Grüneisen系数是多体物理中一个重要的无量纲常数，它刻画了系统内部势场的非简谐性，因此在多种物理系统中都有着广泛应用。近年来低温技术的发展使得冷原子以及低温固体物理实验上精确测定Grüneisen系数成为可能，一系列的实验以及理论工作显示，Grüneisen系数在量子相变点处会随着温度趋于零而发散并具有符号改变，从而对寻找并标定材料的量子相变有着重要意义。更重要的是Grüneisen系数是一种多体系统相互作用效应的宏观表征，它与其它可观测热力学量（如与二体关联紧密联系的Tan关系等）之间的联系能够告诉我们量子多体系统相互作用的微观信息，因此亟需我们深入地研究。.申请人计划通过严格解，解析地研究一系列一维冷原子及固体物理系统中的Grüneisen系数，探讨其热力学和量子临界性质，阐明其在量子气体中和Tan关系等热力学量之间的联系，为冷原子及低温固体物理实验研究提供参考和指导。

The Grüneisen parameter, which quantifies the degree of anharmonicity of the potential governing the physical properties of a system, is an important dimensionless constant in many body physics and is widely studied in various fields of physics. The development of techniques in experiments make it possible to precisely measure the Grüneisen parameter in cold and ultra-cold systems. A series of recent studies, both in theories and experiments, indicate that the Grüneisen parameter diverges and has a sign change at the quantum critical points, which will have a large impact on the studies of quantum critical materials. And more importantly, the Grüneisen parameter is the macroscopic representation of the interaction effect of the system, therefore its relationships with other important measurable thermodynamic quantities (the Tan contact which is closely related to the two-body scattering processes in the many-body systems, for example) can tell us more information about important microcosmic properties of the many-body system. Therefore further study is needed to improve our comprehensive understanding of the Grüneisen parameter. ..We intent to study the Grüneisen parameter in various of 1-D integrable models analytically and systematically via the Bethe Ansatz method, and try to find out the universal properties of the Grüneisen parameter in one dimension and establish intrinsic relations to the Tan contact and other important thermodynamic quantities. Our result will provide theoretical inputs into experimental studies of cold atoms and low temperature solid state physics.