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Register a new userSlow dynamics and subdiffusion in a non-Hamiltonian system with long-range forces
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Inspired by one--dimensional light--particle systems, the dynamics of a non-Hamiltonian system with long--range forces is investigated. While the molecular dynamics does not reach an equilibrium state, it may be approximated in the thermodynamic limit by a Vlasov equation that does possess stable stationary solutions. This implies that on a macroscopic scale, the molecular dynamics evolves on a slow timescale that diverges with the system size. At the single-particle level, the evolution is driven by incoherent interaction between the particles, which may be effectively modeled by a noise, leading to a Brownian-like dynamics of the momentum. Because this self-generated diffusion process depends on the particle distribution, the associated Fokker-Planck equation is nonlinear, and a subdiffusive behavior of the momentum fluctuation emerges, in agreement with numerics.
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Temperature inversion due to velocity filtration, a mechanism originally proposed to explain the heating of the solar corona, is demonstrated to occur also in a simple paradigmatic model with long-range interactions, the Hamiltonian mean-field model. Using molecular dynamics simulations, we show that when the system settles into an inhomogeneous quasi-stationary state in which the velocity distribution has suprathermal tails, the temperature and density profiles are anticorrelated: denser parts of the system are colder than dilute ones. We argue that this may be a generic property of long-range interacting systems.
We consider a velocity field with linear viscous interactions defined on a one dimensional lattice. Brownian baths with different parameters can be coupled to the boundary sites and to the bulk sites, determining different kinds of non-equilibrium steady states or free-cooling dynamics. Analytical results for spatial and temporal correlations are provided by analytical diagonalisation of the systems equations in the infinite size limit. We demonstrate that spatial correlations are scale-free and time-scales become exceedingly long when the system is driven only at the boundaries. On the contrary, in the case a bath is coupled to the bulk sites too, an exponential correlation decay is found with a finite characteristic length. This is also true in the free cooling regime, but in this case the correlation length grows diffusively in time. We discuss the crucial role of boundary driving for long-range correlations and slow time-scales, proposing an analogy between this simplified dynamical model and dense vibro-fluidized granular materials. Several generalizations and connections with the statistical physics of active matter are also suggested.
We have derived a fractional Fokker-Planck equation for subdiffusion in a general space-and- time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.
We present an effective evolution equation for a coarse-grained distribution function of a long-range-interacting system preserving the symplectic structure of the non-collisional Boltzmann, or Vlasov, equation. We first derive a general form of such an equation based on symmetry considerations only. Then, we explicitly derive the equation for one-dimensional systems, finding that it has the form predicted on general grounds. Finally, we use such an equation to predict the dependence of the damping times on the coarse-graining scale and numerically check it for some one-dimensional models, including the Hamiltonian Mean Field (HMF) model, a scalar field with quartic interaction, a 1-d self-gravitating system, and the Self-Gravitating Ring (SGR).
We consider the Ising model on a small-world network, where the long-range interaction strength $J_2$ is in general different from the local interaction strength $J_1$, and examine its relaxation behaviors as well as phase transitions. As $J_2/J_1$ is raised from zero, the critical temperature also increases, manifesting contributions of long-range interactions to ordering. However, it becomes saturated eventually at large values of $J_2/J_1$ and the system is found to display very slow relaxation, revealing that ordering dynamics is inhibited rather than facilitated by strong long-range interactions. To circumvent this problem, we propose a modified updating algorithm in Monte Carlo simulations, assisting the system to reach equilibrium quickly.
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