Bachelard R, Dauxois T, De Ninno G, Ruffo S, Staniscia F
University of Nova Gorica, School of Applied Sciences, Ajdovcina, Slovenia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Jun;83(6 Pt 1):061132. doi: 10.1103/PhysRevE.83.061132. Epub 2011 Jun 21.
We show that, in the continuum limit, the dynamics of Hamiltonian systems defined on a lattice with long-range couplings is well described by the Vlasov equation. This equation can be linearized around the homogeneous state, and a dispersion relation, which depends explicitly on the Fourier modes of the lattice, can be derived. This allows one to compute the stability thresholds of the homogeneous state, which turns out to depend on the mode number. When this state is unstable, the growth rates are also functions of the mode number. Explicit calculations are performed for the α-Hamiltonian mean field model with 0≤α<1, for which the mean-field mode is always found to dominate the exponential growth. The theoretical predictions are successfully compared with numerical simulations performed on a finite lattice.
我们表明，在连续极限下，定义在具有长程耦合的晶格上的哈密顿系统的动力学可以由弗拉索夫方程很好地描述。该方程可以在均匀态附近线性化，并且可以导出一个明确依赖于晶格傅里叶模式的色散关系。这使得人们能够计算均匀态的稳定性阈值，结果发现该阈值取决于模式数。当该状态不稳定时，增长率也是模式数的函数。对0≤α<1的α - 哈密顿平均场模型进行了显式计算，发现平均场模式总是主导指数增长。理论预测与在有限晶格上进行的数值模拟成功地进行了比较。
