※ 引述《yyc2008 (YYC丫逼)》之銘言：
: 大家好
: 我想請問一題決定函數形式的題目
: 是來自於物理上性質良好的函數
: Show that if one assumes the functional form
: T = Σ[x_i f(x_i)]
: i
: Σx_i = 1
: i
: where f(x) is some function of x, then the requirement that
: T is extensive implies f(x) = ?
: 感謝回答
T is extensive means T(x_1, x_2,... a x_i, x_i+1, ...) = a T(x_1,....)
By Euler's homogeneous function theorem,
http://en.wikipedia.org/wiki/Euler%27s_homogeneous_function_theorem#Properties
x . grad T = T
where x = (x_1, x_2, ...)
x . grad T = Σ_k x_k @T/@x_k
= Σ_k x_k @/@x_k ( Σ_i x_i f(x_i) )
= Σ_k x_k Σ_i ( @x_i/@x_k f(x_i) + x_i Σ_j @f/@x_j @x_j/@x_k )
= Σ_k x_k Σ_i ( del_ik f(x_i) + x_i Σ_j @f/@x_j del_jk )
= Σ_k x_k ( f(x_k) + @f/@x_k)
Also T = Σ_k x_k f(x_k) by definition,
so the requirement that T is extensive is that Σ_k @f/@x_k = 0
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