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### fix ave/correlate command

Syntax:

    fix ID group-ID ave/correlate Nevery Nrepeat Nfreq value1 value2 ... keyword args ...

- ID, group-ID are documented in fix command
- ave/correlate = style name of this fix command
- Nevery = use input values every this many timesteps
- Nrepeat = # of correlation time windows to accumulate
- Nfreq = calculate tine window averages every this many timesteps
- one or more input values can be listed
- value = c_ID, c_ID[N], f_ID, f_ID[N], v_name

        c_ID = global scalar calculated by a compute with ID
        c_ID[I] = Ith component of global vector calculated by a compute with ID
        f_ID = global scalar calculated by a fix with ID
        f_ID[I] = Ith component of global vector calculated by a fix with ID
        v_name = global value calculated by an equal-style variable with name

- zero or more keyword/arg pairs may be appended
- keyword = type or ave or start or prefactor or file or title1 or title2 or title3

        type arg = auto or upper or lower or auto/upper or auto/lower or full
          auto = correlate each value with itself
          upper = correlate each value with each succeeding value
          lower = correlate each value with each preceding value
          auto/upper = auto + upper
          auto/lower = auto + lower
          full = correlate each value with every other value, including itself = auto + upper + lower
        ave args = one or running
          one = zero the correlation accumulation every Nfreq steps
          running = accumulate correlations continuously
        start args = Nstart
          Nstart = start accumulating correlations on this timestep
        prefactor args = value
          value = prefactor to scale all the correlation data by
        file arg = filename
          filename = name of file to output correlation data to
        title1 arg = string
          string = text to print as 1st line of output file
        title2 arg = string
          string = text to print as 2nd line of output file
        title3 arg = string
          string = text to print as 3rd line of output file

Examples:

    fix 1 all ave/correlate 5 100 1000 c_myTemp file temp.correlate
    fix 1 all ave/correlate 1 50 10000 &
              c_thermo_press[1] c_thermo_press[2] c_thermo_press[3] &
    	  type upper ave running title1 "My correlation data"

Description:

Use one or more global scalar values as inputs every few timesteps, calculate time correlations bewteen them at varying time intervals, and average the correlation data over longer timescales. The resulting correlation values can be time integrated by variables or used by other output commands such as thermo_style custom, and can also be written to a file.

The group specified with this command is ignored. However, note that specified values may represent calculations performed by computes and fixes which store their own "group" definitions.

Each listed value can be the result of a compute or fix or the evaluation of an equal-style variable. In each case, the compute, fix, or variable must produce a global quantity, not a per-atom or local quantity. If you wish to spatial- or time-average or histogram per-atom quantities from a compute, fix, or variable, then see the fix ave/spatial, fix ave/atom, or fix ave/histo commands. If you wish to sum a per-atom quantity into a single global quantity, see the compute reduce command.

Computes that produce global quantities are those which do not have the word atom in their style name. Only a few fixes produce global quantities. See the doc pages for individual fixes for info on which ones produce such values. Variables of style equal are the only ones that can be used with this fix. Variables of style atom cannot be used, since they produce per-atom values.

The input values must either be all scalars. What kinds of correlations between input values are calculated is determined by the type keyword as discussed below.

The Nevery, Nrepeat, and Nfreq arguments specify on what timesteps the input values will be used to calculate correlation data. The input values are sampled every Nevery timesteps. The correlation data for the preceding samples is computed on timesteps that are a multiple of Nfreq. Consider a set of samples from some initial time up to an output timestep. The initial time could be the beginning of the simulation or the last output time; see the ave keyword for options. For the set of samples, the correlation value Cij is calculated as:

    Cij(delta) = ave(Vi(t)*Vj(t+delta))

which is the correlation value between input values Vi and Vj, separated by time delta. Note that the second value Vj in the pair is always the one sampled at the later time. The ave() represents an average over every pair of samples in the set that are separated by time delta. The maximum delta used is of size (Nrepeat-1)*Nevery. Thus the correlation between a pair of input values yields Nrepeat correlation datums:

    Cij(0), Cij(Nevery), Cij(2*Nevery), ..., Cij((Nrepeat-1)*Nevery)

For example, if Nevery=5, Nrepeat=6, and Nfreq=100, then values on timesteps 0,5,10,15,...,100 will be used to compute the final averages on timestep 100. Six averages will be computed: Cij(0), Cij(5), Cij(10), Cij(15), Cij(20), and Cij(25). Cij(10) on timestep 100 will be the average of 19 samples, namely Vi(0)*Vj(10), Vi(5)*Vj(15), Vi(10)*V j20), Vi(15)*Vj(25), ..., Vi(85)*Vj(95), Vi(90)*Vj(100).

Nfreq must be a multiple of Nevery; Nevery and Nrepeat must be non-zero. Also, if the ave keyword is set to one which is the default, then Nfreq >= (Nrepeat-1)*Nevery is required.

If a value begins with "c_", a compute ID must follow which has been previously defined in the input script. If no bracketed term is appended, the global scalar calculated by the compute is used. If a bracketed term is appended, the Ith element of the global vector calculated by the compute is used.

Note that there is a compute reduce command which can sum per-atom quantities into a global scalar or vector which can thus be accessed by fix ave/correlate. Or it can be a compute defined not in your input script, but by thermodynamic output or other fixes such as fix nvt or fix temp/rescale. See the doc pages for these commands which give the IDs of these computes. Users can also write code for their own compute styles and add them to LAMMPS.

If a value begins with "f_", a fix ID must follow which has been previously defined in the input script. If no bracketed term is appended, the global scalar calculated by the fix is used. If a bracketed term is appended, the Ith element of the global vector calculated by the fix is used.

Note that some fixes only produce their values on certain timesteps, which must be compatible with Nevery, else an error will result. Users can also write code for their own fix styles and add them to LAMMPS.

If a value begins with "v_", a variable name must follow which has been previously defined in the input script. Only equal-style variables can be referenced. See the variable command for details. Note that variables of style equal define a formula which can reference individual atom properties or thermodynamic keywords, or they can invoke other computes, fixes, or variables when they are evaluated, so this is a very general means of specifying quantities to time correlate.

Additional optional keywords also affect the operation of this fix.

The type keyword determines which pairs of input values are correlated with each other. For N input values Vi, for i = 1 to N, let the number of pairs = Npair. Note that the second value in the pair Vi(t)*Vj(t+delta) is always the one sampled at the later time.

- If type is set to auto then each input value is correlated with itself. I.e. Cii = Vi*Vi, for i = 1 to N, so Npair = N.
- If type is set to upper then each input value is correlated with every succeeding value. I.e. Cij = Vi*Vj, for i < j, so Npair = N*(N-1)/2.
- If type is set to lower then each input value is correlated with every preceeding value. I.e. Cij = Vi*Vj, for i > j, so Npair = N*(N-1)/2.
- If type is set to auto/upper then each input value is correlated with itself and every succeeding value. I.e. Cij = Vi*Vj, for i >= j, so Npair = N*(N+1)/2.
- If type is set to auto/lower then each input value is correlated with itself and every preceding value. I.e. Cij = Vi*Vj, for i <= j, so Npair = N*(N+1)/2.
- If type is set to full then each input value is correlated with itself and every other value. I.e. Cij = Vi*Vj, for i,j = 1,N so Npair = N^2.

The ave keyword determines what happens to the accumulation of correlation samples every Nfreq timesteps. If the ave setting is one, then the accumulation is restarted or zeroed every Nfreq timesteps. Thus the outputs on successive Nfreq timesteps are essentially independent of each other. The exception is that the Cij(0) = Vi(T)*Vj(T) value at a timestep T, where T is a multiple of Nfreq, contributes to the correlation output both at time T and at time T+Nfreq.

If the ave setting is running, then the accumulation is never zeroed. Thus the output of correlation data at any timestep is the average over samples accumulated every Nevery steps since the fix was defined. it can only be restarted by deleting the fix via the unfix command, or by re-defining the fix by re-specifying it.

