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A new analytical approach to the determination of the hard-sphere gas free energy P Clippe and R Evrard This content was downloaded from IP address 141.226.8.98 on 12/09/2019 at 18:22 New analytical approach for transition to slow 3-D turbulence J Foukzon1, E Men'kova2, A Potapov3 1Department of mathematics, Israel Institute of Technology, Haifa, Israel E-mail: jaykovfoukzon@list.ru 2All-Russian Research Institute for Optical and Physical Measurements, Moscow, Russia E-mail: E_Menkova@mail.ru 3Kotel'nikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences, Moscow, Russia E-mail: potapov@cplire.ru Abstract. Analytical non-perturbative study of the three-dimensional nonlinear stochastic partial differential equation with additive thermal noise analogous to that proposed by Nikolaevskii V.N. 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 says 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 causing it to make a direct transition from a spatially uniform to a turbulent state. 1. Introduction In the presented work 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. Statistical descriptions of dynamical chaos and investigations of noise effects on chaotic regimes are studied. Proposed approach also allows estimate the influence of additive (thermal) fluctuations on the formation processes of developed turbulence modes in essentially nonlinear processes like electroconvection and others. A principal role of the influence of thermal fluctuations on the dynamics of some types of dissipative systems in the approximate environs of derivation rapid of a short-wave instability was ascertained. Important physical results following from Theorem 2.1 are numerically illustrated by example of 1D stochastic model system: 1 To whom any correspondence should be addressed. 4th International Conference on Mathematical Modeling in Physical Sciences (IC-MSquare2015) IOP Publishing Journal of Physics: Conference Series 633 (2015) 012097 doi:10.1088/1742-6596/633/1/012097 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 (1.1) Systematic study of a different type of chaos at onset ''soft-mode turbulence'' based on numerical integration of the simplest Nikolaevskii model has been executed and analyzed by many authors [1]-[3]. 2. Main Theoretical Results We study the stochastic partial differential equation (1.1) in the sense of Colombeau generalized functions [4]. Theorem 2.1. [5]-[6]. (Strong Large Deviation Principle for Colombeau-Ito's SPDE) (I) Let , be a solution of the Colombeau-Ito's SPDE [5]: (2.1) (2.2) Here: (1) is the Colombeau algebra of Colombeau generalized functions and (2) (3) is a smoothed with respect to white noise. (II) Assume that Colombeau-Ito's SDE (2.1)-(2.2) is a strongly dissipative [5]. (III) Let be the solutions of the linear PDE: (2.3) (2.4) Then Definition 2.1. (Differential Master Equation) The linear PDE [5]-[6] (2.5) (2.6) 4th International Conference on Mathematical Modeling in Physical Sciences (IC-MSquare2015) IOP Publishing Journal of Physics: Conference Series 633 (2015) 012097 doi:10.1088/1742-6596/633/1/012097 2 Definition 2.2. (Transcendental Master Equation) The transcendental equation [5]-[6] (2.7) we will call as the transcendental master equation. We note that concrete structure of the Nikolaevskii chaos is determined by the solution variety by transcendental master equation (2.7). Master equation (2.7) is only determined by the way of 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 the equation (2.1). Assume that for almost all points (in the sense of Lebesgue measure on ), there exist a function such that Then we will say that a function is a quasidetermined solution (QD-solution of the equation (2.1). Assume that there exist a set that is positive Lebesgue measure, i.e. and 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). For each point we define a set by the condition: Definition 3.2. 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: We will say that the set-valued function is a quasidetermined chaotic solution (QD-chaotic solution) of the quantum N-model. Figure 1. Evolution of QD-chaotic solution in time at point 4th International Conference on Mathematical Modeling in Physical Sciences (IC-MSquare2015) IOP Publishing Journal of Physics: Conference Series 633 (2015) 012097 doi:10.1088/1742-6596/633/1/012097 3 Theorem 3.1. Assume that . Then for all values of parameters such that quantum N-model (2.1) has the QD-chaotic solutions. Definition 3.3. For each point we define the functions such that: Function is called upper bound of the QD-quantum chaos at point function is called lower bound of the QD-quantum chaos at point function is called width of the QD-quantum chaos at point . Definition 3.4. For each point we define the functions such that: Theorem 3.2. (Criterion of QD-quantum chaos in Euclidian quantum N-model) Assume that Then Euclidian quantum N-model has QD-quantum chaos. 4. Quasi-determined quantum chaos and physical turbulence nature 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 we have used some appropriate functions [6]. Definition 4.1. The normalized local auto-correlation function is defined by the formula (4.1) Now let us consider Euclidian quantum N-model corresponding to the classical dynamics. Corresponding differential master equations (2.5)-(2.6) are (4.2) (4.3) Corresponding transcendental master equations (2.7) are (4.4) (4.5) We assume now that The result of calculation of the corresponding function using master equation (4.5) is presented in figure 1. Let us calculate now the corresponding normalized local auto-correlation function The result of calculation is presented in figure 2. 4th International Conference on Mathematical Modeling in Physical Sciences (IC-MSquare2015) IOP Publishing Journal of Physics: Conference Series 633 (2015) 012097 doi:10.1088/1742-6596/633/1/012097 4 Figure 2. Normalized local auto-correlation function 5. 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 dramatically destabilizes the ground state of the system thus causing it to make a direct transition from a spatially uniform to a turbulent state. 6. References [1] Tribelsky M I, Tsuboi K 1996 New scenario to transition to slow turbulence Phys. Rev. Lett. 76 1631 [2] Toral R, Xiong J D, Gunton and Xi H W 2003 Wavelet description of the Nikolaevskii model Journ. Phys. A 36 1323 [3] Haowen Xi, Toral R, Gunton D, Tribelsky M I 2003 Extensive chaos in the Nikolaevskii model Phys. Rev. E 61 R17 [4] Oberguggenberger M, 2001 Generalized functions in nonlinear models-a survey Nonlinear Analysis: Theory, Methods & Applications Proceedings of the Third World Congress of Nonlinear Analysts 47, Issue 8, 5029–5040 [5] Foukzon J 2014 Communications in Applied Sciences 2, No 2 230-363 [6] Foukzon J 2015 New scenario for transition to slow 3-D turbulence Journal of Applied Mathematics and Physics 3 371-389. doi: 10.4236/jamp.2015.33048 4th International Conference on Mathematical Modeling in Physical Sciences (IC-MSquare2015) IOP Publishing Journal of Physics: Conference Series 633 (2015) 012097 doi:10.1088/1742-6596/633/1/012097