We consider an N-particle system given by the gradient dynamics $dX(t)= -N\nabla H_N (X(t)) dt + \sigma d\beta(t)$ in $(\mathbb{R}^d)^N$, where $\sigma$ is a positive constant.
We assume that for $x=(x_1,...x_N)$ $H_N (x) = \int V(y) d\mu_N(x)(y) + \frac{1}{2} \int\int W(y,z) d\mu_N(x)(y) d\mu_N(x)(z) $ where $\mu_N$ is the empirical distribution of the model, and the functions $V,W$ are an external and an interacting potential.
If $H_N$ possesses all the necessary properties so that the system has unique invariant Gibbs distribution $P_N$, which is a probability measure on $(\mathbb{R}^d)^N$. However, I was told that the limit $\lim \limits_{N\rightarrow \infty}P_N$ does not necesserily exist ( I don't know in which sense, but I guess in the weak topology convergence).
The problem is that I cannot find this statement and I really need it. How is this related to phase transitions ? Could someone give some more info on this or give a reference with this result ? Any help is appreciated.