New scenario for transition to slow 3D turbulence

Analytical non-perturbative study of the three-dimensional nonlinear stochastic partial differential equation with additive thermal noise, analogous to that proposed by V.N. Nikolaevskii [1]-[5]to describe longitudinal seismic waves, is presented. The equation has a threshold of short-wave instability and symmetry, providing long wave dynamics. New mechanism of quantum chaos generating in nonlinear dynamical systems with infinite number of degrees of freedom is proposed. The hypothesis is said, that physical turbulence could be identified with quantum chaos of considered type. It is shown that the additive thermal noise destabilizes dramatically the ground state of the Nikolaevskii system thus causing it to make a direct transition from a spatially uniform to a turbulent state.


Remark1.2.
Other non trivial problem stays from noise roundoff error in computer computation using floating point arithmetic [16]- [20]. In any computer simulation the numerical solution is fraught with truncation by roundoff errors introduced by finite-precision calculation of trajectories of dynamical systems, where roundoff errors or other noise can introduce new behavior and this problem is a very more pronounced in the case of chaotic dynamical systems, because the trajectories of such systems exhibit extensive dependence on initial conditions. As a result, a small random truncation or roundoff error, made computational error at any step of computation will tend to be large magnified by future computational of the system [17].

Remark1.3.
As it well known, if the digitized or rounded quantity is allowed to occupy the nearest of a large number of levels whose smallest separation is b c , then, provided that the original quantity is large compared to b c and is reasonably well behaved, the effect of the quantization or rounding may be treated as additive random noise [18].Bennett has shown that such additive noise is nearly white, with mean squared value of b c > /12 [19].However the complete uniform white-noise model to be valid in the sense of weak convergence of probabilistic measures as the lattice step tends to zero if the matrices of realization of the system in the state space satisfy certain non resonance conditions and the finite-dimensional distributions of the input signal are absolutely continuous [19].
The method deprived of these essential lacks in general case has been offered by the author in papers [23]- [27].
Remark1.4 Remark1.4 Remark1.4 Remark1.4 . . . . Thus from consideration above it is clear that numerical integration procedure of the1D Nikolaevskii model (1.6)-(1.7) executed in papers [2]- [7] in fact dealing with stochastic model (1.4)-(1.5). There is an erroneous the point of view, that a white noise with enough small intensity does not bring any significant contributions in turbulent modes, see for example [3]. By this wrong assumptions the results of the numerical integration procedure of the1D Nikolaevskii model (1.6)-(1.7) were mistakenly considered and interpreted as a very exact modeling the slow turbulence within purely non stochastic Nikolaevskii model (1.6)-(1.7). Accordingly wrong conclusions about that temperature noises does not influence slow turbulence have been proposed in [3].However in [27] has shown non-perturbatively that that a white noise with enough small intensity can to bring significant contributions in turbulent modes and even to change this modes dramatically.
At the present time it is generally recognized that turbulence in its developed phase has essentially singular spatially-temporal structure. Such a singular conduct is impossible to describe adequately by the means of some model system of equations of a finite dimensionality. In this point a classical theory of chaos is able to describe only small part of turbulence phenomenon in liquid and another analogous s of dynamical systems. The results of non-perturbative modeling of super-chaotic modes, obtained in the present paper allow us to put out a quite probable hypothesis: developed turbulence in the real physical systems with infinite number of degrees of freedom is a quantum super-chaos, at that the quantitative characteristics of this super-chaos, is completely determined by non-perturbative contribution of additive (thermal) fluctuations in the corresponding classical system dynamics [18]- [20].   (8,9) − EFG (8, 9, j) = 0, (2.1) 8 ∈ ℝ n , 6 7 (8, 0, :, j) = 0, G (8,9) The main difficulty with the stochastic Nikolaevskii equation is that the solutions do not take values in an function space but in generalized function space. Thus it is necessary to give meaning to the non-linear terms 5 O s 6 7 > (8, 9, :, j), W = 1, … , h because the usual product makes no sense for arbitrary distributions. We deal with product of distributions via regularizations, i.e., we approximate the distributions by appropriate way and pass to the limit. In this paper we use the approximation of the distributions by approach of Colombeau generalized functions [28]. Notation Notation Notation Notation 2.1. 2.1. 2.1. 2.1. We denote by t(ℝ n × ℝ v ) the space of the infinitely differentiable functions with compact support in ℝ n × ℝ v and by t w (ℝ n × ℝ v ) its dual space. Let ℭ = (Ω, Σ, μ) be a probability space. We denote by } the space of all functions ~: Ω → t w (ℝ n × ℝ v ) such that 〈~, •〉 is a random variable for all • ∈ t(ℝ n × ℝ v ).   . [30].We shall use the following designations. If š ∈ ™(ℝ n ) it representatives will be denoted by › oe , their values on • = •• Ž (8)ž,Š ∈ (0,1] will be denoted by › oe (•) and it point values at 8 ∈ ℝ n will be denoted › oe (•, 8 . [30]. Let Ÿ c = Ÿ c (ℝ n ) be the set of all • ∈ X(ℝ n ) such that• •(8) '8 = 1. Let ℭ = (Ω, Σ, μ) be a probability space. Colombeau random generalized function this is a map š: Ω → ™(ℝ n ) such that there is representing function › oe : Ÿ c × ℝ n × Ω with the properties: almost surely in j ∈ Ω, the function • → › oe (•, . , j) belongs to ℰ˜[ℝ n ] and is a representative of š; . [30]. The Colombeau algebra of Colombeau random generalized function is denoted by ™ Ω (ℝ n ).

Definition Definition Definition
is called upper bound of the QD QD QD QD-quantum chaos at point is called lower bound of the QD QD QD QD-quantum chaos at point is called width of theQD QD QD QD-quantum chaos at point (8,9). Then we will say that Euclidian quantum N-model has QD QD QD QD-quantum chaos of the asymptotically quantum chaos of the asymptotically infinite @ H 10, § H 10 B , â H 1.
at point (8,9). at point (8,9). ) asymptotically finite width quantum chaos of the asymptotically infinite width In generally accepted at the present time hypothesis what physical turbulence in the dynamical systems with freedom really is, the physical turbulence is associated with tors, 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 a very fast decrease of local auto-correlation function-2.
in the dynamical systems with turbulence is associated with a strange properties of stochasticity: h exponential dispersion of the reproduction; everywhere the density on the attractor -2.--9.
In contrast with canonical numerical simulation, by using Theorem2.1 it is possible to study non-perturbativelythe influence of thermal additive fluctuations on classical dynamics, which in the consideredcase 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 process6 7 (8, 9, :, j)on parameter F.
In order to obtain the characteristics of this turbulence, which is a very similarly tolocal auto-correlation function (3.1) we define bellow some appropriate functions. Here by cardƒé † we denote the cardinality of a finite seté,i.e., the number of its elements.  (8,9)Þis ordered be increase of its elements. We introduce thefunction ℜ ß l (8,9), © = 1, … , Ë (8,9)which value at point (8,9), equals the ©-th element of a set Ûℜ ß (8,9) Corresponding transcendental master equation In order to obtain the character of the phase transition (first-order or second-order on parameters :, â) from a spatially uniform to a turbulent stateat instant 9 ≈ 0one can to use the master equation (   From Eq. (5.10)-(5.11) follows second order discontinuity of the quantity ℑ(:, 8,9) at instant 9 = 0. Therefore the system causing it to make a direct transition from a spatially uniform state 6 7≈c (8, 0, :) = 0 to a turbulent state in an analogous fashion to the second-order phase transition in quasi-equilibrium systems.

7.Conclusion Conclusion Conclusion 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.