具有退化率粒子系统的流体动力学极限
Hydrodynamic Limit for a Particle System with degenerate rates
摘要：
研究了具有退化率的保守粒子系统的流体动力学极限，即最近邻交换率对于某些构型为零的保守粒子系统。这些模型属于{ \sl动力学约束晶格气体}（KCLG）类，其在物理文献中作为液体/玻璃化转变的简单模型被引入并被深入研究。由于KCLG速率的简并性，存在{ \sl阻塞组态}，这些组态在动力学下不演化，并且具有固定粒子数的组态的超平面一般可以分解为不同的不可约集.因此，熵和相对熵方法都不能直接应用于证明水动力极限。特别是，在证明保证局部收敛到平衡点的一个和两个块引理时，应特别注意。我们表明，初始配置文件足够光滑，有界远离零和一，宏观密度分布为我们的KCLG的扩散时间尺度下的多孔介质方程的演变。然后，我们证明了相同的结果更一般的配置文件的轻微扰动动力学通过添加跳跃的对称简单排除。后者的作用是消除速率的简并性，同时它们被适当地减慢，以便不改变宏观行为。得到了该扰动模型的平衡涨落和谱隙的大小。
英文摘要：
We study the hydrodynamic limit for some conservative particle systems with degenerate rates, namely with nearest neighbor exchange rates which vanish for certain configurations. These models belong to the class of {\sl kinetically constrained lattice gases} (KCLG) which have been introduced and intensively studied in physics literature as simple models for the liquid/glass transition. Due to the degeneracy of rates for KCLG there exists {\sl blocked configurations} which do not evolve under the dynamics and in general the hyperplanes of configurations with a fixed number of particles can be decomposed into different irreducible sets. As a consequence, both the Entropy and Relative Entropy method cannot be straightforwardly applied to prove the hydrodynamic limit. In particular, some care should be put when proving the One and Two block Lemmas which guarantee local convergence to equilibrium. We show that, for initial profiles smooth enough and bounded away from zero and one, the macroscopic density profile for our KCLG evolves under the diffusive time scaling according to the porous medium equation. Then we prove the same result for more general profiles for a slightly perturbed dynamics obtained by adding jumps of the Symmetric Simple Exclusion. The role of the latter is to remove the degeneracy of rates and at the same time they are properly slowed down in order not to change the macroscopic behavior. The equilibrium fluctuations and the magnitude of the spectral gap for this perturbed model are also obtained.
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