I'm studying a statistical mechanics problem and I have two conserved quantities :
$E = \sum_{k=0}^{K_M} \left[ a_1^2(k) + a_2^2(k) + b_1^2(k) + b_2^2(k)\right]$
$H = \sum_{k=0}^{K_M} 2 k \left[ a_1^2(k) - a_2^2(k) + b_1^2(k) - b_2^2(k)\right]$ $$ E = \sum_{k=0}^{M} \left[ a_1^2(k) + a_2^2(k) + b_1^2(k) + b_2^2(k)\right] $$ $$ H = \sum_{k=0}^{M} 2 k \left[ a_1^2(k) - a_2^2(k) + b_1^2(k) - b_2^2(k)\right] $$
Is there a way to know analytically the probability density of the system of { $ a_1(k) ; a_2(k) ; b_1(k) ; b_2(k) $ }$ \{ a_1(k) ; a_2(k) ; b_1(k) ; b_2(k) \} $ to have $a_1(k) = A$ ?
