230
C. Tang (唐晨宇) and Y. Wang (王延颋)
reflect the real physics of the canonical ensemble in the momentum space. However,
it can normally provide correct statistics in the position space.
Berendsen Thermostat
Instead of rescaling the system temperature directly to the designated one, the
Berendsen thermostat [14] allows the system to relax to the designated temperature with a characteristic relaxation time τ T in multiple steps. The definition of λ in
Eq. (6.17) is replaced by
λ =
1 +
t
τ T
T
T 0
− 1
1
2
(5.6.18)
which is equivalent to:
dT
dt
=
T 0 − T
τ T
(5.6.19)
This thermostat introduces a certain degree of fluctuation but the physics in the
momentum space is still incorrect.
Andersen Thermostat
The Andersen thermostat proposed by Andersen [15] can achieve the proper simulation of the momentum space in the canonical ensemble by introducing stochastic
collisions of thermostat to particles.
If two successive collisions are uncorrelated, it is obvious that the times the collisions happen during time interval t with a collision frequency λ follow the Poisson
distribution:
P(t; λ) = λ exp(−λt)
(5.6.20)
Therefore, a typical procedure to apply the Andersen thermostat in implementing
MD simulations has the following steps:
1. calculate the regular MD integration of Newton’s Equations of Motion with a
time interval of t;
2. a set of particles are chosen randomly with the probability of choosing each
particle to be λλt;
3. for each chosen particle, a velocity is generated randomly according to the
Maxwell-Boltzmann distribution under temperature T and assigned to the particle
to replace its current velocity.
C. Tang (唐晨宇) and Y. Wang (王延颋)
reflect the real physics of the canonical ensemble in the momentum space. However,
it can normally provide correct statistics in the position space.
Berendsen Thermostat
Instead of rescaling the system temperature directly to the designated one, the
Berendsen thermostat [14] allows the system to relax to the designated temperature with a characteristic relaxation time τ T in multiple steps. The definition of λ in
Eq. (6.17) is replaced by
λ =
1 +
t
τ T
T
T 0
− 1
1
2
(5.6.18)
which is equivalent to:
dT
dt
=
T 0 − T
τ T
(5.6.19)
This thermostat introduces a certain degree of fluctuation but the physics in the
momentum space is still incorrect.
Andersen Thermostat
The Andersen thermostat proposed by Andersen [15] can achieve the proper simulation of the momentum space in the canonical ensemble by introducing stochastic
collisions of thermostat to particles.
If two successive collisions are uncorrelated, it is obvious that the times the collisions happen during time interval t with a collision frequency λ follow the Poisson
distribution:
P(t; λ) = λ exp(−λt)
(5.6.20)
Therefore, a typical procedure to apply the Andersen thermostat in implementing
MD simulations has the following steps:
1. calculate the regular MD integration of Newton’s Equations of Motion with a
time interval of t;
2. a set of particles are chosen randomly with the probability of choosing each
particle to be λλt;
3. for each chosen particle, a velocity is generated randomly according to the
Maxwell-Boltzmann distribution under temperature T and assigned to the particle
to replace its current velocity.