The LMBdStep classes used by both the pscf_pc and pscf_pg programs implements the Leimkuhler Matthews Brownian dynamics step algorithm.

Two closely analogous classes named Pscf::Rpc::LMBdStep and Pscf::Rpg::LMBdStep 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 LMBdStep 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 Leimkuhler Matthews (LM) algorithm is described and tested in these references:

B. Vorselaars, J. Chemical Physics, 158 114117 (2023) [ https://doi.org/10.1063/5.0131183 ]
B. Leimkuhler and C. Matthews, Applied Mathematics Research Express, Issue 1, pages 34-56 (2013) [ https://doi.org/10.1093/amrx/abs010 ]
B. Leimkuhler and C. Matthews, J. Chemical Physics, vol. 138, 174102 (2013) [ https://doi.org/10.1063/1.4802990 ]

The LM step algorithm differs from an explicit Euler algorithm in that it uses a random displacement that is given by a sum of random numbers generated at two different time steps, this step and the previous step. As a result, random displacements at neighboring steps are correlated.

Standard AB System : First consider the common case of a standard AB system with two types of monomer. The LM algorithm for the change of the field \( \omega_{0}({\bf r}) = \omega_{-}({\bf r}) \) at grid point location \( {\bf r} \) and time step \( n \) is given by

\[ \omega_{-}^{(n+1)}({\bf r}) = \omega_{-}^{(n)}({\bf r}) - \gamma \frac{\delta H}{\delta \omega_{-}({\bf r})} + \frac{1}{2} \left [ g^{(n)}({\bf r}) + g^{(n-1)}({\bf r}) \right ] \]

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 index \( n \) 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.

General : We now consider the general algorithm for 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

\[ \omega_{i}^{(n+1)}({\bf r}) = \omega_{i}^{(n)}({\bf r}) - \gamma \frac{\delta H}{\delta \omega_{i}({\bf r})} + \frac{1}{2} \left [ g^{(n)}({\bf r}) + g^{(n-1)}({\bf r}) \right ] \quad, \]

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 need 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:

LMBdStep{
  mobility     1e-3
}

The format of this block is simply

LMBdStep{
   mobility      float
}

The mobility parameter is requred.