Editorial Office:

Management:

R. S. Oyarzabal

Technical Support:

D. H. Diaz

M. A. Gomez

W. Abrahão

G. Oliveira

Publisher by

Knobook Pub

doi: 10.6062/jcis.2008.01.01.0003(Free PDF)

Juliana M. Berbert, Rodrigo S. Gonzalez and Alexandre S. Martinez

Consider N points randomly and uniformly distributed in a d-dimensional hypercube. A walker explores this disordered medium going to the nearest site, which has not been visited in the last µ (memory) steps. The walker trajectory is composed of a transient part and a periodic part (cycles). In this case, travelers can or cannot explore all available space, given rise to a crossover at critical memory, for one-dimensional systems µ1 = ln2 N, between localized and extended regimes. The deterministic rule can be softened to consider more realistic situations with the inclusion of a stochastic parameter T (temperature). In this case, the walker movement is defined by a probability density function (PDF) that is parameterized by T and a cost function, which increases as the distance among sites increases. As the temperature increases, the walker can escape from cycles and extend the exploration. Here we review the analytical results obtained for a system with arbitrary dimensionality d for µ = 0 and one-dimensional systems with µ = 1. Also we suggest an extension of this system to study the influence of the temperature on the critical memory.

Stochastic walk, deterministic walk, non-Markovian processes, glass transition.

[1] LIMA GF, MARTINEZ AS & KINOUCHI O. 2001. Deterministic Walks in Random Media. Phys. Rev. Lett., 87: 010603. doi: 10.1103/PhysRevLett.87.010603

[2] STANLEY HE & BULDYREV SV. 2001. Statistical physics - The salesman and the tourist. Nature (London), 413: 373-374. doi: 10.1038/35096668

[3] MARTINEZ AS, KINOUCHI O & RISAU-GUSMAN S. 2004. Exploratory behavior, trap models, and glass transitions. Phys. Rev. E, 69: 017101. doi: 10.1103/PhysRevE.69.017101

[4] RISAU-GUSMAN S, MARTINEZ AS & KINOUCHI O. 2003. Escaping from cycles through a glass transition. Phys. Rev. E, 68: 016104. doi: 10.1103/PhysRevE.68.016104

[5] VISWANATHAN GM, AFANASYEV V, BULDYREV S, MURPHY EJ, PRINCE PA & STANLEY HE. 1996. L'evy flight search patterns of wandering albatrosses. Nature (London), 381: 413-415. doi: 10.1038/381413a0

[6] VISWANATHAN GM, BULDYREV S, HAVLIN S, DA LUZ MGE, RAPOSOL EP & STANLEY HE. 1999. Optimizing the success of random searches. Nature (London), 401: 911-914. doi: 10.1038/44831

[7] BOUCHAUD J-P. 1992. Weak ergodicity breaking and aging in disordered-systems. J. Phys. I (France), 2: 1705-1713. doi: 10.1051/jp1:1992238

[8] DYRE JC. 1995. Energy master equation: a low-temperature approximation to Bassler's random-walk model. Phys. Rev. B, 51: 12276-12294. doi: 10.1103/PhysRevB.51.12276

[9] MONTHUS C & BOUCHAUD J-P. 1996. Models of traps and glass phenomenology. J. Phys. A, 29: 3847-3869. doi: 10.1088/0305-4470/29/14/012

[10] RINN B, MAASS P & BOUCHAUD J-P. 2001. Hopping in the glass configuration space: subaging and generalized scaling laws. Phys. Rev. B, 64: 104417.

[11] BERTIN EM & BOUCHAUD J-P. 2003. Subdiffusion and localization in the one-dimensional trap model. Phys. Rev. E, 67: 026128. doi: 10.1103/PhysRevE.67.026128

[12] DENNY RA, REICHMAN DR & BOUCHAUD J-P. 2003. Trap models and slow dynamics in supercooled liquids. Phys. Rev. Lett., 90: 025503. doi: 10.1103/PhysRevLett.90.025503

[13] ARRUDA TJ, GONZ'ALEZ RS, TERC.ARIOL CAS & MARTINEZ AS. 2008. Arithmetical and geometrical means of generalized logarithmic and exponential functions: generalized sum and product operators. Phys. Lett. A, 372: 2578-2582. doi: 10.1016/j.physleta.2007.12.020

[14] MARTINEZ AS, GONZ'ALEZ RS & TERC.ARIOL CAS. 2008. Continuous growth models in terms of generalized logarithm and exponential functions. Physica A, 387: 5679-5687. doi: 10.1016/j.physa.2008.06.015

[15] KINOUCHI O, MARTINEZ AS, LIMA GF, LOURENC.O GM & RISAUGUSMAN S. 2002. Deterministic walks in random networks: an application to thesaurus graphs. Physica A, 315: 665-676. doi: 10.1016/S0378-4371(02)00972-X

[16] TERC.ARIOL CAS & MARTINEZ AS. 2005. Analytical results for the statistical distribution related to a memoryless deterministic walk: imensionality effect and mean-field models. Phys. Rev. E, 72: 021103. doi: 10.1103/PhysRevE.72.021103

[17] TERC.ARIOL CAS & MARTINEZ AS. 2008. The influence of memory in deterministic walks in random media: analytical calculation within a mean field approximation. Phys. Rev. E, 78: 031111. doi: 10.1103/PhysRevE.78.031111

Combining wavelets and linear spectral mixture model for MODIS satellite sensor time-series analysis

doi: 10.6062/jcis.2008.01.01.0005

Freitas and Shimabukuro(Free PDF) Riddled basins in complex physical and biological systems

doi: 10.6062/jcis.2009.01.02.0009

Viana et al.(Free PDF) Use of ordinary Kriging algorithm and wavelet analysis to understanding the turbidity behavior in an Amazon floodplain

doi: 10.6062/jcis.2008.01.01.0006

Alcantara.(Free PDF) A new multi-particle collision algorithm for optimization in a high performance environment

doi: 10.6062/jcis.2008.01.01.0001

Luz et al.((Free PDF)