   LOCAL_AVERAGE
 This is part of the multicolvar module

Calculate averages over spherical regions centered on atoms

As is explained in this video certain multicolvars calculate one scalar quantity or one vector for each of the atoms in the system. For example COORDINATIONNUMBER measures the coordination number of each of the atoms in the system and Q4 measures the fourth order Steinhardt parameter for each of the atoms in the system. These quantities provide tell us something about the disposition of the atoms in the first coordination sphere of each of the atoms of interest. Lechner and Dellago  have suggested that one can probe local order in a system by taking the average value of such symmetry functions over the atoms within a spherical cutoff of each of these atoms in the systems. When this is done with Steinhardt parameters they claim this gives a coordinate that is better able to distinguish solid and liquid configurations of Lennard-Jones atoms.

You can calculate such locally averaged quantities within plumed by using the LOCAL_AVERAGE command. This command calculates the following atom-centered quantities:

$s_i = \frac{ c_i + \sum_j \sigma(r_{ij})c_j }{ 1 + \sum_j \sigma(r_{ij}) }$

where the $$c_i$$ and $$c_j$$ values can be for any one of the symmetry functions that can be calculated using plumed multicolvars. The function $$\sigma( r_{ij} )$$ is a switchingfunction that acts on the distance between atoms $$i$$ and $$j$$. Lechner and Dellago suggest that the parameters of this function should be set so that it the function is equal to one when atom $$j$$ is in the first coordination sphere of atom $$i$$ and is zero otherwise.

The $$s_i$$ quantities calculated using the above command can be again thought of as atom-centered symmetry functions. They thus operate much like multicolvars. You can thus calculate properties of the distribution of $$s_i$$ values using MEAN, LESS_THAN, HISTOGRAM and so on. You can also probe the value of these averaged variables in regions of the box by using the command in tandem with the AROUND command.

Examples

This example input calculates the coordination numbers for all the atoms in the system. These coordination numbers are then averaged over spherical regions. The number of averaged coordination numbers that are greater than 4 is then output to a file.

Click on the labels of the actions for more information on what each action computes d1: COORDINATIONNUMBER SPECIESthis keyword is used for colvars such as coordination number. =1-64 D_0 could not find this keyword =1.3 R_0 could not find this keyword =0.2
la: LOCAL_AVERAGE SPECIESthis keyword is used for colvars such as coordination number. =d1 SWITCHThis keyword is used if you want to employ an alternative to the continuous switching
function defined above. ={RATIONAL D_0=1.3 R_0=0.2}  MORE_THANcalculate the number of variables more than a certain target value. ={RATIONAL R_0=4}
PRINT ARGthe input for this action is the scalar output from one or more other actions. =la.* FILEthe name of the file on which to output these quantities =colvar


This example input calculates the $$q_4$$ (see Q4) vectors for each of the atoms in the system. These vectors are then averaged component by component over a spherical region. The average value for this quantity is then output to a file. This calculates the quantities that were used in the paper by Lechner and Dellago 

Click on the labels of the actions for more information on what each action computes q4: Q4 SPECIESthis keyword is used for colvars such as coordination number. =1-64 SWITCHThis keyword is used if you want to employ an alternative to the continuous switching
function defined above. ={RATIONAL D_0=1.3 R_0=0.2}
la: LOCAL_AVERAGE SPECIESthis keyword is used for colvars such as coordination number. =q4 SWITCHThis keyword is used if you want to employ an alternative to the continuous switching
function defined above. ={RATIONAL D_0=1.3 R_0=0.2}  MEANtake the mean of these variables.
PRINT ARGthe input for this action is the scalar output from one or more other actions. =la.* FILEthe name of the file on which to output these quantities =colvar

Glossary of keywords and components
Description of components

When the label of this action is used as the input for a second you are not referring to a scalar quantity as you are in regular collective variables. The label is used to reference the full set of quantities calculated by the action. This is usual when using MultiColvar functions. Generally when doing this the previously calculated multicolvar will be referenced using the DATA keyword rather than ARG.

This Action can be used to calculate the following scalar quantities directly. These quantities are calculated by employing the keywords listed below. These quantities can then be referenced elsewhere in the input file by using this Action's label followed by a dot and the name of the quantity. Some of them can be calculated multiple times with different parameters. In this case the quantities calculated can be referenced elsewhere in the input by using the name of the quantity followed by a numerical identifier e.g. label.lessthan-1, label.lessthan-2 etc. When doing this and, for clarity we have made it so that the user can set a particular label for each of the components. As such by using the LABEL keyword in the description of the keyword input you can customize the component name

