6. Integrators and thermostats

6.1. Particle integration and propagation

The main integration scheme of ESPResSo is the velocity Verlet algorithm. A steepest descent algorithm is used to minimize the system.

Additional integration schemes are available, which can be coupled to thermostats to enable Langevin dynamics, Brownian dynamics, Stokesian dynamics, dissipative particle dynamics, and simulations in the NpT ensemble.

6.2. Integrators

To run the integrator call the method system.integrate.run():

system.integrator.run(number_of_steps, recalc_forces=False, reuse_forces=False)

where number_of_steps is the number of time steps the integrator should perform.

6.2.1. Velocity Verlet algorithm

espressomd.integrate.IntegratorHandle.set_vv()

The equations of motion for the trajectory of point-like particles read

\[\begin{split}\dot v_i(t) = F_i(\{x_j\},v_i,t)/m_i \\ \dot x_i(t) = v_i(t),\end{split}\]

where \(x_i\), \(v_i\), \(m_i\) are position, velocity and mass of particle \(i\) and \(F_i(\{x_j\},v_i,t)\) the forces acting on it. These forces comprise all interactions with other particles and external fields as well as non-deterministic contributions described in Thermostats.

For numerical integration, this equation is discretized to the following steps ([Rapaport, 2004] eqs. 3.5.8 - 3.5.10):

  1. Calculate the velocity at the half step

    \[v(t+dt/2) = v(t) + \frac{F(x(t),v(t-dt/2),t)}{m} dt/2\]

  2. Calculate the new position

    \[x(t+dt) = x(t) + v(t+dt/2) dt\]

  3. Calculate the force based on the new position

    \[F = F(x(t+dt), v(t+dt/2), t+dt)\]

  4. Calculate the new velocity

    \[v(t+dt) = v(t+dt/2) + \frac{F(x(t+dt),t+dt)}{m} dt/2\]

Note that this implementation of the velocity Verlet algorithm reuses forces in step 1. That is, they are computed once in step 3, but used twice, in step 4 and in step 1 of the next iteration. In the first time step after setting up, there are no forces present yet. Therefore, ESPResSo has to compute them before the first time step. That has two consequences: first, random forces are redrawn, resulting in a narrower distribution of the random forces, which we compensate by stretching. Second, coupling forces of e.g. the lattice-Boltzmann fluid cannot be computed and are therefore lacking in the first half time step. In order to minimize these effects, ESPResSo has a quite conservative heuristics to decide whether a change makes it necessary to recompute forces before the first time step. Therefore, calling 100 times espressomd.integrate.Integrator.run() with steps=1 does the same as with steps=100, apart from some small calling overhead.

However, for checkpointing, there is no way for ESPResSo to tell that the forces that you read back in actually match the parameters that are set. Therefore, ESPResSo would recompute the forces before the first time step, which makes it essentially impossible to checkpoint LB simulations, where it is vital to keep the coupling forces. To work around this, there is an additional parameter reuse_forces, which tells integrate to not recalculate the forces for the first time step, but use that the values still stored with the particles. Use this only if you are absolutely sure that the forces stored match your current setup!

The opposite problem occurs when timing interactions: In this case, one would like to recompute the forces, despite the fact that they are already correctly calculated. To this aim, the option recalc_forces can be used to enforce force recalculation.

6.2.2. Isotropic NpT integrator

espressomd.integrate.IntegratorHandle.set_isotropic_npt()

As the NpT thermostat alters the way the equations of motion are integrated, it is discussed here and only a brief summary is given in Thermostats.

To activate the NpT integrator, use set_isotropic_npt() with parameters:

  • ext_pressure: The external pressure

  • piston: The mass of the applied piston

  • direction: Flags to enable/disable box dimensions to be subject to fluctuations. By default, all directions are enabled.

Additionally, a NpT thermostat has to be set by set_npt() with parameters:

  • kT: Thermal energy of the heat bath

  • gamma0: Friction coefficient of the bath

  • gammav: Artificial friction coefficient for the volume fluctuations.

