PROBLEMS ON THE HYDRODYNAMIC LIMIT AND RELATED TOPICS

2003年，我们引入了具有两个守恒量的格子气体模型，称为零程排斥模型，得到了该模型的谱隙估计[cf。零距离排斥动力学的谱间隙估计，大阪数学。141(第4号)(2004年)]。该模型是非梯度型的，其自旋值是无界的，因此在研究该模型的流体动力学极限时出现了许多有趣的问题。所得到的估计虽然在气体密度上不像许多已知模型那样是均匀的，但对于处理有关其流体动力学极限的问题似乎足够准确。事实上，根据我们的估计，我们在2004年证明了围绕平衡中流体动力标度的波动收敛于一个被描述为无限维Ornstein-Uhlenbeck过程的过程[cf.零程排除过程的平衡涨落，期刊，统计，物理。(2004)]。我们在证明有限大小的过程序列的紧性时遇到了一定的困难，并用时间反转过程设计了一个技巧来解决这个问题。

In 2003 introducing a lattice gas model, called zero-range-exclusion model, which possesses two conservative quantities, we obtained an estimate of the spectral gap for the model [cf. An estimate of the spectral gap for zero-range-exclusion dynamics, Osaka Jour.Math. 141(no.4) (2004)]. This model is of non-gradient type and its spin values are unbounded so that there arise various interesting problems in the investigation of the hydrodynamic limit for the model.The estimate obtained, though not uniform in the density of the gas as those for many known models are, seems sufficiently accurate for dealing with the problems concerning its hydrodynamic limit. In fact based on our estimate we proved in 2004 that the fluctuations around the hydrodynamic scaling in the equilibrium converge to a process that is characterized as an infinite dimensional Ornstein-Uhlenbeck prosess [cf. Equilibrium fluctuations for zero-range-exclusion processes, Jour.Stat.Phys. (2004)]. We encountered a certain difficulty for proving the tightness of a sequence of the processes of finite size and resolved it by devising a trick by using a time reversed process.