Brownian motion with restoring drift: Micro-canonical ensemble and the thermodynamic limit (Q1865607)

The author considers a problem of micro-canonical conditioning in the setting of diffusion. The problem stems from an old work done by \textit{H. P. McKean} and \textit{K. L. Vaninsky} [Commun. Math. Phys. 160, No. 3, 615-630 (1994; Zbl 0792.60077)], where statistical mechanics for nonlinear wave equations is studied. See also \textit{H. P. McKean} [ibid. 168, No. 3, 479-491 (1995; Zbl 0821.60069)]. For a potential \(F : R^d \rightarrow R\), the Schrödinger operator with ground state \(\psi\) is given by \(- G_0 = \frac{1}{2} \Delta - F\), and a conjugation provides with the corresponding diffusion generator \(G =\) \(\frac{1}{2} \Delta + ( \nabla \psi / \psi) \cdot \nabla\) with invariant density \(\psi^2\). Here the diffusion motion \(X(t)\) is made micro-canonical by first conditioning the path to be periodic: \(X(0) = X(L)\), and by secondly conditioning on the empirical mean-square (or particle number): \(\frac{1}{L} \int_0^L |X(t) |^2 dt \simeq D\). The thermodynamic limit is to take \(L \nearrow \infty\) with \(D\) remaining fixed. The goal of this paper is to show that for \(F(x) / |x |^2 \searrow 0\), a sort of phase transition takes place especially in the case \(D > D_0\), where \(D_0 =\) \(\int_{R^d} |x |^2 \psi^2(x) dx\) is the canonical mean-square. In the above mentioned paper by McKean and Vaninsky it is shown that for \(F(x) / |x |^2 \nearrow \infty\) with \(|x |\nearrow \infty\), the same type of diffusion appears in the thermodynamic limit, but with drift arising from the shifted potential \(F + c |x |^2\), where \(c\) is a constant such that the limiting mean-square equals \(D\). It is interesting to note that their method predicts a similar result for \(F(x) / |x |^2 \searrow 0\) as far as \(D < D_0\), while the opposite case \(D > D_0\) has remained unsolved.