This analyzer calculates the derivative of total free energy \( F \) with respect to parameter \(c_0 = 1/v \) in a fixed unit cell, for a system of fixed system composition, and fixed values of all other model parameters.

See also: ConcentrationDerivative (Class API)

Theory

The required thermodynamic derivative is given by an ensemble average

\[ \frac{\partial F}{\partial c_0} = \langle \Psi \rangle \quad\quad\quad\quad \Psi = \frac{\partial H}{\partial c_{0}} + \frac{\partial \ln N}{\partial c_0} \quad, \]

in which

\[ \frac{\partial H}{\partial c_0} = \frac{H}{c_0} \quad\quad\quad\quad \frac{\partial \ln N}{\partial c_{0}} = - \frac{G}{2 c_0} \quad. \]

Here, \( G \) is the total number of grid points in the computational mesh.

Parameter file

Parameter file format:

ConcentrationDerivative{
  interval*          int    (1 by default)
  outputFileName     string
  nSamplePerOutput*  int    (1 by default)
}

Meanings of the parameters:

Label	Description
interval	number of steps between data samples
outputFileName	name of output file
nSamplePerOutput*	Number of sampled values per block average output to file

Output

In the following, {outputFileName} is used within descriptions of file names to represent the string value of the outputFileName parameter, to which different suffixes are added.

Output during a simulation or trajectory file analysis :

If nSamplePerOutput > 1, then block average values \( H' \), averaged over blocks of nSamplePerOutput sequential samples, are output to the file {outputFileName}.dat every interval * nSamplerPerBlock simulation steps or trajectory file frames.

If nSamplePerOutput = (or if this optional parameter is omitted), then values of \( H' \) will output every interval simulation steps, with no block averaging.

If nSamplerPerBlock = 0, then {outputFileName}.dat data file is not created.

Final output :

At the end of a simulation or analysis operation:

The sequence average of \( H' \) and estimated error on this average are output to the file {outputFileName}.ave
Details of the error analysis are output to the file {outputFileName}.aer

See the discussion of error analysis for further details.