Abstract Plasma electromagnetic (EM) kinetic simulation faces two unavoidable difficulties. One arises from the fact that Maxwell equations determine a simultaneity relation between transverse electric field <?CDATA ${\vec{E}}_{{Tr}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mo>⃗</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi mathvariant="italic">Tr</mml:mi> </mml:mrow> </mml:msub> </mml:math> and local growth rates of probability distribution function (PDF) <?CDATA $f$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>f</mml:mi> </mml:math> in Vlasov-Maxwell (V-M) simulation in Eulerian approach, so does that between <?CDATA ${E}_{{Tr}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="italic">Tr</mml:mi> </mml:mrow> </mml:msub> </mml:math> and Lagrangian particles’ time-varying rates which refers to displacement of <?CDATA $f$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>f</mml:mi> </mml:math> -element in <?CDATA $6D$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>6</mml:mn> <mml:mi>D</mml:mi> </mml:math> phase space in V-M simulation in Lagrangian approach, and macroparticles’ accelerating rates in Particle-in-Cell (PIC) simulation. These simultaneous with <?CDATA ${E}_{{Tr}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="italic">Tr</mml:mi> </mml:mrow> </mml:msub> </mml:math> are termed as bottom objects’ growth rates (BOGRs) in this work. Directly solving the BOGRs needs to diagonalize a large full matrix, and hence often be approximated. The other arises from the fact that Lagrangian particles’ time-histories should uniformly converge with respect to the time-step. This severe requirement is difficult to be satisfied and hence some of Lagrangian particles’ time-histories lose fidelity. We propose a strict alternative method free from two difficulties. The initial-value problem of V-M system can be interpreted by <?CDATA $6D$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>6</mml:mn> <mml:mi>D</mml:mi> </mml:math> phase space allowed deformation of an initial <?CDATA $f$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>f</mml:mi> </mml:math> -profile. By virtue of a more compact description of <?CDATA $f,$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> </mml:math> in which a conditional probability density function (C-PDF) well reflects some macroscopic conservation laws behind the V-M system, we can interpret the initial-value problem of <?CDATA $f$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>f</mml:mi> </mml:math> in terms of <?CDATA $6D$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>6</mml:mn> <mml:mi>D</mml:mi> </mml:math> standing-wave oscillation in the C-PDF. A key subtle difference between microscopic scalar field and microscopic vector field can also lead to a similar scheme of particle simulation. PACS: 52.65.-y.

6D phase space collective modes in Vlasov-Maxwell system

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