均匀驱动的复杂颗粒气体系统的动力学特征研究
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摘要
本文研究了均匀驱动的一维和二维复杂颗粒气体系统的动力学特性,主要讨论了系统从初态到稳态的演变过程中平均能量随时间的演化以及稳态的动力学特征(包括系统的整体颗粒温度、整体压力、非高斯的速度分布、空间密度分布、空间密度和速度的相关性、碰撞之间颗粒自由程和时间的分布、碰撞频率及其演化)。旨在扩大和深化对复杂颗粒系统的认识,使得对颗粒系统的理论研究不仅仅局限于单一颗粒系统和少组分的混合颗粒系统。
首先,本文只研究了两种复杂颗粒系统的粒径分布特征。其一是粒径为分形分布的颗粒系统,用颗粒的粒径分形维数D来描述系统中颗粒粒径分布的不均匀程度。分形维数D越大,粒径分布越不均匀。其二是粒径为准高斯分布的颗粒系统,当颗粒的平均粒径(或均值)μ不变时,用标准偏差σ来描述系统中颗粒粒径分布的不均匀程度。σ越大,粒径越不均匀;反之,亦然。本文研究的重点在于建立均匀驱动的复杂颗粒系统的动力学模型,研究颗粒粒径分布的不均匀性对系统的动力学特征的影响。
随后,建立了粒径为分形分布的准一维颗粒气体的动力学模型,其中每个颗粒受高斯白噪声驱动,研究了系统稳态的动力学特性。首先,定义了多组分的复杂混合颗粒系统的组分颗粒温度、整体颗粒温度和整体压力,并给出了明确的数学表达式。然后,通过Monte Carlo模拟发现分别随着颗粒碰撞的弹性恢复系数e的减小或颗粒粒径分布的分形维数D的增大,系统的整体颗粒温度和整体压力不断减小;颗粒的速度分布更加偏离高斯分布,例如更高的峰态、更大的第四累积量a_2、和更肥胖的尾部;相邻颗粒的间距分布更加偏离弹性情况的理论预测,表现为较小间距和较大空隙的更多的分布,即颗粒空间密度更加成团化。
接着,建立了粒径为准高斯分布的一维颗粒气体的动力学模型,其中每个颗粒受高斯白噪声驱动。在相同的非弹性条件下,运用Monte Carlo模拟,首次研究了颗粒粒径分布的标准偏差σ对系统动力学特征的影响。当驱使颗粒布朗运动的驰豫时间τ远大于颗粒碰撞的平均间隔时间cτ时,系统的平均能量随时间以指数的形式衰减并趋向稳定的渐进值,系统最终到达稳态;并且,标准偏差σ越大,趋向稳态的能量驰豫时间τ_B越短。当系统处于稳态时,随着标准偏差σ的增大,系统的速度分布更加偏离高斯分布,例如更高的峰态和更肥胖的高速尾部;空间密度更加成团,系统的有效熵H_M/H_M~+减小;空间密度和速度的相关性更强,如空间密度相关函数在原点附近显示出更高的峰值、速度的空间相关函数在颗粒分隔间距小时更小。
最后,本文将一维复杂颗粒气体的研究扩展到二维系统。建立了粒径为分形分布的二维颗粒气体的动力学模型,每个颗粒受高斯白噪声的驱动,并限制在一个二维周期性边界的水平方格子中运动。在相同的非弹性条件下,运用Monte Carlo模拟方法研究了颗粒粒径分布的分形维数D对系统稳态动力学特征的影响。随着分形维数D的增大,碰撞之间颗粒的自由程和时间的分布显著地偏离弹性情况的理论预测,具有短的自由程和时间分布的过密集的峰;碰撞频率增大,但不依赖于时间;速度分布显著地偏离高斯分布,例如更高的峰态、更加翘起的高速尾部,但是非高斯的速度分布一般不具有普适性的解析表达式;速度的空间相关性明显增强,在颗粒分隔小间距时垂直的速度相关性大约只有平行的速度相关性的一半,两者都是颗粒间距r L的幂律衰减函数、并且都具有长程性;颗粒碰撞接触时,碰撞后的平行的速度相关性是碰撞前的平行的速度相关性的两倍多,两者近似为分形维数D的线性函数。
本文的研究表明,无论是粒径为分形分布还是粒径为准高斯分布的系统,在非弹性碰撞过程中由于颗粒粒径分布越不均匀或弹性恢复系数e的减小导致系统的能量耗散越多,是产生上述奇特现象的原因。
In this paper, we studied dynamic properties of polydisperse granular gases driven by Gaussian white noise in one and two dimensions, focuing on the evolution of average energy and the steady-state dynamic behaviors (include the global granular temperature, global pressure, non-Gaussian velocity distribution, spatial density distribution, spatial correlations of density and velocities, distributions of path lengths and free times between collisions, collision rate). The particles employed in our study are assigned granularity with fractal size distribution and quasi Gaussian size distribution. For the polydisperse granular systen with fractal size distribution, the inhomogeneity of the particle size distribution is characterized by a fractal dimension D . The higher D implies more inhomogeneity in the particle size distribution. However, for the polydisperse granular systen with quasi Gaussian size distribution, the inhomogeneity of the particle size distribution can be measured by the standard deviationσat the same mean valueμ. The larger value ofσindicates greater inhomogeneity in the particle size distribution.
