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# fix brownian command

# fix brownian/sphere command

# fix brownian/asphere 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 or rotation_style or rotation_temp or planar_rotation

      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
      rotation_style values = geometric or projection for brownian/sphere
         geometric = geometric, rotation-based integration scheme
         projection = projection-based integration scheme
      rotation_temp values = T for brownian/sphere and brownian/asphere
         T = rotation temperature, which can be different than temp when out of equilibrium
      planar_rotation values = none (constrains rotational diffusion to be in xy plane if in 3D)

## 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/sphere 1.0 19581092 gamma_t 1.0 gamma_r 0.3 rng gaussian rotation_style projection
    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

\[d\mathbf{r} = \boldsymbol{\gamma}_t^{-1}\mathbf{F}dt + \sqrt{2k_B T}\boldsymbol{\gamma}_t^{-1/2}d\mathbf{W}_t,\]

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

The overdamped rotational motion for the spherical and ellipsoidal particles results from random rotations instead of translations, which are chosen in such a way that the motion reproduces the equilibrium Boltzmann distribution for the case of conservative torques (see (Hoefling), (Ilie), and (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. The equation of motion for the dipole is

\[\boldsymbol{\mu}(t+dt) = \mathrm{R}(\boldsymbol{\omega} dt) \, \boldsymbol{\mu}(t) \,,\]

where \(\mathrm{R}(\boldsymbol{\omega} dt)\) is a rotation matrix with axis \(\boldsymbol{n} = \boldsymbol{\omega} / |\boldsymbol{\omega}|\) and angle \(\theta = |\boldsymbol{\omega}| dt\) (see (Hoefling)). For small angles, the action of the rotation matrix can be cast into a tangential increment \(\boldsymbol{\omega} \times \boldsymbol{\mu}dt\) and subsequent projection to preserve the magnitude \(|\boldsymbol{\mu}(t)|\) of the dipole (see (Ilie)):

\[\boldsymbol{\mu}(t+dt) = |\boldsymbol{\mu}(t)| \, \frac{\boldsymbol{\mu}(t) + \boldsymbol{\omega} \times \boldsymbol{\mu}dt }{|\boldsymbol{\mu}(t) + \boldsymbol{\omega} \times \boldsymbol{\mu}dt|}\,.\]

For suitable time step \(dt\), both expressions were shown to correctly reproduce the Boltzmann distribution of orientations and rotational diffusion moments when

\[\boldsymbol{\omega} = \frac{\mathbf{T}}{\gamma_r} + \sqrt{\frac{2 k_B T_{rot}}{\gamma_r}\frac{d\mathbf{W}}{dt}},\]

with \(d\mathbf{W}\) being a random number with zero mean and variance \(dt\) and \(T_{rot}\) is rotation_temp. The geometric integration scheme, however, accepts time steps that can be an order of magnitude larger (see (Hoefling)).

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. In this case, the rotational equation of motion is updated via the quaternion

\[\mathbf{q}(t+dt) = \frac{\mathbf{q}(t) + d\mathbf{q}}{\lVert\mathbf{q}(t) + d\mathbf{q}\rVert}\]

which correctly reproduces a Boltzmann distribution of orientations and rotational diffusion moments [see (Ilie)] when the quaternion step is given by

\[d\mathbf{q} = \boldsymbol{\Psi}\boldsymbol{\omega}dt\]

where \(\boldsymbol{\Psi}\) has rows \((-q_1,-q_2,-q_3)\), \((q_0,-q_3,q_2)\), \((q_3,q_0,-q_1)\), and \((-q_2,q_1,q_0)\). \(\boldsymbol{\omega}\) is evaluated in the body frame of reference where the friction tensor is diagonal. See (Delong) for more details of a similar algorithm.

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. For example, 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 sphere’s diameter). A similar (though more complex) relationship holds for ellipsoids and rod-like particles. The translational diffusion and rotational diffusion are given by temp/gamma_t and rotation_temp/gamma_r.

Note

Temperature computation using the compute temp will not correctly compute the 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 x–y plane), and the third value of gamma_t_eigen must be inf (only diffusion in the x–y 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).

Added in version 11Feb2026.

If the rotation_style keyword is used with the geometric value, then the geometric, rotation-based integration scheme ((Hoefling)) is used. If the keyword is used with the projection value, the linearized, projection-based scheme ((Ilie)) is used. Only compatible with brownian/sphere.

Note: rotation_style projection reproduces the legacy behavior (the former default).

If the rotation_temp keyword is used, then the rotational diffusion will occur at this prescribed temperature instead of temp. Only compatible with brownian/sphere and brownian/asphere.

If the planar_rotation keyword is used, then rotation is constrained to the x– y plane in a 3D simulation. Only compatible with brownian/sphere and brownian/asphere in 3D.

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 BROWNIAN package. It is only enabled if LAMMPS was built with that package. See the Build package doc page for more info.

## Related commands

fix propel/self, fix langevin, fix nve/sphere,

## Default

Changed in version 11Feb2026.

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

(Hoefling) \(\mathrm{H\ddot{o}fling}\) and Straube, Physical Review Research, 7, 043034 (2025).

(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).