I Phase space integral in noninteracting dipole system

AI Thread Summary
The discussion centers on a system of N noninteracting electric point dipoles in an external field, focusing on the Lagrangian and Hamiltonian formulations. It raises a question about the integration of cyclic momenta conjugate to the spherical angles, specifically whether they should be included in the partition function calculation. Concerns are expressed regarding the potential divergence of the integral due to the momenta being defined over positive real values. The conversation also touches on the use of polar coordinates in both coordinate and momentum spaces to analyze the phase space. Clarification is sought on the treatment of these momenta in the context of statistical mechanics and partition functions.
raisins
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Hi all,

Consider a system of ##N## noninteracting, identical electric point dipoles (dipole moment ##\vec{\mu}##) subjected to an external field ##\vec{E}=E\hat{z}##. The Lagrangian for this system is
$$L=T-V=\sum_{i=1}^N\left\{\frac{m\dot{\vec{r}_i}^2}{2}+\vec{\mu}_i\cdot\vec{E}\right\}=\sum_{i=1}^N\left\{\frac{m\dot{\vec{r}_i}^2}{2}+E\mu\cos\theta_i\right\}$$
and the Hamiltonian is
$$H=\sum_{i=1}^N\left\{\frac{\vec{p}_i^2}{2m}-E\mu\cos\theta_i\right\}$$
As I see it, 5 sets of generalized coordinates ##\left(\left\{\vec{r}_i\right\},\left\{\theta_i\right\},\left\{\phi_i\right\}\right)##, where ##\theta_i,\phi_i## are the usual spherical angles, are needed to describe this system. Now, the momenta conjugate to ##\theta_i,\phi_i## (call them ##p_{\theta_i},p_{\phi_i}##) are both cyclic, since ##\dot{\theta}_i,\dot{\phi}_i## appear nowhere in the Lagrangian. But do we not still have to integrate over them to find the partition function; ie.
$$Z=\frac{1}{N!}\int \prod_{i=1}^N\frac{d^3\vec{r}_id^3\vec{p}_i}{h^3}\,\frac{d\theta_idp_{\theta_i}}{h}\,\frac{d\phi_idp_{\phi_i}}{h}e^{-\beta H}$$
But ##p_{\theta_i},p_{\phi_i}\in[0,\infty)## so doesn't that integral blow up? Am I wrong in thinking ##p_{\theta_i},p_{\phi_i}## can take any real, positive value? Or, because they're cyclic, do we just omit them from the integration?

Any help would be appreciated, thank you!
 
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I am not familiar with momenta conjugate. Taking polar coordinate for both coordinate and momenta spaces
dv=r^2 sin^2\theta dr d\theta d\phi
dV=R^2 sin^2\Theta dR d\Theta d\Phi
where I noted small r for coordinate space and capital R for momentum space.
So in total number of states in infinitesimal phase space elements is
\frac{\prod_{i=1}^N dv_idV_i}{h^{3N}}
 
Last edited:
raisins said:
But ##p_{\theta_i},p_{\phi_i}\in[0,\infty)## s
From where this comes from?
 
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