First, we presented a dynamic model of a quasi one-dimensional polydisperse granular mixture with fractal size distribution, in which the particles are driven by Gaussian white noise, and studied the steady-state dynamic properties of the sysytem. Firstly, we define the partial and global granular temperature and global pressure of the mixture. By Monte Carlo simulations, we found that, with the increase of D , the global granular temperature and the kinetic pressure decrease, the velocity distribution deviates more obviously from the Gaussian one ( such as the higher kurtosis, the larger fourth cumulant a2 and the fatter tails), and distribution of interparticle spacing deviates more obviously from the elastic form, i.e., the particles cluster more pronouncedly at the same value of the restitution coefficient e (0 < e< 1). On the other hand, as the restitution coefficiente decreases, the dynamic behavior has the similar evolution as above at the fixed D .
Second, we presented a dynamic model of a one-dimensional granular gas with quasi Gaussian size distribution, in which the rods are thermalized by a viscosity heat bath. By Monte Carlo simulations, the effect of the dispersion of the quasi Gaussian size distribution on dynamic behavior of the system is investigated in the same inelasticity case. When the typical relaxation timeτof the driving Brownian process is longer than the mean collision timeτ_c, the average energy of the system decays exponentially with time towards a stable asymptotic value, and the energy relaxation timeτBto a nonequilibrium steady state becomes shorter with increasing values ofσ. In the steady state, asσincreases, the velocity distribution deviates more obviously from the Gaussian one, such as the higher kurtosis and the fatter tails, the spatial density distribution becomes more clusterized, the statistical entropy H_M/H_M~+ of the system decreases, the spatial correlations of density and velocities become stronger (such as the two-particle correlation function C ( x )shows higher peak and is a power-law form decay near the origin, the average velocity Cv ( x )is smaller and a power-law form increase for small x ).
Finally, we present a dynamical model of two-dimensional polydisperse granular gases with fractal size distribution, in which the smooth hard disks are engaged in a two-dimensional horizontal rectangular box and driven by standard white noise. By Monte Carlo simulations, we find the inhomogeneity of the disk size distribution has great influence on the steady-state dynamic properties. With the increase of the fractal dimension D , the distributions of path lengths and free times between collisions deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins. The collision rate increases with D , but it is independent of time. Meanwhile, the tails of the velocity distribution functions rise more significantly above a Gaussian as D increases, but the non-Gaussian velocity distribution functions do not demonstrate any apparent universal form for any value of D. The spatial velocity correlations are apparently stronger with the increase of D. The perpendicular correlations are about one-half of the parallel correlations, and the two correlations are a power-law decay function of dimensionless distance and are long range. Moreover, the parallel velocity correlations of postcollisional state at contact are more than twice as large as the precollisional correlations, and both of them show almost linear behavior of the fractal dimension D.
