AVERAGE

This is part of the analysis module

Calculate the ensemble average of a collective variable

The ensemble average for a non-periodic, collective variable, \(s\) is given by the following expression:

\[ \langle s \rangle = \frac{ \sum_{t'=0}^t w(t') s(t') }{ \sum_{t'=0}^t w(t') } \]

Here the sum runs over a the trajectory and \(s(t')\) is used to denote the value of the collective variable at time \(t'\). The final quantity evaluated is a weighted average as the weights, \(w(t')\), allow us to negate the effect any bias might have on the region of phase space sampled by the system. This is discussed in the section of the manual on Analysis.

When the variable is periodic (e.g. TORSION) and has a value, \(s\), in \(a \le s \le b\) the ensemble average is evaluated using:

\[ \langle s \rangle = a + \frac{b - a}{2\pi} \arctan \left[ \frac{ \sum_{t'=0}^t w(t') \sin\left( \frac{2\pi [s(t')-a]}{b - a} \right) }{ \sum_{t'=0}^t w(t') \cos\left( \frac{2\pi [s(t')-a]}{b - a} \right) } \right] \]

Examples

The following example calculates the ensemble average for the distance between atoms 1 and 2 and output this to a file called COLVAR. In this example it is assumed that no bias is acting on the system and that the weights, \(w(t')\) in the formulas above can thus all be set equal to one.

Click on the labels of the actions for more information on what each action computes

contract shortcuts

d1: DISTANCE 
ATOMS
the pair of atom that we are calculating the distance between. 
=1,2 The DISTANCE action with label d1 calculates a single scalar value
d1a: AVERAGE 
ARG
compulsory keyword 
the quantity that is being averaged 
=d1 The AVERAGE action with label d1a calculates a single scalar value
PRINT 
ARG
the input for this action is the scalar output from one or more other actions. 
=d1a 
FILE
the name of the file on which to output these quantities 
=colvar 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the quantities of interest should be output 
=100 The PRINT action with label 

d1: DISTANCE 
ATOMS
the pair of atom that we are calculating the distance between. 
=1,2 The DISTANCE action with label d1 calculates a single scalar value
d1a_weight: ONES 
SIZE
compulsory keyword 
the number of ones that you would like to create 
=1 The ONES action with label d1a_weight calculates a single scalar value
d1a_denom: ACCUMULATE 
ARG
the input for this action is the scalar output from one or more other actions. 
=d1a_weight 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the data should be collected and added to the quantity being
averaged 
=1 
CLEAR
compulsory keyword ( default=0 )
the frequency with which to clear all the accumulated data. 
=0 The ACCUMULATE action with label d1a_denom calculates a single scalar value
d1a_prod: CUSTOM 
ARG
the input to this function. 
=d1,d1a_weight 
FUNC
compulsory keyword 
the function you wish to evaluate 
=x*y 
PERIODIC
compulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. 
=NO The CUSTOM action with label d1a_prod calculates a single scalar value
d1a_numer: ACCUMULATE 
ARG
the input for this action is the scalar output from one or more other actions. 
=d1a_prod 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the data should be collected and added to the quantity being
averaged 
=1 
CLEAR
compulsory keyword ( default=0 )
the frequency with which to clear all the accumulated data. 
=0 The ACCUMULATE action with label d1a_numer calculates a single scalar value
d1a: CUSTOM 
ARG
the input to this function. 
=d1a_numer,d1a_denom 
FUNC
compulsory keyword 
the function you wish to evaluate 
=x/y 
PERIODIC
compulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. 
=NO The CUSTOM action with label d1a calculates a single scalar value
PRINT 
ARG
the input for this action is the scalar output from one or more other actions. 
=d1a 
FILE
the name of the file on which to output these quantities 
=colvar 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the quantities of interest should be output 
=100 The PRINT action with label 

The following example calculates the ensemble average for the torsional angle involving atoms 1, 2, 3 and 4. At variance with the previous example this quantity is periodic so the second formula in the above introduction is used to calculate the average. Furthermore, by using the CLEAR keyword we have specified that block averages are to be calculated. Consequently, after 100 steps all the information acquired thus far in the simulation is forgotten and the process of averaging is begun again. The quantities output in the colvar file are thus the block averages taken over the first 100 frames of the trajectory, the block average over the second 100 frames of trajectory and so on.

