Complex systems consisting of N agents can be investigated from the aspect of principal fluctuation modes of agents. From the correlations between agents, an N × N correlation matrix C can be obtained. The princi...Complex systems consisting of N agents can be investigated from the aspect of principal fluctuation modes of agents. From the correlations between agents, an N × N correlation matrix C can be obtained. The principal fluctuation modes are defined by the eigenvectors of C. Near the critical point of a complex system, we anticipate that the principal fluctuation modes have the critical behaviors similar to that of the susceptibity. With the Ising model on a two-dimensional square lattice as an example, the critical behaviors of principal fluctuation modes have been studied. The eigenvalues of the first 9 principal fluctuation modes have been invesitigated. Our Monte Carlo data demonstrate that these eigenvalues of the system with size L and the reduced temperature t follow a finite-size scaling form λn(L, t) = Lγ/νf n(t L^(1/ν)), where γ is critical exponent of susceptibility and ν is the critical exponent of the correlation length. Using eigenvalues λ1, λ2 and λ6, we get the finite-size scaling form of the second moment correlation length ξ(L, t) = Lξ(tL^(1/ν)).It is shown that the second moment correlation length in the two-dimensional square lattice is anisotropic.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Supported by the National Natural Science Foundation of China under Grant Nos.11121403 and 11504384
文摘Complex systems consisting of N agents can be investigated from the aspect of principal fluctuation modes of agents. From the correlations between agents, an N × N correlation matrix C can be obtained. The principal fluctuation modes are defined by the eigenvectors of C. Near the critical point of a complex system, we anticipate that the principal fluctuation modes have the critical behaviors similar to that of the susceptibity. With the Ising model on a two-dimensional square lattice as an example, the critical behaviors of principal fluctuation modes have been studied. The eigenvalues of the first 9 principal fluctuation modes have been invesitigated. Our Monte Carlo data demonstrate that these eigenvalues of the system with size L and the reduced temperature t follow a finite-size scaling form λn(L, t) = Lγ/νf n(t L^(1/ν)), where γ is critical exponent of susceptibility and ν is the critical exponent of the correlation length. Using eigenvalues λ1, λ2 and λ6, we get the finite-size scaling form of the second moment correlation length ξ(L, t) = Lξ(tL^(1/ν)).It is shown that the second moment correlation length in the two-dimensional square lattice is anisotropic.
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.
