1) for fixed the function is a jointly measurable on;

2) almost surely in, the function belongs to and is a representative of U;

Notation 2.3. [30] . The Colombeau algebra of Colombeau random generalized function is denoted by.

Definition 2.5. Let be a probability space. Classically, a generalized stochastic process on is a weakly measurable map denoted by. If, then

3) is a measurable with respect to and

4) Smooth with respect to and hence jointly measurable.

5) Also.

6) Therefore qualifies as an representing function for an element of.

7) In this way we have an imbedding.

Definition 2.6. Denote by the space of rapidly decreasing smooth functions on. Let with 1), 2) S-the Borel σ-algebra generated by the weak topology. Therefore there is unique probability measure on such that

for all. White noise with the support in T is the generalized process

such that: 1), 2), 3).

Viewed as a Colombeau random generalized function, it has a representative (denoting on variables in by):, which vanishes if t is less than minus the diameter of the support of. Therefore is a zero on in. Note that its variance is the Colombeau constant:

Definition 2.7. Smoothed with respect to white noise the representative with

, such that, and.

Theorem 2.1. [25] . (Large Deviation Principle for SPDE) (I) Let, be solution of the Colombeau-Ito’s SPDE [26] :

(2.3)

(2.4)

Here: 1) the Colombeau algebra of Colombeau generalized functions and.

2)

3) is a smoothed with respect to white noise.

(II) Let, be solution of the Colombeau-Ito’s SDE [26] :

(2.5)

(2.6)

(2.7)

Here Equations (2.5)-(2.7) is obtained from Equations (2.3)-(2.4) by spatial discretization on finite lattice if and is a latticed Laplacian [31] -[33] .

(III) Assume that Colombeau-Ito’s SDE (2.5)-(2.7) is a strongly dissipative [26] .

(IV) Let be the solutions of the linear PDE:

(2.8)

. (2.9)

Then

(2.10)

Proof. The proof based on Strong Large Deviations Principles (SLDP-Theorem) for Colombeau-Ito’s solution of the Colombeau-Ito’s SDE, see [26] , Theorem 6. By SLDP-Theorem one obtain directly the differential master equation (see [26] , Equation (90)) for Colombeau-Ito’s SDE (2.5)-(2.7):

(2.11)

(2.12)

We set now Then from Equations (2.11)-(2.12) we obtain

(2.13)

(2.14)

From Equations (2.5)-(2.7) and Equations (2.13)-(2.14) by SLDP-Theorem (see see [26] , inequality (89)) we obtain the inequality

(2.15)

Let us consider now the identity

(2.16)

From the identity (2.16) by integration we obtain the identity

(2.17)

From the identity (2.17) by using the triangle inequality we obtain the inequality

(2.18)

From the inequality (2.18) by the inequality (2.15) for all we obtain the inequality

(2.19)

In the limit from the inequality (2.19) we obtain the inequality

In the limit from the inequality we obtain the inequality

(2.20)

We note that

(2.21)

Therefore from (2.20) and (2.21) we obtain the inequality

(2.22)

In the limit from Equations (2.13)-(2.14) for any fixed, , we obtain the differential master equation for Colombeau-Ito’s SPDE (2.3)-(2.4)

(2.23)

(2.24)

Therefore from the inequality (2.22) follows the inequality

(2.25)

In the limit from differential Equations (2.23)-(2.24) we obtain the differential Equations (2.8)-(2.9) and it is easy to see that

(2.26)

From the inequality (2.25) one obtain the inequality

(2.27)

From the inequality (2.27) and Equation (2.26) finally we obtain the inequality

(2.28)

The inequality (2.28) finalized the proof.

Definition 2.7. (The Differential Master Equation) The linear PDE:

(2.29)

(2.30)

we will call as the differential master equation.

Definition 2.8. (The Transcendental Master Equation) The transcendental equation

(2.31)

we will call as the transcendental master equation.

Remark 2.2. We note that concrete structure of the Nikolaevskii chaos is determined by the solutions variety by transcendental master Equation (2.31). Master Equation (2.31) is determines by the only way some many-valued function which is the main constructive object, determining the characteristics of quantum chaos in the corresponding model of Euclidian quantum field theory.