Click on the labels of the actions for more information on what each action computes

contract shortcuts

t1: TORSION 
ATOMS
the four atoms involved in the torsional angle 
=1,2,3,4 The TORSION action with label t1 calculates a single scalar value
t1a: AVERAGE 
ARG
compulsory keyword 
the quantity that is being averaged 
=t1 
CLEAR
compulsory keyword ( default=0 )
the frequency with whihc to clear the data that is being averaged 
=100 The AVERAGE action with label t1a calculates a single scalar value
PRINT 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1a 
FILE
the name of the file on which to output these quantities 
=colvar 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the quantities of interest should be output 
=100 The PRINT action with label 

t1: TORSION 
ATOMS
the four atoms involved in the torsional angle 
=1,2,3,4 The TORSION action with label t1 calculates a single scalar value
t1a_weight: ONES 
SIZE
compulsory keyword 
the number of ones that you would like to create 
=1 The ONES action with label t1a_weight calculates a single scalar value
t1a_denom: ACCUMULATE 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1a_weight 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the data should be collected and added to the quantity being
averaged 
=1 
CLEAR
compulsory keyword ( default=0 )
the frequency with which to clear all the accumulated data. 
=100 The ACCUMULATE action with label t1a_denom calculates a single scalar value
t1a_sin: CUSTOM 
ARG
the input to this function. 
=t1,t1a_weight 
FUNC
compulsory keyword 
the function you wish to evaluate 
=y*sin((x--pi)/((pi--pi)/(pi+pi))) 
PERIODIC
compulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. 
=NO The CUSTOM action with label t1a_sin calculates a single scalar value
t1a_cos: CUSTOM 
ARG
the input to this function. 
=t1,t1a_weight 
FUNC
compulsory keyword 
the function you wish to evaluate 
=y*cos((x--pi)/((pi--pi)/(pi+pi))) 
PERIODIC
compulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. 
=NO The CUSTOM action with label t1a_cos calculates a single scalar value
t1a_sinsum: ACCUMULATE 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1a_sin 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the data should be collected and added to the quantity being
averaged 
=1 
CLEAR
compulsory keyword ( default=0 )
the frequency with which to clear all the accumulated data. 
=100 The ACCUMULATE action with label t1a_sinsum calculates a single scalar value
t1a_cossum: ACCUMULATE 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1a_cos 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the data should be collected and added to the quantity being
averaged 
=1 
CLEAR
compulsory keyword ( default=0 )
the frequency with which to clear all the accumulated data. 
=100 The ACCUMULATE action with label t1a_cossum calculates a single scalar value
t1a: CUSTOM 
ARG
the input to this function. 
=t1a_sinsum,t1a_cossum,t1a_denom 
FUNC
compulsory keyword 
the function you wish to evaluate 
=-pi+((pi--pi)/(pi+pi))*atan2(x/z,y/z) 
PERIODIC
compulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. 
=-pi,pi The CUSTOM action with label t1a calculates a single scalar value
PRINT 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1a 
FILE
the name of the file on which to output these quantities 
=colvar 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the quantities of interest should be output 
=100 The PRINT action with label 

This third example incorporates a bias. Notice that the effect the bias has on the ensemble average is removed by taking advantage of the REWEIGHT_BIAS method. The final ensemble averages output to the file are thus block ensemble averages for the unbiased canonical ensemble at a temperature of 300 K.

Click on the labels of the actions for more information on what each action computes

contract shortcuts

t1: TORSION 
ATOMS
the four atoms involved in the torsional angle 
=1,2,3,4 The TORSION action with label t1 calculates a single scalar value
RESTRAINT 
ARG
the arguments on which the bias is acting. 
=t1 
AT
compulsory keyword 
the position of the restraint 
=pi 
KAPPA
compulsory keyword ( default=0.0 )
specifies that the restraint is harmonic and what the values of the force constants
on each of the variables are 
=100. The RESTRAINT action with label 
ww: REWEIGHT_BIAS 
TEMP
the system temperature. 
=300 The REWEIGHT_BIAS action with label ww calculates a single scalar value
t1a: AVERAGE 
ARG
compulsory keyword 
the quantity that is being averaged 
=t1 
LOGWEIGHTS
the logarithm of the quantity to use as the weights when calculating averages 
=ww 
CLEAR
compulsory keyword ( default=0 )
the frequency with whihc to clear the data that is being averaged 
=100 The AVERAGE action with label t1a calculates a single scalar value
PRINT 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1a 
FILE
the name of the file on which to output these quantities 
=colvar 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the quantities of interest should be output 
=100 The PRINT action with label 

