2,147 research outputs found
Fast simulation of large-scale growth models
We give an algorithm that computes the final state of certain growth models
without computing all intermediate states. Our technique is based on a "least
action principle" which characterizes the odometer function of the growth
process. Starting from an approximation for the odometer, we successively
correct under- and overestimates and provably arrive at the correct final
state.
Internal diffusion-limited aggregation (IDLA) is one of the models amenable
to our technique. The boundary fluctuations in IDLA were recently proved to be
at most logarithmic in the size of the growth cluster, but the constant in
front of the logarithm is still not known. As an application of our method, we
calculate the size of fluctuations over two orders of magnitude beyond previous
simulations, and use the results to estimate this constant.Comment: 27 pages, 9 figures. To appear in Random Structures & Algorithm
Absorbing-state phase transitions in fixed-energy sandpiles
We study sandpile models as closed systems, with conserved energy density
playing the role of an external parameter. The critical energy density,
, marks a nonequilibrium phase transition between active and absorbing
states. Several fixed-energy sandpiles are studied in extensive simulations of
stationary and transient properties, as well as the dynamics of roughening in
an interface-height representation. Our primary goal is to identify the
universality classes of such models, in hopes of assessing the validity of two
recently proposed approaches to sandpiles: a phenomenological continuum
Langevin description with absorbing states, and a mapping to driven interface
dynamics in random media. Our results strongly suggest that there are at least
three distinct universality classes for sandpiles.Comment: 41 pages, 23 figure
Emergence of the second law out of reversible dynamics
Abstract If one demystifies entropy the second law of thermodynamics comes out as an emergent property entirely based on the simple dynamic mechanical laws that govern the motion and energies of system parts on a micro-scale. The emergence of the second law is illustrated in this paper through the development of a new, very simple and highly efficient technique to compare time-averaged energies in isolated conservative linear large scale dynamical systems. Entropy is replaced by a notion that is much more transparent and more or less dual called ectropy. Ectropy has been introduced before but we further modify the notion of ectropy such that the unit in which it is expressed becomes the unit of energy. The second law of thermodynamics in terms of ectropy states that ectropy decreases with time on a large enough time-scale and has an absolute minimum equal to zero. Zero ectropy corresponds to energy equipartition. Basically we show that by enlarging the dimension of an isolated conservative linear dynamical system and the dimension of the system parts over which we consider time-averaged energy partition, the tendency towards equipartition increases while equipartition is achieved in the limit. This illustrates that the second law is an emergent property of these systems. Finally from our large scale linear dynamic model we clarify Loschmidt’s paradox concerning the irreversible behavior of ectropy obtained from the reversible dynamic laws that govern motion and energy at the micro-scal
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