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Математические заметки, 2017, том 101, выпуск 3, статья опубликована в англоязычной версии журнала
(Mi mzm11619)
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Эта публикация цитируется в 1 научной статье (всего в 1 статье)
Статьи, опубликованные в английской версии журнала
A Generalization of a Classical Number-Theoretic Problem, Condensate of Zeros, and Phase Transition to an Amorphous Solid
V. P. Maslov National Research University Higher School of Economics, Moscow, Russia
Аннотация:
Regularization of the Bose–Einstein distribution
using a parastatistical correction,
i.e.,
by means of the Gentile statistics,
is carried out.
It is shown that the regularization result
asymptotically coincides
with the Erdős formula obtained by using Ramanujan's formula
for the number of variants of the partition of an integer
into summands.
The Hartley entropy
regarded
as
the logarithm of the number of variants
defined by Ramanujan's exact formula
asymptotically
coincides
with the polylogarithm
associated with the entropy of the Bose–Einstein distribution.
The fact that these formulas coincide
makes it possible to extend the entropy
to the domain of the Fermi–Dirac distribution
with minus sign.
Further, the formulas
for the distribution
are extended to fractional dimension
and also
to dimension $1$,
which corresponds to the Waring problem.
The relationship between the resulting formulas
and the liquid
corresponding to the case of nonpolar molecules is described
and the law of phase transition of liquid
to an amorphous solid
under negative pressure
is discussed.
Also the connection of the resulting formulas
with
the gold reserve
in economics is considered.
Ключевые слова:
number of degrees of freedom, Bose–Einstein distribution, Fermi–Dirac distribution,
number theory, Erdős formula, Gentile statistics,
parastatistics, Hartley entropy, phase transition to an amorphous solid.
Поступило: 02.12.2016
Образец цитирования:
V. P. Maslov, “A Generalization of a Classical Number-Theoretic Problem, Condensate of Zeros, and Phase Transition to an Amorphous Solid”, Math. Notes, 101:3 (2017), 488–496
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/mzm11619
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