Hydrologic Science and Engineering, Colorado School of Mines, Golden, Colorado 80401, USA.
J Chem Phys. 2013 Apr 7;138(13):131101. doi: 10.1063/1.4800799.
The connection between the governing equations of chemical reaction and the underlying stochastic processes of particle collision and transformation have been developed previously along two end-member conditions: perfectly mixed and maximally diffusion-limited. The complete governing equation recognizes that in the perfectly mixed case, the particle (i.e., molecular or macro-particle) number state evolution is markovian, but that spatial self-organization of reactants decreases the probability of reactant pairs finding themselves co-located. This decreased probability manifests itself as a subordination of the clock time: as reactant concentrations become spatially variable (unmixed), the time required for reactants to find each other increases and the random operational time that particles spend in the active reaction process is less than the clock time. For example, in the system A + B → [empty-set], a simple approximate calculation for the return time of a brownian motion to a moving boundary allows a calculation of the operational time density, and the total solution is a subordination integral of the perfectly-mixed solution with a modified inverse gaussian subordinator. The system transitions from the well-mixed solution to the asymptotic diffusion-limited solution that decays as t(-d∕4) in d-dimensions.
化学反应的控制方程与粒子碰撞和转化的潜在随机过程之间的联系，此前已经沿着两种极端条件发展：完全混合和最大扩散限制。完整的控制方程认识到，在完全混合的情况下，粒子（即分子或宏观粒子）数状态的演化是马尔可夫的，但反应物的空间自组织降低了反应物对发现自己处于共定位的概率。这种概率的降低表现为对时钟时间的从属：随着反应物浓度在空间上变得可变（未混合），反应物找到彼此所需的时间增加，粒子在活性反应过程中随机运行的时间小于时钟时间。例如，在系统 A + B → [空集]中，布朗运动返回移动边界的返回时间的简单近似计算允许计算操作时间密度，并且总解是具有修改后的逆高斯从属的完全混合解的从属积分。该系统从良好混合的解过渡到渐近扩散限制的解，在 d 维中按 t(-d∕4)衰减。
