The probability distribution of the largest local Liapunov exponent is evaluated for a classical An cluster at different values of the internal energy £", for a set of increasing values of the length in which the trajectory is partitioned. These distributions can be directly related to the evolution of ergodic behavior, particularly to how it exhibits distinctive, separable time scales which depend strongly on the energy of the system. Therefore, even though the inequivalence of ergodicity and chaos prohibits a Liapunov exponent itself from being a quantitative index of ergodicity, we find that the sample distributions used to evaluate Liapunov exponents nevertheless can be used for this purpose.PACS numbers: 05.45.+b, 36.40.+dThe Ar3 cluster has recently proved to be useful as a prototypical system in which to study details of phase changes in clusters, especially in the context of the question of how small systems explore their phase space [1-3]. The classical, conservative three-body LennardJones system has been found to be chaotic even at low energies, where the power spectrum displays largely normal-mode structure. The degree of chaotic behavior as measured by the Kolmogorov entropy is nonzero at an energy corresponding to a mean temperature as low as 2 K. The K entropy increases with energy, then decreases sharply in the range of entry into and passage through the saddle regions of the linear configurations of the potential surface, and finally increases again at still higher energies [l].At energies just high enough to allow passage over the linear saddle, the phase space seems to separate into a region of highly chaotic behavior that represents the motion in the well, and a region with much more "ordered" dynamics, that represents the motion across the saddle. In fact, above the transition energy the short-term average kinetic energy has a bimodal distribution that seems to correlate with the local values of the K entropy [1,3].In previous work on the ergodicity of clusters, no examination was made of the systematic evolution from shortterm to long-term behavior and only the statistic of global Liapunov exponents was examined. We suspected, as had others who used simple models as tests [4,5], that one could learn about this evolution by examining the probability distribution of the approximate values of the largest Liapunov exponent as a function of the duration of the interval used to obtain the time average, which is equivalent to the length of the averaging intervals into which the molecular-dynamic trajectory is partitioned. Those distributions make a powerful instrument to gain insight into the intriguing questions of the separation of phase space into high-chaos and low-chaos regions, and how the extent of ergodicity evolves in time for a fewbody system.To describe the classical Ar3 cluster we take the Lennard-Jones Hamiltonian 3 n 2 3 a 12 a > r,j < 61 (1) with AW 358 39.45 amu, a = 3.4 A, and e = 1.67x 10" 14 erg. To generate the trajectories we use the following molecular-dynamics algorithm: r,-(...

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It is shown using large-scale simulations that the density profile in diffusion-limited aggregation in two dimensions satisfies a scaling form (multiscaling) of the type g(r, R) = r "+ l"l lC(r/R), where r is the distance from the origin, R is the radius of gyration, and C(x) is an amplitude. Contrary to standard scaling, which predicts D(x) =const, here D(x) is found to depend continuously on x.Growth phenomena are a subject of very active current research, due to their widespread occurrence in a variety of different physical systems.In particular, considerable attention has been focused on the diffusionlimited-aggregation (DLA) model, which, although simple, appears to exhibit the essential features of many growth phenomena such as electrodeposition, dielectric breakdown, viscous fingers, crystal growth in aqueous solution, and neuron growth.One of the features, well established in DLA, is multifractality.This consists of associating a measure to each perimeter site of the aggregate, usually given by the harmonic measure or growth probability p, and in decomposing the total aggregate of radius R in infinitely many fractal sets each characterized by a singularI:ty o. = -logic p/ logio R with fractal dimension usually referred to as f(n). i2 This continuity of fractal dimensions characterizes the class of universality to which the growth model belongs. To calculate or measure this continuity of fractal dimensions, the scaling behavior of the growth probability distribution has to be considered. This quantity is usually difIicult to evaluate by means of analytical and numerical calculations or by direct experimental obser vations.A different quantity, which is more accessible to calculations and direct experimental observations, is the density profile g(r, R), which is defined as g(r, R)d"r = dN, where dN is the number of particles in the infinitesimal d-dimensional volume d"r at distance r from the origin, and the dependence on the total number of particles N is expressed via the radius of gyration R = R(N) .In the theory of DLA it is usually assumed, by analogy with critical phenomena, that g(r, R) satisfies the standard scaling form (2) where C(z) is a scaling function and D is the fractal dimension of the aggregate. However, very recently it has been suggested that the scaling structure of DLA clusters is much richer than that allowed by standard scaling and that (2) ought to be replaced by the multiscaling form Apparently the two forms may seem to coincide in the limit R oo. This is true if r is finite; however, if both r and R diverge with their ratio z = r/R fixed, the two forms diA'er considerably.In fact, the scaling form (2) gives g(r, R) R & &, while the multiscaling form gives g(r, R) R i~D~&l . Namely, the density profile (3) exhibits a continuity of power-law decay exponents, one for each value of z; consequently there is a different local fractal dimension D(z) for each shell corresponding to a given value of z. The fractal dimension of the infinite aggregate is obtained well inside the frozen regio...

We introduce a simple model describing the evolution of a population of information-carrying macromolecules. We discuss the asymptotic dependence of the variability of the population on different parameters, representing the severity or the fluctuations of the environment. We show the existence of a transition separating a neutralist evolutionary regime from a trapped one. We investigate the dependence of the evolutionary behavior of the population on the correlation properties of the fitness landscape.

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