我們發展出一個碰撞機率分佈來模擬布朗運動，而這個方法只需知道小球的速度分佈函數f(v)。推導出的碰撞機率分佈P(v,b,△t;V,T,c)，表示一個具有速度V的布朗粒子(Brownian particle)，在小球的溫度為T和濃度為c的情況，將會在時間△t之後，被一個具有速度v的小球從質心距b的位置撞到的機率。為了驗證模型的正確性，我們將f(v)以Maxwell‐Boltzmann分佈在平衡溫度T代入，則布朗粒子的速度分佈也必須符合Maxwell‐Boltzmann分佈；而布朗粒子的擴散係數則必須符合Einstein‐Smoluchowski關係D=k_B*T/γ。觀察一個布朗粒子在小球有溫度梯度(或是濃度梯度)下運動，發現布朗粒子喜歡待在溫度低(或是濃度高)的區域。而且如果梯度越大，這個差異會越加明顯，平均的位置梯度會正比於兩端的溫度差(或是濃度差)。最後我們會研究兩顆布朗粒子中間以彈簧連接的複合布朗粒子(multi‐Brownian‐particle)在梯度場下的運動行為。 We have developed a method which uses the collision probability distribution from the velocity distribution function f(v) of small particles to simulate Brownian motion. Small particles are regarded as a dilute gas. We derive P(v,b,△t;V,T,c) which represents the probability of one Brownian particle with velocity V colliding with one small particle with the speed v, the impact parameter b, and the time interval △t at temperature T and concentration c. The dynamics of one Brownian particle is tested by taking f(v) as the Maxwell-Boltzmann distribution at an equilibrium temperature T. The velocity distribution of one Brownian particle is confirmed to follow the Maxwell distribution and the diffusion constant is consistent with the Einstein-Smoluchowski relation, D=k_B*T/γ. We have observed that the motion of one Brownian particle in a system of small particles with temperature (or concentration) gradient likes to stay in the lower temperature (or higher concentration) region. The difference would increase along with the increase of each gradient. We find that the average position gradients are proportional to the temperature (or concentration) differences. At last, we will also study the details of the motion of the multi-Brownian-particle which is composited with Brownian particles that are conecting with nearest neighbors by springs in the gradient fields.
