailed distributed, which slow the overall transport process. We study the statistical properties of the first passage time of biased processes in different models, and employ the big jump principle that shows the dominance of the maximum trapping time on the first passage time. Inspired by the restart paradigm, we demonstrate that the removal of this maximum significantly expedites transport. As the disorder increases, the system enters a phase where the removal shows a dramatic effect. Our results show how we may speed up transport in strongly disordered systems exploiting scale invariance. In contrast to the disordered systems studied here, the removal principle has essentially no effect in homogeneous systems; this indicates that improving the conductance of a poorly conducting system is, theoretically, relatively easy as compared to a homogeneous system.

cond-mat.stat-mech↗

arXiv2020

Modeling non-Fickian solute transport due to mass transfer and physical heterogeneity on arbitrary groundwater velocity fields

We present a hybrid approach to groundwater transport modeling, "CTRW-on-a-streamline", that allows continuous-time random walk (CTRW) particle tracking on large-scale, explicitly-delineated heterogeneous groundwater velocity fields. The combination of a non-Fickian transport model (in this case, the CTRW) with general heterogeneous velocity fields represents an advance of the current state of the art, in which non-Fickian transport models or heterogeneous velocity fields are employed, but generally not both. We present a general method for doing this particle tracking that fully separates the model parameters characterizing macroscopic flow, subscale advective heterogeneity, and mobile-immobile mass transfer, such that each can be directly specified a priori from available data. The method is formalized and connections to classic CTRW and subordination approaches are made. Numerical corroboration is presented.

physics.comp-ph↗

arXiv2013

Origins of Anomalous Transport in Disordered Media: Structural and Dynamic Controls

We quantitatively identify the origin of anomalous transport in a representative model of a heterogeneous system---tracer migration in the complex flow patterns of a lognormally distributed hydraulic conductivity ($K$) field. The transport, determined by a particle tracking technique, is characterized by breakthrough curves; the ensemble averaged curves document anomalous transport in this system, which is entirely accounted for by a truncated power-law distribution of local transition times $ψ(t)$ within the framework of a continuous time random walk. Unique to this study is the linking of $ψ(t)$ directly to the system heterogeneity. We assess the statistics of the dominant preferred pathways by forming a particle-visitation weighted histogram $\{wK\}$. Converting the ln($K$) dependence of $\{wK\}$ into time yields the equivalence of $\{wK\}$ and $ψ(t)$, and shows the part of $\{wK\}$ that forms the power-law of $ψ(t)$, which is the origin of anomalous transport. We also derive an expression defining the power law exponent in terms of the $\{wK\}$ parameters. This equivalence is a remarkable result, particularly given the correlated $K$-field, the complexity of the flow field and the statistics of the particle transitions.

physics.flu-dyn↗

arXiv2013

Record-breaking statistics for random walks in the presence of measurement error and noise

We address the question of distance record-setting by a random walker in the presence of measurement error, $δ$, and additive noise, $γ$ and show that the mean number of (upper) records up to $n$ steps still grows universally as $< R_n> \sim n^{1/2}$ for large $n$ for all jump distributions, including Lévy flights, and for all $δ$ and $γ$. In contrast to the universal growth exponent of 1/2, the pace of record setting, measured by the pre-factor of $n^{1/2}$, depends on $δ$ and $γ$. In the absence of noise ($γ=0$), the pre-factor $S(δ)$ is evaluated explicitly for arbitrary jump distributions and it decreases monotonically with increasing $δ$ whereas, in case of perfect measurement $(δ=0)$, the corresponding pre-factor $T(γ)$ increases with $γ$. Our analytical results are supported by extensive numerical simulations and qualitatively similar results are found in two and three dimensions.

physics.data-an↗

arXiv2002

Physical Pictures of Transport in Heterogeneous Media: Advection-Dispersion, Random Walk and Fractional Derivative Formulations

The basic conceptual picture and theoretical basis for development of transport equations in porous media are examined. The general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations, for single realizations and for ensemble averages of the domain. The application of these transport equations is focused on accounting for the appearance of non-Fickian (anomalous) transport behavior. The general ensemble-averaged transport equation is shown to be equivalent to a continuous time random walk (CTRW) and reduces to the conventional forms of the advection-dispersion equation (ADE) under highly restrictive conditions. Fractional derivative formulations of the transport equations, both temporal and spatial, emerge as special cases of the CTRW. In particular, the use in this context of L{é}vy flights is critically examined. In order to determine chemical transport in field-scale situations, the CTRW approach is generalized to non-stationary systems. We outline a practical numerical scheme, similar to those used with extended geological models, to account for the often important effects of unresolved heterogeneities.

cond-mat.stat-mech↗

arXiv1996

Theory of continuum percolation III. Low density expansion

We use a previously introduced mapping between the continuum percolation model and the Potts fluid (a system of interacting s-states spins which are free to move in the continuum) to derive the low density expansion of the pair connectedness and the mean cluster size. We prove that given an adequate identification of functions, the result is equivalent to the density expansion derived from a completely different point of view by Coniglio et al. [J. Phys A 10, 1123 (1977)] to describe physical clustering in a gas. We then apply our expansion to a system of hypercubes with a hard core interaction. The calculated critical density is within approximately 5% of the results of simulations, and is thus much more precise than previous theoretical results which were based on integral equations. We suggest that this is because integral equations smooth out overly the partition function (i.e., they describe predominantly its analytical part), while our method targets instead the part which describes the phase transition (i.e., the singular part).

cond-mat.stat-mech↗