### Abstract

We study the spectrum of the operator Lf(Q) = - ∑x∈ℤ^{d} (∂^{2}f/∂q^{2}_{x}) (Q) - β ∑x∈ℤ^{d} (∂H/∂q_{x}) (Q) (∂f/∂q_{x}) (Q), Q = {q_{x}}, generating an infinite-dimensional diffusion process Ξ(t), in space L_{2} (ℝ^{ℤd}, dv(Q)). Here v is a "natural" Ξ(t)-invariant measure on ℝ^{ℤd} which is a Gibbs distribution corresponding to a (formal) Hamiltonian H of an anharmonic crystal, with a value of the inverse temperature β > 0. For β small enough, we establish the existence of an L-invariant subspace H_{1} ⊂ L_{2}(ℝ^{ℤd}, dv(Q)) such that L | H_{1} has a distinctive character related to a "quasi-particle" picture. In particular, L | H_{1} has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point K_{1} > 0 giving the smallest non-zero eigenvalue of a limiting problem associated with β = 0. An immediate corollary of our result is an exponentially fast L_{2}-convergence to equilibrium for the process Ξ(t) for small values of β.

Original language | English (US) |
---|---|

Pages (from-to) | 463-489 |

Number of pages | 27 |

Journal | Communications In Mathematical Physics |

Volume | 206 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1999 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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*Communications In Mathematical Physics*, vol. 206, no. 2, pp. 463-489. https://doi.org/10.1007/s002200050714

**On the spectrum of the generator of an infinite system of interacting diffusions.** / Minlos, R. A.; Soukhov, Iouri M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the spectrum of the generator of an infinite system of interacting diffusions

AU - Minlos, R. A.

AU - Soukhov, Iouri M.

PY - 1999/1/1

Y1 - 1999/1/1

N2 - We study the spectrum of the operator Lf(Q) = - ∑x∈ℤd (∂2f/∂q2x) (Q) - β ∑x∈ℤd (∂H/∂qx) (Q) (∂f/∂qx) (Q), Q = {qx}, generating an infinite-dimensional diffusion process Ξ(t), in space L2 (ℝℤd, dv(Q)). Here v is a "natural" Ξ(t)-invariant measure on ℝℤd which is a Gibbs distribution corresponding to a (formal) Hamiltonian H of an anharmonic crystal, with a value of the inverse temperature β > 0. For β small enough, we establish the existence of an L-invariant subspace H1 ⊂ L2(ℝℤd, dv(Q)) such that L | H1 has a distinctive character related to a "quasi-particle" picture. In particular, L | H1 has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point K1 > 0 giving the smallest non-zero eigenvalue of a limiting problem associated with β = 0. An immediate corollary of our result is an exponentially fast L2-convergence to equilibrium for the process Ξ(t) for small values of β.

AB - We study the spectrum of the operator Lf(Q) = - ∑x∈ℤd (∂2f/∂q2x) (Q) - β ∑x∈ℤd (∂H/∂qx) (Q) (∂f/∂qx) (Q), Q = {qx}, generating an infinite-dimensional diffusion process Ξ(t), in space L2 (ℝℤd, dv(Q)). Here v is a "natural" Ξ(t)-invariant measure on ℝℤd which is a Gibbs distribution corresponding to a (formal) Hamiltonian H of an anharmonic crystal, with a value of the inverse temperature β > 0. For β small enough, we establish the existence of an L-invariant subspace H1 ⊂ L2(ℝℤd, dv(Q)) such that L | H1 has a distinctive character related to a "quasi-particle" picture. In particular, L | H1 has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point K1 > 0 giving the smallest non-zero eigenvalue of a limiting problem associated with β = 0. An immediate corollary of our result is an exponentially fast L2-convergence to equilibrium for the process Ξ(t) for small values of β.

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UR - http://www.scopus.com/inward/citedby.url?scp=0033471711&partnerID=8YFLogxK

U2 - 10.1007/s002200050714

DO - 10.1007/s002200050714

M3 - Article

AN - SCOPUS:0033471711

VL - 206

SP - 463

EP - 489

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -