#### 克服模拟的时间尺度的局限

Kinetic Monte Carlo attempts to overcome this limitation by exploiting the fact that the long-time dynamics of this kind of system typically consists of diffusive jumps from state to state. Rather than following the trajectory through every vibrational period, these state-to-state transitions are treated directly

kMC从某种程度上就是对MD的一种粗化，将关注点从原子粗化到体系，将原子轨迹粗化到体系组态的跃迁，那么模拟的时间跨度就将从原子振动的尺度提高到组态跃迁的尺度，这是因为这种处理方法摈弃了与体系穿越势垒无关的微小振动，而只着眼于体系的组态变化。因此，虽然不能描绘原子的运动轨迹，但是作为体系演化，其“组态轨迹”仍然是正确的。

for each possible escape pathway to an adjacent basin, there is a rate constant $k_{ij}$ that characterizes the probability, per unit time, that it escapes to that state $j$

#### kMC的时间步长

$k_{ij}$ 代表体系从组态 $i$ 逃逸到 $j$ 的速率，则发生跃迁的概率为 $k_{tot} = \sum_{j}^{}{k_{ij}}$
Gillespie给出的假设：

Suppose that the system is in configuration $c$. The probability that a particular enabled reaction $c \rightarrow c^{‘}$ occurs in an infinitesimal period $\delta t$ is given by $k_{c \rightarrow c^{‘}}*\delta t$.

$$P(T>= t + \delta t) = P(T>= t)(1 - k_{tot}\delta t)$$

$$\frac{P(T>= t + \delta t) - P(T>= t)}{\delta t} = -k_{tot}P(T>= t)$$

$$P(T>= t) = exp(-k_{tot}t)$$

$$p_{survive} = exp(-k_{tot}t)$$

$$p_{transition} = 1 - p_{survive} = 1 - exp(-k_{tot}t)$$

$$\tau = \int_{0}^{\infty} tp(t)=\frac{1}{k_{tot}}$$

$$t\_{draw} = -\frac{ln(\rho)}{k}$$

#### 步长的另一种推导方法

$$p_{n}(k, \Delta t) = (k\Delta t)^{n}e^{-k\Delta t}/n!$$

$$p_{0}(k, \Delta t) = e^{-k\Delta t}$$

$$\Delta t = \frac{-ln(r)}{k}$$