BROWNIAN-MOTION WITH RESTORING DRIFT - THE PETIT AND MICRO-CANONICAL ENSEMBLES

Let f(Q) be odd and positive near +infinity. Then the non-linear wave equation partial derivative2Q/partial derivative t2 - partial derivative2Q/partial derivative x2 + f(Q) = 0, considered on the circle 0 less-than-or-equal-to x < L, can be written in Hamiltonian form Q. = partial derivative H/partial derivative P, P. = -partial derivative H/partial derivative Q with [GRAPHICS] the corresponding flow preserves the (suitably interpreted) ''petit ensemble'' e(-H) d(infinity)Qd(infinity)P; and, for L up infinity, Q settles down to the stationary diffusion with infinitesimal operator 1/2 partial derivative2/partial derivative Q2 + m(Q) partial derivative/partial derivative Q, m being the logarithmic derivative of the ground state of -d2/dQ2\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 [GRAPHICS] and making L up infinity with fixed D = N/L. Now, for F(Q)/Q2 --> infinity, the same type of diffusion appears, but with drift arising from the modified potential F(Q) + cQ2, c being chosen so that the mean of Q2 is the assigned number D. The proof employs Doblin's method of ''loops'' [1937] and steepest descent. The same is true-for F(Q) = m2Q2, only now the proof is elementary. The outcome is also the same if F(Q)/Q2 --> 0, provided D is smaller than the petit canonical mean of Q2; for D larger than this mean, the matter is more subtle and the outcome is unknown.