En route to fusion: confinement state as a waveleton
aa r X i v : . [ phy s i c s . p l a s m - ph ] M a r En route to fusion: confinement state as a waveleton
Antonina N. Fedorova, Michael G. Zeitlin
IPME RAS, St. Petersburg, V.O. Bolshoj pr., 61, 199178, Russiahttp://math.ipme.ru/zeitlin.html, http://mp.ipme.ru/zeitlin.htmlE-mail: [email protected], [email protected]
Abstract.
A fast and efficient numerical-analytical approach is proposed for description ofcomplex behaviour in non-equilibrium ensembles in the BBGKY framework. We constructthe multiscale representation for hierarchy of partition functions by means of the variationalapproach and multiresolution decomposition. Numerical modeling shows the creation of variousinternal structures from fundamental localized (eigen)modes. These patterns determine thebehaviour of plasma. The localized pattern (waveleton) is a model for energy confinement state(fusion) in plasma. “A magnetically confined plasma cannotbe in thermodinamical equilibrium”
Unknown author ... Folklore
1. Introduction
It is well known that fusion problem in plasma physics could be solved neither experimentallynor theoretically during last fifty year, so it seems that there are the serious obstacles whichprevent real progress in the problem of real fusion as the main subject in the area [1], [2]. Ofcourse, it may be a result of some unknown no-go theorem(s) but it seems that the currenttheoretical level demonstrates that not all possibilities, at least on the level of theoretical andmatematical modeling, are exhausted. Definitely, the first thing which we need to change is aframework of generic mathematical methods which can help to improve the current state of thetheory. Our postulates (conjectures) are as follows [3]: A) The fusion problem (at least at the first step) must be considered as a problem inside the(non) equilibrium ensemble in the full phase space. It means, at least, that:
A1) our dynamical variables are partitions (partition functions, hierarchy of N-pointspartition functions),
A2) it is impossible to fix a priori the concrete distribution function and postulate it (e.g.Maxwell-like or other concrete (gaussian-like or even not) distributions) but, on the contrary,the proper distrubution(s) must be the solutions of proper (stochastic) dynamical problem(s),e.g., it may be the well-known framework of BBGKY hierarchy of kinetic equations or somethingsimilar. So, the full set of dynamical variables must include partitions also. B) Fusion state = (meta) stable state (with minimum entropy and zero measure) in the spaceof partitions on the whole phase space in which most of energy of the system is concentratedin the relatively small area (preferable with measure zero) of the whole domain of definition inthe phase space during the time period which is enough to take reasonable part of it outside forpossible usage. From the formal/mathematical point of view it means that: fusion state must be localized (first of all, in the phase space), B2) we need a set of building blocks, localized basic states or eigenmodes which can provide
B3) the creation of localized pattern which can be considered as a possible model for plasmain a fusion state. Such pattern must be:
B4) (meta) stable and controllable, because of obvious reasons. So, the main courses are:
C1) to present smart localized building blocks which may be not only useful from point ofview of analytical statements, such as the best possible localization, fast convergence, sparseoperators representation, etc, but also exist as real physical fundamental modes,
C2) to construct various possible patterns with special attention to localized pattern whichcan be considered as a needful thing in analysis of fusion;
C3) after points C1 and C2 in ensemble (BBGKY) framework to consider some standardreductions to Vlasov-like and RMS-like equations (following the set-up from well-known results)which may be useful also. These particular cases may be important as from physical point ofview as some illustration of general consideration.The lines above are motivated by our attempts to analyze the hidden internal contents of thephrase mentioned in the epigraph of this paper: “A magnetically confined plasma cannot be inthermodinamical equilibrium.” Also, it should be noted that our results below can be applied toany scenario (fusion, ignition, etc): we describe pattern formation in arbitrary non-equilibriumensembles.
