The Limit Distribution of a Number Theoretic Function Arising from a Problem in Statistical Mechanics

Thermodynamics of the Farey Fraction Spin Chain

“…The "density of states" (number of configurations with a given energy) for the FFSC not well-behaved. In fact it is known rigorously that this quantity, summed over all chain lengths, has a limit distribution [3].…”

“…Despite this, there are many examples of one-dimensional systems that do exhibit a phase transition. The Farey Fraction Spin Chain (FFSC) [1] is one such case, which has attracted interest from both physicists and mathematicians [2,3,4]. (Since this work uses some methods that may be unfamiliar to the latter, we include a paragraph at the end of this section outlining our results from a mathematical viewpoint.…”

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“…For the Farey spin chain, an equation with the same form as (36) may also be written, with the same leading divergent behavior. Here the limit of the function corresponding to φ k (n) is not known, though some related information is available [3].…”

“…One is the Farey fraction spin chain, a one-dimensional statistical model first proposed by two of the authors [1]. This work has spawned a number of further studies, by both physicists and number theorists [2,3,4] One can define the model as a periodic chain of sites with two possible spin states (A or B)…”

Спиновые Цепочки И Задача Арнольда О Статистиках Гаусса - Кузьмина Для Квадратичных Иррациональностей

“…Введение В работе [1] была предложена теоретико-числовая модель спиновых цепочек, основанная на рядах Фарея (дальнейшие результаты см. в [2]- [4]). В этой мо-дели конечной цепочке из спинов, каждый из которых может быть направлен вверх (↑) или вниз (↓), ставится в соответствие произведение матриц по правилу ↑= A, ↓= B. Например,…”