Thermostats in Molecular Dynamics

Max Soloviov

Thursday, September 19,

Why using thermostat?

➡ Match experimental conditions

➡ Study temperature dependent processes

➡ Simulations of non-equilibrium systems

➡ Enhance dynamics by means of temperature (simulated annealing, metadynamics)

NVE NVTmicrocanonical canonical

NPT

Thursday, September 19,

Thermostat is a modification of the newtonian MD scheme applied in order to mimic the canonical ensemble.

What is a thermostat?

Thursday, September 19,

Thermostat is a modification of the newtonian MD scheme applied in order to mimic the canonical ensemble.

What is a thermostat?

Thermostats modulate the energy passing through the boundaries of the system.

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Langevin

dpi

dt=

NX

j

(Fij | qi � qj |)� �pi +Ri(t)

Stochastic force

Friction coefficient- deterministic

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- Strong coupling (isokinetic/Gaussian) scale system variable to give the exact preset derived value

- Stochastic (Langevin)contain a system variable to predefine distribution function

- Weak coupling (Berendsen)scale system variable in direction of desired derived variable

- Extended system (Nosé–Hoover)extend system’s degrees of freedom to include temperature

or pressure terms

How do we control system variables?

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Strong Coupling (Scaling Velocities)

dpi

dt=

NX

j

(Fij | qi � qj |)� ↵pi

P (vi,↵) =

✓m

2⇡kBT

◆ 12

e

✓�

mv2i,↵

2kBT

◆Velocities are described by Maxwell-Bolzmann distribution

Adjust instantaneous temperature by scaling all velocities

L.V.Woodcock, (1971) Chem. Phys. Letter., 10: 257-261.

↵ =�P pi

mi

dUdqiP p2

imi

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Strong Coupling (Scaling Velocities)

+ Straightforward to implement

- Results do not correspond to any ensemble, no proper temperature fluctuations.

- No localized correlation

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Stochastic (Langevin)

dpi

dt=

NX

j

(Fij | qi � qj |)� �pi +Ri(t)

Motion of large particles through a continuum of smaller ones

Viscous drag force proportional to velocity

Smaller particles give random pushes to large particle

⇢(�p) =1p2⇡�

e

✓� |�p|2

2�2

◆

�2 = 2�mikBT

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+ Samples from canonical ensemble, i.e., the behavior is properly thermal for temperature T.

+ Can use a larger time step due to the damping factor

+ Each particle is coupled to a local heat bath. This can remove heat trapped in localized modes.

- Difficult to implement drag for non-spherical particles: γ related to particle radius

- Momentum transfer lost due to randomness (no diffusion coefficients etc)

Stochastic (Langevin)

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Berendsen H.J.C et al (1984) J Chem. Phys. 81:3684–3690

Weak coupling (Berendsen)

dpi

dt=

NX

j

(Fij | qi � qj |)�pi

⌧T

✓T0

T� 1

◆

dT

dt=

1

⌧(T0 � T )

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+ Deterministic

+ Robust (has shown minor deviations from canonical distribution)

- Not canonical. For 0<τ<∞ samples an unusual “weak- coupling” ensemble (underestimates temperature fluctuations)

Weak coupling (Berendsen)

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Extended system (Nosé–Hoover)

Hoover (1985) Phys. Rev. A 31:1695 S. Nosé (1984) J. Chem. Phys. 81:511

S. Nosé (1984) Mol. Phys. 52:255

The Q can be considered to be some fictional “heat bath mass”.

Effective mass Q associated with s determines thermostat strength.

Microcanonical dynamics with such a extended system results in canonical properties.

H =NX

i

NX

j>i

(Fij | qi � qj |) +NX

i

p2

2mis2+

⇠2Q

2� (3N + 1)kBT ln s

Thursday, September 19,

+ Deterministic and time-reversible

+ Increasing Q lengthens decay time of response to instantaneous temperature jump

- Does not guarantee that the ensemble is canonical (unless Nosé-Hoover chains are used, where multiple heat baths (multiple degrees of freedom s) are linked to enhance temperature equilibration)

Extended system (Nosé–Hoover)

Glenn J. Martyna et al (1992) J. Chem. Phys. 97:2635

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Example of temperature controls

The temperature response of a Lennard–Jones fluid under control of three thermostats (solid line: Langevin; dotted line: weak-coupling; dashed line: Nose–Hoover) after a step change in the reference temperature (Hess, 2002a, van der Spoel et al., 2005.)

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Conclusions

• When using Berendsen thermostat, use it for equilibration only.

• Nosé-Hoover is good. Nosé-Hoover chains is the best.

• Stochastic thermostats can’t be used when the local dynamics is critical.

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References

• Daan Frenkel & Berend Smit“Understanding Molecular Dynamics: From Algorithms to Applications”, 2nd edition

• Philippe H. Hünenberder “Thermostat Algorithms for Molecular Dynamics Simulations”, Adv. Polym. Sci. (2005) 173: 105 - 149

• Alden Johnson, Teresa Johnson, Aimee Khan“Thermostats in Molecular Dynamics Simulations”http://www.math.umass.edu/~markos/697SC/ThermostatMD_Final.pdf

• Yanxiang Zhao“Brief introduction to the thermostats”http://www.math.ucsd.edu/~y1zhao/ResearchNotes/ResearchNote007Thermostat.pdf

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Thank you for your attention

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