# fix brownian/sphere command¶

## Syntax¶

fix ID group-ID style_name temp seed keyword args

• ID, group-ID are documented in fix command

• style_name = brownian or brownian/sphere or brownian/asphere

• temp = temperature

• seed = random number generator seed

• one or more keyword/value pairs may be appended

• keyword = rng or dipole or gamma_r_eigen or gamma_t_eigen or gamma_r or gamma_t

rng value = uniform or gaussian or none
uniform = use uniform random number generator
gaussian = use gaussian random number generator
none = turn off noise
dipole value = mux and muy and muz for brownian/asphere
mux, muy, and muz = update orientation of dipole having direction (mux,*muy*,*muz*) in body frame of rigid body
gamma_r_eigen values = gr1 and gr2 and gr3 for brownian/asphere
gr1, gr2, and gr3 = diagonal entries of body frame rotational friction tensor
gamma_r values = gr for brownian/sphere
gr = magnitude of the (isotropic) rotational friction tensor
gamma_t_eigen values = gt1 and gt2 and gt3 for brownian/asphere
gt1, gt2, and gt3 = diagonal entries of body frame translational friction tensor
gamma_t values = gt for brownian and brownian/sphere
gt = magnitude of the (isotropic) translational friction tensor

## Examples¶

fix 1 all brownian 1.0 12908410 gamma_t 1.0
fix 1 all brownian 1.0 12908410 gamma_t 3.0 rng gaussian
fix 1 all brownian/sphere 1.0 1294019 gamma_t 3.0 gamma_r 1.0
fix 1 all brownian/sphere 1.0 19581092 gamma_t 1.0 gamma_r 0.3  rng none
fix 1 all brownian/asphere 1.0 1294019 gamma_t_eigen 1.0 2.0 3.0 gamma_r_eigen 4.0 7.0 8.0 rng gaussian
fix 1 all brownian/asphere 1.0 1294019 gamma_t_eigen 1.0 2.0 3.0 gamma_r_eigen 4.0 7.0 8.0 dipole 1.0 0.0 0.0


## Description¶

Perform Brownian Dynamics time integration to update position, velocity, dipole orientation (for spheres) and quaternion orientation (for ellipsoids, with optional dipole update as well) of all particles in the fix group in each timestep. Brownian Dynamics uses Newton’s laws of motion in the limit that inertial forces are negligible compared to viscous forces. The stochastic equation of motion for the center of mass positions is

$d\mathbf{r} = \mathbf{\gamma}_t^{-1}\mathbf{F}dt+\sqrt{2k_BT}\mathbf{\gamma}_t^{-1/2}d\mathbf{W}_t,$

in the lab-frame (i.e. $$\mathbf{\gamma}_t$$ is not diagonal, but only depends on orientation and so the noise is still additive).

The rotational motion for the spherical and ellipsoidal particles is not as simple an expression, but is chosen to replicate the Boltzmann distribution for the case of conservative torques (see (Ilie) or (Delong)).

For the style brownian, only the positions of the particles are updated. This is therefore suitable for point particle simulations.

For the style brownian/sphere, the positions of the particles are updated, and a dipole slaved to the spherical orientation is also updated. This style therefore requires the hybrid atom style atom_style dipole and atom_style sphere.

For the style brownian/asphere, the center of mass positions and the quaternions of ellipsoidal particles are updated. This fix style is suitable for equations of motion where the rotational and translational friction tensors can be diagonalized in a certain (body) reference frame.

Note

This integrator does not by default assume a relationship between the rotational and translational friction tensors, though such a relationship should exist in the case of no-slip boundary conditions between the particles and the surrounding (implicit) solvent. E.g. in the case of spherical particles, the condition $$\gamma_t=3\gamma_r/\sigma^2$$ must be explicitly accounted for by setting gamma_t to 3x and gamma_r to x (where $$\sigma$$ is the spherical diameter). A similar (though more complex) relationship holds for ellipsoids and rod-like particles.

Note

Temperature computation using the compute temp will not correctly compute temperature of these overdamped dynamics since we are explicitly neglecting inertial effects. Furthermore, this time integrator does not add the stochastic terms or viscous terms to the force and/or torques. Rather, they are just added in to the equations of motion to update the degrees of freedom.

If the rng keyword is used with the uniform value, then the noise is generated from a uniform distribution (see (Dunweg) for why this works). This is the same method of noise generation as used in fix_langevin.

If the rng keyword is used with the gaussian value, then the noise is generated from a gaussian distribution. Typically this added complexity is unnecessary, and one should be fine using the uniform value for reasons argued in (Dunweg).

If the rng keyword is used with the none value, then the noise terms are set to zero.

The gamma_t keyword sets the (isotropic) translational viscous damping. Required for (and only compatible with) brownian and brownian/sphere. The units of gamma_t are mass/time.

The gamma_r keyword sets the (isotropic) rotational viscous damping. Required for (and only compatible with) brownian/sphere. The units of gamma_r are mass*length**2/time.

The gamma_r_eigen, and gamma_t_eigen keywords are the eigenvalues of the rotational and viscous damping tensors (having the same units as their isotropic counterparts). Required for (and only compatible with) brownian/asphere. For a 2D system, the first two values of gamma_r_eigen must be inf (only rotation in xy plane), and the third value of gamma_t_eigen must be inf (only diffusion in xy plane).

If the dipole keyword is used, then the dipole moments of the particles are updated as described above. Only compatible with brownian/asphere (as brownian/sphere updates dipoles automatically).

Note

For style brownian/asphere, the components gamma_t_eigen =(x,x,x) and gamma_r_eigen = (y,y,y), the dynamics will replicate those of the brownian/sphere style with gamma_t = x and gamma_r = y.

## Restart, fix_modify, output, run start/stop, minimize info¶

No information about this fix is written to binary restart files. No global or per-atom quantities are stored by this fix for access by various output commands.

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¶

The style brownian/sphere fix requires that atoms store torque and angular velocity (omega) as defined by the atom_style sphere command. The style brownian/asphere fix requires that atoms store torque and quaternions as defined by the atom_style ellipsoid command. If the dipole keyword is used, they must also store a dipole moment as defined by the atom_style dipole command.

This fix is part of the USER-BROWNIAN package. It is only enabled if LAMMPS was built with that package. See the Build package doc page for more info.

## Default¶

The default for rng is uniform. The default for the rotational and translational friction tensors are the identity tensor.

(Ilie) Ilie, Briels, den Otter, Journal of Chemical Physics, 142, 114103 (2015).

(Delong) Delong, Usabiaga, Donev, Journal of Chemical Physics. 143, 144107 (2015)

(Dunweg) Dunweg and Paul, Int J of Modern Physics C, 2, 817-27 (1991).