How to estimate the number of self-avoiding walks over 10^100? Use random walks

Counting the number of N-step self-avoiding walks (SAWs) on a lattice is one of the most difficult problems of enumerative combinatorics. Once we give up calculating the exact number of them, however, we have a chance to apply powerful computational methods of statistical mechanics to this problem. In this paper, we develop a statistical enumeration method for SAWs using the multicanonical Monte Carlo method. A key part of this method is to expand the configuration space of SAWs to random walks, the exact number of which is known. Using this method, we estimate a number of N-step SAWs on a square lattice, c_N, up to N=256. The value of c_256 is 5.6(1)*10^108 (the number in the parentheses is the statistical error of the last digit) and this is larger than one googol (10^100).Comment: 5 pages, 3 figures, 1 table, to appear in proceedings of YSMSPIP in Senda

The Audiovisual Tau Effect in Infancy

Perceived spatial intervals between successive flashes can be distorted by varying the temporal intervals between them (the “tau effect”). A previous study showed that a tau effect for visual flashes could be induced when they were accompanied by auditory beeps with varied temporal intervals (an audiovisual tau effect).We conducted two experiments to investigate whether the audiovisual tau effect occurs in infancy. Forty-eight infants aged 5–8 months took part in this study. In Experiment 1, infants were familiarized with audiovisual stimuli consisting of three pairs of two flashes and three beeps. The onsets of the first and third pairs of flashes were respectively matched to those of the first and third beeps. The onset of the second pair of flashes was separated from that of the second beep by 150 ms. Following the familiarization phase, infants were exposed to a test stimulus composed of two vertical arrays of three static flashes with different spatial intervals. We hypothesized that if the audiovisual tau effect occurred in infancy then infants would preferentially look at the flash array with spatial intervals that would be expected to be different from the perceived spatial intervals between flashes they were exposed to in the familiarization phase. The results of Experiment 1 supported this hypothesis. In Experiment 2, the first and third beeps were removed from the familiarization stimuli, resulting in the disappearance of the audiovisual tau effect. This indicates that the modulation of temporal intervals among flashes by beeps was essential for the audiovisual tau effect to occur (Experiment 2).These results suggest that the cross-modal processing that underlies the audiovisual tau effect occurs even in early infancy. In particular, the results indicate that audiovisual modulation of temporal intervals emerges by 5–8 months of age

Dataset _ Shirai and Imura (submitted to 'Psychological Science' for publication)

<p>These data are raw data of Shirai & Imura (submitted to 'Psychological Science' for publication).</p

Movies showing heat maps of gaze patterns in response to an expansion flow pattern in infants and adults

Supplemental materials for a submitted manuscript (XX). Movies showing heat maps of gaze patterns in response to an expansion flow pattern with low dot speed (5.8 deg/s) in all age groups (<i>N</i> = 20 for each group). <br

Negative Energetic Elasticity of Lattice Polymer Chain in Solvent

Negative internal energetic contribution to elastic modulus (negative energetic elasticity) has recently been observed in polymer gels. This finding challenges the conventional notion that the elastic moduli of rubberlike materials are determined mainly by entropic elasticity. However, the microscopic origin of negative energetic elasticity has not yet been clarified. Here, we consider the $n$-step interacting self-avoiding walk on a cubic lattice as a model of a single polymer chain (a subchain of a network in a polymer gel) in a solvent. We show the occurrence of negative energetic elasticity based on an exact enumeration up to $n=20$ and analytic expressions for arbitrary $n$ in three cases where the chain is highly stretched. Furthermore, we demonstrate that the negative energetic elasticity of this model originates from the attractive polymer-solvent interaction, which locally stiffens the chain and conversely softens the stiffness of the entire chain. This model qualitatively reproduces the temperature dependence of negative energetic elasticity observed in the polymer-gel experiments, indicating that the analysis of a single chain can explain the properties of negative energetic elasticity in polymer gels.Comment: 6 pages, 5 figure