Determination of the spectral gap in the Kac model for physical momentum and energy conserving collisions
| dc.creator | Carlen, Eric A. | |
| dc.creator | Geronimo, Jeffry S. | |
| dc.creator | Loss, Michael | |
| dc.date | 2007-05-25 | |
| dc.date | 2007-06-23 | |
| dc.date.accessioned | 2026-07-07T08:11:44Z | |
| dc.date.available | 2026-07-07T08:11:44Z | |
| dc.description | The Kac model describes the local evolution of a gas of $N$ particles with three dimensional velocities by a random walk in which the steps correspond to binary collisions that conserve momentum as well as energy. The state space of this walk is a sphere of dimension $3N - 4$. The Kac conjecture concerns the spectral gap in the one step transition operator $Q$ for this walk. In this paper, we compute the exact spectral gap. As in previous work by Carlen, Carvalho and Loss where a lower bound on the spectral gap was proved, we use a method that relates the spectral properties of $Q$ to the spectral properties of a simpler operator $P$, which is simply an average of certain non commuting projections. The new feature is that we show how to use a knowledge of certain eigenfunctions and eigenvalues of $P$ to determine spectral properties of $Q$, instead of simply using the spectral gap for $P$ to bound the spectral gap for $Q$, inductively in $N$, as in previous work. The methods developed here can be applied to many other high--dimensional stochastic process, as we shall explain. We also use some deep results on Jacobi polynomials to obtain the required spectral information on $P$, and we show how the identity through which Jacobi polynomials enter our problem may be used to obtain new bounds on Jacobi polynomials. | |
| dc.description | This June 21 version contains a new theorem significantly extending the scope of the results, and more eplanation of how the method works | |
| dc.identifier | https://arxiv.org/abs/0705.3729 | |
| dc.identifier | http://arxiv.org/abs/0705.3729 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132213 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Functional Analysis | |
| dc.subject | 39B42; 82C40; 14A42 | |
| dc.title | Determination of the spectral gap in the Kac model for physical momentum and energy conserving collisions | |
| dc.type | text |