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S. F. Edwards

Publications and source records attributed to S. F. Edwards.

14 recordsLinked to original sources
arXiv2008

Trajectory end point distribution of a test particle in a turbulent liquid

In a recent paper the mean square displacement (MSD), , of a particle carried by a turbulent liquid over time T has been shown to be proportional to T^6/5, meaning that the motion of the particle is slightly super-diffusive. In some cases, though, it might be important to have more information than that provided by that law. An example would be the distribution of pollutants as a function of time by turbulent flow. Here small amounts of material reaching relatively large distances are of importance. This motivates our interest in the full distribution of the location of particles swept by the fluid as a function of time. The distribution depends on the distance through the dimensionless quantity X^2=R^2/ . We find that for small X, the distribution P(R,T) is proportional to exp(-aX^2) but at its tail when X is large it behaves as exp(-bX^2/3) . pacs numbers-02.50.Ey, 05.40.Fb

cond-mat.stat-mech↗
arXiv2008

Does a particle swept by a turbulent liquid diffuse?

Since the famous 1926 paper by Richardson, the relative diffusion of two particles in a turbulent liquid has attracted a lot of interest. The motion of a single particle on the other hand is usually considered not to be especially interesting. The widely accepted picture is that the velocity of the particle has short-range correlations in time, resulting in motion that is diffusive on time scales large compared to the correlation time. We find, however, that the correlation time is infinite and that the square displacement, F, is not linear in the traversed time, T, which would correspond to diffusion, but rather F is proportional to T^6/5. Namely, the motion is slightly super diffusive.

cond-mat.stat-mech↗
arXiv2006

Trajectory end point distribution of a test particle in the atmosphere

The classic meteorological law of diffusion in the atmosphere was given experimentally, by Richardson in 1926, whose result that the mean squared distance =cT^3, the time cubed, is in accord with the scaling theory of Komogorov [ Obukhov (1941)]. In some cases it might be important to have more information than that provided by Richardson's law. An example would be the distribution of pollutants in time by turbulent flow. Here small amounts of material reaching relatively large distances are of importance. This motivates our interest in the full distribution of the location of particles swept by the fluid as a function of time. The distribution depends on the distance through the dimensionless quantity X^2=R^2/ . Using the Kolmogorov picture, we find that for small X, the distribution is proportional to exp(-aX^2) and exp(-bX^4/3) at its tail when X is large.

cond-mat.stat-mech↗
arXiv2004

Persisting roughness when deposition stops

Useful theories for growth of surfaces under random deposition of material have been developed by several authors. The simplest theory is that introduced by Edwards and Wilkinson (EW), which is linear and soluble. Its non linear generalization by Kardar, Parisi and Zhang (KPZ), resulted in many subsequent studies. Yet both theories EW and KPZ contain an unphysical feature. When deposition of material is stopped both theories predict that as time tends to infinity, the surface becomes flat. In fact, of course, the final surface is not flat, but simply has no gradients larger than the gradient related to the angle of repose. We modify the EW and KPZ to accommodate this feature and study the consequences for the simpler system which is a modification of the EW equation. In spite of the fact that the equation describing the evolution of the surface is not linear, we find that the steady state in the presence of noise is not very different in the long wave length limit from that of the linear EW. The situation is quite different from that of EW when deposition stops. Initially there is still some rearrangement of the surface but that stops as everywhere on the surface the gradient is less than that related to the angle of repose. The most interesting feature observed after deposition stops is the emergence of history-dependent steady state distributions.

cond-mat.stat-mech↗
arXiv2002

Statistical Mechanics in Collective Coordinates

We study the transformation of the statistical mechanics of N particles to the statistical mechanics of fields, that are the collective coordinates, describing the system. We give an explicit expression for the functional Fourier transform of the Jacobian of the transformation from particle to collective coordinate and derive the Fokker-Planck equation in terms of the collective coordinates. Simple approximations, leading to Debye-Huckel theory and to the hard sphere Percus-Yevick equation are discussed.

cond-mat.stat-mech↗
arXiv2001

The nature of the long time decay at a second order transition point

We show that at a second order phase transition, of ϕ^4 like system, a necessary condition for streched exponential decay of the time structure factor is obeyed. Using the ideas presented in this proof a crude estimate of the decay of the structure factor is obtained and shown to yield stretched exponential decay under very reasonable conditions.

