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PSCF provides an extensible set of C++ classes for analysis of field theoretic simulations. These are referred to here as analyzers classes, or simply analyzers.

Overview

Each PSCF analyzer class performs a specified analysis and/or data output operation at some regular interval during a simulation (e.g., every 10 or 100 simulation steps). Analyzer classes can be designed to periodically compute physical quantities of interest, output data to file, or compute statistical properties such as averages and variances, or any combination of these types of operation. The PSCF parameter file format allows the user to enable a set of zero or more analyzers in order to specify which such operations should be performed during a BD or MC simulation, and how often each should be performed.

The same set of analyzer classes may also be used during a separate postprocess analysis operation that may be performed using the ANALYZE command. Postprocess analysis of field trajectory files is discussed in more detail on a separate page.

Algorithms

All of the analyzer classes that are provided with the current version of PSCF are listed below, along with a brief description of the purpose of each class. Further details can be obtained by clicking on the link provided for each such class:

StepLogger: Record progress of a simulation by periodically writing the number of completed BD or MC steps to standard output (e.g., the screen).
TrajectoryWriter : Periodically write the current configuration of monomer chemical potential fields (w-fields), in r-grid format, as one frame of a field trajectory file.
ConcentrationWriter : Periodically write all monomer concentration fields (c-fields) to a concentration trajectory file, in r-grid format.
HamiltonianAnalyzer : Compute the field theoretic Hamiltonian \( H \) and its components \( H_{\rm id} \) and \( H_{\rm f} \).
BinaryStructureFactor: Compute the structure factor of a system with two monomer types.
MaxOrderParameter : Identify the Fourier coefficient of maximum absolute value, for use in identifying spontaneous phase transitions.
FourthOrderParameter: Compute an alternative order parameter that is useful for identifying spontaneous phase transitions.
BinaryChiDerivative: Compute the derivative of free energy with respect to changes in the chi parameter of an AB (two-monomer) system.
CubicLengthDerivative Compute the derivative of total free energy with respect to the length of one edge of a cubic box.
ConcentrationDerivative Compute the derivative of total free energy with respect to changes in total monomer concentation (the inverse of vMonomer).
PerturbationDerivative: Compute the derivative of free energy with respect to the strength of a perturbation (the lambda parameter).

Categories:

Data Output : The first three classes listed above (StepLogger, TrajectoryWriter, and ConcentrationWriter) simply write readily available information to a file or standard output (e.g., the screen) at regular intervals, rather than doing any nontrivial data analysis.

Thermodynamics and Structure : The HamiltonianAnalyzer and BinaryStructureFactor classes compute and evaluate average values for two basic thermodynamic and structural properties.

Identifying Phase Transitions : The MaxOrderParameter and FourthOrderParameter) classes are designed specifically as tools for identifying spontaneous phase transitions. Specifically, they are useful for identifying spontaneous disorder-to-order (crystallization) or order-to-disorder (melting) transition that occur during a simulation.

Free Energy Derivatives : The last four analyzers listed above (BinaryChiDerivative, CubicLengthDerivative, ConcentrationDerivative, and PerturbationDerivative) all have names that end with the suffix "Derivative", and are all designed to compute derivatives of the free energy with respect to a model parameter, for use in thermodynamic integration calculations of free energy differences. The PerturbationDerivative analyzer can be used to implement Einstein crystal free energy calculations.

Output files

Most analyzer classes that compute a single physical quantity or a small number of related quantities provide information in two forms:

Averages of a sequence of values are computed during a simulation or postprocess analysis and reported at the end of the simulation or analysis operation, along with an estimate of the error on the average. The average and error are often reported in a file with file name suffix .ave. Further details of the error analysis are are reported in a file with file name sufix .aer, as discussed below.
A sequence of values may optionally be output to a file that can contain values of individual sampled values or block averages (i.e., values of the average over "blocks" containing some a user-specified number of consecutive sampled values). The name of this file usually ends with a file name suffix .dat.

The option to output sequences of block averages rather than all sampled values is provided to allow a user to reduce the number of file output operations required for quantities that are computed frequently. The number of sampled values per block average output to a .dat file is usually given by a parameter named nSamplePerOutput. Setting nSamplerPerOutput = 1 yields individual sampled outputs. Setting nSamplePerOutput = 0 suppresses creation of the .dat file.

For users who examine source code : Class templates for analyzer classes that compute a sequence average for a single physical quantity are usually derived from a specialization of the Rp::AverageAnalyzer class template. This template defines conventions for parameter and output files for such classes, in which the shared base name for all output files produced by a given analyzer is given by a parameter named "outputFileName". Classes that compute sequence averages for several related quantities (such as HamiltonianAnalyzer) may instead be derived from specializations of Rp::AverageListAnalyzer.

Statistical error analysis

Classes that compute sequence averages of one or more physical quantities generally report an estimate of the statistical error on each such average. The reported error is as a estimate of the standard deviation of the probability distribution for the sequence average. To estimate this quantity, one must into account the fact that nearby values in the underlying time sequence are often strongly correlated, rather than independent trials. Estimates of errors on sequence averages reported by PSCF are computed using a method introduced by H. Flyvbjerg and H.G. Petersen [J. Chem. Phys 91, 461 (1989)], which is based on the analysis of sequences of block-averaged values. The version of this algorithm used in PSCF is implemented by the Util::AverageState and Util::Average classes.

In this algorithm, one divides a long sequence of \( N \) sampled values of physical variables into blocks that each contain \( L \) consecutive sampled values, for multiple different choices of value for the block length \( L \). The implementation used by PSCF uses values of \( L = 2^{n} \) for all integer values of \( n = 0, 1, 2, \ldots \) for which \( L \). For each value of \( L \), we construct a sequence of approximately \( L / N \) block averages, in which each value in the resulting sequence is the average of values within one block. For each such block-averaged sequence, we compute a naive estimate of the error on the average by analyzing the sequence as if block averaged values were uncorrelated. The assumption that block average values are uncorrelated is generally incorrect for short block lengths (small \( n \)), but is expected to become approximately correct for sufficiently long blocks within a sufficiently long primary sequence. Estimates of the error on the average obtained using different values of \( L \) that are all long enough to yield uncorrelated block averages are expected to give equivalent error estimates in the limit of an infinite sequence. In practice, for a long but finite sequence, if estimates of the error on the average obtained from different values of \( n \) exhibit a "plateau" in which consistent estimates are obtained from several different levels, then this plateau value is expected to provides a reliable estimate of the true error. This strategy is implemented in PSCF by the Util::Average::blockingError function, which compares naive estimates obtained with different values of \( n \) for which \( L \leq N / 16 \) and attempts to automatically identify and estimate a plateau value.

Analyzers that report sequence averages at the end of a simulation or postprocess analysis generally report just sequence averages and an associated error estimates in a file with file name suffix .ave. More detailed information about the naive error estimates obtained from different levels of block-averaging is usually written to a file with a file name suffix .aer.

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