The Master Equation and the Pauli Equation in the Fuzzy Time Model

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.


Introduction
Hamiltonian mechanics (the mechanics of conservative systems) is reversible in time. The world based on Hamiltonian mechanics is either orbitally stable, or, in the presence of hyperbolic points, highly sensitive to initial conditions. One of the options to solve this problem has been proposed in the works of I. Prigozhin and his school [1] [2]. The essence of the approach involves considering highly unstable conservative systems (so-called "K-flows"), for which the Poincaré section of the total phase space is analyzed.
Phase space regions are associated with special objects-partitions n γ . In the particular case when the system evolution can be transformed to a discrete transformation, for example, "baker's transformation" of phase space unit square, partitions are functions that take the value of ±1 on the left and the right sides of the square.
The sequential action of evolution operator 1 n n U γ γ + = describes the state of the system after n cycles of evolution. It can be shown that there is operator T, for which functions n γ are eigenfunctions with infinitely singular eigenvalue n.
Further, existence of Hermitian operator Λ , which is a nonnegative decreasing function of T, is postulated. It is this operator that determines connection . In this case it is possible to show the validity of the second law of thermodynamics from microscopic principles.
Problems and contradictions of this approach are quite obvious: firstly, the approach is realized in the phase space Poincaré section, but not in the full phase space; secondly, reducibility of all unstable systems to transformations of this kind is not obvious; finally, the main thing is that the explicit form of operator Λ is not obtained. This somewhat depreciates the approach.
In the proposed study, the author assumes a different concept. The approach is based on the revision of the concept of time as a point on the real axis to the concept of fuzzy time as a set of real numbers with a finite, but not equal to one, membership function, i.e. to the fuzzy time concept. The operation of defuzzification (weighing) with respect to measure ( ) , which is hereinafter referred to as macrotime, difference t T τ = − , which is called micro-or fuzzy time.
In the limiting case (classical interpretation of time), measure The characteristic scale of fuzzy time 0 τ can be defined as: 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 at ( ) Thus, time fuzziness + interaction make the world highly irreversible and "derive" the second law of thermodynamics from more general principles.
The proposed study is devoted to a consistent presentation of this approach.
Another approach which is associated with the quantum delocalization of the time proposed in [3] [4] [5].

Elements of Theory of Fuzzy Sets
Let us consider fuzzy set is the measure of membership of element x to set Ω . As defined, the membership measure is For two sets A, B the following assertion holds: A B Let us confine to the case of even measures ( ) ( ) for any integer powers n at x → ±∞ . By the intersection of two odd sets A and B we mean odd set C with member- , by the union of two odd sets A and B we assume odd set C with membership measure Other generalizations of fuzzy sets' union or intersection operators are possible, e.g. Jager et al. (  If set Ω is a set of casual events, then set is the expected measure of membership of event x to set Ω or, in the continuous case, It is obvious that always ( ) ( ) The expected fuzzy event value (the operation of defuzzification) looks like: Hence an essential remark follows. If some event is likely, then it is possible, but if possible, it is not necessarily likely [6].
As a particular example of defuzzification let us consider a particle moving in fuzzy time t and not interacting with other particles. Suppose that at the initial macroscopic instant of time 0 T = it had non-fuzzy coordinate 0 0 x = and velocity v. Let the measure of fuzzy time membership look like: Then, at the macroscopic instant of time T, defuzzified distribution density looks like:

Principle of Fuzzy Causality
Let X be a fuzzy set with measure ( ) fuzzy interval with level η [7]- [12]. This definition allows us to formulate the principle of fuzzy causality. Let set X be time axis t.
Definition. Events attributed to fuzzy instants of time 1 t and 2 t at one or similar spatial points can be connected by an unconditional causal link with level η , if the fuzzy interval between them is almost positive. Otherwise they are connected by a conditional causal link with level η . In ordinary space with non-fuzzy time, this principle goes into the ordinary principle of causality at velocities much lower than the light velocity.
The principle of fuzzy causality leads to an interesting result: for an unconditional fuzzy causality there is a classical analogue-the regular causality principle realized in fuzzy time.
A conditional causality does not have classical analogues: the "past" in it can depend on the "future", the very concept of the "past" and the "future" has no definite meaning. It is possible to talk about the "past" and the "future" only after the defuzzification process. Schematically, this principle is illustrated in Figure 1.

Defuzzification of the Liouville Equation
Let us consider a system consisting of N material points in space Ω of dimension Let us assume that the points' velocities v are much smaller than the light velocity, the interaction between them is pairwise ( ) The system Hamiltonian has the standard form: j j p x is momentum and coordinate of point j, ε is small parameter.
The validity of dynamics equations in fuzzy time is postulated: Let us seek the solution of this equation by the small parameter expansion method ε .
Operator 0 iS is Hermitian, its own functions are:

Diagram Technique
Let us consider the value of ( ) e n n i p k t and present it in the form of a product of two prediagrams: In this expression, as before, n is the particle number, n k is the wave vector, n p is momentum, T is macro-time and τ is fuzzy time: t T τ = + .
To begin with, let us confine to integration of Equation (3) in fuzzy time for the homogeneous case, using all 0 n k = .
The first essential term that appears in the integration is of order 2 ε and the following diagram corresponds to it:    Let us consider the following orders with respect to ε of perturbation theory.
Some diagrams and their orders are presented in Table 1. The value of N C = Ω is the concentration of particles. Since we are interested in behavior for large macro-times T, we will keep the terms having the maximal order with respect to macro-time T and the minimal order with respect to small parameter ε. If these orders are equal, then we leave the terms of minimal order with respect to C.
In other words, we keep diagram 2, omitting 3 (the same order with respect to ε, but different with respect to C). Further, we keep diagram 4, omitting 5, etc. In addition, it is necessary to take into account iterations of diagrams 1 -5, i.e. diagram 6 and those similar to it.
Taking into account (3), let us write in an explicit form, for example, the contribution due to defuzzification into a diagram of type 2:  Table 1. Diagrams of integration in fuzzy time of Equation (2). The type of the diagram, its order with respect to perturbation parameter ε, number of particles N, concentration C and the macro-time T are indicated.
Diagram Order However it is obvious that the series obtained in Table 1

Quantum Case. Fuzzy Pauli Equation
For a quantum mechanical system it is necessary to use a half-set of phase variables and a density operator instead of distribution density ( ) [ ] The only difference is in the fuzziness of time t and, as a consequence, the need to defuzzify this expression.
In momentum space this equation can be written as: In this expression:

of Applied Mathematics and Physics
Here we introduce designation is even and therefore the integral is positive, entropy increases with the system's evolution in macro- It is essential that there is no requirement to the infinity of system Ω → ∞ provided that const N C = = Ω , i.e. the second law of thermodynamics is a consequence of just the fuzziness of time in a system with interactions. In the quantum case the situation is similar.

Discussion
Let us consider correction In the quantum case, the effect should be observed for high-energy processes: Finally, it is necessary to note one more aspect that is related to the results obtained. It can be shown that the algebra of non-commuting operators is isomorphic to the algebra of fuzzy numbers, so an alternative (or more familiar) approach is to consider time as the Hermitian operator that does not commute with the operator of evolution. The results will be similar.

Conclusions
1) The principle of fuzzy causality is formulated, generalizing the principle of causality to the system evolution in fuzzy time.
2) The master equation for the fuzzy time model is obtained, both in the classical and in the quantum cases.
3) It is shown that the second law of thermodynamics is a consequence of the system evolution in fuzzy time.