论文标题
具有排除的KAC模型
A Kac Model with Exclusion
论文作者
论文摘要
我们考虑一个具有能源保护和排除规则的一个维度KAC模型：修复许多粒子$ n $，以及一个能量$ e> 0 $。让每个粒子都有一个能量$ x_j \ geq 0 $，带有$ \ sum_ {j = 1}^n x_j = e $。对于某些$ε$，允许的配置$（x_1，\ dots，x_n）$是满足$ | x_i -x_j | \geqε$用于所有$ i \ neq j $。在此过程的每个步骤中，一对$（i，j）$的粒子是随机选择的，然后它们“碰撞”，并且在产生新能量的总能量$ x_i + x_j $之间，它们产生了新的能量$ x^*_ I $ $ x^*_ I $和$ x^*_ j $与$ x^*_ i + x^*_ I + x^*y IS useviention usection noctional cortional cortional clubtion对于新的能量。这个过程与fermions的KAC模型具有一定的相似之处，其中排除代表了Pauli排除原则的效果。但是，这里的“非量化”排除规则仅在间隙上只有一个下限，引入了有趣的新颖特征，并且需要对KAC的混乱进行详细的概念，以得出一个进化方程，以进化为这一过程的重新经验措施的进化，如我们在这里所示。
We consider a one dimension Kac model with conservation of energy and an exclusion rule: Fix a number of particles $n$, and an energy $E>0$. Let each of the particles have an energy $x_j \geq 0$, with $\sum_{j=1}^n x_j = E$. For some $ε$, the allowed configurations $(x_1,\dots,x_n)$ are those that satisfy $|x_i - x_j| \geq ε$ for all $i\neq j$. At each step of the process, a pair $(i,j)$ of particles is selected uniformly at random, and then they "collide", and there is a repartition of their total energy $x_i + x_j$ between them producing new energies $x^*_i$ and $x^*_j$ with $x^*_i + x^*_j = x_i + x_j$, but with the restriction that exclusion rule is still observed for the new pair of energies. This process bears some resemblance to Kac models for Fermions in which the exclusion represents the effects of the Pauli exclusion principle. However, the "non-quantized" exclusion rule here, with only a lower bound on the gaps, introduces interesting novel features, and a detailed notion of Kac's chaos is required to derive an evolution equation for the evolution of rescaled empirical measures for the process, as we show here.
