Achieving Equilibrium in Molecular Dynamics: Temperature and Pressure Control in NVT and NPT Ensembles

When investigating the behavior of microscopic systems using in silico methods, it is critically important to define the physical conditions under which the system evolves. While energy is conserved in an isolated system (microcanonical ensemble), real experimental conditions are typically open to heat and work exchange with the surrounding environment. Therefore, achieving consistency between computational models and experimental reality relies on the precise control of macroscopic variables such as temperature and pressure (1).

In this article, the NVT and NPT ensembles—commonly used to maintain a system under desired thermodynamic conditions—along with their fundamental control equations, are examined.

NVT Method (Temperature Control)

In the canonical ensemble (NVT), the number of particles (N), volume (V), and temperature (T) are held constant, while the internal pressure and total energy of the system are allowed to fluctuate. Temperature control is based on regulating the kinetic energy of the system. For a system consisting of N particles, the instantaneous temperature is related to the Equipartition Theorem as follows:

Here, m denotes the particle mass, v  the particle velocity, and kB the Boltzmann constant. The NVT method maintains the average kinetic energy of the system at the target temperature by modifying atomic velocities.

Berendsen Thermostat: This thermostat weakly couples the system to an external heat bath through a characteristic relaxation time. The rate of temperature change is controlled by the following equation (2):

It drives the system rapidly toward the target temperature (making it ideal for the heating or equilibration phase) and suppresses statistical fluctuations.

Nosé–Hoover Thermostat:This thermostat introduces a dynamic variable into the equations of motion that acts similarly to a friction coefficient. Because it correctly generates the canonical (Boltzmann) distribution, it is considered the standard choice during the data collection or production phase (3).

Figure 1. Comparative schematics of stochastic and deterministic thermostat/barostat algorithms. Variable cell–shape barostat (Parrinello–Rahman), weak coupling method (Berendsen), and extended-system thermostat (Nosé–Hoover). (Image designed using artificial intelligence.)

NPT Method (Pressure and Temperature Control)

In the isothermal–isobaric ensemble (NPT), the number of particles (N), pressure (P), and temperature (T) are kept constant. The system volume (V) behaves as a dynamic variable, adjusting to equalize the internal pressure with the external pressure. Most chemical and biological processes occur under constant pressure conditions (typically 1 atm). By enabling the system to reach the correct density, the NPT method allows accurate modeling of intermolecular interactions and phase behavior.

Control Algorithms (Barostats): To maintain constant pressure, algorithms that modify the system volume are employed:

Berendsen Barostat:This barostat rescales the coordinates and simulation box dimensions using a scaling factor to minimize the pressure difference (2):

Here, κ denotes the isothermal compressibility, and τ p is the pressure relaxation time. This barostat is effective during density equilibration stages.

Parrinello–Rahman Barostat: This method treats the simulation cell vectors as additional degrees of freedom incorporated into the Lagrangian formalism. It allows not only the volume but also the shape of the simulation box to fluctuate. It is considered the most accurate approach for phase transition studies and crystal structure analyses (4).

Control Stages and Application Order

1. NVT Control (Heating): Initially, the system volume is kept fixed while the system is brought into thermal equilibrium. This step helps prevent structural distortions.

2. NPT Control (Equilibration): Once the temperature has stabilized, pressure control is activated. The volume is released, allowing the system to relax to the correct density.

3. Steady State (Production): After the system has reached equilibrium in terms of both temperature and pressure/density, the most accurate algorithms—such as Nosé–Hoover combined with Parrinello–Rahman—are employed under NPT (or NVT, depending on the system) conditions for data collection and analysis.

Figure 2. Time profile of the NVT and NPT stages in a standard molecular dynamics (MD) simulation protocol. (The data used in the figure were obtained by the authors.)

In molecular dynamics simulations, temperature and pressure control play a critical role in accurately representing the physical reality of the system. In this context, the NVT and NPT ensembles emerge as complementary tools employed at different stages of a simulation. While the NVT approach enables the system to be brought to the target temperature in a controlled manner, the NPT method allows volume fluctuations, thereby ensuring realistic density and pressure conditions. The choice of thermostat and barostat algorithms affects not only numerical stability but also the accuracy of the resulting statistical distributions. Therefore, the deliberate and purpose-driven use of different control algorithms during rapid equilibration and precise data production phases is of paramount importance. A well-designed NVT–NPT protocol enables reliable analysis of molecular interactions, structural changes, and thermodynamic properties.

References

1. Allen, M. P., & Tildesley, D. J. (2017). Computer Simulation of Liquids. Oxford University Press. 

2. Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., DiNola, A., & Haak, J. R. (1984). Molecular dynamics with coupling to an external bath. The Journal of Chemical Physics, 81(8), 3684–3690. https://doi.org/10.1063/1.448118

3. Nosé, S. (1984). A unified formulation of the constant temperature molecular dynamics methods. The Journal of Chemical Physics, 81(1), 511–519. https://doi.org/10.1063/1.447334

4. Parrinello, M., & Rahman, A. (1981). Polymorphic transitions in single crystals: A new molecular dynamics method. Journal of Applied Physics, 52(12), 7182–7190. https://doi.org/10.1063/1.328693