# compute viscosity/cos command

## Syntax

``` LAMMPS
compute ID group-ID viscosity/cos
```

-   ID, group-ID are documented in [compute](compute) command
-   viscosity/cos = style name of this compute command

## Examples

``` LAMMPS
units    real
compute  cos all viscosity/cos
variable V equal c_cos[7]
variable A equal 0.02E-5  # A/fs^2
variable density equal density
variable lz equal lz
variable reciprocalViscosity equal v_V/${A}/v_density*39.4784/v_lz/v_lz*100  # 1/(Pa*s)
```

## Description

Define a computation that calculates the velocity amplitude of a group
of atoms with an cosine-shaped velocity profile and the temperature of
them after subtracting out the velocity profile before computing the
kinetic energy. A compute of this style can be used by any command that
computes a temperature (e.g., [thermo_modify](thermo_modify), [fix
npt](fix_nh)).

This command together with
[fix_accelerate/cos\<fix_accelerate_cos\>]{.title-ref} enables viscosity
calculation with periodic perturbation method, as described by
[Hess\<Hess1\>]{.title-ref}. An acceleration along the $x$-direction is
applied to the simulation system by using
[fix_accelerate/cos\<fix_accelerate_cos\>]{.title-ref} command. The
acceleration is a periodic function along the $z$-direction:

$$a_{x}(z) = A \cos \left(\frac{2 \pi z}{l_{z}}\right)$$

where $A$ is the acceleration amplitude, $l_z$ is the $z$-length of the
simulation box. At steady state, the acceleration generates a velocity
profile:

$$v_{x}(z) = V \cos \left(\frac{2 \pi z}{l_{z}}\right)$$

The generated velocity amplitude $V$ is related to the shear viscosity
$\eta$ by

$$V = \frac{A \rho}{\eta}\left(\frac{l_{z}}{2 \pi}\right)^{2},$$

and it can be obtained from ensemble average of the velocity profile via

$$V = \frac{\sum\limits_i 2 m_{i} v_{i, x} \cos \left(\frac{2 \pi z_i}{l_{z}}\right)}{\sum\limits_i m_{i}}$$

where $m_i$, $v_{i,x}$ and $z_i$ are the mass, $x$-component velocity,
and $z$-coordinate of a particle, respectively.

After the cosine-shaped collective velocity in the $x$-direction has
been subtracted for each atom, the temperature is calculated by the
formula

$$\text{KE} = \frac{\text{dim}}{2} N k_B T,$$

where KE is the total kinetic energy of the group of atoms (sum of
$\frac12 m v^2$), dim = 2 or 3 is the dimensionality of the simulation,
$N$ is the number of atoms in the group, $k_B$ is the Boltzmann
constant, and $T$ is the absolute temperature.

A kinetic energy tensor, stored as a six-element vector, is also
calculated by this compute for use in the computation of a pressure
tensor. The formula for the components of the tensor is the same as the
above formula, except that $v^2$ is replaced by $v_x v_y$ for the $xy$
component, and so on. The six components of the vector are ordered $xx$,
$yy$, $zz$, $xy$, $xz$, $yz$.

The number of atoms contributing to the temperature is assumed to be
constant for the duration of the run; use the *dynamic* option of the
[compute_modify](compute_modify) command if this is not the case.
However, in order to get meaningful result, the group ID of this compute
should be all.

The removal of the cosine-shaped velocity component by this command is
essentially computing the temperature after a \"bias\" has been removed
from the velocity of the atoms. If this compute is used with a fix
command that performs thermostatting then this bias will be subtracted
from each atom, thermostatting of the remaining thermal velocity will be
performed, and the bias will be added back in. Thermostatting fixes that
work in this way include [fix nvt](fix_nh), [fix
temp/rescale](fix_temp_rescale), [fix
temp/berendsen](fix_temp_berendsen), and [fix langevin](fix_langevin).

This compute subtracts out degrees of freedom due to fixes that
constrain molecular motion, such as [fix shake](fix_shake) and [fix
rigid](fix_rigid). This means that the temperature of groups of atoms
that include these constraints will be computed correctly. If needed,
the subtracted degrees of freedom can be altered using the *extra*
option of the [compute_modify](compute_modify) command.

See the [Howto thermostat](Howto_thermostat) page for a discussion of
different ways to compute temperature and perform thermostatting.

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

## Output info

This compute calculates a global scalar (the temperature) and a global
vector of length 7, which can be accessed by indices 1\--7. The first
six elements of the vector are the KE tensor, and the seventh is the
cosine-shaped velocity amplitude $V$, which can be used to calculate the
reciprocal viscosity, as shown in the example. These values can be used
by any command that uses global scalar or vector values from a compute
as input. See the [Howto output](Howto_output) page for an overview of
LAMMPS output options.

The scalar value calculated by this compute is \"intensive\". The first
six elements of vector values are \"extensive\", and the seventh element
of vector values is \"intensive\".

The scalar value will be in temperature [units](units). The first six
elements of vector values will be in energy [units](units). The seventh
element of vector value will be in velocity [units](units).

## Restrictions

This command is only available when LAMMPS was built with the MISC
package. Since this compute depends on [fix
accelerate/cos](fix_accelerate_cos) which can only work for 3d systems,
it cannot be used for 2d systems.

## Related commands

[fix accelerate/cos](fix_accelerate_cos)

## Default

> none

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

::: {#Hess1}
**(Hess)** Hess, B. The Journal of Chemical Physics 2002, 116 (1),
209-217.
:::