cond-mat.stat-mech↗
arXiv2001

The missing stress-geometry equation in granular media

The simplest solvable problem of stress transmission through a static granular material is when the grains are perfectly rigid and have an average coordination number of $\bar{z}=d+1$. Under these conditions there exists an analysis of stress which is independent of the analysis of strain and the $d$ equations of force balance $\nabla_{j} σ_{ij}({\vec r}) = g_{i}({\vec r})$ have to be supported by $\frac{d(d-1)}{2}$ equations. These equations are of purely geometric origin. A method of deriving them has been proposed in an earlier paper. In this paper alternative derivations are discussed and the problem of the "missing equations" is posed as a geometrical puzzle which has yet to find a systematic solution as against sensible but fundamentally arbitrary approaches.

cond-mat.stat-mech↗
arXiv2001

Stretched Exponential Decay in the Edwards-Wilkinson Model

We consider the exactly soluble Edwards-Wilkinson Model in one dimension and demonstrate explicitly, that it is possible to construct a field, that does not depend explicitly on time, such that the corresponding time dependent correlation function,, is dominated at long times by a stretched exponential decay. The difference between this and the stretched exponential decay present in truly non linear systems is discussed.

cond-mat.stat-mech↗
arXiv2000

Streched exponential in non-linear stochastic filed theories

We consider the time dependent two point function, <ϕ_q (t) ϕ_-q (0)> in non-linear stochastic field theories, for which the KPZ equation serves as a prototype. In particular we consider the small q's and long times such that ω_q t>>1 (ω_q being the corresponding decay rate). We find that, since the generic case has ω_q \propto q^μfor small q where μ>1, the decay of the two point function is given by a streched exponential in ω_qt multiplied by a factor of t, <ϕ_q (t) ϕ_-q (0)> \propto t^{β_d}exp[-γ(ω_qt)^{1/μ}], where β_d=(d-1)/2μ, d is the dimensionality of space and γa dimensionless constant.

cond-mat.stat-mech↗
arXiv2000

Dynamic mechanical response of polymer networks

The dynamic-mechanical response of flexible polymer networks is studied in the framework of tube model, in the limit of small affine deformations, using the approach based on Rayleighian dissipation function. The dynamic complex modulus G* is calculated from the analysis of a network strand relaxation to the new equilibrium conformation around the distorted primitive path. Chain equilibration is achieved via a sliding motion of polymer segments along the tube, eliminating the inhomogeneity of the polymer density caused by the deformation. The characteristic relaxation time of this motion separates the low-frequency limit of the complex modulus from the high-frequency one, where the main role is played by chain entanglements, analogous to the rubber plateau in melts. The dependence of storage and loss moduli, G' and G'', on crosslink and entanglement densities gives an interpolation between polymer melts and crosslinked networks. We discuss the experimental implications of the rather short relaxation time and the slow square-root variation of the moduli and the loss factor tan at higher frequencies.

cond-mat.soft↗
arXiv1999

Statistical Mechanics of Stress Transmission in Disordered Granular Arrays

We give a statistical-mechanical theory of stress transmission in disordered arrays of rigid grains with perfect friction. Starting from the equations of microscopic force and torque balance we derive the fundamental equations of stress equilibrium. We illustrate the validity of our approach by solving the stress distribution of a homogeneous and isotropic array.

cond-mat.dis-nn↗
arXiv1998

Statistical Mechanics of Vibration-Induced Compaction of Powders

We propose a theory which describes the density relaxation of loosely packed, cohesionless granular material under mechanical tapping. Using the compactivity concept we develope a formalism of statistical mechanics which allows us to calculate the density of a powder as a function of time and compactivity. A simple fluctuation-dissipation relation which relates compactivity to the amplitude and frequency of a tapping is proposed. Experimental data of E.R.Nowak et al. [{\it Powder Technology} 94, 79 (1997) ] show how density of initially deposited in a fluffy state powder evolves under carefully controlled tapping towards a random close packing (RCP) density. Ramping the vibration amplitude repeatedly up and back down again reveals the existence of reversible and irreversible branches in the response. In the framework of our approach the reversible branch (along which the RCP density is obtained) corresponds to the steady state solution of the Fokker-Planck equation whereas the irreversible one is represented by a superposition of "excited states" eigenfunctions. These two regimes of response are analyzed theoretically and a qualitative explanation of the hysteresis curve is offered.

cond-mat.dis-nn↗
arXiv1996

The Dynamics of a Meandering River

We present a statistical model of a meandering river on an alluvial plane which is motivated by the physical non-linear dynamics of the river channel migration and by describing heterogeneity of the terrain by noise. We study the dynamics analytically and numerically. The motion of the river channel is unstable and we show that by inclusion of the formation of ox-bow lakes, the system may be stabilised. We then calculate the steady state and show that it is in agreement with simulations and measurements of field data.

cond-mat↗