So this question has been bugging me because I can't begin to start it. The question is, prove that [tex]\Omega[/tex], the number of microstates of the combination of two physical states in thermal contact is a Gaussian of the energy of one of the states. [tex]\Omega[/tex] is given here as [tex]\Omega_1*\Omega_2[/tex] where those are the respective number of microstates in the first and second systems, respectively. They are functions of the # of particles, Energy of the state, and Volume, of course, and, knowing the total energy E, we can express [tex]\Omega_2(E_2) = \Omega_2(E-E_1)[/tex] thus we only have one variable which effects [tex]\Omega[/tex]
I've got no real attempt at a solution, I'm hoping I can get a starting off point from you guys, because I can't seem to find where I can establish that [tex]\Omega[/tex] is a Gaussian.
The actual question reads thusly:
1(a) : Show that, for two large systems in thermal contact, the number [tex]\Omega[/tex] can be expressed as a Gaussian in the variable [tex]E_1[/tex]. Determine the root-mean-square deviation of [tex]E_1[/tex] from the mean value in terms of other quantities pertaining to the problem.
1(b) Make an explicit evaluation of the root-mean-square deviation in the special case where the systems are classical gases.
I'd really appreciate some help, I just feel that I can't get off the ground here.
The book is Pathria's Statistical Mechanics