md.methods¶

Overview

`Brownian`	Brownian dynamics.
`Langevin`	Langevin dynamics.
`NPT`	NPT Integration via MTK barostat-thermostat.
`NVE`	NVE Integration via Velocity-Verlet
`NVT`	NVT Integration via the Nosé-Hoover thermostat.

Details

class hoomd.md.methods.Brownian(filter, kT, alpha=None)¶

Brownian dynamics.

Parameters

filter (hoomd.filter.ParticleFilter) – Subset of particles to apply this method to.
kT (hoomd.variant.Variant or float) – Temperature of the simulation (in energy units).
alpha (float) – When set, use \(\alpha d_i\) for the drag coefficient where \(d_i\) is particle diameter. Defaults to None.

Brownian integrates particles forward in time according to the overdamped Langevin equations of motion, sometimes called Brownian dynamics, or the diffusive limit.

\[ \begin{align}\begin{aligned}\frac{d\vec{x}}{dt} = \frac{\vec{F}_\mathrm{C} + \vec{F}_\mathrm{R}}{\gamma}\\\langle \vec{F}_\mathrm{R} \rangle = 0\\\langle |\vec{F}_\mathrm{R}|^2 \rangle = 2 d k T \gamma / \delta t\\\langle \vec{v}(t) \rangle = 0\\\langle |\vec{v}(t)|^2 \rangle = d k T / m\end{aligned}\end{align} \]

where \(\vec{F}_\mathrm{C}\) is the force on the particle from all potentials and constraint forces, \(\gamma\) is the drag coefficient, \(\vec{F}_\mathrm{R}\) is a uniform random force, \(\vec{v}\) is the particle’s velocity, and \(d\) is the dimensionality of the system. The magnitude of the random force is chosen via the fluctuation-dissipation theorem to be consistent with the specified drag and temperature, \(T\). When \(kT=0\), the random force \(\vec{F}_\mathrm{R}=0\).

Brownian uses the integrator from I. Snook, The Langevin and Generalised Langevin Approach to the Dynamics of Atomic, Polymeric and Colloidal Systems , 2007, section 6.2.5, with the exception that \(\vec{F}_\mathrm{R}\) is drawn from a uniform random number distribution.

In Brownian dynamics, particle velocities are completely decoupled from positions. At each time step, Brownian draws a new velocity distribution consistent with the current set temperature so that hoomd.compute.thermo will report appropriate temperatures and pressures if logged or needed by other commands.

Brownian dynamics neglects the acceleration term in the Langevin equation. This assumption is valid when overdamped: \(\frac{m}{\gamma} \ll \delta t\). Use Langevin if your system is not overdamped.

You can specify \(\gamma\) in two ways:

Specify \(\alpha\) which scales the particle diameter to \(\gamma = \alpha d_i\). The units of \(\alpha\) are mass / distance / time.
After the method object is created, specify the attribute gamma and gamma_r for rotational damping or random torque to assign them directly, with independent values for each particle type in the system.

Examples:

brownian = hoomd.md.methods.Brownian(filter=hoomd.filter.All(), kT=0.2,
alpha=1.0)
integrator = hoomd.md.Integrator(dt=0.001, methods=[brownian],
forces=[lj])

Examples of using gamma pr gamma_r on drag coefficient:

brownian = hoomd.md.methods.Brownian(filter=hoomd.filter.All(), kT=0.2)
brownian.gamma.default = 2.0
brownian.gamma_r.default = [1.0, 2.0, 3.0]

filter¶

Subset of particles to apply this method to.

Type: hoomd.filter.ParticleFilter

kT¶

Temperature of the simulation (in energy units).

Type: hoomd.variant.Variant

alpha¶

When set, use \(\alpha d_i\) for the drag coefficient where \(d_i\) is particle diameter. Defaults to None.

Type: float

gamma¶

The drag coefficient can be directly set instead of the ratio of particle diameter (\(\gamma = \alpha d_i\)). The type of gamma parameter is either positive float or zero.