2. Description
At this stage our main goal is an attempt of classification and construction of a possible zooof nontrivial (meta) stable states/patterns: high-localized (nonlinear) eigenmodes, complex(chaotic-like or entangled) patterns, localized (stable) patterns (waveletons). We will use itlater for fusion description, modeling and control. In our opinion localized (meta)stable pattern(waveleton) is the proper image for fusion state in plasma (energy confinement).Our constructions can be applied to the hierarchy of N -particle distribution function,satisfying the standard BBGKY hierarchy ( υ is the volume): ∂F s ∂t + L s F s = 1 υ Z dµ s +1 s X i =1 L i,s +1 F s +1 . (1)Our key point is the proper nonperturbative generalization of the previous perturbativemultiscale approaches (like Bogolubov/virial expansions). The infinite hierarchy of distributionfunctions is: F = { F , F ( x ; t ) , . . . , F N ( x , . . . , x N ; t ) , . . . } ,F p ( x , . . . , x p ; t ) ∈ H p , H = R, H p = L ( R p ) ,F ∈ H ∞ = H ⊕ H ⊕ · · · ⊕ H p ⊕ . . . (2)with the natural Fock space like norm (guaranteeing the positivity of the full measure):( F, F ) = F + X i Z F i ( x , . . . , x i ; t ) i Y ℓ =1 µ ℓ . (3)Multiresolution decomposition (filtration) naturally and efficiently introduces the infinitesequence (tower) of the underlying hidden scales, which is a sequence of increasing closedsubspaces V j ∈ L ( R ) [4]: ...V − ⊂ V − ⊂ V ⊂ V ⊂ V ⊂ ... (4)ur variational approach [3] reduces the initial problem to the problem of solution of functionalequations at the first stage and some algebraic problems at the second one. Let L be an arbitrary(non)linear differential/integral operator with matrix dimension d (finite or infinite), whichacts on some set of functions from L (Ω ⊗ n ): Ψ ≡ Ψ( t, x , x , . . . ) = (cid:16) Ψ ( t, x , x , . . . ) , . . . ,Ψ d ( t, x , x , . . . ) (cid:17) , x i ∈ Ω ⊂ R , n is the number of particles: L Ψ ≡ L ( Q, t, x i )Ψ( t, x i ) = 0 ,Q ≡ Q d ,d ,d ,... ( t, x , x , . . . , ∂/∂t, ∂/∂x , ∂/∂x , . . . , Z µ k )= d ,d ,d ,... X i ,i ,i , ··· =1 q i i i ... ( t, x , x , . . . ) (cid:16) ∂∂t (cid:17) i (cid:16) ∂∂x (cid:17) i (cid:16) ∂∂x (cid:17) i · · · Z µ k . (5)Let us consider now the N mode approximation for the solution as the following ansatz:Ψ N ( t, x , x , . . . ) = N X i ,i ,i , ··· =1 a i i i ... A i ⊗ B i ⊗ C i . . . ( t, x , x , . . . ) . (6)We shall determine the expansion coefficients from the following conditions: ℓ Nk ,k ,k ,... ≡ Z ( L Ψ N ) A k ( t ) B k ( x ) C k ( x )d t d x d x · · · = 0 . (7)As a result the solution has the following multiscale/multiresolution decomposition via nonlinearhigh-localized eigenmodes [4]: F ( t, x , x , . . . ) = X ( i,j ) ∈ Z a ij U i ⊗ V j ( t, x , x , . . . ) ,V j ( t ) = V j,slowN ( t ) + X l ≥ N V jl ( ω l t ) , ω l ∼ l , (8) U i ( x s ) = U i,slowM ( x s ) + X m ≥ M U im ( k sm x s ) , k sm ∼ m . So, we may move from the coarse scales of resolution (coarse graining) to the finest ones forobtaining more detailed information about the dynamical process. In this way one obtains con-tributions to the full solution from each scale of resolution or each time/space scale or fromeach nonlinear eigenmode. It should be noted that such representations give the best possiblelocalization properties in the corresponding (phase) space/time coordinates. Numerical calcula-tions are based on compactly supported wavelets and related wavelet families and on evaluationof the accuracy on the level N of the corresponding cut-off of the full system w.r.t. the norm (3): k F N +1 − F N k ≤ ε (9)Numerical modeling shows the creation of various complex structures from localized modes,which are related to (meta)stable or unstable type of behaviour and the corresponding patterns(waveletons) formation (Figs. 1, 2). Reduced algebraic structure (7), Generalized DispersionRelations, provide the pure algebraic control of stability/unstability scenario. So, we consideredthe construction for controllable (meta) stable waveleton configuration representing a reasonableapproximation for the possible realizable confinement state. . Conclusions Let us summarize our main results:
Physical Conjectures:P1
State of fusion (confinement of energy) in plasma physics may and need be considered fromthe point of view of non-equilibrium statistical physics. According to this BBGKY frameworklooks naturally as first iteration. Main dynamical variables are partitions. P2 Basic high localized nonlinear eigenmodes are real physical modes important for fusionmodeling. Intermode multiscale interactions create various complex patterns from thesefundamental building blocks, and determine the behaviour of plasma (Figs. 1, 2). High localized(meta) stable patterns (waveletons), considered as long-living fluctuations, are proper imagesfor plasma in fusion state (Fig. 2).
Mathematical framework:M1
The problems under consideration, like BBGKY hierarchies or their reductions areconsidered as pseudodifferential hierarchies in the framework of proper family of methods unifiedby effective multiresolution approach or local nonlinear harmonic analysis on the orbits ofrepresentations of hidden underlying symmetry of properly chosen functional space [3]. momentum (p)coordinate (q) d i s t r i bu t i on f un c t i on F ( q , p ) Figure 1.
Trash: Chaotic Partition. momentum (p)coordinate (q) d i s t r i bu t i on f un c t i on F ( q , p ) Figure 2.
Goal: Waveleton/Fusion State. M2 Formulas (8) based on Generalized Dispersion Relations (GDR) (7) provide exact multiscalerepresentation for all dynamical variables (partitions, first of all) in the basis of high-localizednonlinear (eigen)modes. Numerical realizations in this framework are maximally effective fromthe point of view of complexity of all algorithms inside. GDR provide the way for the statecontrol on the pure algebraical level.
Realizability:
According to this approach, it is possible on formal level, in principle, to control ensemblebehaviour and to realize the localization of energy (confinement state) inside the waveletonconfigurations created from a few fundamental modes only during self-organization via possible(external) algebraical control (Figs. 1, 2) [3].
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