A code snippet would look like:

import espressomd

system = espressomd.System(box_l=[1, 1, 1])
system.thermostat.set_npt(kT=1.0, gamma0=1.0, gammav=1.0, seed=42)
system.integrator.set_isotropic_npt(ext_pressure=1.0, piston=1.0)

The physical meaning of these parameters is described below:

The relaxation towards a desired pressure \(P\) (parameter ext_pressure) is enabled by treating the box volume \(V\) as a degree of freedom with corresponding momentum \(\Pi = Q\dot{V}\), where \(Q\) (parameter piston) is an artificial piston mass. Which box dimensions are affected to change the volume can be controlled by a list of boolean flags for parameter direction. An additional energy \(H_V = 1/(2Q)\Pi + PV\) associated with the volume is postulated. This results in a “force” on the box such that

\[\dot{\Pi} = \mathcal{P} - P\]

where

\[\mathcal{P} = \frac{1}{Vd} \sum_{i,j} f_{ij}x_{ij} + \frac{1}{Vd} \sum_i m_i v_i^2\]

Here \(\mathcal{P}\) is the instantaneous pressure, \(d\) the dimension of the system (number of flags set by direction), \(f_{ij}\) the short range interaction force between particles \(i\) and \(j\) and \(x_{ij}= x_j - x_i\).

In addition to this deterministic force, a friction \(-\frac{\gamma^V}{Q}\Pi(t)\) and noise \(\sqrt{k_B T \gamma^V} \eta(t)\) are added for the box volume dynamics and the particle dynamics. This introduces three new parameters: The friction coefficient for the box \(\gamma^V\) (parameter gammav), the friction coefficient of the particles \(\gamma^0\) (parameter gamma0) and the thermal energy \(k_BT\) (parameter kT). For a discussion of these terms and their discretisation, see Langevin thermostat, which uses the same approach, but only for particles. As a result of box geometry changes, the particle positions and velocities have to be rescaled during integration.

The discretisation consists of the following steps (see [Kolb and Dünweg, 1999] for a full derivation of the algorithm):

  1. Calculate the particle velocities at the half step

    \[v'(t+dt/2) = v(t) + \frac{F(x(t),v(t-dt/2),t)}{m} dt/2\]

  2. Calculate the instantaneous pressure and “volume momentum”

    \[\mathcal{P} = \mathcal{P}(x(t),V(t),f(x(t)), v'(t+dt/2))\]
    \[\Pi(t+dt/2) = \Pi(t) + (\mathcal{P}-P) dt/2 -\frac{\gamma^V}{Q}\Pi(t) dt/2 + \sqrt{k_B T \gamma^V dt} \overline{\eta}\]

  3. Calculate box volume and scaling parameter \(L\) at half step and full step, scale the simulation box accordingly

    \[V(t+dt/2) = V(t) + \frac{\Pi(t+dt/2)}{Q} dt/2\]
    \[L(t+dt/2) = V(t+dt/2)^{1/d}\]
    \[V(t+dt) = V(t+dt/2) + \frac{\Pi(t+dt/2)}{Q} dt/2\]
    \[L(t+dt) = V(t+dt)^{1/d}\]

  4. Update particle positions and scale velocities

    \[x(t+dt) = \frac{L(t+dt)}{L(t)} \left[ x(t) + \frac{L^2(t)}{L^2(t+dt/2)} v(t+dt/2) dt \right]\]
    \[v(t+dt/2) = \frac{L(t)}{L(t+dt)} v'(t+dt/2)\]

  5. Calculate forces, instantaneous pressure and “volume momentum”

    \[F = F(x(t+dt),v(t+dt/2),t)\]
    \[\mathcal{P} = \mathcal{P}(x(t+dt),V(t+dt),f(x(t+dt)), v(t+dt/2))\]
    \[\Pi(t+dt) = \Pi(t+dt/2) + (\mathcal{P}-P) dt/2 -\frac{\gamma^V}{Q}\Pi(t+dt/2) dt/2 + \sqrt{k_B T \gamma^V dt} \overline{\eta}\]

    with uncorrelated numbers \(\overline{\eta}\) drawn from a random uniform process \(\eta(t)\)

  6. Update the velocities

    \[v(t+dt) = v(t+dt/2) + \frac{F(t+dt)}{m} dt/2\]

Notes:

  • The NpT algorithm is only tested for all 3 directions enabled for scaling. Usage of direction is considered an experimental feature.

  • In step 4, only those coordinates are scaled for which direction is set.

  • For the instantaneous pressure, the same limitations of applicability hold as described in Pressure.

  • The particle forces \(F\) include interactions as well as a friction (\(\gamma^0\)) and noise term (\(\sqrt{k_B T \gamma^0 dt} \overline{\eta}\)) analogous to the terms in the Langevin thermostat.

  • The particle forces are only calculated in step 5 and then reused in step 1 of the next iteration. See Velocity Verlet algorithm for the implications of that.

  • The NpT algorithm doesn’t support Lees–Edwards boundary conditions.

  • The NpT algorithm doesn’t support propagation of angular velocities.

6.2.3. Steepest descent

espressomd.integrate.IntegratorHandle.set_steepest_descent()

This feature is used to propagate each particle by a small distance parallel to the force acting on it. When only conservative forces for which a potential exists are in use, this is equivalent to a steepest descent energy minimization. A common application is removing overlap between randomly placed particles.

Please note that the behavior is undefined if a thermostat is activated, in which case the integrator will generate an error. The integrator runs the following steepest descent algorithm:

\[\vec{r}_{i+1} = \vec{r}_i + \min(\gamma \vec{F}_i, \vec{r}_{\text{max_displacement}}),\]

while the maximal force/torque is bigger than f_max or for at most steps times. The energy is relaxed by gamma, while the change per coordinate per step is limited to max_displacement. The combination of gamma and max_displacement can be used to get a poor man’s adaptive update. Rotational degrees of freedom are treated similarly: each particle is rotated around an axis parallel to the torque acting on the particle, with max_displacement interpreted as the maximal rotation angle. Please be aware of the fact that this needs not to converge to a local minimum in periodic boundary conditions. Translational and rotational coordinates that are fixed using the fix and rotation attribute of particles are not altered.

Usage example:

system.integrator.set_steepest_descent(
    f_max=0, gamma=0.1, max_displacement=0.1)
system.integrator.run(20)   # maximal number of steps
system.integrator.set_vv()  # to switch back to velocity Verlet

6.2.3.1. Using a custom convergence criterion

The f_max parameter can be set to zero to prevent the integrator from halting when a specific force/torque is reached. The integration can then be carried out in a loop with a custom convergence criterion:

min_sigma = 1  # size of the smallest particle
max_sigma = 5  # size of the largest particle
min_dist = 0.0
system.integrator.set_steepest_descent(f_max=0, gamma=10,
                                       max_displacement=min_sigma * 0.01)
# gradient descent until particles are separated by at least max_sigma
while min_dist < max_sigma:
    min_dist = system.analysis.min_dist()
    system.integrator.run(10)
system.integrator.set_vv()

When writing a custom convergence criterion based on forces or torques, keep in mind that particles whose motion and rotation are fixed in space along some or all axes with fix or rotation need to be filtered from the force/torque observable used in the custom convergence criterion. Since these two properties can be cast to boolean values, they can be used as masks to remove forces/torques that are ignored by the integrator:

particles = system.part.all()
max_force = np.max(np.linalg.norm(particles.f * np.logical_not(particles.fix), axis=1))
max_torque = np.max(np.linalg.norm(particles.torque_lab * np.logical_not(particles.rotation), axis=1))

Virtual sites can also be an issue since the force on the virtual site is transferred to the target particle at the beginning of the integration loop. The correct forces need to be re-calculated after running the integration:

def convergence_criterion(forces):
    '''Function that decides when the gradient descent has converged'''
    return ...
p1 = system.part.add(pos=[0, 0, 0], type=1)
p2 = system.part.add(pos=[0, 0, 0.1], type=1)
p2.vs_auto_relate_to(p1)
system.integrator.set_steepest_descent(f_max=800, gamma=1.0, max_displacement=0.01)
while convergence_criterion(system.part.all().f):
    system.integrator.run(10)
    system.integrator.run(0, recalc_forces=True)  # re-calculate forces from virtual sites
system.integrator.set_vv()

