An inhomogeneous Landau-like equation

The problem of the mixing of two plasmas is very difficult to solve if the methods of statistical mechanics are used. When both plasmas are homogeneous in space, and the limit of weak coupling is appropriate, the Landau equation describes the motion. An equation, similar to the Landau equation, is developed for the case when one of the plasmas is initially inhomogeneous in space. The equation is a power series in time and involves wave vectors in space. The short-time-after-mixing limit is considered, which restricts the number of wave vectors. A sample problem is presented of the short time development of an infinite-temperature spike interacting with a homogeneous plasma. The spike is found to move very rapidly from the origin owing to the long range of the forces. Extension of the time development past the initial short time requires consideration of destructive wave vectors in addition to the wave vector generation considered here.

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The quantum mechanical Landau equation

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1983.0088 ◽

1983 ◽

Vol 388 (1795) ◽

pp. 393-400 ◽

Cited By ~ 1

Keyword(s):

Landau Equation ◽

Time Development ◽

Feynman Diagrams ◽

Quantum Mechanical ◽

Number Density ◽

Short Discussion ◽

Mechanical Effects ◽

Quantum Mechanical Effects ◽

Apparent Similarity ◽

Short Time

The significance of quantum mechanical effects on the short time development of the Landau equation is examined. The analysis is undertaken using the density diagram technique of Prigogine and Balescu. Owing to the apparent similarity that exists between the density diagrams and the more familiar Feynman diagrams derived from generalized Green functions a short discussion of both techniques is presented. The solution is undertaken for small h/m and does not necessarily apply to electrons. It is found that quantum mechanical effects may become significant at a number density around 10 14 , which is the lower bound in density that an earlier analysis suggested. Non-Markovian effects may be significant.

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The effect of non-Markovian terms on the Landau equation

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1980.0086 ◽

1980 ◽

Vol 371 (1746) ◽

pp. 381-392 ◽

Cited By ~ 2

Keyword(s):

Time Series ◽

Lower Bound ◽

Landau Equation ◽

Plasma Potential ◽

Time Span ◽

Time Interval ◽

Number Density ◽

Time Effects ◽

Homogeneous Plasma ◽

Short Time

The Landau equation can be derived as the sum of an expansion series in time provided certain non-Markovian short-time terms are neglected. The time series could be very useful for the solution of a variety of problems. It is necessary to estimate the time span over which the neglected term s are significant. The non-Markovian terms for the case of a homogeneous plasma are evaluated using a technique developed by Prigogine and Balescu. For typical values of plasma potential cut-offs the conditions under which the short-time terms may be ignored are estimated. It is found that for electrons of number density less than 1010 and protons of number density less than 1014 it may be possible to ignore the short-time effects. It is conjectured that for many situations the non-Markovian effects will be important for a time interval which has a lower bound t 0 and an upper bound several orders of magnitude greater than t .

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