Type: TypeParameter[ particle type, float ]

gamma_r¶

The rotational drag coefficient can be set. The type of gamma_r parameter is a tuple of three float. The type of each element of tuple is either positive float or zero.

Type: TypeParameter[ particle type, [ float, float, float ] ]

class hoomd.md.methods.Langevin(filter, kT, alpha=None, tally_reservoir_energy=False)¶

Langevin dynamics.

Parameters

filter (hoomd.filter.ParticleFilter) – Subset of particles to apply this method to.
kT (hoomd.variant.Variant or float) – Temperature of the simulation (in energy units).
alpha (float) – When set, use \(\alpha d_i\) for the drag coefficient where \(d_i\) is particle diameter. Defaults to None.
tally_reservoir_energy (bool) – If true, the energy exchange between the thermal reservoir and the particles is tracked. Total energy conservation can then be monitored by adding langevin_reservoir_energy_groupname to the logged quantities. Defaults to False.

Translational degrees of freedom

Langevin integrates particles forward in time according to the Langevin equations of motion:

\[ \begin{align}\begin{aligned}m \frac{d\vec{v}}{dt} = \vec{F}_\mathrm{C} - \gamma \cdot \vec{v} + \vec{F}_\mathrm{R}\\\langle \vec{F}_\mathrm{R} \rangle = 0\\\langle |\vec{F}_\mathrm{R}|^2 \rangle = 2 d kT \gamma / \delta t\end{aligned}\end{align} \]

where \(\vec{F}_\mathrm{C}\) is the force on the particle from all potentials and constraint forces, \(\gamma\) is the drag coefficient, \(\vec{v}\) is the particle’s velocity, \(\vec{F}_\mathrm{R}\) is a uniform random force, and \(d\) is the dimensionality of the system (2 or 3). The magnitude of the random force is chosen via the fluctuation-dissipation theorem to be consistent with the specified drag and temperature, \(T\). When \(kT=0\), the random force \(\vec{F}_\mathrm{R}=0\).

Langevin dynamics includes the acceleration term in the Langevin equation and is useful for gently thermalizing systems using a small gamma. This assumption is valid when underdamped: \(\frac{m}{\gamma} \gg \delta t\). Use Brownian if your system is not underdamped.

Langevin uses the same integrator as NVE with the additional force term \(- \gamma \cdot \vec{v} + \vec{F}_\mathrm{R}\). The random force \(\vec{F}_\mathrm{R}\) is drawn from a uniform random number distribution.

You can specify \(\gamma\) in two ways:

Specify \(\alpha\) which scales the particle diameter to \(\gamma = \alpha d_i\). The units of \(\alpha\) are mass / distance / time.
After the method object is created, specify the attribute gamma and gamma_r for rotational damping or random torque to assign them directly, with independent values for each particle type in the system.

Warning

When restarting a simulation, the energy of the reservoir will be reset to zero.

Examples:

langevin = hoomd.md.methods.Langevin(filter=hoomd.filter.All(), kT=0.2,
alpha=1.0)
integrator = hoomd.md.Integrator(dt=0.001, methods=[langevin],
forces=[lj])

Examples of using gamma or gamma_r on drag coefficient:

langevin = hoomd.md.methods.Langevin(filter=hoomd.filter.All(), kT=0.2)
langevin.gamma.default = 2.0
langevin.gamma_r.default = [1.0,2.0,3.0]

filter¶

Subset of particles to apply this method to.

Type: hoomd.filter.ParticleFilter

kT¶

Temperature of the simulation (in energy units).

Type: hoomd.variant.Variant

alpha¶

When set, use \(\alpha d_i\) for the drag coefficient where \(d_i\) is particle diameter. Defaults to None.

Type: float

gamma¶

The drag coefficient can be directly set instead of the ratio of particle diameter (\(\gamma = \alpha d_i\)). The type of gamma parameter is either positive float or zero.

Type: TypeParameter[ particle type, float ]

gamma_r¶

The rotational drag coefficient can be set. The type of gamma_r parameter is a tuple of three float. The type of each element of tuple is either positive float or zero.

