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C. Tang (唐晨宇) and Y. Wang (王延颋)
It should be noted from the detailed balance condition that the transition
probability from configuration n to o, or π(n → o), follows an equation that
N Q (o)π(o → n) = N Q (n)π(n → o)
(5.5.12)
In practice, we can construct π(o → n) with transition matrix α(o → n), which is
the underlying matrix for the corresponding Markov chain, and accepting probability
acc(o → n) so that:
π(o → n) = α(o → n) × acc(o → n)
(5.5.13)
If matrix α is symmetric, we can know that
acc(o → n)
acc(n → o)
=
N Q (n)
N Q (o)
= exp{−β[U (n) − U (o)]}
(5.5.14)
We can choose acc(o → n) to fulfill the condition Eq. (5.14) by assigning
acc(o → n) =
N Q (n)/N Q (o) if N Q (n) < N Q (o)
1 if N Q (n) ≥ N Q (o)
(5.5.15)
So that the transition probability from state o to state n is given by
π(o → n) =
α(o → n) if N Q (n) ≥ N Q (o)
α(o → n)[N (n)/N (o)] if N Q (n) < N Q (o)
π(o → o) = 1 −
o =n
π(o → n)
(5.5.16)
To decide whether to accept such a trial move or not, we can generate a random
number X from a uniform distribution between [0, 1]. We can then accept the trial
move if X is less than or equals to acc(o → n) and reject it otherwise. This gives the
eventual sequence of how we can perform an MC simulation.
The procedure of an MC simulation with the Metropolis algorithm can then be
given as follows. First, generate a new “trial” configuration by making a perturbation
to the present configuration. Then, accept the new configuration based on the ratio
of the probabilities for the new and old configurations, according to the Metropolis
algorithm. If the trial is rejected, the present configuration is taken as the next one in
the Markov chain. The above steps are repeated many times, and instantaneous data
and configurations are sampled for data analysis.
From what we have discussed we can have a snapshot of the characteristics of
MC simulation. By default, the MC simulation simulates a system under the NVT
ensemble. The potential energy is calculated without force, which saves some computational complexity and time. It is mostly used for simulating equilibrium systems,
C. Tang (唐晨宇) and Y. Wang (王延颋)
It should be noted from the detailed balance condition that the transition
probability from configuration n to o, or π(n → o), follows an equation that
N Q (o)π(o → n) = N Q (n)π(n → o)
(5.5.12)
In practice, we can construct π(o → n) with transition matrix α(o → n), which is
the underlying matrix for the corresponding Markov chain, and accepting probability
acc(o → n) so that:
π(o → n) = α(o → n) × acc(o → n)
(5.5.13)
If matrix α is symmetric, we can know that
acc(o → n)
acc(n → o)
=
N Q (n)
N Q (o)
= exp{−β[U (n) − U (o)]}
(5.5.14)
We can choose acc(o → n) to fulfill the condition Eq. (5.14) by assigning
acc(o → n) =
N Q (n)/N Q (o) if N Q (n) < N Q (o)
1 if N Q (n) ≥ N Q (o)
(5.5.15)
So that the transition probability from state o to state n is given by
π(o → n) =
α(o → n) if N Q (n) ≥ N Q (o)
α(o → n)[N (n)/N (o)] if N Q (n) < N Q (o)
π(o → o) = 1 −
o =n
π(o → n)
(5.5.16)
To decide whether to accept such a trial move or not, we can generate a random
number X from a uniform distribution between [0, 1]. We can then accept the trial
move if X is less than or equals to acc(o → n) and reject it otherwise. This gives the
eventual sequence of how we can perform an MC simulation.
The procedure of an MC simulation with the Metropolis algorithm can then be
given as follows. First, generate a new “trial” configuration by making a perturbation
to the present configuration. Then, accept the new configuration based on the ratio
of the probabilities for the new and old configurations, according to the Metropolis
algorithm. If the trial is rejected, the present configuration is taken as the next one in
the Markov chain. The above steps are repeated many times, and instantaneous data
and configurations are sampled for data analysis.
From what we have discussed we can have a snapshot of the characteristics of
MC simulation. By default, the MC simulation simulates a system under the NVT
ensemble. The potential energy is calculated without force, which saves some computational complexity and time. It is mostly used for simulating equilibrium systems,