arXiv subjects

J. M. Luck

Publications and source records attributed to J. M. Luck.

At least 55 records · Page 3Linked to original sources

arXiv2007

Dynamical diversity and metastability in a hindered granular column near jamming

Granular media jam into a panoply of metastable states. The way in which these states are achieved depends on the nature of local and global constraints on grains; here we investigate this issue by means of a non-equilibrium stochastic model of a hindered granular column near its jamming limit. Grains feel the constraints of grains above and below them differently, depending on their position. A rich phase diagram with four dynamical phases (ballistic, activated, logarithmic and glassy) is revealed. The statistics of the jamming time and of the metastable states reached as attractors of the zero-temperature dynamics is investigated in each of these phases. Of particular interest is the glassy phase, where intermittency and a strong deviation from Edwards' flatness are manifest.

cond-mat.stat-mech↗

arXiv2007

Structure of the stationary state of the asymmetric target process

We introduce a novel migration process, the target process. This process is dual to the zero-range process (ZRP) in the sense that, while for the ZRP the rate of transfer of a particle only depends on the occupation of the departure site, it only depends on the occupation of the arrival site for the target process. More precisely, duality associates to a given ZRP a unique target process, and vice-versa. If the dynamics is symmetric, i.e., in the absence of a bias, both processes have the same stationary-state product measure. In this work we focus our interest on the situation where the latter measure exhibits a continuous condensation transition at some finite critical density $ρ_c$, irrespective of the dimensionality. The novelty comes from the case of asymmetric dynamics, where the target process has a nontrivial fluctuating stationary state, whose characteristics depend on the dimensionality. In one dimension, the system remains homogeneous at any finite density. An alternating scenario however prevails in the high-density regime: typical configurations consist of long alternating sequences of highly occupied and less occupied sites. The local density of the latter is equal to $ρ_c$ and their occupation distribution is critical. In dimension two and above, the asymmetric target process exhibits a phase transition at a threshold density $ρ_0$ much larger than $ρ_c$. The system is homogeneous at any density below $ρ_0$, whereas for higher densities it exhibits an extended condensate elongated along the direction of the mean current, on top of a critical background with density $ρ_c$.

cond-mat.stat-mech↗

arXiv2006

Nonequilibrium stationary states with Gibbs measure for two or three species of interacting particles

We construct explicit examples of one-dimensional driven diffusive systems for two and three species of interacting particles, defined by asymmetric dynamical rules which do not obey detailed balance, but whose nonequilibrium stationary-state measure coincides with a prescribed equilibrium Gibbs measure. For simplicity, the measures considered in this construction only involve nearest-neighbor interactions. For two species, the dynamics thus obtained generically has five free parameters, and does not obey pairwise balance in general. The latter property is satisfied only by the totally asymmetric dynamics and the partially asymmetric dynamics with uniform bias, i.e., the cases originally considered by Katz, Lebowitz, and Spohn. For three species of interacting particles, with nearest-neighbor interactions between particles of the same species, the totally asymmetric dynamics thus obtained has two free parameters, and obeys pairwise balance. These models are put in perspective with other examples of driven diffusive systems. The emerging picture is that asymmetric (nonequilibrium) stochastic dynamics leading to a given stationary-state measure are far more constrained (in terms of numbers of free parameters) than the corresponding symmetric (equilibrium) dynamics.

cond-mat.stat-mech↗

arXiv2005

Ground-state structure and dynamics in a toy model for granular compaction

We report on a toy model for the glassy compaction dynamics of granular systems, introduced and investigated in collaboration with Anita Mehta and Peter Stadler. A stochastic dynamics is defined on a column of grains. Grains are anisotropic and possess a discrete orientational degree of freedom. Gravity induces long-range directional interactions down the column. The key control parameter of the model, $ε$, is a representation of granular shape. Rational and irrational values of $ε$ correspond to very different kinds of behavior, both in statics (structure of ground states) and in low-temperature dynamics (retrieval of ground states).

cond-mat.stat-mech↗

arXiv2005

Dynamics of the condensate in zero-range processes

For stochastic processes leading to condensation, the condensate, once it is formed, performs an ergodic stationary-state motion over the system. We analyse this motion, and especially its characteristic time, for zero-range processes. The characteristic time is found to grow with the system size much faster than the diffusive timescale, but not exponentially fast. This holds both in the mean-field geometry and on finite-dimensional lattices. In the generic situation where the critical mass distribution follows a power law, the characteristic time grows as a power of the system size.

