Brownian motion with restoring drift: The petit and micro-canonical ensembles

Abstract

Letf(Q) be odd and positive near +∞. Then the non-linear wave equation ∂² Q/∂t ²−∂² Q/∂x ²−f(Q)=0, considered on the circle 0≤x<L, can be written in Hamiltonian formQ ^⊙=∂H/∂P, P ^⊙=−∂H/∂Q with

$$P = Q^\cdot and H = \tfrac{1}{2}\mathop \smallint \limits_0^L (Q')^2 + \mathop \smallint \limits_0^L F(Q) + \tfrac{1}{2}\mathop \smallint \limits_0^L P^2 ;$$

the corresponding flow preserves the (suitably interpreted) “petit ensemble”e ^−H d ^∞ Qd ^∞ P; and forL↓∞,Q settles down to the stationary diffusion with infinitesimal operator 1/2 ∂²/∂Q ²+m(Q)∂/∂Q,m being the logarithmic derivative of the ground state of −d ²/dQ ² ‖F(Q). This diffusion is the “Brownian motion with restoring drift”; see McKean-Vaninsky [1993(1)]. For reasons suggested by the paper of Lebowitz-Rose-Speer [1988] on NLS, it is interesting to study the “micro-canonical ensemble” obtained by restricting to the sphere$\int\limits_0^L {Q^2 } = N$ and makingL↓∞ with fixedD=N/L. Now, forF(Q)/Q ²→∞, the same type of diffusion appears, but with drift arising from the modified potentialF(Q)+cQ ²,c being chosen so that the mean ofQ ² is the assigned numberD. The proof employs Döblin's method of “loops” [1937] and steepest descent. The same is true forF(Q)=m ² Q ², only now the proof is elementary. The outcome is also the same ifF(Q)/Q ²→0, providedD is smaller than the petit canonical mean ofQ ²; forD larger than this mean, the matter is more subtle and the outcome is unknown.