What is the topology of the phase space of $N$ indistinguishable particles?

Question

In classical mechanics, we often wish to calculate a partition function of, say, an $N$-particle system $$Z = \int_\Omega {\mathrm e}^{- \frac{ E(\vec x) }{ k_\text{B} T} } \, \mathrm d^{6N} \vec x .$$ This integral is taken over a $6N$-dimensional phase space $\Omega$, with elements of type $\vec x = (\vec q_1, \dots, \vec q_N, \vec p_1, \dots, \vec p_N)$.

Because particles are indistinguishable, the energy depends on $\vec x$ in such a way that permuting any two position components (as well as, simultaneously, the corresponding momentum components) changes nothing.

I have the feeling, however, that you could bake this information directly into the topology of the phase space you integrate over, so that two points $\vec x_1, \vec x_2$ with particles permuted land in the same point of phase space. What would this space be, topologically?

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I apologise in advance if this question already has an answer; I couldn't find it. This answer discusses the topology of phase space given Hamiltonian constraints, but this is not what I am looking for, as it doesn't talk about the Gibbs correction. This talks about the Gibbs indistinguishability factor being an approximation, but no mention of the topological perspective.

Answer 1

One important reference on this question is the Leinaas & Myrheim paper "On the Theory of Identical Particles", pdf here, which considers the configuration spaces of identical particles under different symmetry conditions. Configuration space is the position part of phase space. For example, in this figure

they identify the points $(x_1,x_2)$ and $(x_2,x_1)$ of two particles on a line, meaning that only half of the configuration plane is available, with trajectories being reflected off the $x_1=x_2$ line.

While I am not entirely sure they explicitly answer your question, it should provide the framework supporting your intuition. In particular, the symmetries of your particles will be important. Particles with non-trivial exchange symmetry (e.g. fermions) will, for example, live in spaces that are not simply connected, leading to their picking up of an exchange phase.