ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 316–318. © Pleiades Publishing, Ltd., 2018.
On the Transition of a Mesoscopic System
to a Macroscopic System
V. P. Maslov
National Research University Higher School of Economics, Moscow, Russia
Received December 20, 2017
Keywords: mesoscopic physics, van der Waals model, Gentile statistics, Bose–Einstein statis-
tics, Fermi–Dirac statistics, GDP, gold reserve.
It is well known that mesoscopic physics is the part of the physics of condensed media that deals
with the properties of systems on a scale intermediate between the macroscopic and the microscopic
scale. A mesoscopic object in mathematics appeared in partition theory when it was formally carried
over from the integers to the rational numbers (for the abstract analytic theory of numbers, see  and
the bibliography therein).
The mesoscopic approach developed by the author involves values from N = −1 to N =+1.Letus
ﬁnd out values of N for which mesoscopic physics passes into the van der Waals macroscopic model .
In the author’s theory, mesoscopy is closely related to parastatistics (Gentile statistics) and poly-
logarithms. It is well known that parastatistics involves the parameter K, which deﬁnes the maximal
number of particles located at the given energy level. If the number of particles at the given energy level
is arbitrarily large, then the formulas of parastatistics coincide with the Bose–Einstein distribution. If
the number of particles K is at most 1, then the formulas of parastatistics coincide with the well-known
Self-consistent equations applicable in the intermediate case between Gentile statistics and
Bose–Einstein statistics were obtained by the author in  and . For the number of particles N,
they are of the form (see )
N = W (Li
where a is the activity, D =2γ +2is the number of degrees of freedom, W = V (λ
, λ is a constant
depending on the mass, V is the volume, and T is the temperature.
As a result of the expansion in terms of small N, Eq. (1) takes the form
(a) − W log(a)Li
(a) − 1
(a) − 2log(a)Li
The value of the parameter W deﬁnes the scaling of the dimensionless parameter a (the activity) as
well as the number of mesoobjects for |N|≤1.
The mesoscopic distribution (2) is bounded below by the number N = −1, but is not bounded above;
therefore, it can be extended to any value of N, of course, without the need to expand in terms of small N;
this leads from formula (1) to formula (2). At the same time, it turns out that, for N>2, the mesoscopic
distribution coincides, to a high accuracy, with the van der Waals macroscopic distribution.
The mesoscopic object examined in  may contain a large number of mesoparticles. This depends
on the number W , related to the scaling of a.For|N|≤1, the transition from the mesoscopic model
to the van der Waals macroscopic model occurs for N>2. In other words, for N>2, the formulas of
The article was submitted by the author for the English version of the journal.