TrialComputeAdd

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, \(\mu + \ln(\Lambda^d)/\beta \rightarrow \mu\).
For configurational bias, consider multiple trial positions and select one.
Forward event
[reverse event]
Probability, \(\pi_{on}\)
[reverse probability, \(\pi_{no}\)]
Generate k positions in V. Probability that x_n is in k.
[Select particle of type t]
\(k/V\)
\([\frac{1}{N_t + 1}]\)
Pick x_n in k positions with probability P_k.
[Remove selected particle]
\(P_k\)
\([1]\)
Accept
[Accept]
\(min(1, \chi)\)
\([min(1, 1/\chi)]\)
\(\frac{k P_k}{\Lambda^{dN} V}e^{\beta U_o + \beta\mu_t N_t} min(1, \chi) = \frac{1}{\Lambda^{d(N+1)} (N_t+1)}e^{\beta (U_o + U) + \beta\mu_t (N_t+1)}min(1, 1/\chi)\)
where \(U\) is the interaction energy of the new site with the existing sites and \(U_o\) is the energy of the original configuration.
\(\chi = \frac{V}{k P_k(N_t+1)\Lambda^d}e^{\beta U + \beta\mu_t}\)
If the probability of picking a position is chosen by the Rosenbluth factor,
\(P_k = e^{\beta U}/\sum_i^k e^{\beta U_i}\).
Thus, the acceptance is given by
\(\chi = \frac{V \sum_i^k e^{\beta U_i}}{k(N_t+1)\Lambda^d}e^{\beta\mu_t}\)
For dualcut configurational bias, the new trials are instead chosen from a reference potential, \(U^r\), that is ideally much faster to compute than the full potential but still contains samplingrelevant terms (e.g., excluded volume in a dense system).
\(P_k = e^{\beta U^r}/\sum_i^k e^{\beta U_i^r}\)
\(\chi = \frac{V \sum_i^k e^{\beta U_i^r}}{k(N_t+1)\Lambda^d}e^{\beta(U  U^r) + \beta\mu_t}\)
Note that these equations consider only a singlesite particle.
For the deletion trial, the forward and reverse moves are switched.
Forward event
[reverse event]
Probability, \(\pi_{on}\)
[reverse probability, \(\pi_{no}\)]
Select particle of type t.
[Generate k positions in V. Probability that x_o is in k.]
\(\frac{1}{N_t}\)
\([k/V]\)
Remove selected particle
[Pick x_o in k positions with probability P_k.]
\(1\)
\([P_k]\)
Accept
[Accept]
\(min(1, \chi)\)
\([min(1, 1/\chi)]\)
\(\frac{1}{N_t \Lambda^{dN}}e^{\beta U_o + \beta\mu_t N_t} min(1, \chi) = \frac{k P_k}{V \Lambda^{d(N1)}}e^{\beta (U_o  U) + \beta\mu_t (N_t1)}min(1, 1/\chi)\)
where \(U\) is the interaction energy of the removed site with the existing sites and \(U_o\) is the energy of the original configuration.
\(\chi = \frac{k P_k \Lambda^d}{V N_t}e^{\beta U  \beta\mu_t}\)
If the probability of picking a position is chosen by the Rosenbluth factor,
\(P_k = e^{\beta U}/\sum_i^k e^{\beta U_i}\).
Note that one of the k positions is \(x_o\). Thus, the acceptance is given by
\(\chi = \frac{k \Lambda^d}{V N_t \sum_i^k e^{\beta U_i}}e^{\beta\mu_t}\)