The algorithm can also be used for energy minimization:

# minimize until energy difference < 5% or energy < 1e-3
system.integrator.set_steepest_descent(f_max=0, gamma=1.0, max_displacement=0.01)
relative_energy_change = float('inf')
relative_energy_change_threshold = 0.05
energy_threshold = 1e-3
energy_old = system.analysis.energy()['total']
print(f'Energy: {energy_old:.2e}')
for i in range(20):
    system.integrator.run(50)
    energy = system.analysis.energy()['total']
    print(f'Energy: {energy:.2e}')
    relative_energy_change = (energy_old - energy) / energy_old
    if relative_energy_change < relative_energy_change_threshold or energy < energy_threshold:
        break
    energy_old = energy
else:
    print(f'Energy minimization did not converge in {i + 1} cycles')
system.integrator.set_vv()

Please note that not all features support energy calculation. For example IBM and OIF do not implement energy calculation for mesh surface deformation.

6.2.4. Brownian Dynamics

Brownian Dynamics integrator [Schlick, 2010]. See details in Brownian thermostat.

6.2.5. Stokesian Dynamics

Note

Requires STOKESIAN_DYNAMICS external feature, enabled with -DWITH_STOKESIAN_DYNAMICS=ON.

espressomd.integrate.IntegratorHandle.set_stokesian_dynamics()

The Stokesian Dynamics method is used to model the behavior of spherical particles in a viscous fluid. It is targeted at systems with very low Reynolds numbers. In such systems, particles come to a rest almost immediately as soon as any force on them is removed. In other words, motion has no memory of the past.

The integration scheme is relatively simple. Only the particles’ positions, radii and forces (including torques) are needed to compute the momentary velocities (including angular velocities). The particle positions are integrated by the simple Euler scheme.

The computation of the velocities is an approximation with good results in the far field. The Stokesian Dynamics method is only available for open systems, i.e. no periodic boundary conditions are supported. The box size has no effect either.

The Stokesian Dynamics method is outlined in [Durlofsky et al., 1987].

The following minimal example illustrates how to use the SDM in ESPResSo:

import espressomd
system = espressomd.System(box_l=[1.0, 1.0, 1.0])
system.periodicity = [False, False, False]
system.time_step = 0.01
system.cell_system.skin = 0.4
system.part.add(pos=[0, 0, 0], rotation=[1, 0, 0])
system.integrator.set_stokesian_dynamics(viscosity=1.0, radii={0: 1.0})
system.integrator.run(100)

Because there is no force on the particle yet, nothing will move. You will need to add your own actors to the system. The parameter radii is a dictionary that maps particle types to different radii. viscosity is the dynamic viscosity of the ambient infinite fluid. There are additional optional parameters for set_stokesian_dynamics(). For more information, see espressomd.integrate.IntegratorHandle.set_stokesian_dynamics().

Note that this setup represents a system at zero temperature. In order to thermalize the system, the SD thermostat needs to be activated (see Stokesian thermostat).

6.2.5.1. Important

The particles must be prevented from overlapping. It is mathematically allowed for the particles to overlap to a certain degree. However, once the distance of the sphere centers is less than 2/3 of the sphere diameter, the mobility matrix is no longer positive definite and the Stokesian Dynamics integrator will fail. Therefore, the particle centers must be kept apart from each other by a strongly repulsive potential, for example the WCA potential that is set to the appropriate particle radius (for more information about the available interaction types see Non-bonded interactions).

The current implementation of SD only includes the far field approximation. The near field (so-called lubrication) correction is planned. For now, Stokesian Dynamics provides a good approximation of the hydrodynamics in dilute systems where the average distance between particles is several sphere diameters.

6.3. Thermostats

To add a thermostat, call the appropriate setter:

system.thermostat.set_langevin(kT=1.0, gamma=1.0, seed=41)

The different thermostats available in ESPResSo will be described in the following subsections.

