被動粒子在靜止液體中藉由隨機的熱擾動展現出無時空關聯性的布朗運動。藉此背景分子隨機的熱擾動與能量耗散的平衡，使其擴散係數與系統溫度成正比且與此被動粒子的阻力成反比，而此阻力又線性成正比於此被動粒子的尺寸，正如眾所皆知的史托克斯-愛因斯坦關係所描述。
與前述靜止液體相比，自推進桿狀粒子懸浮液是一非平衡系統。其能量由自推進桿狀粒子注入且耗散於高黏滯的背景裡。藉由體積不相容效應與桿狀粒子之非等向體積分佈所造成排列效應及該粒子自推進的特性的交互影響，高濃度自推進桿狀粒子懸浮液中可展現紊流般的行為，並產生廣泛空間尺度的渦流與噴流。於先前文獻中，小尺寸的被動粒子主要作為顯跡物以調查背景流場與其粒子的異常擴散行為。然而，隨著被動粒子尺寸增加，使得紊流中小尺度的貢獻被壓制。這不同尺寸被動圓筏如何於此自推進桿狀粒子紊流中擴散仍然是個有趣且未被探討的議題。
在此論文中，我們藉由二維的數值模擬研究前述未探討的議題。以知增加自推進粒子的體佔比會導致此懸浮液由主動粒子間將近無關聯性的稀釋態轉變至具有多重尺度渦流與噴流的主動紊流態。在稀釋態中，被動粒子在小時間尺度展現過擴散 (super-diffusion) 且在大時間尺度有正常擴散行為。增加被動粒子的尺寸使得有效擴散係數接近常數，因自推進桿狀粒子與被動粒子的碰撞率增加而抵消來自背景液體所造成的阻力。然而在主動紊流態中，於我們可觀察的時間尺度內，被動粒子完全展現過擴散行為。其有效擴散係數隨著此被動粒子的尺寸增加而減少，跟僅由熱擾動所造成之影響比起來減少的較緩慢。即史托克斯-愛因斯坦關係無法概括被動粒子在自推進桿狀粒子懸浮液的擴散行為。

Under stochastic thermal kicks, passive particles in a quiescent liquid background exhibit random walk type Brownian motion without spatiotemporal correlations. The balance between the energy pumped by the stochastic thermal kick on background molecules and the energy dissipated to the background leads to the well known Stokes-Einstein relation. Namely, the diffusivity of the passive particle is proportional to the ratio of the temperature to the drag of passive particles, and the drag coefficient linearly increases with the particle diameter.
On the other hand, the liquid with active rod-like particle suspensions is a different non-equilibrium system in which the energy is pumped by each self-propelling rod and dissipated to the highly viscous background. Through the interplay of the alignment effects from the anisotropic exclusive volume of active rods and the self-propelling from rods, the dense rod suspensions exhibit turbulence like behavior with coherent vortices and jets over a broad range of spatiotemporal scales. In the previous studies, small size passive particles have been mainly used as tracers for investigating the background flow field and its anomalous diffusion. However, increasing the passive particle size tends to suppress the response to the small scale drives from the background turbulent flow. How the passive circular rafts with different sizes diffuse in the active particle turbulence still remains an interesting unexplored issue.
In this thesis, the above unexplored issue is addressed through 2D numerical simulation. It is found that, increasing the volume fraction of active rods leads to the transition from the dilute state with nearly uncorrelated rod motion to the active turbulent state with multi-scale coherent vortices and jets. In the dilute state, the passive particle exhibit super diffusion at small time scale and normal diffusion at large time scale. The effective diffusivity remains nearly constant with increasing passive particle size, because the increasing collision rate associated with the increasing number of background rods compensates the increasing drag from the background viscous liquid. However, in the active turbulent state, the passive particle exhibits super diffusion over the entire tested time scale. The effective diffusion coefficient of the passive particle decreases with increasing size of passive particles, but less rapidly than those in the quiescent liquid background driven solely by thermal agitation. Namely, the Stokes-Einstein relation cannot be generalized to the passive particle diffusion in the active particle background.

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