Allison mixtures: Where random digits obey thermodynamic principles

Abstract

Parrondo's paradox is a well-known situation where losing strategies can be combined to win, and is based on a thermodynamic Brownian ratchet principle. We demonstrate a new extension of Parrondo's paradox called the Allison mixture where it is possible to paradoxically generate random sequences with nonzero autocovariance, out of sub-sequences of zero autocovariance. We explain the effect with a ratchet analogy and show the behavior of random digits follows a thermodynamic analogy. As Kish-based cipher techniques rely on the Second Law for their security, this raises the tantalizing open question of whether the Allison mixture can be exploited for new Kish-inspired digital security schemes.

Towards an information-theoretic model of the Allison mixture stochastic process

“…Substituting the stationary and transition probabilities into the entropy, we find the single-step autoinformation, stated without further detail in the following lemma. , and so these three previously-described [2] conditions for decorrelation of the sampling process imply zero mutual information and therefore genuine independence.…”

“…Knowing the means of the input processes U t and V t , the single-step autocovariance is given by the following theorem, given by [2] and which we restate without proof:…”

Paradoxical Survival: Examining the Parrondo Effect across Biology

“…The paradox was first conceptualized as an abstraction of the phenomenon of flashing Brownian ratchets, [2][3][4][5][6][7][8][9] wherein diffusive particles exhibit unexpected drift when exposed to alternating periodic potentials. It has since been applied across a multitude of neighboring disciplines in the physical sciences and engineering-related fields, [10,11] such as diffusive and granular flow dynamics, [12][13][14] information thermodynamics, [15][16][17][18] chaos theory, [19][20][21][22][23][24][25] switching problems, [26][27][28] and quantum phenomena; [29][30][31][32][33][34][35][36][37][38] but a plethora of exciting applications has also been found in biology, which this paper focuses upon. Parrondo's paradox has nourished a synergistic interdisciplinary effort in which the existing mathematical and gametheoretic work is driving its rapid application in biology, where it has inspired new ideas and technical approaches.…”

Allison mixture and the two-envelope problem

“…Another interesting variant of Parrondo's paradox is the Allison mixture [33][34][35][36], where random mixing of two random sequences creates autocorrelation [36]. This is an area which has not been fully explored yet.…”

“…[8] and [37] for more information on the Allison mixture. There have been discussions to link the Allison mixture to applications in encryption and optimization of file compression [33].…”