3. Criterion of the Existence Quantum Chaos in Euclidian Quantum N-Model

Definition 3.1. Let be the solution of Equation (2.1). Assume that for almost all points (in the sense of Lebesgue-measureon), there exist a function such that

(3.1)

Then we will say that a function is a quasi-determined solution (QD-solution of Equation (2.1)).

Definition 3.2. Assume that there exist a set that is positive Lebesgue-measure, i.e., and

(3.2)

i.e., imply that the limit: does not exist.

Then we will say that Euclidian quantum N-model has the quasi-determined Euclidian quantum chaos (QD- quantum chaos).

Definition 3.3. For each point we define a set by the condition:

(3.3)

Definition 3.4. Assume that Euclidian quantum N-model (2.1) has the Euclidian QD-quantum chaos.

For each point we define a set-valued function by the condition:

(3.4)

We will say that the set-valued function is a quasi-determined chaotic solution (QD-chaotic solution) of the quantum N-model (In Figure 1 and Figure 2).

Theorem 3.1. Assume that Then for all values of parameters such that, , , , , quantum N-model (2.1) has the QD- chaotic solutions.

Definition 3.5. For each point we define the functions such that:

Figure 1. Evolution of QD-chaotic solution in time at point, , , ,.

Figure 2. The spatial structure of QD-chaotic solution at instant, , ,.

1)

2)

3)

Definition 3.7.

1) Function is called upper bound of the QD-quantum chaos at point

2) Function is called lower bound of the QD-quantum chaos at point

3) Function is called width of the QD-quantum chaos at point.

Definition 3.8. Assume now that

(3.5)

Then we will say that Euclidian quantum N-model has QD-quantum chaos of the asymptotically finite width at point.

Definition 3.9. Assume now that

(3.6)

Then we will say that Euclidian quantum N-model has QD-quantum chaos of the asymptotically infinite width at point (in Figure 3, Figure 4)

Definition 3.10. For each point we define the functions such that:

1)

2)

3)

Theorem 3.2. For each point is satisfied the inequality

(3.7)

Proof. Immediately follows by Theorem 2.1 and Definitions 3.5, 3.10.

Theorem 3.3. (Criterion of QD-quantum chaos in Euclidian quantum N-model)

Assume that

(3.8)

Then Euclidian quantum N-model has QD-quantum chaos.

Proof. Immediately follows by the Inequality (3.7) and Definition 3.2.

Figure 3. The QD-quantum chaos of the asymptotically infinite width at point, , , ,.

Figure 4. The fine structure of the QD-quantum chaos of the asymptotically infinite width at point x = 3, , , , ,. T = 10^{4}, ∆t = 10^{−}^{3}.

4. Quasi-Determined Quantum Chaos and Physical Turbulence Nature

In generally accepted at the present time hypothesis what physical turbulence in the dynamical systems with an infinite number of degrees of freedom really is, the physical turbulence is associated with a strange attractors, on which the phase trajectories of dynamical system reveal the known properties of stochasticity: a very high dependence on the initial conditions, which is associated with exponential dispersion of the initially close trajectories and brings to their non-reproduction; everywhere the density on the attractor almost of all the trajectories a very fast decrease of local auto-correlation function [2] -[9]

(4.1)

Here

In contrast with canonical numerical simulation, by using Theorem 2.1 it is possible to study non-perturba- tively the influence of thermal additive fluctuations on classical dynamics, which in the considered case is described by Equation (4.1).

The physical nature of quasi-determined chaos is simple and mathematically is associated with discontinuously of the trajectories of the stochastic process on parameter.

In order to obtain the characteristics of this turbulence, which is a very similarly to local auto-correlation function (3.1) we define bellow some appropriate functions.

Definition 4.1. The numbering function of quantum chaos in Euclidian quantum N-model is defined by

(4.2)

Here by we denote the cardinality of a finite set X, i.e., the number of its elements.

Definition 4.2. Assume now that a set is ordered be increased of its elements. We introduce the function, which value at point, equals the i-th element of a set.

