论文标题
一类无限多个主方程生成的半群的光谱定理
A spectral theorem for the semigroup generated by a class of infinitely many master equations
论文作者
论文摘要
在本文中，我们研究了在统计力学中平衡方法分析时产生的无限方程系统的无限发电机的光谱特性。因此，所研究的系统由无限的许多一阶微分方程组成，这些方程的时间是易于描述具有离散点频谱的差异操作员之间的跳跃的概率演变。特征态之间的过渡速率以满足所谓的详细平衡条件的方式选择，因此，对于大量的初始条件，给定系统具有一个全局解决方案，该解决方案将指数迅速收敛于Gibbs类型的时间独立概率。在调查中的问题的一个特定特征和挑战是，在自然功能空间中，初始值问题良好的自然功能空间中，无穷小的发电机被实现为非正常和非耗散的紧凑型操作员，因此，与有限的频谱相比，该光谱没有在零特征值周围表现出光谱差距。
In this article we investigate the spectral properties of the infinitesimal generator of an infinite system of master equations arising in the analysis of the approach to equilibrium in statistical mechanics. The system under investigation thus consists of infinitely many first-order differential equations governing the time evolution of probabilities susceptible of describing jumps between the eigenstates of a differential operator with a discrete point spectrum. The transition rates between eigenstates are chosen in such a way that the so-called detailed balance conditions are satisfied, so that for a large class of initial conditions the given system possesses a global solution which converges exponentially rapidly toward a time independent probability of Gibbs type. A particular feature and a challenge of the problem under investigation is that in the natural functional space where the initial-value problem is well-posed, the infinitesimal generator is realized as a non normal and non dissipative compact operator, whose spectrum therefore does not exhibit a spectral gap around the zero eigenvalue in contrast to the finite-dimensional case.
