Tuesday, November 17, 2009

Random walks or how models drown in a sea of mathematical theory.

What I suggest is that in a persistent way, a system may exhibit historical behaviour, instead of recurrence.

David Ruelle a question to Yasha Sinai.


Another interesting paper from Demetris Koutsoyiannis shows the divergence of skill,between modelers and mathematical physicists.

According to the traditional notion of randomness and uncertainty, natural phenomena are separated into two mutually exclusive components, random (or stochastic) and deterministic. Within this dichotomous logic, the deterministic part supposedly represents cause-effect relationships and, thus, is physics and science (the “good”), whereas randomness has little relationship with science and no relationship with understanding (the “evil”). We argue that such views should be reconsidered by admitting that uncertainty is an intrinsic property of nature, that causality implies dependence of natural processes in time, thus suggesting predictability, but even the tiniest uncertainty (e.g., in initial conditions) may result in unpredictability after a certain time horizon. On these premises it is possible to shape a consistent stochastic representation of natural processes, in which predictability (suggested by deterministic laws) and unpredictability (randomness) coexist and are not separable or additive components. Deciding which of the two dominates is simply a matter of specifying the time horizon of the prediction. Long horizons of prediction are inevitably associated with high uncertainty, whose quantification relies on understanding the long-term stochastic properties of the processes.

http://www.hydrol-earth-syst-sci-discuss.net/6/6611/2009/hessd-6-6611-2009.pdf

"Long horizons of prediction are inevitably associated with high uncertainty,"

Indeed as DK cites the Kolmogorov-Chaiten problem of convergence-divergence due to slow-fast oscillators where relaxation times of the oscillator may be hiddem in the mists of time such as spin glasses etc. or randomness may be simply exhibiting historical in stead of recurrent behavior as rigorously proven by Yasha Sinai.

eg Ya G Sinai (1982) Limit behaviour of one-dimensional random walks in random environments,

Some more examination of this paper and the previous post is in order.

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