t1: TORSION 
ATOMS
the four atoms involved in the torsional angle 
=1,2,3,4 The TORSION action with label t1 calculates a single scalar value
@2: RESTRAINT_SCALAR 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1 
AT
compulsory keyword 
the position of the restraint 
=pi 
KAPPA
compulsory keyword ( default=0.0 )
specifies that the restraint is harmonic and what the values of the force constants
on each of the variables are 
=100. The RESTRAINT_SCALAR action with label @2 calculates the following quantities:

 Quantity   
 Description  
@2.bias
the instantaneous value of the bias potential
@2.force2
the instantaneous value of the squared force due to this bias potential


ww: REWEIGHT_BIAS 
TEMP
the system temperature. 
=300 The REWEIGHT_BIAS action with label ww calculates a single scalar value
t1a_wsum: COMBINE 
ARG
the input to this function. 
=ww 
PERIODIC
compulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. 
=NO The COMBINE action with label t1a_wsum calculates a single scalar value
t1a_weight: CUSTOM 
ARG
the input to this function. 
=t1a_wsum 
FUNC
compulsory keyword 
the function you wish to evaluate 
=exp(x) 
PERIODIC
compulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. 
=NO The CUSTOM action with label t1a_weight calculates a single scalar value
t1a_denom: ACCUMULATE 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1a_weight 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the data should be collected and added to the quantity being
averaged 
=1 
CLEAR
compulsory keyword ( default=0 )
the frequency with which to clear all the accumulated data. 
=100 The ACCUMULATE action with label t1a_denom calculates a single scalar value
t1a_sin: CUSTOM 
ARG
the input to this function. 
=t1,t1a_weight 
FUNC
compulsory keyword 
the function you wish to evaluate 
=y*sin((x--pi)/((pi--pi)/(pi+pi))) 
PERIODIC
compulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. 
=NO The CUSTOM action with label t1a_sin calculates a single scalar value
t1a_cos: CUSTOM 
ARG
the input to this function. 
=t1,t1a_weight 
FUNC
compulsory keyword 
the function you wish to evaluate 
=y*cos((x--pi)/((pi--pi)/(pi+pi))) 
PERIODIC
compulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. 
=NO The CUSTOM action with label t1a_cos calculates a single scalar value
t1a_sinsum: ACCUMULATE 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1a_sin 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the data should be collected and added to the quantity being
averaged 
=1 
CLEAR
compulsory keyword ( default=0 )
the frequency with which to clear all the accumulated data. 
=100 The ACCUMULATE action with label t1a_sinsum calculates a single scalar value
t1a_cossum: ACCUMULATE 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1a_cos 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the data should be collected and added to the quantity being
averaged 
=1 
CLEAR
compulsory keyword ( default=0 )
the frequency with which to clear all the accumulated data. 
=100 The ACCUMULATE action with label t1a_cossum calculates a single scalar value
t1a: CUSTOM 
ARG
the input to this function. 
=t1a_sinsum,t1a_cossum,t1a_denom 
FUNC
compulsory keyword 
the function you wish to evaluate 
=-pi+((pi--pi)/(pi+pi))*atan2(x/z,y/z) 
PERIODIC
compulsory keyword 
if the output of your function is periodic then you should specify the periodicity
of the function. 
=-pi,pi The CUSTOM action with label t1a calculates a single scalar value
PRINT 
ARG
the input for this action is the scalar output from one or more other actions. 
=t1a 
FILE
the name of the file on which to output these quantities 
=colvar 
STRIDE
compulsory keyword ( default=1 )
the frequency with which the quantities of interest should be output 
=100 The PRINT action with label 

Glossary of keywords and components

Compulsory keywords

ARG	the quantity that is being averaged
STRIDE	( default=1 ) the frequency with which to store data for averaging
CLEAR	( default=0 ) the frequency with whihc to clear the data that is being averaged

Options

LOGWEIGHTS

the logarithm of the quantity to use as the weights when calculating averages