The ExplictBdStep classes used by both the pscf_pc and pscf_pg programs implements a simple explicit Euler Brownian dynamics step algorithm.

Two closely analogous classes named Pscf::Rpc::ExplicitBdStep and Pscf::Rpg::ExplicitBdStep that have the same class name but that are defined in different namespaces are used to implement this the same step algorithm for different programs. The ExplicitBdStep class defined in the Rpc namespace is used in the pscf_pc program (using CPU hardware) and the class defined in the Rpg namespace is used in the pscf_pg program (using GPU acceleration). These two analogous classes implement the same step algorithm and use the same parameter file format, which are documented below.

Step Algorithm

A general discussion of conventions used in PSCF for BD step algorithms is given here.

The explicit Euler BD step algorithm for the common case of a standard AB system with two types of monomer is described by an update scheme for the field \( \omega_{0}({\bf r}) = \omega_{-}({\bf r}) \) at grid point location \( {\bf r} \) and time step \( n \) that is given by

\[ \omega_{-}^{(n+1)}({\bf r}) = \omega_{-}^{(n)}({\bf r}) - \gamma \frac{\delta H}{\delta \omega_{-}({\bf r})} + g^{(n)}({\bf r}) \]

where \( \gamma \) is the BD mobility parameter,

\[ \frac{\delta H}{\delta \omega_{-}({\bf r})} = \frac{1}{v} \left [ \frac{2\omega_{-}({\bf r})}{\chi_{b}} - \Phi_{-}({\bf r}) \right ] \]

is the generalized force evaluated using the field configuration at time step \( n \), and \( g^{(n)}({\bf r}) \) is a random field displacement. The random displacement \( g^{(n)}({\bf r}) \) is chosen independently at each grid point and each BD step from a Gaussian distribution with a vanishing mean value, \( \overline{g^{(n)}({\bf r})} = 0 \), and a variance

\[ \overline{[g^{(n)}({\bf r})]^2} = \gamma \Delta V \quad. \]

Here, \( \Delta V \) is the volume per grid point of the computational mesh, given by \( \Delta V = V/G \), where \( V \) is the total system volume and \( G \) is the number of grid points on the mesh.

This algorithm can be generalized to a system with \( M \) monomer types for which the first \( L \) eigenvalues of the projected chi matrix are negative, with \( L \leq M - 1 \). In this case, each field component \( \omega_{i}({\bf r}) \) associated with eigenvalue \( \lambda_{i} < 0 \) of the projected \( \chi \) matrix, for \( i = 0, \ldots, L - 1 \), is updated via a scheme

\[ \omega_{i}^{(n+1)}({\bf r}) = \omega_{i}^{(n)}({\bf r}) - \gamma \frac{\delta H}{\delta \omega_{i}({\bf r})} + g_{i}^{(n)}({\bf r}) \quad. \]

with a separate random noise components for each such field component. The random noise components \( g_{i}^{(n)}({\bf r})\) with eigenvector indices \( i = 0, \ldots, L - 1 \) are chosen independently for each index value \( i \), each grid position \( {\bf r} \) and each time step \( n \) from a Gaussian distribution of zero mean and variance \( \overline{[g^{(n)}_{i}({\bf r})]^2} = \gamma \Delta V \).

Parameter File

The only parameter needed by this algorithm is a required parameter named "mobility", which gives the value of parameter denoted by \( \gamma \) in the above equations. A typical example of the contents of the block is shown below. below:

ExplicitBdStep{
  mobility     5.0e-3
}

The format of this block is thus simply

ExplicitBdStep{
   mobility      float
}

The mobility parameter is required.