Any general link between the average minimum distance of i.i.d. random variables and the distribution from which these random variables are drawn?

Question

Suppose that $\rho(x)$ is a probability density function supported on $\mathbb{R}$ (or a subset of $\mathbb{R}$), and we generate $N$ i.i.d. random variables $\{X_i\}_{1\leq i\leq N}$ with law $\rho(x)\,\mathrm{d}x$, I am wondering is there a general expression of $$\mathbb{E}\left[\min_{i,j\in [1,n]: i\neq j} |X_i - X_j|\right]$$ in terms of $\rho$ (or functionals of $\rho$)? This post considers the situation when $\rho(x) = \frac{1}{a}\mathbf{1}_{x\in [0,a]}$, I am wondering how would the expression look like for a 'generic' $\rho$, if such expression exists at all. Thank you!