TrialComputeAdd¶
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class
TrialComputeAdd
: public feasst::TrialCompute¶ Attempt to add a particle.
The derivation of the acceptance criteria follows a similar procedure as descibed in TrialComputeMove, except with the following differences.
The limiting distribution in the grand canonical ensemble is
\(\pi_i \propto \frac{e^{-\beta U + \beta \mu_t N_t}}{\Lambda^{dN}}\)
where \(\mu_t\) is the chemical potential of particles of type t, \(\Lambda\) is the de Broglie wavelength, \(N_t\) is the number of particles of type t and d is the dimension.
The transition probabilities are as follows, assuming that this move is coupled with a trial that removes particles with the same selection weight.
Forward event
[reverse event]
Probability, \(\pi_{on}\)
[reverse probability, \(\pi_{no}\)]
Select insert trial
[Select remove trial]
\(1/w\)
\([1/w]\)
Place particle of type t
[Delete particle type t]
\(1/V\)
\(\left[\frac{1}{N_t+1}\right]\)
Accept
[Accept]
\(min(1, \chi)\)
\([min(1, 1/\chi)]\)
Application of local detailed balance yields the acceptance probability, \(\chi\).
\(\frac{e^{-\beta U_o + \beta\mu_t N_t}}{\Lambda^{dN}w V }min(1, \chi) = \frac{e^{-\beta U_n + \beta\mu_t (N_t+1)}}{\Lambda^{d(N+1)}w (N_t+1)} min(1, 1/\chi)\)
\(\chi = \frac{V e^{-\beta\Delta U + \beta\mu_t}}{(N_t+1)\Lambda^d}\)
Note that the number of particles, \(N_t\) is from the perspective of the old state. Thus, if the particle has already been added during computation of \(\chi\), then \(N_t + 1 \rightarrow N_t\). The same applies for
TrialComputeRemove
.The de Broglie wavelength, \(\Lambda^d\), is absorbed into the definition of \(\mu\) for convenience, \(\mu + \ln(\Lambda^d)/\beta \rightarrow \mu\).
Public Functions
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void
perturb_and_acceptance
(Criteria *criteria, System *system, Acceptance *acceptance, std::vector<TrialStage*> *stages, Random *random)¶ Perform the Perturbations and determine acceptance.
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void