Polytechnic university of Valencia, Valencia, Spain
al-Farabi Kazakh National University, Almaty, Kazakhstan
ASKARULY ABDIADIL
INVESTIGATION OF DYNAMIC PROPERTIES
OF NON-IDEAL ONE COMPONENT PLASMAS
BY THE METHOD OF MOMENTS
For the degree of Doctor of Philosophy in Mathematics
at the
POLYTECHNIC UNIVERSITY OF VALENCIA
Supervisors:
I.M. Tkachenko Górski
Yu.V. Arkhipov
2010
RESUMEN DE LA TESIS
Dynamic characteristics of strongly coupled one-component plasmas are to be studied within the moment approach with local constraints by an algorithm similar to that of Schur. Some simulations of two-component plasmas are analyzed using sum rules and other exact relations.
One of the main problems of plasma physics is to obtain an expression for the dielectric function determining screening effects, dispersion relations and other dynamic characteristics, such as conductivity, reflectivity, etc. The dielectric function can be derived from the linear-response theory [1], using the methods of the kinetic theory or hydrodynamics [2] and by means of perturbation expansion of the Kubo formula [3]. On the other hand, the dielectric function can be deduced on the basis of the method of moments [4]. All methods mentioned above are mostly applicable in a limited range of variation of plasma parameters where some perturbation expansion can be used. There are no such restrictions on the plasma parameters in the method of moments [5,6] which permits to reconstruct any Nevanlinna class function by its convergent power moments. In Physics these functions are called response functions which due to the causality principle satisfy the Kramers-Kronig relations, e.g., the plasma inverse dielectric function.
Another dynamic characteristic, i.e. the dynamic structure factor which is related, via the fluctuation dissipation theorem, to the imaginary part of the inverse dielectric function, can be extracted from the experimental data [7]. Thus, from both the practical and mathematical points of view, the study of the dynamic structure factor is important.
There exist several approaches to the investigation of the dynamic structure factor. Beyond experimental and theoretical methods, some simulation techniques based on the first principles of mechanics and statistical physics, can be applied. The results of the pioneering work [8] remain in a good agreement with the modern data of [9].
In this thesis the one-component plasma dynamic structure factor is modeled by the first three terms of its asymptotic decomposition at high frequencies and its values at a few interpolation points on the real axis. This will make the dynamic structure factor to be a non-rational function whose extension onto the upper half-plane of the complex frequency is holomorphic with a nonnegative imaginary part and with a continuous restriction on the real axis. The Schur-like algorithm [10] free parameter of the non-rational model function is obtained from the Shannon entropy maximization procedure. The results satisfy automatically not only the sum rules (the power moments) and other exact relations but, also, interpolate between the chosen frequencies. An agreement is obtained with available simulation data on the plasma dynamic properties. The method permits to take into account the energy dissipation processes so that the results of alternative theoretical approaches are included into the moment scheme and are complemented. The algorithm could be applied in a broader setting.
Additionally, the classical molecular dynamic simulation data on the charge-charge dynamic structure factor of two-component plasmas modeled in [11] and in [12] is analyzed using the sum rules and other exact relations. The convergent power moments of the imaginary part of this model system dielectric function is expressed in terms of its partial static structure factors computed by the method of hypernetted chains using the Deutsch-like and other effective potentials. High-frequency asymptotic behavior of the dielectric function are specified to include the effects of inverse bremsstrahlung.
Finally, we apply the approach to the modeling of optical properties of moderately coupled plasmas. On the basis of the Hölder inequalities, the monotonicity, predicted by the classical Drude-Lorentz model, is studied. And, at last, the comparison between the method of moments and the classical method of continuous fractions is carried out.
References.
1. Yu.V. Arkhipov, F.B. Bayimbetov, A.E. Davletov, K.V. Starikov, Pseudopotential theory of dense high-temperature plasma (Kazakh University, Almaty, 2002) in Russian.
2. S. Ichimaru, Statistical Plasma Physics: Condensed Plasmas (Addison-Wesley, New York, 1994) Vol. 1,2.
3. G. Mahan, Many-Particle Physics (Plenum, New York, 1981).
4. Yu.V. Arkhipov, A. Askaruly, D. Ballester, A.E. Davletov, I.M. Tkachenko, G. Zwicknagel, Phys. Rev. E 76, 026403, 2007.
5. M.G. Krein, A.A. Nudel'man, The Markov moment problem and extremal problems, Trans. of Math. Monographs, 50, Amer. Math. Soc.,Providence, R.I.,1977.
6. N.I. Akhiezer, The Classical Moment Problem, Hafner Publishing Company, N.Y., 1965.
7. S.H. Glenzer, R. Redmer, Rev.Mod.Phys. 81, 1625, 2009.
8. J.-P. Hansen, I. R. McDonald, E.L. Pollock, Phys. Rev. A. 11, 1025, 1975..
9. A. Wierling, T. Pschiwul, G. Zwicknagel, Physics of Plasmas, 9 (12), 4871, 2002.
10. V.M. Alcober, I.M. Tkachenko, M. Urrea, Construction of solutions of the Hamburger-Löwner interpolation problem for Nevanlinna class functions, In: Integral Methods in Science and Engineering, Volume 2, Computational Methods, Edited by C. Constanda, Mª. Eugenia Pérez, Ch. 2, pp. 11–20, 2009, Birkhäuser Verlag, Basel, Switzerland.
11. J.-P. Hansen, I.R. McDonald, Phys. Rev. A., 23, 2041, 1981.
12. G. Zwicknagel, Th. Pschiwul, Contrib. Plasma Phys., 43, No. 5-6, 393, 2003.