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- Massively parallel algorithms and non-equilibrium surface growth
- Extremal-point densities of non-equilibrium interfaces
- Tagged diffusion and its applications
- Random hopping models and electrophoresis of polymers with impurities
- The persistence problem
- The delayed self-avoiding walk
- Thermodynamic formalism for chaotic maps
- Thermodynamics of spin chains via the Ruelle-Araki transfer operator

We study the asymptotic scaling properties of a massively parallel algorithm for discrete-event simulations where the discrete events are Poisson arrivals. The evolution of the simulated time horizon is analogous to a non-equilibrium surface. Monte Carlo simulations and a coarse-grained approximation indicate that the macroscopic landscape in the steady state is governed by the Edwards-Wilkinson Hamiltonian. Since the efficiency of the algorithm corresponds to the density of local minima in the associated surface, our results imply that the algorithm is asymptotically scalable. The figure below presents the density plots for the nearest-neighbor two-slope distribution. (a) L=10000. (b) L=3, exact.

**G. Korniss, Z. Toroczkai, M.A. Novotny, and P.A. Rikvold,***From massively parallel algorithms and fluctuating time horizons**to non-equilibrium surface growth,***Phys. Rev. Lett.,****84, 1351 (2000)****G. Korniss, M. A. Novotny, Z. Toroczkai and P. A. Rikvold,***Nonequilibrium Surface Growth and Scalability of Parallel Algorithms for Large Asynchronous Systems,***Computer Simulated Studies in Condensed-Matter Physics XIII, Springer Proceedings in Physics, Vol. 86, Eds. D. P. Landau, S.P. Lewis and H.-B. Schütler, pp. 183-188 (2001)**

The aim of statistical mechanics is to relate the macroscopic
observables to the microscopic properties of the system. Before
performing any such derivation one always has to specify the
spectrum of length-scales the analysis will comprise: while
'macroscopic' is usually defined in a unique way by the
every-day-life length scale, the 'microscopic' is never so
obvious, and the choice of the best lower-end scale is highly
non-universal, it is system dependent, usually left to our
physical 'intuition', or it is set by the limitations of the
experimental instrumentation. It is obvious that in order to
derive the laws of the gaseous matter we do not need to use the
physics of quarks, it is enough to start from an effective
microscopic model (or Hamiltonian) on the level of molecular
interactions. Once a lower lenght scale is set on which we can
define an effective microscopic dynamics, it becomes meaningful
to ask questions about local properties *at this length scale*,
e.g. nearest neighbor correlations, extremal-point densities,
etc.. These quantities are obviously not universal, however their
*behavior *against the variation of the length scales can
present qualitative and universal features*. *Here we
study the dynamics of macroscopically rough surfaces via
investigating an intriguing miscroscopic quantity: the density of
extrema (minima) and its finite size effects. We derive a large
number of exact analytical results about these quantities for a
large class of non-equilibrium surface fluctuations described by
both pure relaxational processes and linear Langevin equations,
and solid-on-solid (SOS) lattice-growth models. Via an analogy
between a simple SOS lattice-growth model and the time horizon of
conservative parallel algorithms these results allow us to prove
the asymptotic scalability of parallel computing when using such
algorithms, i.e., the fact that the efficiency of such a
computation does not vanish with increasing the number of
processing elements, but it has a lower non-trivial bound.

**Z. Toroczkai, G. Korniss, S. Das Sarma, and R. K. P. Zia,***Extremal-Point densities of interface fluctuations,***Phys. Rev. E, 62, 276 (2000)****G. Korniss, Z. Toroczkai, M.A. Novotny, and P.A. Rikvold,***From massively parallel algorithms and fluctuating time horizons**to non-equilibrium surface growth**,***Phys. Rev. Lett., 84, 1351 (2000)**

We analyze the lattice walk performed by a tagged member (the
red disk in the figure below) of an infinite 'sea' of particles
(the light blue disks) filling a d-dimensional lattice, in the
presence of a single vacancy (the empty node). The vacancy is
allowed to be occupied with probability 1/2*d* by any of
its 2*d* nearest neighbors, so that it executs a Brownian
walk. The only interaction between the particles is hard core
exclusion, and particle-particle exchange is forbidden. Thus, the
tagged particle, differing from the others only by its tag, moves
only when it exchanges places with the hole. In this sense, it is
a random walk 'driven' by the Brownian vacancy. The probability
distributions for its displacement and for the number of steps
taken, after *n*-steps of the vacancy, are derived.
Neither is a Gaussian! We also show that the only nontrivial
dimension where the walk is recurrent is in *d*=2. As an
application, we compute the expected energy shift caused by a
Brownian vacancy in a model for an extreme anisotropic binary
alloy. Our studies also include a Monte-Carlo study and a
mean-field analysis for interface erosion caused by mobile
vacancies.

**Z. Toroczkai,***The Brownian vacancy driven walk**,***Int. J. Mod. Phys. B****11****, 3343 (1997)****R.K.P. Zia and Z. Toroczkai,***Random walk with a hop-over site: a novel approach to tagged diffusion and its applications**,***J. Phys. A: Math.Gen.****31****, 9667 (1998)****Z. Toroczkai, G. Korniss, B. Schmittmann, and R.K.P. Zia,***Brownian-vacancy mediated disordering dynamics**,***Europhys. Lett.****40****, 281 (1997)**

We study a random walk on a one-dimensional periodic lattice
with *arbitrary* hopping rates. Further, the lattice
contains a single mobile, directional impurity (defect bond),
across which the rate is fixed at another arbitrary value. Due to
the defect, translational invariance is broken, even if all other
rates are identical. The structure of Master equations lead
naturally to the introduction of a new entity, associated with
the walker-impurity pair which we coin as 'quasi-walker'.
Analytic solution for the distributions in the steady state limit
is obtained. The velocities and diffusion constants for both the
random walker and impurity are given, being simply related to
that of the quasi-particle through physically meaningful
equations. As an application, we extend the Duke-Rubinstein
reptation model of gel electrophoresis (figure below) to include
polymers with impurities (neutral monomers represented as open
circles) and give the exact distribution of the steady state.

