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The Holtsmark-Continuum Model for the Statistical Description of a Plasma

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The Holtsmark-Continuum Model for the Statistical Description of a Plasma On fluctuations in plasmas : the Holtsmark-continuum model for the statistical description of a plasma Citation for published version (APA): Dalenoort, G. J. (1970). On fluctuations in plasmas : the Holtsmark-continuum model for the statistical description of a plasma. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR108607 DOI: 10.6100/IR108607 Document status and date: Published: 01/01/1970 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 29. Sep. 2021 Gerhard J. Dalenoort On Fluctuations in Plasmas -· • • ' .e On Fluctuations in Plasmas ON FLUCTUATIONS IN PLASMAS THE. HOLTSMARK-CONTINUUM MODEL FOR THE STATISTICAL DESCRIPTION OF A PLASMA PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. DR. IFI. A. A. TH. M. VAN TRIER, HOOGLERAAR IN DE AFDELING DER ELEKTROTECHNIEK, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDl:DIGEN OP DONDERDAG 21 MEI 1970 DES NA MIDDAGS TE 4 UUR DOOR GERHARD JOHAN DALENOORT GEBOREN TE GARUT 1970 OFFSET RIJNHUIZEN-JUTPHAAS DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE' PROMOTOR PROF. DR. H. BREMMER iii Enfin on y arrive presque toujours When finally a work comes to an end, The man who did it, Can but hardly think of all Who somehow contributed. iv Acknowledqement This work was performed as part of the research progrJmme of the association agreement of Euratom and the "Sticqting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver~weten­ schappelijk Onderzoek" (ZWO) and Euratom. Colofon Omslag Jenny Dalenoord Typewerk en correctie Gonny Scholman en andere leden van het Secretariaat van Rijnhuizen Produktie en technische H. Hulsebos en uitwerking Tekenkamer Rijnhuizen Inhoud deel I General Introduction pp vii-xii deel II Short-range Corrections to the Probability Distributions of the Electric Microfield in a Plasma, Physica .!2_ (1969) 283-304 pp 283-304 deel !II The Holtsmark-continuum Model for the Statistical Description of a Plasma, bestemd voor publikatie pp 1-103 Samenvatting in de Nederlandse taal Curriculum vitae vii Part I, General Introduction The characteristic feature of a gas containing charged par­ ticles, i.e. a completely or incompletely ionized gas, is the simultaneous interaction of all the particles. The main inter­ action between - not too fast moving - particles is the Coulomb interaction, the force between two charges e being equal to e 2 /r! In the - idealized - model of a uniform, infinite plasma the interaction of any particle with distant particles is in principle as important as that with its iw.mediate neighbours. This can be clearly seen from the force on a particle that is due to the ether particles: where cr{*) indicates the charge density. This force is infinite because the volume of the spherical shell increases with the square of its radius (we assume that the average value of a is constant and the plasma infinite); due to spontaneous fluctu­ ations cr(*) differs randomly from zero. (In general one con­ siders quasi-neutral plasmas, which have the tendency to re­ store any excess of charge, but due to inertia the neutraliza­ tion of this charge overshoots, whence the irnportance of the time-dependence of plasma fluctuations). There are two important reasons for the long-range charac­ ter of the Coulomb interactions. Firstly, the decrease with distance, exactly balanced by the increase of the volume of the spherical shell; it has this property in common with the force of gravity. It seems likely that there exists a connec­ tion with the fact that the decrease with distance of the in­ tensity of the radiation emitted by a source is-also inversely proportional to that distance squared. This phenomenon is the source of profound speculation. The second important reason for the long-range character is the absence of any parameter in the dependence on the distance(1f? *) nwnbers between brackets refer to the list of references at the end of this general introduction. viii in contrast to an_interaction like e.g. the Yukawa one (the latter contains an exponential exp(-r/\) with a range À). From these considerations one may infer the importance of a dynamica! description of the plasma, in which two natural lengths present themselves. The first one, the Weisskopf radi­ us, also called Landau radius, is given by: k being Boltzmann's constant and T the absolute temperature: the other length is the Debye length: À - r kT ) D - l81Tne2 e being the elementary charge and n the densities of electrons and protons, here taken equal. The Weisskopf radius is the distance at which two equal charges e, with a relative kinetic energy equal kT, can ap­ proach each other. It often serves as a lower cutoff in statie models for ionized gases. As yet there seems to be no general­ ly accepted dynamica! model which can replace such a procedure; the problem is so difficult that one must be satisfied with rather rough approximations. The Debye length can be illustrated by saying that the plas­ ma is polarized in the following sense: the positive and nega­ tive particles are, on the average, arranged such as to show a slight tendency towards an alternate ordering of positive and negative charges, like in a crystal. The effect is enormously smaller than in a crystal and therefore the comparison must not be taken as a very good one. This ordering makes its appearance in the screening of the Coulomb interactions; the potential at a distance r from a charge e (formally to be considered as an external charge) in a plasma is, for not too small distances, on the average equal to: exp(-r/À ) 0 eo r the range of this potential is the Debye length À0 • Fo~ a characteristic plasma the number of particles in a sphere with ix radiu::;; "n may be of the order of 1000; if this nu!l'.ber becc,mes too small 10 say) it is not justified to speak al:.out a plas- ma at all, at least not as a plasma in the common sense. It turns out tha~ the additional charge in the Debye sphere {radius 'o} around e equals - e , i.e. it just neutralizes 0 0 the charge e • Again, we must stress that this is only true .Q!! 0 the average; this statement obtains special weight if one con­ siders that the spo~taneous fluctuations in the number of par­ ticles in the Debye sphere are of the order of the root of that number; thus in the above case of the order of 11000 ~ 30, much larger than one. Hence , one must be very careful with the con­ cept of screening <l•P· 85). For instance, it is a common method to take collective inter­ actions into account by considering particles which interact through a screened Coulomb potential, based on the concept of Debye screening. One may expect, and so it actually happens, that this procedure will lead to corrections (on the results of models without interactions) which have the proper trend, i.e. which correspond to a diminishing of the amplitude of sponta­ neous (or thermal) fluctuations. However, one may object that this is not a fundamental approach; in a proper theory, based on fundamental principles, the screening itself must also come out of the model. The situation is similar to the procedure sometimes followed in the evaluation of expressions to be av­ eraged over velocities: instead of properly averaging the ex­ pression one replaces everywhere the velocity by its averaqe. Obviously, this is a very rough approach for expressions which are non-linear in the velocity. Nevertheless, there is some justification for this - so­ called - superposition principle: a superposition.of indepen­ dent dressed test particles for the description of the statis­ tica! properties of the plasma (also: quasi particles, a name more common in the quantum mechanica! many-body problero). Thompson was the first to introduce this concept into plasma physics (see e.g. _!); Rostoker CD has shown that many results from rigorous theories can be derived with the aid of the super­ position principle.
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