Generalized Exclusion Processes: Transport Coefficients

A class of generalized exclusion processes parametrized by the maximal occupancy, $k\geq 1$, is investigated. For these processes with symmetric nearest-neighbor hopping, we compute the diffusion coefficient and show that it is independent on the spatial dimension. In the extreme cases of $k=1$ (simple symmetric exclusion process) and $k=\infty$ (non-interacting symmetric random walks) the diffusion coefficient is constant; for $2\leq k<\infty$, the diffusion coefficient depends on the density and the maximal occupancy $k$. We also study the evolution of a tagged particle. It exhibits a diffusive behavior which is characterized by the coefficient of self-diffusion which we probe numerically.Comment: v1: 9 pages, 6 figures. v2: + 2 references. v3: 10 pages, 7 figures, published versio

Microscopic Transport Theory of Nuclear Processes

We formulate a microscopic theory of the decay of a compound nucleus through fission which generalizes earlier microscopic approaches of fission dynamics performed in the framework of the adiabatic hypothesis. It is based on the constrained Hartree-Fock-Bogoliubov procedure and the Generator Coordinate Method, and requires an effective nucleon-nucleon interaction as the only input quantity. The basic assumption is that the slow evolution of the nuclear shape must be treated explicitely, whereas the rapidly time-dependent intrinsic excitations can be treated by statistical approximations. More precisely, we introduce a reference density which represents the slow evolution of the nuclear shape by a reduced density matrix and the state of intrinsic excitations by a canonical distribution at each given shape of the nucleus. The shape of the nuclear density distribution is described by parameters ("generator coordinates"), not by "superabundant" degrees of freedom introduced in addition to the complete set of nucleonic degrees of freedom. We first derive a rigorous equation of motion for the reference density and, subsequently, simplify this equation on the basis of the Markov approximation. The temperature which appears in the canonical distribution is determined by the requirement that, at each time t, the reference density should correctly reproduce the mean excitation energy at given values of the shape parameters. The resulting equation for the "local" temperature must be solved together with the equations of motion obtained for the reduced density matrix.Comment: 33 pages, accepted in Nucl. Phys.

Transport and Reaction Processes in Soil

In order to register agrochemicals in Europe it is necessary to have a detailed understanding of the processes in the environment that break down agrochemicals. The existing framework for environmental assessment includes a consideration of soil water movement and microbial breakdown of products in soil and these processes are relatively understood and represented in models. However the breakdown of agrochemicals by the action of light incident on the soil surface by a process termed photolysis is not so well represented in models of environmental fate. The problem brought by Syngenta (one of the worlds leading agrochemical companies) to the workshop was how to include the effects of light degradation of chemicals into predictive models of environmental fate. Photolysis is known to occur in a very thin layer at the surface of soil. The workshop was asked to consider how the very rough nature of the upper surface of a ploughed field might affect the degradation of chemicals by sunlight. The discussions were directed down two avenues: - firstly to determine how the very small distances over which photolysis occurs might be adequately incorporated into models of transport in soils and, - secondly to consider how the rough surface might modify the illumination of the surface and hence alter degradation. The rate of degradation by photolysis is measured in the laboratory by illuminating a thin, typically about 1 or 2 mm, layer of soil with very strong xenon lamps. The amount of chemical is measured at various intervals and is fitted to a first-order process. Field experiments where the chemical is sprayed on a bare field show evidence of photolysis indicated by biphasic degradation patterns and the presence of breakdown products only formed by photolysis. This report addresses methods for mathematically modelling the action of photolysis on particular relevant chemical species. We start with a general discussion of mechanisms that transport chemicals within soil §2. There is an existing computational model exploited by Syngenta for such modelling and we discuss how this performs and the predictions that can be derived using it §3. The particular mechanism of photolysis is then considered. One aspect of this mechanism that is investigated is how the roughness of the surface of the soil could be adequately incorporated into the modelling. Some results relating to this are presented §4.2. Some of the original experimental data used to derive aspects of the model of photolysis are revisited and a simple model of the process presented and shown to fit the data very well §5. By considering photolysis with a constant diffusion coefficient various analytical results are derived and general behaviour of the system outlined. This simple model is then applied to real field-based data and shown to give very good fit when simply extended to account for the moisture variations by utilising moisture dependent diffusion coefficients derived from the existing computational model §5.3. Some consequences of the simple model are then discussed §6

Stochastic exclusion processes versus coherent transport

Stochastic exclusion processes play an integral role in the physics of non-equilibrium statistical mechanics. These models are Markovian processes, described by a classical master equation. In this paper a quantum mechanical version of a stochastic hopping process in one dimension is formulated in terms of a quantum master equation. This allows the investigation of coherent and stochastic evolution in the same formal framework. The focus lies on the non-equilibrium steady state. Two stochastic model systems are considered, the totally asymmetric exclusion process and the fully symmetric exclusion process. The steady state transport properties of these models is compared to the case with additional coherent evolution, generated by the $XX$-Hamiltonian

Collision number statistics for transport processes

Many physical observables can be represented as a particle spending some random time within a given domain. For a broad class of transport-dominated processes, we detail how it is possible to express the moments of the number of particle collisions in an arbitrary volume in terms of repeated convolutions of the ensemble equilibrium distribution. This approach is shown to generalize the celebrated Kac formula for the moments of residence times, which is recovered in the diffusion limit. Some practical applications are illustrated for bounded, unbounded and absorbing domains.Comment: 4 pages, 4 figure

Comment on "Generalized exclusion processes: Transport coefficients"

In a recent paper Arita et al. [Phys. Rev. E 90, 052108 (2014)] consider the transport properties of a class of generalized exclusion processes. Analytical expressions for the transport-diffusion coefficient are derived by ignoring correlations. It is claimed that these expressions become exact in the hydrodynamic limit. In this Comment, we point out that (i) the influence of correlations upon the diffusion does not vanish in the hydrodynamic limit, and (ii) the expressions for the self- and transport diffusion derived by Arita et al. are special cases of results derived in [Phys. Rev. Lett. 111, 110601 (2013)].Comment: (citation added, published version

Fractional transport equations for Levy stable processes

The influence functional method of Feynman and Vernon is used to obtain a quantum master equation for a Brownian system subjected to a Levy stable random force. The corresponding classical transport equations for the Wigner function are then derived, both in the limit of weak and strong friction. These are fractional extensions of the Klein-Kramers and the Smoluchowski equations. It is shown that the fractional character acquired by the position in the Smoluchowski equation follows from the fractional character of the momentum in the Klein-Kramers equation. Connections among fractional transport equations recently proposed are clarified.Comment: 4 page

Burgers velocity fields and dynamical transport processes

We explore a connection of the forced Burgers equation with the Schr\"{o}dinger (diffusive) interpolating dynamics in the presence of deterministic external forces. This entails an exploration of the consistency conditions that allow to interpret dispersion of passive contaminants in the Burgers flow as a Markovian diffusion process. In general, the usage of a continuity equation $\partial_t\rho =-\nabla (\vec{v}\rho)$, where $\vec{v}=\vec{v}(\vec{x},t)$ stands for the Burgers field and $\rho$ is the density of transported matter, is at variance with the explicit diffusion scenario. Under these circumstances, we give a complete characterisation of the diffusive matter transport that is governed by Burgers velocity fields. The result extends both to the approximate description of the transport driven by an incompressible fluid and to motions in an infinitely compressible medium.Comment: Latex fil