We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length.The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means o...We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length.The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means of phase portraits,Lyapunov exponents,and Poincarémaps.Based on several periodic orbits with different sizes and shapes,they are encoded systematically with two letters or four letters for two different sets of parameters.The periodic orbits outside the attractor with complex topology are discovered by accident.In addition,the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by different methods respectively.In this process,the rule of orbital period changing with parameters is also investigated.The calculation and classification method of periodic orbits in this study can be widely used in other similar low-dimensional dissipative systems.展开更多
The phonon thermal contribution to the melting temperature of nano-particles is inspected. The discrete summation of phonon states and its corresponding integration form as an approximation for a nano-particle or for ...The phonon thermal contribution to the melting temperature of nano-particles is inspected. The discrete summation of phonon states and its corresponding integration form as an approximation for a nano-particle or for a bulk system have been analyzed. The discrete phonon energy levels of pure size effect and the wave-vector shifts of boundary conditions are investigated in detail. Unlike in macroscopic thermodynamics, the integration volume of zero-mode of phonon for a nano-particle is not zero, and it plays an important role in pure size effect and boundary condition effect. We find that a nano-particle will have a rising melting temperature due to purely finite size effect; a lower melting temperature bound exists for a nano-particle in various environments, and the melting temperature of a nano-particle with free boundary condition reaches this lower bound. We suggest an easy procedure to estimation the melting temperature, in which the zero-mode contribution will be excluded, and only several bulk quantities will be used as input. We would like to emphasize that the quantum effect of discrete energy levels in nano-particles, which is not present in early thermodynamic studies on finite size corrections to melting temperature in small systems, should be included in future researches.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12205257,11647085,and11647086)the Shanxi Province Science Foundation for Youths(Grant No.201901D211252)+1 种基金Fundamental Research Program of Shanxi Province(Grant No.202203021221095)the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi of China(Grant Nos.2019L0505,2019L0554,and 2019L0572)。
文摘We use the variational method to extract the short periodic orbits of the Qi system within a certain topological length.The chaotic dynamical behaviors of the Qi system with five equilibria are analyzed by the means of phase portraits,Lyapunov exponents,and Poincarémaps.Based on several periodic orbits with different sizes and shapes,they are encoded systematically with two letters or four letters for two different sets of parameters.The periodic orbits outside the attractor with complex topology are discovered by accident.In addition,the bifurcations of cycles and the bifurcations of equilibria in the Qi system are explored by different methods respectively.In this process,the rule of orbital period changing with parameters is also investigated.The calculation and classification method of periodic orbits in this study can be widely used in other similar low-dimensional dissipative systems.
基金Supported by National Natural Science Foundation of China under Grant No.1121403
文摘The phonon thermal contribution to the melting temperature of nano-particles is inspected. The discrete summation of phonon states and its corresponding integration form as an approximation for a nano-particle or for a bulk system have been analyzed. The discrete phonon energy levels of pure size effect and the wave-vector shifts of boundary conditions are investigated in detail. Unlike in macroscopic thermodynamics, the integration volume of zero-mode of phonon for a nano-particle is not zero, and it plays an important role in pure size effect and boundary condition effect. We find that a nano-particle will have a rising melting temperature due to purely finite size effect; a lower melting temperature bound exists for a nano-particle in various environments, and the melting temperature of a nano-particle with free boundary condition reaches this lower bound. We suggest an easy procedure to estimation the melting temperature, in which the zero-mode contribution will be excluded, and only several bulk quantities will be used as input. We would like to emphasize that the quantum effect of discrete energy levels in nano-particles, which is not present in early thermodynamic studies on finite size corrections to melting temperature in small systems, should be included in future researches.
