where ¯ β stands for the average of β over f ( β ): 〈 β 〉 β = ¯ β . The PDF averaged over f ( β ) is given by with The PDF of harmonic oscillators in the local equilibrium state with a given inverse temperature β is expressed by The PDF averaged over f ( β ) is given by with It is easy to see that PDFs given by Eqs. (85) and (89) have the same structure as those given by Eqs. (16) and (29) derived by the MEMs. For example, Eq. (85) for ideal gases may be rewritten as with or equivalently p ( N $^{)}$( p ) = ∫ ∞ 0 Π ( N $^{)}$( p ; β ) f ( β ) dβ, (84) = 1 Z [ 1 + ( 2 ¯ β n ) N ∑ i =1 p 2 i 2 m ] − ( DN + n ) / 2 + , (85) Z = V N N ! ( πmn h 2 ¯ β ) DN/ 2 Γ( n/ 2) Γ( DN/ 2 + n/ 2) . (86) Π ( N $^{)}$( p , x ; β ) = ( hωβ 2 π ) DN e − $_{β}$∑ N i $_{=1}$( p 2 $_{i}$/ 2 m + mω $^{2}$r 2 $_{i}$/ $^{2)}$. (87) p ( N $^{)}$( p , x ) = ∫ ∞ 0 Π ( N $^{)}$( p , x ; β ) f ( β ) dβ, (88) = 1 Z [ 1 + ( 2 ¯ β n ) N ∑ i =1 ( p 2 i 2 m + mω 2 r 2 i 2 ) ] − ( DN + n/ 2) + . (89) Z = ( πn hω ¯ β ) DN Γ( n/ 2) Γ( DN + n/ 2) . (90) p ( N $^{)}$( p ) ∝ [ 1 − (1 − q ) ( DN + n n ) ¯ β N ∑ i =1 p 2 i 2 m ] 1 / $^{(1}$− q ) + , (91) q = 1 + 2 DN + n , (92) p ( N $^{)}$( p ) ∝ [ 1 − ( ˜ q − 1 ˜ q )( DN + n − 2 n ) ¯ β N ∑ i =1 p 2 i 2 m ] 1 / (˜ $^{q}$− 1) + , (93) 15