Research in Statistical Physics

建议在非平衡统计力学与动力系统理论之间的联系方面继续进行首席研究员和合作者的工作。这项工作的中心思想是将流体系统的输运和其他非平衡性质，如熵的产生，与流体组成粒子的潜在混沌动力学联系起来。PI和合作者在过去几年的工作表明，玻尔兹曼方程技术和非平衡统计力学的相关方法不仅可以有效地应用于计算流体的输运性质，而且可以计算它们的混沌性质。这种统一的方法使人们能够使输运和混沌性质之间的联系变得特别清楚。对模型系统的研究导致了先进方法的发展，这些方法允许对所有粒子都能运动的更现实系统的混沌特性进行详细的分析计算。例如，计算硬盘或硬球体的稀气体的Kolmogorov-Sinai熵是可能的。这些系统在数学上仍然相当复杂，其混沌行为的许多有趣特征与粒子碰撞的半色散性质有关。本文建议：(a)继续研究计算密集洛伦兹气体系统混沌性质的解析方法；(b)对更实际的系统进行类似的计算，即对与短程力相互作用的稀和中等密度的粒子气体进行计算；(c)继续PI和合作者的工作，研究不可逆热力学的不可逆熵产生与在动力系统理论方法中自然出现的看似密切相关的量之间的联系；(d)将这些方法和结果推广到量子情况。后一点非常重要，因为：(a)自然界基本上是量子力学的，而不是经典的；(b)在人们对量子系统如何产生经典混沌有任何清晰的理解之前，需要回答许多深刻而基本的问题；(c)量子系统，如量子点和量子反点，具有很大的物理和实际意义。

It is proposed to continue the work of the Principal Investigator and collaborators on the connections between non-equilibrium statistical mechanics and dynamical systems theory. The central idea of this work is to relate transport and other nonequilibrium properties, such as entropy production, of fluid systems to the underlying chaotic dynamics of the constituent particles of the fluid. Work by the PI and collaborators over the past few years has shown that Boltzmann equation techniques and related methods of nonequilibrium statistical mechanics can be fruitfully applied to calculate not only transport properties of fluids but their chaotic properties as well. This unified approach allows one to make the connections between transport and chaotic properties especially clear. Work on model systems has led to the development of advanced methods which allow detailed analytic calculations of the chaotic properties of more realistic systems where all of the particles can move. It has been possible, for example, to compute the Kolmogorov-Sinai entropy of a dilute gas of hard disks or of hard spheres. These systems are still quite complicated mathematically, and many interesting features of their chaotic behavior are related to the semi-dispersive nature of the particle collisions. It is proposed here: (a) to continue the work on analytic methods to compute the chaotic properties, of dense Lorentz gas systems; (b) to carry out similar calculations for more realistic systems, namely, for dilute and moderately dense gases of particles interacting with short ranged forces; (c) to continue the work of the PI and collaborators on the connection between the irreversible entropy production of irreversible thermodynamics and seemingly closely related quantities that appear naturally in the dynamical systems theory approach to transport; and (d) to extend these methods and results to the quantum case. The latter point is of major importance since: (a) nature is fundamentally quantum mechanical rather than classical, (b) many deep and basic questions need to be answered before one has any clear understanding of how classical chaos emerges from quantum systems, and (c) quantum systems such as quantum dots and quantum anti-dots are of great physical and practical interest.