Thermal Quantum Fields with Spatially Cut-off Interactions in 1+1 Space-time Dimensions

空间截断 $\phi^4_2$ 中的序列指数-动机周期

Sequential exponential-motivic periods in spatially cut-off $ϕ^4_2$

本文研究空间截断 $\phi^4_2$ 的构造性积分，用指数-动机周期序列表示，分析伽罗瓦变换对配分周期发散和固定可观测量极限的影响，并给出无有界实现时极限不成比例的反例。

Tunneling for spatially cut-off $P(ϕ)_2$-Hamiltonians

We study the asymptotic behavior of low-lying eigenvalues of spatially cut-off $P(ϕ)_2$-Hamiltonian under semi-classical limit. The corresponding classical equation of the $P(ϕ)_2$-field is a nonlinear Klein-Gordon equation which is an infinite dimensional Newton's equation. We determine the semi-classical limit of the lowest eigenvalue of the spatially cut-off $P(ϕ)_2$-Hamiltonian in terms of the Hessian of the potential function of the Klein-Gordon equation. Moreover, we prove that the gap of the lowest two eigenvalues goes to 0 exponentially fast under semi-classical limit when the potentia…

Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering every physical collision kernels). These estimates are conditioned to some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to the one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non cutoff and non mollified) hard potentials and moderately soft potentials. In particular, we obtain t…

Designing the sound of a cut-off drum

The spectral action in noncommutative geometry naturally implements an ultraviolet cut-off, by counting the eigenvalues of a (generalized) Dirac operator lower than an energy of unification. Inverting the well known question "how to hear the shape of a drum ?", we ask what drum can be designed by hearing the truncated music of the spectral action ? This makes sense because the same Dirac operator also determines the metric, via Connes distance. The latter thus offers an original way to implement the high-momentum cut-off of the spectral action as a short distance cut-off on space. This is a no…

Singularity of macroscopic variables near boundary for gases with cut-off inverse-power potential

In this article, the boundary singularity for stationary solutions of the linearized Boltzmann equation with cut-off inverse power potential is analyzed. In particular, for cut-off hard-potential cases, we establish the asymptotic approximation for the gradient of the moments. Our analysis indicates the logarithmic singularity of the gradient of the moments.

Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials

We consider the spatially homogeneous Boltzmann equation for regularized soft potentials and Grad's angular cutoff. We prove that uniform (in time) bounds in $L^1 ((1 + |v|^s)dv)$ and $H^k$ norms, $s, k \ge 0$ hold for its solution. The proof is based on the mixture of estimates of polynomial growth in time of those norms together with the quantitative results of relaxation to equilibrium in $L^1$ obtained by the so-called "entropy-entropy production" method in the context of dissipative systems with slowly growing a priori bounds (see reference [14]).