You may combine different thermostats at your own risk by turning them on one by one. The list of active thermostats can be cleared at any time with system.thermostat.turn_off(). Not all combinations of thermostats are allowed, though (see espressomd.thermostat.AssertThermostatType() for details). Some integrators only work with a specific thermostat and throw an error otherwise. Note that there is only one temperature for all thermostats, although for some thermostats like the Langevin thermostat, particles can be assigned individual temperatures.

Since ESPResSo does not enforce a particular unit system, it cannot know about the current value of the Boltzmann constant. Therefore, when specifying the temperature of a thermostat, you actually do not define the temperature, but the value of the thermal energy \(k_B T\) in the current unit system (see the discussion on units, Section On units).

All thermostats have a seed argument that controls the state of the random number generator (Philox Counter-based RNG). This seed is required on first activation of a thermostat, unless stated otherwise. It can be omitted in subsequent calls of the method that activates the same thermostat. The random sequence also depends on the thermostats counters that are incremented after each integration step.

6.3.1. Langevin thermostat

In order to activate the Langevin thermostat the member function set_langevin() of the thermostat class espressomd.thermostat.Thermostat has to be invoked. Best explained in an example:

import espressomd
system = espressomd.System(box_l=[1, 1, 1])
system.thermostat.set_langevin(kT=1.0, gamma=1.0, seed=41)

As explained before the temperature is set as thermal energy \(k_\mathrm{B} T\).

The Langevin thermostat is based on an extension of Newton’s equation of motion to

\[m_i \dot{v}_i(t) = f_i(\{x_j\},v_i,t) - \gamma v_i(t) + \sqrt{2\gamma k_B T} \eta_i(t).\]

Here, \(f_i\) are all deterministic forces from interactions, \(\gamma\) the bare friction coefficient and \(\eta\) a random, “thermal” force. The friction term accounts for dissipation in a surrounding fluid whereas the random force mimics collisions of the particle with solvent molecules at temperature \(T\) and satisfies

\[<\eta(t)> = 0 , <\eta^\alpha_i(t)\eta^\beta_j(t')> = \delta_{\alpha\beta} \delta_{ij}\delta(t-t')\]

(\(<\cdot>\) denotes the ensemble average and \(\alpha,\beta\) are spatial coordinates).

In the ESPResSo implementation of the Langevin thermostat, the additional terms only enter in the force calculation. This reduces the accuracy of the velocity Verlet integrator by one order in \(dt\) because forces are now velocity-dependent.

The random process \(\eta(t)\) is discretized by drawing an uncorrelated random number \(\overline{\eta}\) for each component of all the particle forces. The distribution of \(\overline{\eta}\) is uniform and satisfies

\[<\overline{\eta}> = 0 , <\overline{\eta}\overline{\eta}> = 1/dt\]

If the feature ROTATION is compiled in, the rotational degrees of freedom are also coupled to the thermostat. If only the first two arguments are specified then the friction coefficient for the rotation is set to the same value as that for the translation. A separate rotational friction coefficient can be set by inputting gamma_rotate. The two options allow one to switch the translational and rotational thermalization on or off separately, maintaining the frictional behavior. This can be useful, for instance, in high Péclet number active matter systems, where one wants to thermalize only the rotational degrees of freedom while translational degrees of freedom are affected by the self-propulsion.

The keywords gamma and gamma_rotate can be specified as a scalar, or, with feature PARTICLE_ANISOTROPY compiled in, as the three eigenvalues of the respective friction coefficient tensor. This is enables the simulation of the anisotropic diffusion of anisotropic colloids (rods, etc.).

Using the Langevin thermostat, it is possible to set a temperature and a friction coefficient for every particle individually via the feature THERMOSTAT_PER_PARTICLE. Consult the reference of the part command (chapter Setting up particles) for information on how to achieve this.