Definition 3.3. The mean value function of the chaotic solution at point is defined by

(4.3)

Definition 3.4. The turbulent pulsations function of the chaotic solution at point is defined by

(4.4)

Definition 3.5. The local auto-correlation function is defined by

(4.5)

(4.6)

Definition 3.6. The normalized local auto-correlation function is defined by

(4.7)

Let us consider now 1D Euclidian quantum N-model corresponding to classical dynamics

(4.8)

Corresponding Langevin equation are [34] [35] :

(4.9)

(4.10)

Corresponding differential master Equations (2.29)-(2.30) are

(4.11)

. (4.12)

Corresponding transcendental master Equation (2.31) are

(4.13)

(4.14)

We assume now that. Then from Equation (4.13) for all we obtain

, or (4.14)

(4.15)

The result of calculation using transcendental master Equation (4.15) the corresponding function is presented by Figure 5 and Figure 6.

The result of calculation using master Equation (4.13) the corresponding function is presented by Figure 7 and Figure 8.

Let us calculate now corresponding normalized local auto-correlation function. The result of calculation using Equation (4.7) is presented by Figure 9 and Figure 10.

Figure 5. Evolution of QD-chaotic solution in time, , , , , , Δλ = 0.01.

Figure 6. The spatial structure of QD-chaotic solution at instant, , , , , Δx = 0.1, Δλ = 0.01.

Figure 7. The development of temporal chaotic regime of 1D Euclidian quantum N-model at point, , , , ,.

Figure 8. The development of temporal chaotic regime of 1D Euclidian quantum N-model at point x = 1, , , , δ = 1, p = 1.

Figure 9. Normalized local auto-correlation function, , , , ,.

In paper [7] the mechanism of the onset of chaos and its relationship to the characteristics of the spiral attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg-Landau Equation (4.14). Numerical data are compared with experimental results (in Figure 11).

(4.14)

However, as pointed out above (see Remarks 1.1-1.4) such numerical simulation in fact gives numerical data for stochastic model

Figure 10. Normalized local auto-correlation function., , , ,.

Figure 11. Normalized local auto-correlation function [7] .

(4.15)

5. The Order of the Phase Transition from a Spatially Uniform State to a Turbulent State at Instant

In order to obtain the character of the phase transition (first-order or second-order on parameters) from a spatially uniform to a turbulent state at instant one can to use the transcendental master Equation (2.31) of the form

(5.1)

By differentiation Equation (5.1) one obtain

(5.2)

From Equation (5.2) one obtain

(5.3)

Let us consider now 1D Euclidian quantum N-model given by Equations (4.9)-(4.10). From corresponding transcendental master Equation (4.13) by differentiation Equation (4.13) with respect to variable one obtain

(5.4)

From Equation (5.4) for a sufficiently small one obtain

. (5.5)

From master Equation (4.13) one obtain by differentiation Equation (4.13) with respect to variable one obtain

(5.6)

From Equation (5.6) for a sufficiently small one obtain

(5.7)

Therefore from Equation (5.3), (5.5) and (5.7) one obtain

(5.8)

In the from Equation (5.8) one obtain

(5.9)

and where

.

From Equation (5.9) follows that

(5.10)

. (5.11)

From Equations (5.10)-(5.11) follows second order discontinuity of the quantity at instant. Therefore the system causing it to make a direct transition from a spatially uniform state to a turbulent state in an analogous fashion to the second-order phase transition in quasi-equilibrium systems.

6. Chaotic Regime Generated by Periodical Multi-Modes External Perturbation

Assume now that external periodical force has the following multi-modes form (Figure 12)

Figure 12. Evolution of QD-chaotic solution in time., , , , , , ,.

(6.1)

Corresponding transcendental master equation are (Figure 13)

(6.2)

Let us consider the examples of QD-chaotic solutions with a periodical force (Figure 14):

Figure 13. The spatial structure of QD-chaotic solution at instant t = 10^{3}, , , , , , , , ,.

Figure 14. The spatial structure of QD-chaotic solution at instant t = 5 × 10^{3}, , , , , , , , ,.

(6.3)

7. Conclusion

A non-perturbative analytical approach to the studying of problem of quantum chaos in dynamical systems with infinite number of degrees of freedom is proposed and developed successfully. It is shown that the additive thermal noise destabilizes dramatically the ground state of the system thus causing it to make a direct transition from a spatially uniform to a turbulent state.

Acknowledgements

A reviewer provided important clarifications.

Cite this paper

Jaykov Foukzon,, (2015) New Scenario for Transition to Slow 3-D Turbulence. *Journal of Applied Mathematics and Physics*, **03**, 371-389. doi: 10.4236/jamp.2015.33048

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