Problems of Statistical Theory of Open Systems

Klimontovich, Yu. L.

doi:10.1142/9789814368148_0015

Cited by 26 publications

(76 citation statements)

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“…The theory follows from the long-wave limit of the Boltzmann equation and requires inclusion of the tensors of higher moments. We show that this leads to correct plasmon dispersion.The kinetic equation (1) can be transformed in an infinite chain of coupled equations for the moments of the distribution function [17]. For the first three moments: density n, current j = nv and stress tensor P ij we have…”

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confidence: 99%

“…The kinetic equation (1) can be transformed in an infinite chain of coupled equations for the moments of the distribution function [17]. For the first three moments: density n, current j = nv and stress tensor P ij we have…”

mentioning

confidence: 99%

“…The requirement of the local equilibrium is fulfilled if the characteristic time of the process τ ∼ 1/ω is much longer than the inverse collision frequency 1/ν and the typical length L is greater than the mean free path l ∼ u/ν (u is the average particle velocity) [17] …”

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“…Each equation in (4-6) is coupled to the next one through the higher moment. If conditions (2) are fulfilled, the higher moments are small and the infinite chain of equations can be decoupled (Chapman-Enskog or Grad methods, see [17]). For example, as π ij and Q ijk vanish for locally equilibrium form of the distribution function, they remain small under conditions (2).…”

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confidence: 99%

“…The reason is that in the low frequency range where collisions are important, the second moment of I is always a linear function of π ij [17]. The coefficient ν can thus be considered as a phenomenological parameter, which is connected to viscosity as η = P/ν.…”

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Hydrodynamic theory of an electron gas

1999

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The generalised hydrodynamic theory of an electron gas, which does not rely on an assumption of a local equilibrium, is derived as the long-wave limit of a kinetic equation. Apart from the common hydrodynamics variables the theory includes the tensor fields of the higher moments of the distribution function. In contrast to the Bloch hydrodynamics, the theory leads to the correct plasmon dispersion and in the low frequency limit recovers the Navies-Stocks hydrodynamics. The linear approximation to the generalised hydrodynamics is closely related to the theory of highly viscous fluids.PACS numbers: 71.10. Ca, 05.20.Dd, 47.10.+g, 62.10.+s Hydrodynamic theory of an electron gas was heuristically introduced by Bloch in 1933 [1] as an extension of Thomas-Fermi model. Only macroscopic variableselectron density n(r, t), velocity v(r, t), pressure P , and electrostatic potential ϕ(r, t), enter the theory. The set of equations (continuity equation, Euler and Poisson equations) becomes complete when the equation of state is added, and in the original paper [1] Bloch identified P with the kinetic pressure of a degenerate Fermi gas. The Bloch's hydrodynamic theory (BHT) has been applied to variety of kinetic problems [2,3,4,5,6,7,8,9] with minor improvements (inclusion of exchange, correlation and quantum gradient corrections) [5,9].From the microscopic point of view BHT cannot be a fully consistent theory since it extends the collisiondominated hydrodynamics to the electron gas where the collisionless (Vlasov) limit is most common. For example, in the plasmon dispersion law ω 2 = ω 2 p + v 2 0 q 2 BHT predicts for degenerate electron gas v 2 0 = 1 3 v 2 F instead of a correct result 3 5 v 2 F (v F is the Fermi velocity) [4,6,7,8,11,12,13,14,15] . At arbitrary degeneracy the hydrodynamics gives v 2 0 = v 2 s , where v s is velocity of sound, whereas in the kinetic theory v 2 0 equals to the mean square of the particle velocity < v 2 p > [15]. It has been realized [11,12,13] that this discrepancy originates from the assumption of a local equilibrium , which underlies the common hydrodynamic theory [16]. The assumption allows to reduce the kinetic equation for the distribution function f p (r, t)to equations for macroscopic variables n(r, t),v(r, t) and, in general case, temperature T (r, t). The requirement of the local equilibrium is fulfilled if the characteristic time of the process τ ∼ 1/ω is much longer than the inverse collision frequency 1/ν and the typical length L is greater than the mean free path l ∼ u/ν (u is the average particleIn a zero oder with respect to parameters (2) Eq. (1) reduces to I p [f p ] = 0 which means that f p must have a locally equilibrium form. The stress tensor in the comoving (Lagrange) framebecomes diagonal P ij = P δ ij with P being the local pressure. As a result, the equations for the first three moments of the distribution function i.e. density n, current j = nv and stress tensor P ij , form the closed set of hydrodynamics equations for an ideal liquid. Due to the high frequency of ...

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Hydrodynamic theory of an electron gas

1999

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Introduction to the Theory of Lévy Flights

Chechkin¹,

Metzler

Klafter

et al. 2008

Anomalous Transport

135

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Contributions of Yuri L. Klimontovich to the kinetic theory of nonideal plasmas

Bönitz

Ébeling

Romanovsky

2003

Contrib. Plasma Phys.

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We give a short summary of live and work of Yuri L. Klimontovich (1924Klimontovich ( -2002, in particular we discuss his work on nonideal plasma physics.Finally we arrived at the end of our long way. No doubt, not everything on this long path was sufficiently smooth. Yuri L. Klimontovich, in the conclusions to his book "Statistical Physics" (1982) 1 On the biography of YLK YLK was born on September 28 1924 in Moscow. His mother, Natalia Vladimirovna Vladykina, was from the famous russian noble family Vladykin which is of tataric origin and took service under Ivan the Terrible. At that time they obtained very large land property and became wealthy (what expresses the meaning of the family name). His father also came from a noble family, in second generation. He was arrested under Stalin's terror regime in 1931 and killed after 2 weeks (only 62 years later the family received an official information). Since his mother died soon after, Yuri and his 2 brothers lived with the family of an aunt under very poor conditions. To support the family, Yuri carried luggage at railway stations. He fell ill on tuberculosis and survived a very difficult operation. For this reason he did not serve in the army and studied for a short time at an Engineering institute (college). After the war, in 1946, Yuri began to study physics at Moscow State Lomonosov University (MGU). The adviser of his diploma (master) thesis on "radiative friction" [1] was Prof. V.S. Fursov. The very active young man wanted to become an "aspirant" (graduate student) and write a dissertation. However, this was refused by the authorities since he was considered the son of a "repressirovan" (repressed) father. Shortly after the refusal of "aspirantura" and of employment at the Academy of sciences as well, just by a lucky chance, Yuri met Prof. Nikolai N. Bogoliubov near the Academy who asked him about his plans. Having a very high opinion about Yuri, Bogoliubov took him personally as a PhD student, ignoring the official decision. In 1951 Yuri defended his dissertation. He knew very well that without the support of Fursov (who later served many years as the dean of the Department and Bogoliubov (who later held high positions in the Academy) he never would have had a chance to make a scientific carrier. He always expressed highest opinions about his advisers.Starting physics with some problems given by his advisers, YLK went on to develop quite independently a new theory of plasmas based on his new method called "second quantization in phase space" [3]. Among his coauthors of this period were Silin (work on excitation spectra, [2]), and Ebeling (hydrodynamic approximation).

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Problems of Statistical Theory of Open Systems

Cited by 26 publications

References 0 publications

Hydrodynamic theory of an electron gas

Hydrodynamic theory of an electron gas

Introduction to the Theory of Lévy Flights

Contributions of Yuri L. Klimontovich to the kinetic theory of nonideal plasmas

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