The start keyword specifies what timestep the accumulation of correlation samples will begin on. The default is step 0. Setting it to a larger value can avoid adding non-equilibrated data to the correlation averages.

The prefactor keyword specifies a constant which will be used as a multiplier on the correlation data after it is averaged. It is effectively a scale factor on Vi*Vj, which can be used to account for the size of the time window or other unit conversions.

The file keyword allows a filename to be specified. Every Nfreq steps, an array of correlation data is written to the file. The number of rows is Nrepeat, as described above. The number of columns is the Npair+2, also as described above. Thus the file ends up to be a series of these array sections.

The title1 and title2 and title3 keywords allow specification of the strings that will be printed as the first 3 lines of the output file, assuming the file keyword was used. LAMMPS uses default values for each of these, so they do not need to be specified.

By default, these header lines are as follows:

    # Time-correlated data for fix ID
    # TimeStep Number-of-time-windows
    # Index TimeDelta Ncount valueI*valueJ valueI*valueJ ...

In the first line, ID is replaced with the fix-ID. The second line describes the two values that are printed at the first of each section of output. In the third line the value pairs are replaced with the appropriate fields from the fix ave/correlate command.

Let Sij = a set of time correlation data for input values I and J, namely the Nrepeat values:

    Sij = Cij(0), Cij(Nevery), Cij(2*Nevery), ..., Cij(*Nrepeat-1)*Nevery)

As explained below, these datums are output as one column of a global array, which is effectively the correlation matrix.

The trap function defined for equal-style variables can be used to perform a time integration of this vector of datums, using a trapezoidal rule. This is useful for calculating various quantities which can be derived from time correlation data. If a normalization factor is needed for the time integration, it can be included in the variable formula or via the prefactor keyword.

Restart, fix_modify, output, run start/stop, minimize info:

No information about this fix is written to binary restart files. None of the fix_modify options are relevant to this fix.

This fix computes a global array of values which can be accessed by various output commands. The values can only be accessed on timesteps that are multiples of Nfreq since that is when averaging is performed. The global array has # of rows = Nrepeat and # of columns = Npair+2. The first column has the time delta (in timesteps) between the pairs of input values used to calculate the correlation, as described above. The 2nd column has the number of samples contributing to the correlation average, as described above. The remaining Npair columns are for I,J pairs of the N input values, as determined by the type keyword, as described above.

- For type = auto, the Npair = N columns are ordered: C11, C22, ..., CNN.
- For type = upper, the Npair = N*(N-1)/2 columns are ordered: C12, C13, ..., C1N, C23, ..., C2N, C34, ..., CN-1N.
- For type = lower, the Npair = N*(N-1)/2 columns are ordered: C21, C31, C32, C41, C42, C43, ..., CN1, CN2, ..., CNN-1.
- For type = auto/upper, the Npair = N*(N+1)/2 columns are ordered: C11, C12, C13, ..., C1N, C22, C23, ..., C2N, C33, C34, ..., CN-1N, CNN.
- For type = auto/lower, the Npair = N*(N+1)/2 columns are ordered: C11, C21, C22, C31, C32, C33, C41, ..., C44, CN1, CN2, ..., CNN-1, CNN.
- For type = full, the Npair = N^2 columns are ordered: C11, C12, ..., C1N, C21, C22, ..., C2N, C31, ..., C3N, ..., CN1, ..., CNN-1, CNN.

The array values calculated by this fix are treated as "intensive". If you need to divide them by the number of atoms, you must do this in a later processing step, e.g. when using them in a variable.

No parameter of this fix can be used with the start/stop keywords of the run command. This fix is not invoked during energy minimization.

Restrictions: none

Related commands:

compute, fix ave/time, fix ave/atom, fix ave/spatial, fix ave/histo, variable

Default: none

The option defaults are ave = one, type = auto, start = 0, no file output, title 1,2,3 = strings as described above, and prefactor = 1.0.