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

H. P. McKean, K. L. Vaninsky

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Let f(Q) be odd and positive near +∞. Then the non-linear wave equation ∂2Q/∂t2-∂2Q/∂x2-f(Q)=0, considered on the circle 0≤x<L, can be written in Hamiltonian form Q=∂H/∂P, P=-∂H/∂Q with {Mathematical expression} the corresponding flow preserves the (suitably interpreted) "petit ensemble"e-HdQdP; and for L↓∞, Q settles down to the stationary diffusion with infinitesimal operator 1/2 ∂2/∂Q2+m(Q)∂/∂Q, m being the logarithmic derivative of the ground state of -d2/dQ2{norm of matrix}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 {Mathematical expression} and making L↓∞ with fixed D=N/L. Now, for F(Q)/Q2→∞, 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 Döblin'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.

Original languageEnglish (US)
Pages (from-to)615-630
Number of pages16
JournalCommunications In Mathematical Physics
Issue number3
StatePublished - Mar 1994

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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