The maximum flux transition path (MFTP) zeta = Z(s), s in [0,1], is defiend
by the condition that
$\beta \nabla F + \frac{(D(Z)^{-1}Z_s)_s}{Z_s^TD(Z)^{-1}Z_s} // D(Z)^{-1}Z_s$.

We implemented the semi-implicit simplified string method for calculating 
the MFTP and the simplified string method for calculating the minimum free
energy path (MFEP).
The module mftp.py written in Python does such a job, in which 
"pathrefiner" is the principal function.
The function is invoked on separate processors using mpi4py.
*************************************************************************************
Input of pathrefiner:                                                               *
1. msys     -- (OBJECT): defining the molecular system, including at least two      *
               attributes: system temperature (a real number), and mean_force       *
               (a function), and user-defined attributes (see example CHARMM).      *
2. initpath -- (ARRAY): an initial path in collective variable space with dimension *
               (number of images) times (dimension of collective variables)         *
3. param    -- (DICTIONARY): {"noiter":1, "preiter":0, "tau":0.04, "TOL":0.02,      *
               "method":MFEP_SS/MFTP_ISS}. In general, "noiter" is the maximum      *
               iterations users specify for a continuous run; "preiter" allows      *
               users to continue from the previous run; "tau" is the pseudo time    *
               time square determining the moving speed; "TOL" is the stopping      *
               criterion defined as d = max_{0<=j<=J}|x^current - x^previous|;      *
               "method" means whether to calculate the MFEP or the MFTP. By default *
               is the MFTP.                                                         *
Output of pathrefiner:                                                              *
1. tranpath -- (ARRAY): transition path with same dimension as initpath             *
2. noiter   -- (INTEGER): the actual number of iterations for one continuous run    *
*************************************************************************************

The interface between the developer code "pathrefiner" and user code is
the system dependent function "mean_force".
The function "mean_force" should at least provide the proto-diffusion 
tensor $D$ and the derivative of the free energy: $\nabla F$. 

*******************************************************************************
Input of "mean_force" contains at least:                                      *
1. Z_j (ARRAY): the coordinates of j-th image in collective variable space    *
2. j (INTEGER): the identity number of the image                              *
3. Molecular system parameters, such as temperature, pressure, etc.           *
Output of "mean_force" contains at least:                                     *
1. D_j (ARRAY): proto-diffusion tensor at j-th images                         *
2. nabla F_j (ARRAR): the derivative of the free energy at j-th images        *
Remark: The values of cartesian coordinates at each iteration should be saved.* 
Or else, the input and output of "mean_force" could contain this arguments.   *
In CHARMM, it is manipulated by saving COORDINATE FILES.                      *
*******************************************************************************

We demonstrate how to use the module mftp by three examples: 
1. an artificial problem in which the collective variables are 
   configurations; 
2. alanine dipeptide in vacuum using CHARMM; 
3. alanine dipeptide in vacuum using NAMD.

We provide the module "utility" containing some functions to facilitate the examples.
These functions are:
1. readpath      -- read formatted path data file from disk
2. writepath     -- write path data as formatted file into disk
3. plotpath      -- visualize path

For each example, run:
    mpirun -np x python main.py 
where "x" is the number of processors users provide.