Type: TypeParameter[ particle type, [ float, float , float ]]

class hoomd.md.methods.NPT(filter, kT, tau, S, tauS, couple, box_dof=[True, True, True, False, False, False], rescale_all=False, gamma=0.0)¶

NPT Integration via MTK barostat-thermostat.

Parameters

filter (hoomd.filter.ParticleFilter) – Subset of particles on which to apply this method.
kT (hoomd.variant.Variant or float) – Temperature set point for the thermostat. (in energy units).
tau (float) – Coupling constant for the thermostat (in time units).
S (list [ hoomd.variant.Variant ] or float) – Stress components set point for the barostat (in pressure units). In Voigt notation: \([S_{xx}, S_{yy}, S_{zz}, S_{yz}, S_{xz}, S_{xy}]\). In case of isotropic pressure P (\([p, p, p, 0, 0, 0]\)), use S = p.
tauS (float) – Coupling constant for the barostat (in time units).
couple (str) – Couplings of diagonal elements of the stress tensor, can be “none”, “xy”, “xz”,”yz”, or “all”, default to “all”.
box_dof (list [ bool ]) – Box degrees of freedom with six boolean elements corresponding to x, y, z, xy, xz, yz, each. Default to [True,True,True,False,False,False]). If turned on to True, rescale corresponding lengths or tilt factors and components of particle coordinates and velocities.
rescale_all (bool) – if True, rescale all particles, not only those in the group, Default to False.
gamma (float) – Dimensionless damping factor for the box degrees of freedom, Default to 0.

NPT performs constant pressure, constant temperature simulations, allowing for a fully deformable simulation box.

The integration method is based on the rigorous Martyna-Tobias-Klein equations of motion for NPT. For optimal stability, the update equations leave the phase-space measure invariant and are manifestly time-reversible.

By default, NPT performs integration in a cubic box under hydrostatic pressure by simultaneously rescaling the lengths Lx, Ly and Lz of the simulation box.

NPT can also perform more advanced integration modes. The integration mode is specified by a set of couplings and by specifying the box degrees of freedom that are put under barostat control.

Couplings define which diagonal elements of the pressure tensor \(P_{\alpha,\beta}\) should be averaged over, so that the corresponding box lengths are rescaled by the same amount.

Valid couplings are:

none (all box lengths are updated independently)
xy (Lx and Ly are coupled)
xz (Lx and Lz are coupled)
yz (Ly and Lz are coupled)
all (Lx and Ly (and Lz if 3D) are coupled)

The default coupling is all, i.e. the ratios between all box lengths stay constant.

Degrees of freedom of the box specify which lengths and tilt factors of the box should be updated, and how particle coordinates and velocities should be rescaled. The box_dof tuple controls the way the box is rescaled and updated. The first three elements box_dof[:3] controls whether the x, y, and z box lengths are rescaled and updated, respectively. The last three entries box_dof[3:] control the rescaling or the tilt factors xy, xz, and yz. All options also appropriately rescale particle coordinates and velocities.

By default, the x, y, and z degrees of freedom are updated. [True,True,True,False,False,False]

Note

If any of the diagonal x, y, z degrees of freedom is not being integrated, pressure tensor components along that direction are not considered for the remaining degrees of freedom.

For example:

Specifying all couplings and x, y, and z degrees of freedom amounts to cubic symmetry (default)
Specifying xy couplings and x, y, and z degrees of freedom amounts to tetragonal symmetry.
Specifying no couplings and all degrees of freedom amounts to a fully deformable triclinic unit cell

For the MTK equations of motion, see:

G. J. Martyna, D. J. Tobias, M. L. Klein 1994
M. E. Tuckerman et. al. 2006
T. Yu et. al. 2010
Glaser et. al (2013), unpublished

\(\tau\) is related to the Nosé mass \(Q\) by

\[\tau = \sqrt{\frac{Q}{g k T_0}}\]

where \(g\) is the number of degrees of freedom, and \(k T_0\) is the set point (kT above).

