41 research outputs found
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.Comment: 15 pages, 1 figure. Minor changes and update
Pascal Principle for Diffusion-Controlled Trapping Reactions
"All misfortune of man comes from the fact that he does not stay peacefully
in his room", has once asserted Blaise Pascal. In the present paper we evoke
this statement as the "Pascal principle" in regard to the problem of survival
of an "A" particle, which performs a lattice random walk in presence of a
concentration of randomly moving traps "B", and gets annihilated upon
encounters with any of them. We prove here that at sufficiently large times for
both perfect and imperfect trapping reactions, for arbitrary spatial dimension
"d" and for a rather general class of random walks, the "A" particle survival
probability is less than or equal to the survival probability of an immobile
target in the presence of randomly moving traps.Comment: 4 pages, RevTex, appearing in PR
Lattice theory of trapping reactions with mobile species
We present a stochastic lattice theory describing the kinetic behavior of
trapping reactions , in which both the and particles
perform an independent stochastic motion on a regular hypercubic lattice. Upon
an encounter of an particle with any of the particles, is
annihilated with a finite probability; finite reaction rate is taken into
account by introducing a set of two-state random variables - "gates", imposed
on each particle, such that an open (closed) gate corresponds to a reactive
(passive) state. We evaluate here a formal expression describing the time
evolution of the particle survival probability, which generalizes our
previous results. We prove that for quite a general class of random motion of
the species involved in the reaction process, for infinite or finite number of
traps, and for any time , the particle survival probability is always
larger in case when stays immobile, than in situations when it moves.Comment: 12 pages, appearing in PR
Fluctuations in Stationary non Equilibrium States
In this paper we formulate a dynamical fluctuation theory for stationary non
equilibrium states (SNS) which covers situations in a nonlinear hydrodynamic
regime and is verified explicitly in stochastic models of interacting
particles. In our theory a crucial role is played by the time reversed
dynamics. Our results include the modification of the Onsager-Machlup theory in
the SNS, a general Hamilton-Jacobi equation for the macroscopic entropy and a
non equilibrium, non linear fluctuation dissipation relation valid for a wide
class of systems
Kinetics of stochastically-gated diffusion-limited reactions and geometry of random walk trajectories
In this paper we study the kinetics of diffusion-limited, pseudo-first-order
A + B -> B reactions in situations in which the particles' intrinsic
reactivities vary randomly in time. That is, we suppose that the particles are
bearing "gates" which interchange randomly and independently of each other
between two states - an active state, when the reaction may take place, and a
blocked state, when the reaction is completly inhibited. We consider four
different models, such that the A particle can be either mobile or immobile,
gated or ungated, as well as ungated or gated B particles can be fixed at
random positions or move randomly. All models are formulated on a
-dimensional regular lattice and we suppose that the mobile species perform
independent, homogeneous, discrete-time lattice random walks. The model
involving a single, immobile, ungated target A and a concentration of mobile,
gated B particles is solved exactly. For the remaining three models we
determine exactly, in form of rigorous lower and upper bounds, the large-N
asymptotical behavior of the A particle survival probability. We also realize
that for all four models studied here such a probalibity can be interpreted as
the moment generating function of some functionals of random walk trajectories,
such as, e.g., the number of self-intersections, the number of sites visited
exactly a given number of times, "residence time" on a random array of lattice
sites and etc. Our results thus apply to the asymptotical behavior of the
corresponding generating functions which has not been known as yet.Comment: Latex, 45 pages, 5 ps-figures, submitted to PR
Persistence properties of a system of coagulating and annihilating random walkers
We study a d-dimensional system of diffusing particles that on contact either
annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1).