 Quantity Keyword Description vmean VMEAN the norm of the mean vector. The output component can be referred to elsewhere in the input file by using the label.vmean vsum VSUM the norm of sum of vectors. The output component can be referred to elsewhere in the input file by using the label.vsum between BETWEEN the number/fraction of values within a certain range. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters. lessthan LESS_THAN the number of values less than a target value. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters. mean MEAN the mean value. The output component can be referred to elsewhere in the input file by using the label.mean moment MOMENTS the central moments of the distribution of values. The second moment would be referenced elsewhere in the input file using label.moment-2, the third as label.moment-3, etc. morethan MORE_THAN the number of values more than a target value. This is calculated using one of the formula described in the description of the keyword so as to make it continuous. You can calculate this quantity multiple times using different parameters.
The atoms involved can be specified using
 SPECIES this keyword is used for colvars such as coordination number. In that context it specifies that plumed should calculate one coordination number for each of the atoms specified. Each of these coordination numbers specifies how many of the other specified atoms are within a certain cutoff of the central atom. You can specify the atoms here as another multicolvar action or using a MultiColvarFilter or ActionVolume action. When you do so the quantity is calculated for those atoms specified in the previous multicolvar. This is useful if you would like to calculate the Steinhardt parameter for those atoms that have a coordination number more than four for example
Or alternatively by using
 SPECIESA this keyword is used for colvars such as the coordination number. In that context it species that plumed should calculate one coordination number for each of the atoms specified in SPECIESA. Each of these coordination numbers specifies how many of the atoms specifies using SPECIESB is within the specified cutoff. As with the species keyword the input can also be specified using the label of another multicolvar SPECIESB this keyword is used for colvars such as the coordination number. It must appear with SPECIESA. For a full explanation see the documentation for that keyword
Compulsory keywords
 NN ( default=6 ) The n parameter of the switching function MM ( default=0 ) The m parameter of the switching function; 0 implies 2*NN D_0 ( default=0.0 ) The d_0 parameter of the switching function R_0 The r_0 parameter of the switching function
Options
 NUMERICAL_DERIVATIVES ( default=off ) calculate the derivatives for these quantities numerically NOPBC ( default=off ) ignore the periodic boundary conditions when calculating distances SERIAL ( default=off ) do the calculation in serial. Do not use MPI TIMINGS ( default=off ) output information on the timings of the various parts of the calculation LOWMEM ( default=off ) lower the memory requirements SWITCH This keyword is used if you want to employ an alternative to the continuous switching function defined above. The following provides information on the switchingfunction that are available. When this keyword is present you no longer need the NN, MM, D_0 and R_0 keywords. MEAN take the mean of these variables. The final value can be referenced using label.mean. You can use multiple instances of this keyword i.e. MEAN1, MEAN2, MEAN3... The corresponding values are then referenced using label.mean-1, label.mean-2, label.mean-3... MORE_THAN calculate the number of variables more than a certain target value. This quantity is calculated using $$\sum_i 1.0 - \sigma(s_i)$$, where $$\sigma(s)$$ is a switchingfunction. The final value can be referenced using label.morethan. You can use multiple instances of this keyword i.e. MORE_THAN1, MORE_THAN2, MORE_THAN3... The corresponding values are then referenced using label.morethan-1, label.morethan-2, label.morethan-3... LESS_THAN calculate the number of variables less than a certain target value. This quantity is calculated using $$\sum_i \sigma(s_i)$$, where $$\sigma(s)$$ is a switchingfunction. The final value can be referenced using label.lessthan. You can use multiple instances of this keyword i.e. LESS_THAN1, LESS_THAN2, LESS_THAN3... The corresponding values are then referenced using label.lessthan-1, label.lessthan-2, label.lessthan-3... BETWEEN calculate the number of values that are within a certain range. These quantities are calculated using kernel density estimation as described on histogrambead. The final value can be referenced using label.between. You can use multiple instances of this keyword i.e. BETWEEN1, BETWEEN2, BETWEEN3... The corresponding values are then referenced using label.between-1, label.between-2, label.between-3... HISTOGRAM calculate how many of the values fall in each of the bins of a histogram. This shortcut allows you to calculates NBIN quantities like BETWEEN. The final value can be referenced using label.histogram. You can use multiple instances of this keyword i.e. HISTOGRAM1, HISTOGRAM2, HISTOGRAM3... The corresponding values are then referenced using label.histogram-1, label.histogram-2, label.histogram-3... MOMENTS calculate the moments of the distribution of collective variables. The mth moment of a distribution is calculated using $$\frac{1}{N} \sum_{i=1}^N ( s_i - \overline{s} )^m$$, where $$\overline{s}$$ is the average for the distribution. The moments keyword takes a lists of integers as input or a range. Each integer is a value of $$m$$. The final calculated values can be referenced using moment- $$m$$. You can use the COMPONENT keyword in this action but the syntax is slightly different. If you would like the second and third moments of the third component you would use MOMENTS={COMPONENT=3 MOMENTS=2-3}. The moments would then be referred to using the labels moment-3-2 and moment-3-3. This syntax is also required if you are using numbered MOMENT keywords i.e. MOMENTS1, MOMENTS2... VMEAN calculate the norm of the mean vector. The final value can be referenced using label.vmean. You can use multiple instances of this keyword i.e. VMEAN1, VMEAN2, VMEAN3... The corresponding values are then referenced using label.vmean-1, label.vmean-2, label.vmean-3... VSUM calculate the norm of the sum of vectors. The final value can be referenced using label.vsum. You can use multiple instances of this keyword i.e. VSUM1, VSUM2, VSUM3... The corresponding values are then referenced using label.vsum-1, label.vsum-2, label.vsum-3...