Slow dynamics and subdiffusion in a non-Hamiltonian system with long-range forces

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 fluctuations emerges, in agreement with numerics.

Figure 1

The single-particle $p$ distribution as a function of rescaled time $t/N$ for the dynamics (3) for three different systems size $N$ and for an initial state that is WB with $\mathrm{\Delta }p=0.5$. The data are obtained from numerical integration of the dynamics (3) for different system sizes.

Figure 2

Autocorrelation in time of the magnetization $M\left(t\right)$, computed after a time $100N$, and over a time window $\mathrm{\Delta }t=100$, starting from a WB initial state with $\mathrm{\Delta }p=0.5$. The theory curve has been drawn using the measured width of the momentum distribution $\sigma \approx 1.8$.

Figure 3

Evolution of $\langle {(p-\overline{p})}^2\rangle$ for different system sizes and the subdiffusive behavior prediction, Eq. (13). The system is initially in a WB state with $\mathrm{\Delta }p=0.5$.

Figure 4

Evolution of magnetization for a configuration where the Hamiltonian coupling is stronger than the nonsymmetric one $(\mathrm{\Delta }=-2\kappa$ and $-2g^2\kappa /({\kappa }^2+{\mathrm{\Delta }}^2)=1$, labeled “$\mathrm{H}>\mathrm{nH}$”), and for the opposite configuration $(\mathrm{\Delta }=-\kappa /2$ and $-2g^2\kappa /({\kappa }^2+{\mathrm{\Delta }}^2)=1$, labeled “$\mathrm{nH}>\mathrm{H}$”), for different system sizes. The system is initially in a WB state with $\mathrm{\Delta }p=0.5$.