- #1

P3X-018

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In order to get a "feeling" of the sharpness of the multiplicity function, for a system of 2 solids (A & B) with N quantum oscillators in each, in the high-temeperature limit (that is the total number of energy units q is much larger than N), it is approximated as a Gaussian as [Shroeder, eq. (2.27), 1999],

[tex] \Omega(x) = \Omega_{max}\,e^{-4Nx^2/q^2} [/tex]

Where x comes from that if [itex] q = q_A + q_B[/itex] then [itex] q_A = q/2 + x[/itex] and [itex] q_B = q/2 - x[/itex]. The width parameter of the above equation would be [itex] \sigma = \frac{q}{2\sqrt{2}N} [/itex] around x = 0 (q_A = q_B).

Now how can the function be called "sharp near it's peak", when this width parameter is such a huge number by this assumption that is q >> N?

This was used to explain that only a very small number of macrostates have a reasonable chance of occuring, but how can this be, if the width is so great?

[tex] \Omega(x) = \Omega_{max}\,e^{-4Nx^2/q^2} [/tex]

Where x comes from that if [itex] q = q_A + q_B[/itex] then [itex] q_A = q/2 + x[/itex] and [itex] q_B = q/2 - x[/itex]. The width parameter of the above equation would be [itex] \sigma = \frac{q}{2\sqrt{2}N} [/itex] around x = 0 (q_A = q_B).

Now how can the function be called "sharp near it's peak", when this width parameter is such a huge number by this assumption that is q >> N?

This was used to explain that only a very small number of macrostates have a reasonable chance of occuring, but how can this be, if the width is so great?

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