6.3.2. Brownian thermostat

Brownian thermostat is a formal name of a thermostat enabling the Brownian Dynamics feature (see [Schlick, 2010]) which implies a propagation scheme involving systematic and thermal parts of the classical Ermak-McCammom’s (see [Ermak and McCammon, 1978]) Brownian Dynamics. Currently it is implemented without hydrodynamic interactions, i.e. with a diagonal diffusion tensor. The hydrodynamic interactions feature will be available later as a part of the present Brownian Dynamics or implemented separately within the Stokesian Dynamics.

In order to activate the Brownian thermostat, the member function set_brownian of the thermostat class espressomd.thermostat.Thermostat has to be invoked. The system integrator should be also changed. Best explained in an example:

import espressomd
system = espressomd.System(box_l=[1, 1, 1])
system.thermostat.set_brownian(kT=1.0, gamma=1.0, seed=41)
system.integrator.set_brownian_dynamics()

where gamma (hereinafter \(\gamma\)) is a viscous friction coefficient. In terms of the Python interface and setup, the Brownian thermostat is very similar to the Langevin thermostat. The feature THERMOSTAT_PER_PARTICLE is used to control the per-particle temperature and the friction coefficient setup. The major differences are its internal integrator implementation and other temporal constraints. The integrator is still a symplectic velocity Verlet-like one. It is implemented via a viscous drag part and a random walk of both the position and velocity. Due to a nature of the Brownian Dynamics method, its time step \(\Delta t\) should be large enough compared to the relaxation time \(m/\gamma\) where \(m\) is the particle mass. This requirement is just a conceptual one without specific implementation technical restrictions. Note that with all similarities of Langevin and Brownian Dynamics, the Langevin thermostat temporal constraint is opposite. A velocity is restarting from zero at every step. Formally, the previous step velocity at the beginning of the the \(\Delta t\) interval is dissipated further and does not contribute to the end one as well as to the positional random walk. Another temporal constraint which is valid for both Langevin and Brownian Dynamics: conservative forces should not change significantly over the \(\Delta t\) interval.

The viscous terminal velocity \(\Delta v\) and corresponding positional step \(\Delta r\) are fully driven by conservative forces \(F\):

\[\Delta r = \frac{F \cdot \Delta t}{\gamma}\]
\[\Delta v = \frac{F}{\gamma}\]

A positional random walk variance of each coordinate \(\sigma_p^2\) corresponds to a diffusion within the Wiener process:

\[\sigma_p^2 = 2 \frac{kT}{\gamma} \cdot \Delta t\]

Each velocity component random walk variance \(\sigma_v^2\) is defined by the heat component:

\[\sigma_v^2 = \frac{kT}{m}\]

Note: the velocity random walk is propagated from zero at each step.

A rotational motion is implemented similarly. Note: the rotational Brownian dynamics implementation is compatible with particles which have the isotropic moment of inertia tensor only. Otherwise, the viscous terminal angular velocity is not defined, i.e. it has no constant direction over the time.

6.3.3. Isotropic NpT thermostat

This feature allows to simulate an (on average) homogeneous and isotropic system in the NpT ensemble. In order to use this feature, NPT has to be defined in the myconfig.hpp. Activate the NpT thermostat with the command set_npt() and setup the integrator for the NpT ensemble with set_isotropic_npt().

For example:

import espressomd

system = espressomd.System(box_l=[1, 1, 1])
system.thermostat.set_npt(kT=1.0, gamma0=1.0, gammav=1.0, seed=41)
system.integrator.set_isotropic_npt(ext_pressure=1.0, piston=1.0)

For an explanation of the algorithm involved, see Isotropic NpT integrator.

Be aware that this feature is neither properly examined for all systems nor is it maintained regularly. If you use it and notice strange behavior, please contribute to solving the problem.

6.3.4. Dissipative Particle Dynamics (DPD)

The DPD thermostat adds friction and noise to the particle dynamics like the Langevin thermostat, but these are not applied to every particle individually but instead encoded in a dissipative interaction between particles [Soddemann et al., 2003].

To realize a complete DPD fluid model in ESPResSo, three parts are needed: the DPD thermostat, which controls the temperate, a dissipative interaction between the particles that make up the fluid, see DPD interaction, and a repulsive conservative force, see Hat interaction.

