By Chen T.Q.
This paintings offers the development of an asymptotic approach for fixing the Liouville equation, that is an analogue of the Enskog-Chapman strategy for the Boltzmann equation. as the assumption of molecular chaos has no longer been brought, the macroscopic variables outlined via the mathematics technique of the corresponding microscopic variables are random generally. for this reason, it really is handy for describing the turbulence phenomena. The asymptotic procedure for the Liouville equation unearths a time period displaying the interplay among the temperature and the speed of the fluid flows, that allows you to be misplaced below the belief of molecular chaos.
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2/ NŠ 1 O . 2/ O . 1/ D j 1; 2; ; N j Á p h0j O . 19) naturally forms the eigenspace of PO . To see this, we start with transpositions. 19) are eigenstates of transpositions. Second, as any PO is either even or odd, we easily conclude that PO j 1 ; 2 ; holds generally. 21) satisfies the normalization condition: 0 N 0 j 1; 2; ; Ni D ıN 0 N X NŠ P ı. 10 ; 0 p1 /ı. 2 ; p2 / ı. 22) To prove this, we make use of the identity: O . 10 / O . 1 / O . 2 / O . N/ D ı. 10 ; 1 / O . 2 / C 1 O . N/ ı.
0 / Á O . / O . 0/ O . 0 / O . / D ı. ; 0 /; O . /; O . 0 / D O . /; O . 17) with D ˙1 and ı. r r0 /ı˛˛0 . In addition, we define the ket j0i and bra h0j via the right (left) action of the annihilation (creation) operator, 0 D h0j O . / D O . /j0i ; O . 18) These are the basic ingredients needed to construct the eigenspace of permutations. Next, we introduce the ket j 1 ; 2 ; ; N i and its Hermitian conjugate by 1 O O . N /j0i; Áp . 1/ O . 2/ NŠ 1 O . 2/ O . 1/ D j 1; 2; ; N j Á p h0j O .
X q cOq 'q . 49) q To obtain “coefficients” cOq and cOq , we multiply the two expansions by 'q 0 . / and 'q 0 . 47). Replacing q 0 ! q in the resulting expression, we obtain Z cOq D Z 'q . / O . / d ; cOq D 'q . / O . 47). 52) 36 3 Quantum Mechanics of Identical Particles Next, we obtain the eigenstates of HO 0 in the same representation. 32) to obtain ˆ . 53). Let us focus on fermions. F/ . 55) N jqN i which is known as the Slater determinant. F/ minant [1, 8] that ˆ vanishes when a pair of columns or rows are identical.