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

H. P. McKean, K. L. Vaninsky

Research output: Contribution to journalArticle

Abstract

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
Volume160
Issue number3
DOIs
StatePublished - Mar 1994

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Brownian Motion with Drift
Microcanonical Ensemble
Logarithmic Derivative
Steepest Descent
Nonlinear Wave Equation
descent
norms
wave equations
Ground State
Circle
Ensemble
Odd
Norm
operators
Unknown
ground state
matrices
Operator

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

Brownian motion with restoring drift : The petit and micro-canonical ensembles. / McKean, H. P.; Vaninsky, K. L.

In: Communications in Mathematical Physics, Vol. 160, No. 3, 03.1994, p. 615-630.

Research output: Contribution to journalArticle

McKean, H. P. ; Vaninsky, K. L. / Brownian motion with restoring drift : The petit and micro-canonical ensembles. In: Communications in Mathematical Physics. 1994 ; Vol. 160, No. 3. pp. 615-630.
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abstract = "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-Hd∞Qd∞P; 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 D{\"o}blin'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.",
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