TITLE:
The Master Equation and the Pauli Equation in the Fuzzy Time Model
AUTHORS:
Aleksey M. Avdeenko
KEYWORDS:
Fuzzy Sets, Fuzzy Time, Master Equation, Second Law of Ther-modynamics, Fuzzy Pauli Equation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.6 No.2,
February
13,
2018
ABSTRACT: The approach proposed in the study is based on the revision of the concept of
time as a point on the real axis. It uses the concept of fuzzy time as the set of
real numbers with a finite, but not equal to one, function of membership to
the time set, i.e. the fuzzy time concept. It is postulated that in fuzzy time t the
system dynamics follows from the standard variational principle of the least
action and is ordinary Hamilton-Jacobi mechanics. This validates the passage
to the limit from fuzzy mechanics to ordinary variational conservative mechanics.
The Liouville equation is solved by the method of successive approximations
in the time domain of a much larger characteristic scale of fuzziness,
using interaction as a small parameter. A standard diagram technique
is used. It can be shown that the defuzzification of the Liouville equation inevitably
reduces the reversible part in the description to the irreversible evolutionary
equation. The latter leads to the second law of thermodynamics. Generalization
to the quantum case is possible, i.e. the so-called fuzzy Pauli equation
can be drawn.