Statistical physics: Microcanonical distribution and oscillators

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Hamilton's function in the context of statistical physics combines potential and kinetic energy, expressed as H(x,p;e)=sum(for i=1 to N)(p^2/2m+(mw^2)*x^2/2+qex)=E. The challenge lies in determining the phase volume, Г, which is related to the number of oscillators, where Гn=(Г1)^N. To calculate the phase volume, a proper substitution or transformation in the equations is necessary, but the specific method for this replacement is unclear to the participants. Clarification on how to incorporate the field into the calculations is sought, emphasizing the need for a deeper understanding of the microcanonical distribution. The discussion highlights the complexities of statistical mechanics and the importance of precise definitions in Hamiltonian systems.
Marcustryi
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Homework Statement
An ideal gas, consisting of N harmonic oscillators of mass m and frequency w and charge q, is placed in an external uniform electric field with intensity e. Find the phase volume limited by the energy hypersurface H(x, p; e)=E. Find an expression for the temperature of the oscillator system.
Relevant Equations
H(x, p; e)=E
Hamilton's function in this case is the sum of potential and kinetic energy? But then I don’t remember or don’t understand what to do with e.
I need to find Г, but I don't understand what to do with the field.
 
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H(x,p;e)=sum(for i=1 to N)(p^2/2m+(mw^2)*x^2/2+qex)=E
Гn=(Г1)^N.
But how to calculate the phase volume? what replacement should I make?
 

Thread 'Help needed with Elliptic Integrals'

I want to find the solution to the integral ##\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}## I can see that ##\frac{d^2u}{d\theta^2} = A +Bu+Cu^2## is a Weierstrass elliptic function, which can be generated from ##\Large(\normalsize\frac{du}{d\theta}\Large)\normalsize^2 = c-u^2 +2u^3## (A = 0, B=-1, C=3) So does this make my integral an elliptic integral? I haven't been able to find a table of integrals anywhere which contains an integral of this form so I'm a bit stuck. TerryW
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