# compute entropy/atom command

## Syntax

``` LAMMPS
compute ID group-ID entropy/atom sigma cutoff keyword value ...
```

-   ID, group-ID are documented in [compute](compute) command
-   entropy/atom = style name of this compute command
-   sigma = width of Gaussians used in the $g(r)$ smoothing
-   cutoff = cutoff for the $g(r)$ calculation
-   one or more keyword/value pairs may be appended

<!-- -->

    keyword = *avg* or *local*
      *avg* args = neigh cutoff2
        neigh value = *yes* or *no* = whether to average the pair entropy over neighbors
        cutoff2 = cutoff for the averaging over neighbors
      *local* arg = *yes* or *no* = use the local density around each atom to normalize the g(r)

## Examples

``` LAMMPS
compute 1 all entropy/atom 0.25 5.
compute 1 all entropy/atom 0.25 5. avg yes 5.
compute 1 all entropy/atom 0.125 7.3 avg yes 5.1 local yes
```

## Description

Define a computation that calculates the pair entropy fingerprint for
each atom in the group. The fingerprint is useful to distinguish between
ordered and disordered environments, for instance liquid and solid-like
environments, or glassy and crystalline-like environments. Some
applications could be the identification of grain boundaries, a
melt-solid interface, or a solid cluster emerging from the melt. The
advantage of this parameter over others is that no a priori information
about the solid structure is required.

This parameter for atom i is computed using the following formula from
[(Piaggi)](Piaggi) and [(Nettleton)](Nettleton) ,

$$s_S^i=-2\pi\rho k_B \int\limits_0^{r_m} \left [ g(r) \ln g(r) - g(r) + 1 \right ] r^2 dr$$

where $r$ is a distance, $g(r)$ is the radial distribution function of
atom $i$, and $\rho$ is the density of the system. The $g(r)$ computed
for each atom $i$ can be noisy and therefore it is smoothed using

$$g_m^i(r) = \frac{1}{4 \pi \rho r^2} \sum\limits_{j} \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(r-r_{ij})^2/(2\sigma^2)}$$

where the sum over $j$ goes through the neighbors of atom $i$ and
$\sigma$ is a parameter to control the smoothing.

The input parameters are *sigma* the smoothing parameter $\sigma$, and
the *cutoff* for the calculation of $g(r)$.

If the keyword *avg* has the setting *yes*, then this compute also
averages the parameter over the neighbors of atom $i$ according to

$$\left< s_S^i \right>  = \frac{\sum_j s_S^j + s_S^i}{N + 1},$$

where the sum over $j$ goes over the neighbors of atom $i$ and $N$ is
the number of neighbors. This procedure provides a sharper distinction
between order and disorder environments. In this case the input
parameter *cutoff2* is the cutoff for the averaging over the neighbors
and must also be specified.

If the *avg yes* option is used, the effective cutoff of the neighbor
list should be *cutoff*+\*cutoff2\* and therefore it might be necessary
to increase the skin of the neighbor list with:

    neighbor <skin distance> bin

See [neighbor](neighbor) for details.

If the *local yes* option is used, the $g(r)$ is normalized by the local
density around each atom, that is to say the density around each atom is
the number of neighbors within the neighbor list cutoff divided by the
corresponding volume. This option can be useful when dealing with
inhomogeneous systems such as those that have surfaces.

Here are typical input parameters for fcc aluminum (lattice constant
$4.05~\AA$),

    compute 1 all entropy/atom 0.25 5.7 avg yes 3.7

and for bcc sodium (lattice constant $4.23~\AA$),

    compute 1 all entropy/atom 0.25 7.3 avg yes 5.1

## Output info

By default, this compute calculates the pair entropy value for each atom
as a per-atom vector, which can be accessed by any command that uses
per-atom values from a compute as input. See the [Howto
output](Howto_output) page for an overview of LAMMPS output options.

The pair entropy values have units of the Boltzmann constant. They are
always negative, and lower values (lower entropy) correspond to more
ordered environments.

## Restrictions

This compute is part of the EXTRA-COMPUTE package. It is only enabled if
LAMMPS was built with that package. See the [Build
package](Build_package) page for more info.

## Related commands

[compute cna/atom](compute_cna_atom) [compute
centro/atom](compute_centro_atom)

## Default

The default values for the optional keywords are avg = no and local =
no.

------------------------------------------------------------------------

::: {#Piaggi}
**(Piaggi)** Piaggi and Parrinello, J Chem Phys, 147, 114112 (2017).
:::

::: {#Nettleton}
**(Nettleton)** Nettleton and Green, J Chem Phys, 29, 6 (1958).
:::