In this paper, the study indicates that the energy dissipation due to the more inhomogeneity of the particle size distribution or the smaller restitution coefficient e in the inelastic collisions causes a variety of very peculiar phenomena as above.
首先,本文只研究了两种复杂颗粒系统的粒径分布特征。其一是粒径为分形分布的颗粒系统,用颗粒的粒径分形维数D来描述系统中颗粒粒径分布的不均匀程度。分形维数D越大,粒径分布越不均匀。其二是粒径为准高斯分布的颗粒系统,当颗粒的平均粒径(或均值)μ不变时,用标准偏差σ来描述系统中颗粒粒径分布的不均匀程度。σ越大,粒径越不均匀;反之,亦然。本文研究的重点在于建立均匀驱动的复杂颗粒系统的动力学模型,研究颗粒粒径分布的不均匀性对系统的动力学特征的影响。
随后,建立了粒径为分形分布的准一维颗粒气体的动力学模型,其中每个颗粒受高斯白噪声驱动,研究了系统稳态的动力学特性。首先,定义了多组分的复杂混合颗粒系统的组分颗粒温度、整体颗粒温度和整体压力,并给出了明确的数学表达式。然后,通过Monte Carlo模拟发现分别随着颗粒碰撞的弹性恢复系数e的减小或颗粒粒径分布的分形维数D的增大,系统的整体颗粒温度和整体压力不断减小;颗粒的速度分布更加偏离高斯分布,例如更高的峰态、更大的第四累积量a_2、和更肥胖的尾部;相邻颗粒的间距分布更加偏离弹性情况的理论预测,表现为较小间距和较大空隙的更多的分布,即颗粒空间密度更加成团化。
接着,建立了粒径为准高斯分布的一维颗粒气体的动力学模型,其中每个颗粒受高斯白噪声驱动。在相同的非弹性条件下,运用Monte Carlo模拟,首次研究了颗粒粒径分布的标准偏差σ对系统动力学特征的影响。当驱使颗粒布朗运动的驰豫时间τ远大于颗粒碰撞的平均间隔时间cτ时,系统的平均能量随时间以指数的形式衰减并趋向稳定的渐进值,系统最终到达稳态;并且,标准偏差σ越大,趋向稳态的能量驰豫时间τ_B越短。当系统处于稳态时,随着标准偏差σ的增大,系统的速度分布更加偏离高斯分布,例如更高的峰态和更肥胖的高速尾部;空间密度更加成团,系统的有效熵H_M/H_M~+减小;空间密度和速度的相关性更强,如空间密度相关函数在原点附近显示出更高的峰值、速度的空间相关函数在颗粒分隔间距小时更小。
最后,本文将一维复杂颗粒气体的研究扩展到二维系统。建立了粒径为分形分布的二维颗粒气体的动力学模型,每个颗粒受高斯白噪声的驱动,并限制在一个二维周期性边界的水平方格子中运动。在相同的非弹性条件下,运用Monte Carlo模拟方法研究了颗粒粒径分布的分形维数D对系统稳态动力学特征的影响。随着分形维数D的增大,碰撞之间颗粒的自由程和时间的分布显著地偏离弹性情况的理论预测,具有短的自由程和时间分布的过密集的峰;碰撞频率增大,但不依赖于时间;速度分布显著地偏离高斯分布,例如更高的峰态、更加翘起的高速尾部,但是非高斯的速度分布一般不具有普适性的解析表达式;速度的空间相关性明显增强,在颗粒分隔小间距时垂直的速度相关性大约只有平行的速度相关性的一半,两者都是颗粒间距r L的幂律衰减函数、并且都具有长程性;颗粒碰撞接触时,碰撞后的平行的速度相关性是碰撞前的平行的速度相关性的两倍多,两者近似为分形维数D的线性函数。
本文的研究表明,无论是粒径为分形分布还是粒径为准高斯分布的系统,在非弹性碰撞过程中由于颗粒粒径分布越不均匀或弹性恢复系数e的减小导致系统的能量耗散越多,是产生上述奇特现象的原因。
In this paper, we studied dynamic properties of polydisperse granular gases driven by Gaussian white noise in one and two dimensions, focuing on the evolution of average energy and the steady-state dynamic behaviors (include the global granular temperature, global pressure, non-Gaussian velocity distribution, spatial density distribution, spatial correlations of density and velocities, distributions of path lengths and free times between collisions, collision rate). The particles employed in our study are assigned granularity with fractal size distribution and quasi Gaussian size distribution. For the polydisperse granular systen with fractal size distribution, the inhomogeneity of the particle size distribution is characterized by a fractal dimension D . The higher D implies more inhomogeneity in the particle size distribution. However, for the polydisperse granular systen with quasi Gaussian size distribution, the inhomogeneity of the particle size distribution can be measured by the standard deviationσat the same mean valueμ. The larger value ofσindicates greater inhomogeneity in the particle size distribution.
