Theory · 3 methods

Barostats

What CAPS computes, as it computes it: the equation, its symbols with CAPS's defaults, when to use the method, the source file, the tests that check it, and any departure from the cited method.

Berendsen pressure coupling

The cell and all positions are scaled so that the pressure relaxes exponentially towards the target. It controls the density quickly but does not sample the isothermal–isobaric ensemble: volume fluctuations are too small.

$$\mu = \exp\left(-\frac{\beta\,\Delta t_p\,(P_0 - P)}{3\tau_P}\right), \qquad x \leftarrow \mu x, \qquad h \leftarrow \mu h$$

Berendsen et al. 1984, in CAPS's exponential form

Symbol	Meaning	In CAPS
β	isothermal compressibility	4.5 × 10⁻⁵ atm⁻¹
τ_P	pressure relaxation time	τ_P · default 1000 fs
Δt_p	time between cell updates	10 steps
P	virial pressure (2K + W)/(3V)	computed

When to use it

Early equilibration and the lateral axes of a tensile test (per-axis form). Use stochastic cell rescaling for production NPT.

Source: core/src/dynamics.cpp
Tested by: Dynamics.BarostatRespondsToTargetPressure
Departure from the reference: exponential rather than linear scaling; the volume change is limited to 1 % per update; a per-axis variant scales each axis from P_kk

References

Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., DiNola, A., Haak, J. R., "Molecular dynamics with coupling to an external bath", J. Chem. Phys. 81, 3684–3690 (1984). doi:10.1063/1.448118

Stochastic cell rescaling

A first-order barostat like Berendsen's, with a noise term that restores the correct volume fluctuations of the isothermal–isobaric ensemble. Velocities are scaled by the inverse factor so the kinetic energy stays consistent.

$$\begin{gathered}\mathrm{d}\epsilon = -\frac{\beta}{\tau_P}\left(P_0 - P - \frac{k_\mathrm{B}T}{V}\right)\mathrm{d}t + \sqrt{\frac{2k_\mathrm{B}T\beta}{V\tau_P}}\,\mathrm{d}W \\ \mu = e^{\mathrm{d}\epsilon/3}, \qquad x \leftarrow \mu x, \qquad v \leftarrow v/\mu\end{gathered}$$

Bernetti & Bussi 2020, isotropic

Symbol	Meaning	In CAPS
ε	logarithm of the volume	—
β	compressibility (sets the response, not the result)	4.5 × 10⁻⁵ atm⁻¹
τ_P	relaxation time	τ_P · default 1000 fs
dW	Wiener increment: √dt times a standard Gaussian	mt19937-64 stream from the run's seed

When to use it

The default barostat for NPT production and for the NPT stages of equilibration protocols.

Source: core/src/dynamics.cpp
Tested by: Dynamics.BarostatRespondsToTargetPressure
Departure from the reference: isotropic only; dε limited to ±1 % per update

References

Bernetti, M., Bussi, G., "Pressure control using stochastic cell rescaling", J. Chem. Phys. 153, 114107 (2020). doi:10.1063/5.0020514

Full-shape cell and stress control

The per-axis Berendsen barostat with a target for each axis (a tensile stress is a negative pressure), and the full shape: the cell's tilts also follow the shear stresses toward zero, so a triclinic or sheared cell relaxes its angles as well as its lengths.

$$d\varepsilon_{kk} = -\frac{\beta}{\tau_p}\,(P_k^0 - P_{kk})\,\frac{\Delta t}{3},\qquad d\gamma_{ab} = \frac{\beta}{\tau_p}\,P_{ab}\,\frac{\Delta t}{3}$$

each update at most 0.33 % strain; a tilt past half the cell is flipped to the equivalent cell

Symbol	Meaning	In CAPS
P_k⁰	target pressure of axis k	atm
P_ab	shear component of the pressure tensor	atm
β	compressibility (the barostat's response)	1/atm

When to use it

Crystals and crystalline fibres from a guessed cell, sheared composites, cells after a deformation: the shape relaxes to zero shear stress. LAMMPS: fix npt … tri does the same with Nosé–Hoover.

Source: core/src/dynamics.cpp
Tested by: tests/test_mechanics.cpp Mechanics.StressControlAndCreep (a sheared argon crystal's tilt relaxes and ⟨P_xy⟩ goes to zero)
Departure from the reference: Berendsen: the right averages, not the right fluctuations of the cell