=============
KMC algorithm
=============

* Form and maintain a list of all possible rates :math:`r_j` in system 
  
  .. math::

     r_j\;\mbox{for all}\;\; j=1,2,\ldots,M

  :math:`M` -- number of possible events

* Calculate the cummulative function

  .. math::
        
     R_m=\sum_{j=1}^{m}r_j, \;\;\mbox{for all}\;\; m=1,2,\ldots,M

* Generate random number :math:`\rho_1\in[0,1)`
* Select event :math:`i`

  .. math::
        
     R_{i-1}\leq \rho_1 R_M < R_i

  .. image:: images/kmc_cdf_invert.png
     :scale: 25%
     :align: center

* Carry out event :math:`i`

* Generate random number :math:`\rho_2\in(0,1)`

* Compute time increment :math:`\Delta t`

  .. math::
        
     \Delta t=-\frac{ln(\rho_2)}{R_M}
        
* Update time: :math:`t=t+\Delta t`

* Update list of events and goto next step