First, we presented a dynamic model of a quasi one-dimensional polydisperse granular mixture with fractal size distribution, in which the particles are driven by Gaussian white noise, and studied the steady-state dynamic properties of the sysytem. Firstly, we define the partial and global granular temperature and global pressure of the mixture. By Monte Carlo simulations, we found that, with the increase of D , the global granular temperature and the kinetic pressure decrease, the velocity distribution deviates more obviously from the Gaussian one ( such as the higher kurtosis, the larger fourth cumulant a2 and the fatter tails), and distribution of interparticle spacing deviates more obviously from the elastic form, i.e., the particles cluster more pronouncedly at the same value of the restitution coefficient e (0 < e< 1). On the other hand, as the restitution coefficiente decreases, the dynamic behavior has the similar evolution as above at the fixed D .
Second, we presented a dynamic model of a one-dimensional granular gas with quasi Gaussian size distribution, in which the rods are thermalized by a viscosity heat bath. By Monte Carlo simulations, the effect of the dispersion of the quasi Gaussian size distribution on dynamic behavior of the system is investigated in the same inelasticity case. When the typical relaxation timeτof the driving Brownian process is longer than the mean collision timeτ_c, the average energy of the system decays exponentially with time towards a stable asymptotic value, and the energy relaxation timeτBto a nonequilibrium steady state becomes shorter with increasing values ofσ. In the steady state, asσincreases, the velocity distribution deviates more obviously from the Gaussian one, such as the higher kurtosis and the fatter tails, the spatial density distribution becomes more clusterized, the statistical entropy H_M/H_M~+ of the system decreases, the spatial correlations of density and velocities become stronger (such as the two-particle correlation function C ( x )shows higher peak and is a power-law form decay near the origin, the average velocity Cv ( x )is smaller and a power-law form increase for small x ).
Finally, we present a dynamical model of two-dimensional polydisperse granular gases with fractal size distribution, in which the smooth hard disks are engaged in a two-dimensional horizontal rectangular box and driven by standard white noise. By Monte Carlo simulations, we find the inhomogeneity of the disk size distribution has great influence on the steady-state dynamic properties. With the increase of the fractal dimension D , the distributions of path lengths and free times between collisions deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins. The collision rate increases with D , but it is independent of time. Meanwhile, the tails of the velocity distribution functions rise more significantly above a Gaussian as D increases, but the non-Gaussian velocity distribution functions do not demonstrate any apparent universal form for any value of D. The spatial velocity correlations are apparently stronger with the increase of D. The perpendicular correlations are about one-half of the parallel correlations, and the two correlations are a power-law decay function of dimensionless distance and are long range. Moreover, the parallel velocity correlations of postcollisional state at contact are more than twice as large as the precollisional correlations, and both of them show almost linear behavior of the fractal dimension D.
In this paper, the study indicates that the energy dissipation due to the more inhomogeneity of the particle size distribution or the smaller restitution coefficient e in the inelastic collisions causes a variety of very peculiar phenomena as above.
引文
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