cond-mat.stat-mech↗

arXiv2005

How the rich get richer

In our model, $n$ traders interact with each other and with a central bank; they are taxed on the money they make, some of which is dissipated away by corruption. A generic feature of our model is that the richest trader always wins by 'consuming' all the others: another is the existence of a threshold wealth, below which all traders go bankrupt. The two-trader case is examined in detail,in the socialist and capitalist limits, which generalise easily to $n>2$. In its mean-field incarnation, our model exhibits a two-time-scale glassy dynamics, as well as an astonishing universality.When preference is given to local interactions in finite neighbourhoods,a novel feature emerges: instead of at most one overall winner in the system,finite numbers of winners emerge, each one the overlord of a particular region.The patterns formed by such winners (metastable states) are very much a consequence of initial conditions, so that the fate of the marketplace is ruled by its past history; hysteresis is thus also manifested.

physics.soc-ph↗

arXiv2005

Interacting black holes on the brane: the seeding of binaries

We consider the evolution of sub-horizon-sized black holes which are formed during the high energy phase of the braneworld scenario. These black holes are long-lived due to modified evaporation and accretion of radiation during the radiation dominated era. We argue that an initial mass difference between any two neighbouring black holes is always amplified because of their exchange of energy with the surrounding radiation. We present a scheme of binary formation based on mass differences suggesting that such a scenario could lead to binaries with observable signatures.

astro-ph↗

arXiv2004

Metastability in zero-temperature dynamics: Statistics of attractors

The zero-temperature dynamics of simple models such as Ising ferromagnets provides, as an alternative to the mean-field situation, interesting examples of dynamical systems with many attractors (absorbing configurations, blocked configurations, zero-temperature metastable states). After a brief review of metastability in the mean-field ferromagnet and of the droplet picture, we focus our attention onto zero-temperature single-spin-flip dynamics of ferromagnetic Ising models. The situations leading to metastability are characterized. The statistics and the spatial structure of the attractors thus obtained are investigated, and put in perspective with uniform a priori ensembles. We review the vast amount of exact results available in one dimension, and present original results on the square and honeycomb lattices.

cond-mat.stat-mech↗

arXiv2004

Cooperativity in sandpiles: statistics of bridge geometries

Bridges form dynamically in granular media as a result of spatiotemporal inhomogeneities. We classify bridges as linear and complex, and analyse their geometrical characteristics. In particular, we find that the length distribution of linear bridges is exponential. We then turn to the analysis of the orientational distribution of linear bridges and find that, in three dimensions, they are {\it vertically diffusive but horizontally superdiffusive}; thus, when they exist, long linear bridges form `domes'. Our results are in good accord with Monte Carlo simulations of bridge structure; we make predictions for quantities that are experimentally accessible, and suggest that bridges are very closely related to force chains.

cond-mat.stat-mech↗

arXiv2004

Dynamics at the angle of repose: jamming, bistability, and collapse

When a sandpile relaxes under vibration, it is known that its measured angle of repose is bistable in a range of values bounded by a material-dependent maximal angle of stability; thus, at the same angle of repose, a sandpile can be stationary or avalanching, depending on its history. In the nearly jammed slow dynamical regime, sandpile collapse to a zero angle of repose can also occur, as a rare event. We claim here that fluctuations of {\it dilatancy} (or local density) are the key ingredient that can explain such varied phenomena. In this work, we model the dynamics of the angle of repose and of the density fluctuations, in the presence of external noise, by means of coupled stochastic equations. Among other things, we are able to describe sandpile collapse in terms of an activated process, where an effective temperature (related to the density as well as to the external vibration intensity) competes against the configurational barriers created by the density fluctuations.

cond-mat.stat-mech↗

arXiv2004

A deterministic model of competitive cluster growth: glassy dynamics, metastability and pattern formation

We investigate a model of interacting clusters which compete for growth. For a finite assembly of coupled clusters, the largest one always wins, so that all but this one die out in a finite time. This scenario of `survival of the biggest' still holds in the mean-field limit, where the model exhibits glassy dynamics, with two well separated time scales, corresponding to individual and collective behaviour. The survival probability of a cluster eventually falls off according to the universal law $(\ln t)^{-1/2}$. Beyond mean field, the dynamics exhibits both aging and metastability, with a finite fraction of the clusters surviving forever and forming a non-trivial spatial pattern.