**Z. Toroczkai, and R.K.P. Zia,***Periodic one-dimensional hopping model with one mobile directional impurity**,***J. Stat. Phys.****87****, 545 (1997)****Z. Toroczkai, and R.K.P. Zia,***A model for electrophoresis of polymers with impurities: exact distribution for a steady state,***Phys. Lett. A****217****, 97 (1996)**

Persistence exponents are notoriously hard to calculate even for simple processes such as the deterministic diffusion equation with random initial condition. As a matter of fact in spite of the extensive work and the large number of publications since 1996, a rigorous derivation of the exponent is still missing.

We present a new method for extracting the persistence exponent theta for the diffusion equation, based on the distribution P of 'sign-times'. With the aid of a numerically verified Ansatz for P we derive an exact formula for theta in arbitrary spatial dimension d. Our results are in excellent agreement with previous numerical studies. Furthermore, our results indicate a qualitative change in P above d ~ 36, signalling the existence of a sharp change in the ergodic properties of the diffusion field.

**T. J. Newman and Z. Toroczkai,***Diffusive persistence and the "sign-time" distribution**,***Phys. Rev. E****58****, R2685 (1998)****Z. Toroczkai, T. J. Newman and S. Das Sarma,***Sign-time distributions for interface growth**,***Phys. Rev. E****60****, R1115 (1999)**

A soldier receives the order to mine a field with mines that
have delayed activation. Once a mine is planted it will not go
off when stepped on it, for a finite time interval *L*,
which is called the activation time, or delay time. After the
activation time has elapsed, the mine becomes active and will
blow if touched. Thus our soldier can step without worries on a
mine that has been planted not earlier than *L*.
Unfortunately, the soldier become very drunk for the day he had
to plant the mines. He could barely wander from planting site to
planting site and he could neither remember which planted mines
became active nor where he even planted them. The soldier became
a simple random walker on the field. Naturally the following
question arises: - what is the probability that our soldier is
able to plant *N* mines without being blown to the skies?
In more mathematical terms this problem amounts to the
determination of the statistics of simple random walk paths of
length *N* that do not contain loops longer than *L*.
Obviously the *L*=0 case corresponds to the pure
self-avoiding walk, and the *L* = *N* case
represents the simple random walk without any constraints.

In one dimension the pure self-avoiding walk has a trivial
solution: one straight path to the left and another straight path
to the right. The problem of the delayed self-avoiding walk in
one dimension is still awaiting for a rigorous solution for
general *L *!

Intermittent chaos is investigated by
means of an extended version of the statistical-mechanical
formalism developed by Sato and Honda [Phys. Rev. A **42, **3233
(1990)]. An exact criterion is given to classify intermittent
systems from the point of view of the generated chaotic phases
based on the probability distribution of the laminar lengths
which is an explicitely measurable quantity from the time series.
This criterion provides us with the generalization of the concept
of intermittency which broadens the class of critical phenomena
associated with the spectrum of dynamical entropies. It is shown
that, in contrast to general belief, the presence of the regular
chaos phase (i.e., vanishing Rényi entropies for inverse
temperatures q > 1) is not necessarily related to
intermittency. In fact, the absence of any phase transition or
the appearance of an anomalous chaos phase (i.e., infinite Rényi
entropies for q < 0) is also possible in intermittent systems.
We derive how the pressure, computed from a series of signals of
increasing length, approaches its asymptotic value in the regular
and anomalous phases.

**Z. Toroczkai and Á. Péntek,***Classification criterion for dynamical systems in intermittent chaos,***Phys. Rev. E****48****, 136 (1993)****Z. Toroczkai and Á. Péntek,***Detecting phase transitions in intermittent systems by using the thermodynamical formalism,***Z. Naturforsch.****49a****, 1235 (1994)**

The Kac model, a spin chain with exponentially decreasing long-range interactions, is investigated by means of a simple functional representation of the transfer operator. An analogy between the thermodynamics of spin chains and of 1D chaotic maps allows us to use techniques developed for generalized Frobenius-Perron equations to extract properties of the spin system, such as free energy and the decay rate of the correlation function. Although the Kac chain does not exhibit a phase transition, we find that the correlation decay shows a nonanalytic behavior at some finite temperature. We are also interested in a generalized version of the Kac model where the interaction still decays exponentially but in an oscillating fashion. This leads to the appearance of complicated patterns in the free energy caused by frustration which is a typical effect for disordered systems. By working out the analogy with 1D chaotic maps in more detail, we show how one can construct maps with the same thermodynamics as the spin chain. The associated maps turn out to be not smoothly differentiable, and their derivatives exhibit fractal properties.

**Á. Péntek, Z. Toroczkai, D.H. Mayer, and T. Tél,***Kac Model from a dynamical system's point of view,***Phys. Rev. E****49****, 2026 (1994)****Á. Péntek, Z. Toroczkai, D.H. Mayer, and T. Tél,***A generalized Kac model as a dynamical system**,***Z. Naturforsch.****49a****, 1212 (1994)**

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