On Cauchy-Schwarz inequality and energy fluctuations for effusion and Maxwell distributions in d-dimension

We have analysed the Cauchy-Schwarz inequality for the α-th moment of the speed of the molecules of an ideal gas in d-dimension following the Maxwell distribution of speeds and also for the molecules effusing through the hole. In the later case the distribution is non-Maxwellian. We have also studied the energy fluctuations for both cases in d-dimension and examined the role of the spatial dimensionality of the system on the distribution. (author)