cond-mat.stat-mech↗

arXiv2004

On the statistics of superlocalized states in self-affine disordered potentials

We investigate the statistics of eigenstates in a weak self-affine disordered potential in one dimension, whose Gaussian fluctuations grow with distance with a positive Hurst exponent $H$. Typical eigenstates are superlocalized on samples much larger than a well-defined crossover length, which diverges in the weak-disorder regime. We present a parallel analytical investigation of the statistics of these superlocalized states in the discrete and the continuum formalisms. For the discrete tight-binding model, the effective localization length decays logarithmically with the sample size, and the logarithm of the transmission is marginally self-averaging. For the continuum Schrödinger equation, the superlocalization phenomenon has more drastic effects. The effective localization length decays as a power of the sample length, and the logarithm of the transmission is fully non-self-averaging.

cond-mat.stat-mech↗

arXiv2003

Shaken, not stirred: why gravel packs better than bricks

We explore the effect of shape -- jagged vs. regular -- in the jamming limit of very gently shaken packings. Our measure of shape $\eps$ is the void space occupied by a disordered grain; we show that depending on its number-theoretic nature, two generic behaviours are obtained. Thus, regularly shaped grains (rational $\eps$) have ground states of perfect packing, which are irretrievably lost under zero-temperature shaking; the reverse is the case for jagged grains (irrational $\eps$), where the ground state is only optimally packed, but entirely retrievable. At low temperatures, we find intermittency at the surface, which has recently been seen experimentally.

cond-mat.soft↗

arXiv2003

Anomalous self-diffusion in the ferromagnetic Ising chain with Kawasaki dynamics

We investigate the motion of a tagged spin in a ferromagnetic Ising chain evolving under Kawasaki dynamics. At equilibrium, the displacement is Gaussian, with a variance growing as $A t^{1/2}$. The temperature dependence of the prefactor $A$ is derived exactly. At low temperature, where the static correlation length $ξ$ is large, the mean square displacement grows as $(t/ξ^2)^{2/3}$ in the coarsening regime, i.e., as a finite fraction of the mean square domain length. The case of totally asymmetric dynamics, where $(+)$ (resp. $(-)$) spins move only to the right (resp. to the left), is also considered. In the steady state, the displacement variance grows as $B t^{2/3}$. The temperature dependence of the prefactor $B$ is derived exactly, using the Kardar-Parisi-Zhang theory. At low temperature, the displacement variance grows as $t/ξ^2$ in the coarsening regime, again proportionally to the mean square domain length.

cond-mat.stat-mech↗

arXiv2003

Non-monotonic disorder-induced enhanced tunneling

The quantum-mechanical transmission through a disordered tunnel barrier is investigated analytically in the following regime: (correlation range of the random potential) << (penetration length) << (barrier length). The mean and/or the width of the potential can either be constant, or vary slowly across the barrier. The typical transmission is found to be a non-monotonic function of the disorder strength, increasing at weak disorder, reaching a maximum in the crossover from weak to strong disorder, and decreasing at strong disorder. This work provides a quantitative analysis of the phenomenon of disorder-induced enhanced tunneling, put forward by Freilikher et al. [Phys. Rev. E {\bf 51}, 6301 (1995); B {\bf 53}, 7413 (1996)].

cond-mat.mes-hall↗

arXiv2003

A column of grains in the jamming limit: glassy dynamics in the compaction process

We investigate a stochastic model describing a column of grains in the jamming limit, in the presence of a low vibrational intensity. The key control parameter of the model, $ε$, is a representation of granular shape, related to the reduced void space. Regularity and irregularity in grain shapes, respectively corresponding to rational and irrational values of $ε$, are shown to be centrally important in determining the statics and dynamics of the compaction process.

cond-mat.stat-mech↗

arXiv2003

Why shape matters in granular compaction

We present a stochastic model of dynamically interacting grains in one dimension, in the presence of a low vibrational intensity, to investigate the effect of shape on the statics and dynamics of the compaction process. Regularity and irregularity in grain shapes are shown to be centrally important in determining the statics of close-packing states, as well as the nature of zero- and low-temperature dynamics in this columnar model.

cond-mat.stat-mech↗

arXiv2003

On random symmetric matrices with a constraint: the spectral density of random impedance networks

We derive the mean eigenvalue density for symmetric Gaussian random N x N matrices in the limit of large N, with a constraint implying that the row sum of matrix elements should vanish. The result is shown to be equivalent to a result found recently for the average density of resonances in random impedance networks [Y.V. Fyodorov, J. Phys. A: Math. Gen. 32, 7429 (1999)]. In the case of banded matrices, the analytical results are compared with those extracted from the numerical solution of Kirchhoff equations for quasi one-dimensional random impedance networks.

cond-mat.dis-nn↗