The temperature is set via espressomd.thermostat.Thermostat.set_dpd() which takes kT and seed as arguments.

The friction coefficients and cutoff are controlled via the DPD interaction on a per type-pair basis.

The friction (dissipative) and noise (random) term are coupled via the fluctuation-dissipation theorem. The friction term is a function of the relative velocity of particle pairs. The DPD thermostat is better for dynamics than the Langevin thermostat, since it mimics hydrodynamics in the system.

As a conservative force any interaction potential can be used, see Isotropic non-bonded interactions. A common choice is a force ramp which is implemented as Hat interaction.

A complete example of setting up a DPD fluid and running it to sample the equation of state can be found in /samples/dpd.py.

When using a Lennard-Jones interaction, \({r_\mathrm{cut}} = 2^{\frac{1}{6}} \sigma\) is a good value to choose, so that the thermostat acts on the relative velocities between nearest neighbor particles. Larger cutoffs including next nearest neighbors or even more are unphysical.

Boundary conditions for DPD can be introduced by adding the boundary as a particle constraint, and setting a velocity and a type on it, see espressomd.constraints.Constraint. Then a DPD interaction with the type can be defined, which acts as a boundary condition.

6.3.5. Lattice-Boltzmann thermostat

The Lattice-Boltzmann thermostat acts similar to the Langevin thermostat in that the governing equation for particles is

\[m_i \dot{v}_i(t) = f_i(\{x_j\},v_i,t) - \gamma (v_i(t)-u(x_i(t),t)) + \sqrt{2\gamma k_B T} \eta_i(t).\]

where \(u(x,t)\) is the fluid velocity at position \(x\) and time \(t\). To preserve momentum, an equal and opposite friction force and random force act on the fluid.

Numerically the fluid velocity is determined from the lattice-Boltzmann node velocities by interpolating as described in Interpolating velocities. The backcoupling of friction forces and noise to the fluid is also done by distributing those forces amongst the nearest LB nodes. Details for both the interpolation and the force distribution can be found in [Ahlrichs and Dünweg, 1999] and [Dünweg and Ladd, 2009].

The LB fluid can be used to thermalize particles, while also including their hydrodynamic interactions. The LB thermostat expects an instance of either espressomd.lb.LBFluid or espressomd.lb.LBFluidGPU. Temperature is set via the kT argument of the LB fluid.

The magnitude of the frictional coupling can be adjusted by the parameter gamma. To enable the LB thermostat, use:

import espressomd
import espressomd.lb
system = espressomd.System(box_l=[1, 1, 1])
lbf = espressomd.lb.LBFluid(agrid=1, dens=1, visc=1, tau=0.01)
system.actors.add(lbf)
system.thermostat.set_lb(LB_fluid=lbf, seed=123, gamma=1.5)

No other thermostatting mechanism is necessary then. Please switch off any other thermostat before starting the LB thermostatting mechanism.

The LBM implementation provides a fully thermalized LB fluid, all nonconserved modes, including the pressure tensor, fluctuate correctly according to the given temperature and the relaxation parameters. All fluctuations can be switched off by setting the temperature to 0.

Note

Coupling between LB and MD only happens if the LB thermostat is set with a \(\gamma \ge 0.0\).

6.3.6. Stokesian thermostat

Note

Requires STOKESIAN_DYNAMICS external feature, enabled with -DWITH_STOKESIAN_DYNAMICS=ON.

In order to thermalize a Stokesian Dynamics simulation, the SD thermostat needs to be activated via:

import espressomd
system = espressomd.System(box_l=[1.0, 1.0, 1.0])
system.periodicity = [False, False, False]
system.time_step = 0.01
system.cell_system.skin = 0.4
system.part.add(pos=[0, 0, 0], rotation=[1, 0, 0], ext_force=[0, 0, -1])
system.thermostat.set_stokesian(kT=1.0, seed=43)
system.integrator.set_stokesian_dynamics(viscosity=1.0, radii={0: 1.0})
system.integrator.run(100)

where kT denotes the desired temperature of the system, and seed the seed for the random number generator.