The Berendsen thermostat is a deterministic temperature‑control algorithm employed in molecular dynamics (MD) simulations to regulate the kinetic temperature of a simulated system toward a specified reference temperature $T_0$. It was introduced by H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, and J. R. Haak in 1984 (J. Chem. Phys. 81, 3684).
Operating principle
The thermostat rescales particle velocities at each integration step to mimic weak coupling of the system to an external heat bath. The instantaneous kinetic temperature $T$ of the system is computed from the kinetic energy
$$ T = \frac{2K}{3Nk_{\mathrm{B}}}, $$
where $K$ is the total kinetic energy, $N$ the number of particles, and $k_{\mathrm{B}}$ Boltzmann’s constant.
The velocities $\mathbf{v}_i$ are then multiplied by a scaling factor $\lambda$ given by
$$ \lambda = \sqrt{1 + \frac{\Delta t}{\tau_T}\left(\frac{T_0}{T} - 1\right)}, $$
where $\Delta t$ is the MD time step and $\tau_T$ is a user‑defined coupling constant (relaxation time). The new velocities are
$$ \mathbf{v}_i \leftarrow \lambda , \mathbf{v}_i . $$
If $\tau_T$ is large relative to $\Delta t$, the temperature evolves slowly toward $T_0$; a small $\tau_T$ forces rapid equilibration but may distort the dynamics.
Key characteristics
| Feature | Description |
|---|---|
| Type | Weak‑coupling deterministic thermostat |
| Parameters | Reference temperature $T_0$; coupling time $\tau_T$; integration time step $\Delta t$ |
| Advantages | Simple to implement; efficiently drives the system to the desired temperature; suitable for equilibration phases |
| Limitations | Does not generate a proper canonical (NVT) ensemble; alters the distribution of kinetic energies; may suppress natural temperature fluctuations; unsuitable for production runs where correct ensemble sampling is required |
| Common usage | Initial equilibration of liquids, proteins, and other condensed‑phase systems; often combined with a more rigorous thermostat (e.g., Nosé–Hoover, Langevin) for subsequent production runs |
Relationship to other thermostats
The Berendsen scheme can be viewed as an exponential relaxation of temperature, analogous to a first‑order linear response to a heat bath. Unlike stochastic thermostats (e.g., Langevin) that add random forces, the Berendsen thermostat only rescales velocities, preserving deterministic dynamics. Compared with the Nosé–Hoover thermostat, which conserves an extended Hamiltonian and yields the canonical distribution, the Berendsen thermostat does not correctly sample the statistical ensemble.
Implementation notes
- Choice of $\tau_T$ – Typical values range from 0.1 ps to 2 ps for biomolecular simulations. Too small a $\tau_T$ can cause “over‑damping” of vibrational motions.
- Temperature calculation – The kinetic temperature is usually evaluated over all degrees of freedom, optionally excluding constrained bonds (e.g., SHAKE‑constrained bonds).
- Integration with pressure control – When paired with a Berendsen barostat, the combined “Berendsen coupling” yields rapid equilibration of both temperature and pressure, albeit at the cost of ensemble fidelity.
Historical impact
The Berendsen thermostat, together with the Berendsen barostat, contributed to the widespread adoption of MD simulations in the 1990s by providing a straightforward method for system equilibration. Its simplicity remains attractive for pedagogical examples and for initial equilibration steps in modern simulation protocols.
References
- H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, J. R. Haak, “Molecular dynamics with coupling to an external bath,” J. Chem. Phys., 81, 3684 (1984).
- D. Frenkel, B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, 2nd ed., Academic Press, 2002 (section on thermostats).
The Berendsen thermostat continues to be incorporated in major MD packages such as GROMACS, NAMD, AMBER, and LAMMPS.