The NPT equations of motion include a translational thermostat (with momentum \(\xi\) and position \(\eta\)), a rotational thermostat (with momentum \(\xi_{\mathrm{rot}}\) and position \(\eta_\mathrm{rot}\)), and a barostat tensor \(\nu_{\mathrm{ij}}\). Access these quantities using translational_thermostat_dof, rotational_thermostat_dof, and barostat_dof.

Note

Coupling constant for barostat tauS should be set within appropriate range for pressure and volume to fluctuate in reasonable rate and equilibrate. Too small tauS can cause abrupt fluctuation, whereas too large tauS would take long time to equilibrate. In most of systems, recommended value for tauS is 1000 * dt, where dt is the length of the time step.

Examples:

npt = hoomd.md.methods.NPT(filter=hoomd.filter.All(), tau=1.0, kT=0.65,
tauS = 1.2, S=2.0)
# orthorhombic symmetry
npt = hoomd.md.methods.NPT(filter=hoomd.filter.All(), tau=1.0, kT=0.65,
tauS = 1.2, S=2.0, couple="none")
# tetragonal symmetry
npt = hoomd.md.methods.NPT(filter=hoomd.filter.All(), tau=1.0, kT=0.65,
tauS = 1.2, S=2.0, couple="xy")
# triclinic symmetry
npt = hoomd.md.methods.NPT(filter=hoomd.filter.All(), tau=1.0, kT=0.65,
tauS = 1.2, S=2.0, couple="none", rescale_all=True)
integrator = hoomd.md.Integrator(dt=0.005, methods=[npt], forces=[lj])

filter¶

Subset of particles on which to apply this method.

Type: hoomd.filter.ParticleFilter

kT¶

Temperature set point for the thermostat. (in energy units).

Type: hoomd.variant.Variant

tau¶

Coupling constant for the thermostat (in time units).

Type: float

S¶

Stress components set point for the barostat (in pressure units). In Voigt notation, \([S_{xx}, S_{yy}, S_{zz}, S_{yz}, S_{xz}, S_{xy}]\). Stress can be reset after method object is created. For example, An isoropic pressure can be set by npt.S = 4.

Type: List[hoomd.variant.Variant]

tauS¶

Coupling constant for the barostat (in time units).

Type: float

couple¶

Couplings of diagonal elements of the stress tensor, can be “none”, “xy”, “xz”,”yz”, or “all”.

Type: str

box_dof¶

Box degrees of freedom with six boolean elements corresponding to x, y, z, xy, xz, yz, each.

Type: List[bool]

rescale_all¶

if True, rescale all particles, not only those in the group.

Type: bool

gamma¶

Dimensionless damping factor for the box degrees of freedom.

Type: float

translational_thermostat_dof¶

Additional degrees of freedom for the translational thermostat (\(\xi\), \(\eta\))

Type: tuple[float, float]

rotational_thermostat_dof¶

Additional degrees of freedom for the rotational thermostat (\(\xi_\mathrm{rot}\), \(\eta_\mathrm{rot}\))

Type: tuple[float, float]

barostat_dof¶

Additional degrees of freedom for the barostat (\(\nu_{xx}\), \(\nu_{xy}\), \(\nu_{xz}\), \(\nu_{yy}\), \(\nu_{yz}\), \(\nu_{zz}\))

Type: tuple[float, float, float, float, float, float]

thermalize_thermostat_and_barostat_dof()¶

Set the thermostat and barostat momenta to random values.

thermalize_thermostat_and_barostat_dof sets a random value for the momentum \(\xi\) and the barostat \(\nu_{\mathrm{ij}}\). When Integrator.aniso is True, it also sets a random value for the rotational thermostat momentum \(\xi_{\mathrm{rot}}\). Call thermalize_thermostat_and_barostat_dof to set a new random state for the thermostat and barostat.

Important

You must call Simulation.run before thermalize_thermostat_and_barostat_dof. Call run(steps=0) to prepare a newly created Simulation.