Dear Yuhang,

Regarding your 1st question:

(Working in SI units)

RHS:

Integral = J*J*dt = W/m^2 * W/m^2 * sec

Therefore, V/(k_B*(T^2)) * Integral

= m^3/ [ (J/K) * K^2 ] * ( W/m^2 * W/m^2 * sec)

= W/(m^2 K) = units of thermal conductivity.

Best,

Mario