In 1-dimension, the system models the zero temperature Glauber dynamics of
domain walls in the q-state Potts model. We calculate P(m,t), the probability
that a randomly chosen lattice site contains a particle whose ancestors have
undergone exactly (m-1) coagulations. Using perturbative renormalization group
analysis for d < 2, we show that, if the number of coagulations m is much less
than the typical number M(t), then P(m,t) ~ m^(z/d) t^(-theta), with theta=d Q
+ Q(Q-1/2) epsilon + O(epsilon^2), z=(2Q-1) epsilon + (2 Q-1) (Q-1)(1/2+A Q)
epsilon^2 +O(epsilon^3), where Q=(q-1)/q, epsilon =2-d and A =-0.006. M(t) is
shown to scale as t^(d/2-delta), where delta = d (1 -Q)+(Q-1)(Q-1/2) epsilon+
O(epsilon^2). In two dimensions, we show that P(m,t) ~ ln(t)^(Q(3-2Q))
ln(m)^((2Q-1)^2) t^(-2Q) for m << t^(2 Q-1). The 1-dimensional results
corresponding to epsilon=1 are compared with results from Monte Carlo
simulations.Comment: 12 pages, revtex, 5 figure
Natural boundaries for the Smoluchowski equation and affiliated diffusion processes
The Schr\"{o}dinger problem of deducing the microscopic dynamics from the
input-output statistics data is known to admit a solution in terms of Markov
diffusions. The uniqueness of solution is found linked to the natural
boundaries respected by the underlying random motion. By choosing a reference
Smoluchowski diffusion process, we automatically fix the Feynman-Kac potential
and the field of local accelerations it induces. We generate the family of
affiliated diffusions with the same local dynamics, but different inaccessible
boundaries on finite, semi-infinite and infinite domains. For each diffusion
process a unique Feynman-Kac kernel is obtained by the constrained (Dirichlet
boundary data) Wiener path integration.As a by-product of the discussion, we
give an overview of the problem of inaccessible boundaries for the diffusion
and bring together (sometimes viewed from unexpected angles) results which are
little known, and dispersed in publications from scarcely communicating areas
of mathematics and physics.Comment: Latex file, Phys. Rev. E 49, 3815-3824, (1994
Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
We reconsider the conceptual foundations of the renormalization-group (RG)
formalism, and prove some rigorous theorems on the regularity properties and
possible pathologies of the RG map. Regarding regularity, we show that the RG
map, defined on a suitable space of interactions (= formal Hamiltonians), is
always single-valued and Lipschitz continuous on its domain of definition. This
rules out a recently proposed scenario for the RG description of first-order
phase transitions. On the pathological side, we make rigorous some arguments of
Griffiths, Pearce and Israel, and prove in several cases that the renormalized
measure is not a Gibbs measure for any reasonable interaction. This means that
the RG map is ill-defined, and that the conventional RG description of
first-order phase transitions is not universally valid. For decimation or
Kadanoff transformations applied to the Ising model in dimension ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . We discuss in detail
the distinction between Gibbsian and non-Gibbsian measures, and give a rather
complete catalogue of the known examples. Finally, we discuss the heuristic and
numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also
ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.
A Tight Parallel Repetition Theorem for Partially Simulatable Interactive Arguments via Smooth KL-Divergence
Hardness amplification is a central problem in the study of interactive protocols. While natural parallel repetition transformation is known to reduce the soundness error of some special cases of interactive arguments: three-message protocols (Bellare, Impagliazzo, and Naor [FOCS \u2797]) and public-coin protocols (Hastad, Pass, Wikstrom, and Pietrzak [TCC \u2710], Chung and Lu [TCC \u2710] and Chung and Pass [TCC \u2715]), it fails to do so in the general case (the above Bellare et al.; also Pietrzak and Wikstrom [TCC \u2707]).
The only known round-preserving approach that applies to all interactive arguments is Haitner\u27s random-terminating transformation [SICOMP \u2713], who showed that the parallel repetition of the transformed protocol reduces the soundness error at a weak exponential rate: if the original -round protocol has soundness error , then the -parallel repetition of its random-terminating variant has soundness error (omitting constant factors). Hastad et al. have generalized this result to partially simulatable interactive arguments, showing that the -fold repetition of an -round -simulatable argument of soundness error has soundness error . When applied to random-terminating arguments, the Hastad et al. bound matches that of Haitner.
In this work we prove that parallel repetition of random-terminating arguments reduces the soundness error at a much stronger exponential rate: the soundness error of the parallel repetition is , only an factor from the optimal rate of achievable in public-coin and three-message arguments. The result generalizes to -simulatable arguments, for which we prove a bound of . This is achieved by presenting a tight bound on a relaxed variant of the KL-divergence between the distribution induced by our reduction and its ideal variant, a result whose scope extends beyond parallel repetition proofs. We prove the tightness of the above bound for random-terminating arguments, by presenting a matching protocol
A review of Monte Carlo simulations of polymers with PERM
In this review, we describe applications of the pruned-enriched Rosenbluth
method (PERM), a sequential Monte Carlo algorithm with resampling, to various
problems in polymer physics. PERM produces samples according to any given
prescribed weight distribution, by growing configurations step by step with
controlled bias, and correcting "bad" configurations by "population control".
The latter is implemented, in contrast to other population based algorithms
like e.g. genetic algorithms, by depth-first recursion which avoids storing all
members of the population at the same time in computer memory. The problems we
discuss all concern single polymers (with one exception), but under various
conditions: Homopolymers in good solvents and at the point, semi-stiff
polymers, polymers in confining geometries, stretched polymers undergoing a
forced globule-linear transition, star polymers, bottle brushes, lattice
animals as a model for randomly branched polymers, DNA melting, and finally --
as the only system at low temperatures, lattice heteropolymers as simple models
for protein folding. PERM is for some of these problems the method of choice,
but it can also fail. We discuss how to recognize when a result is reliable,
and we discuss also some types of bias that can be crucial in guiding the
growth into the right directions.Comment: 29 pages, 26 figures, to be published in J. Stat. Phys. (2011