625:
is the flux of trajectories starting before the first interface and going through the last interface. Being a rare event, the flux is very small and practically impossible to compute with a direct simulation. However, using the other interfaces between the states, one can rewrite the flux in terms of
169:
is a random perturbation consistent with system constraints, e.g. conservation of energy and linear and angular momentum. A new trajectory is then simulated from this point, both backward and forward in time until one of the states is reached. Being in a transition region, this will not take long. If
59:
Consider in general a system with two stable states A and B. The system will spend a long time in those states and occasionally jump from one to the other. There are many ways in which the transition can take place. Once a probability is assigned to each of the many pathways, one can construct a
833:
Theoretical considerations show that TIS computations are at least twice as fast as TPS, and computer experiments have shown that the TIS rate constant can converge up to 10 times faster. A reason for this is due to TIS using paths of adjustable length and on average shorter than TPS. Also, TPS
610:
The TPS rate constant calculation can be improved in a variation of the method called
Transition interface sampling (TIS). In this method the transition region is divided in subregions using interfaces. The first interface defines state A and the last state B. The interfaces are not physical
43:
can generate the dynamical trajectories of all the atoms in the system. However, because of the gap in accessible time-scales between simulation and reality, even present supercomputers might require years of simulations to show an event that occurs once per millisecond without some kind of
572:
75:
Given an initial path, TPS provides some algorithms to perturb that path and create a new one. As in all Monte Carlo walks, the new path will then be accepted or rejected in order to have the correct path probability. The procedure is iterated and the ensemble is gradually sampled.
291:
56:. For example, an initially unfolded protein will vibrate for a long time in an open-string configuration before undergoing a transition and fold on itself. The aim of the method is to reproduce precisely those folding moments.
735:
383:
867:). To treat non-stationary systems in which there is time dependence in the dynamics, due either to variation of an external parameter or to evolution of the system itself, then other
777:. By making this interface close enough, the quantity can be computed with a standard simulation, as the crossing event through this interface is not a rare event any more.
27:
of rare events: physical or chemical transitions of a system from one stable state to another that occur too rarely to be observed on a computer timescale. Examples include
1142:
Bolhuis, Peter G.; Chandler, David; Dellago, Christoph; Geissler, Phillip L. (2002). "TRANSITION PATH SAMPLING: Throwing Ropes Over Rough
Mountain Passes, in the Dark".
801:. These probabilities can be computed with a path sampling simulation using the TPS shooting move. A path crossing interface i is perturbed and a new path is
68:
of all transition paths. All the relevant information can then be extracted from the ensemble, such as the reaction mechanism, the transition states, and the
196:
842:), computed by summation of positive and negative terms due to recrossings. TIS instead computes the rate as an effective positive flux, the quantity
872:
1132:
978:
567:{\displaystyle k_{AB}^{TPS}(t)={\frac {d}{dt}}C(t)={\frac {\langle {\dot {h_{B}(t)}}\rangle _{AB}}{\langle h_{B}(t')\rangle _{AB}}}C(t')}
366:
for times of the order of the transition time. Hence once the function is known up to these times, the rate constant is also available.
631:
994:
Van Erp, Titus S.; Moroni, Daniele; Bolhuis, Peter G. (2003). "A novel path sampling method for the calculation of rate constants".
1251:
1266:
934:
Chandler, David (1978). "Statistical mechanics of isomerization dynamics in liquids and the transition state approximation".
1256:
780:
Remarkably, in the formula above there is no Markov assumption of independent transition probabilities. The quantities
1261:
899:; Chandler, David (1998). "Efficient transition path sampling: Application to Lennard-Jones cluster rearrangements".
578:
where the subscript AB denotes an average in the ensemble of paths that start in A and visit B at least once. Time
1047:
Berryman, Joshua T.; Schilling, Tanja (2010). "Sampling rare events in nonequilibrium and nonstationary systems".
851:
is directly computed as an average of only positive terms contributing to the interface transition probabilities.
797:
to indicate that the probabilities are all dependent on the history of the path, all the way from when it left
336:
170:
the new path still connects A to B it is accepted, otherwise it is rejected and the procedure starts again.
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40:
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Dellago, Christoph; Bolhuis, Peter G.; Geissler, Phillip L. (2002). "Transition Path
Sampling".
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286:{\displaystyle C(t)={\frac {\langle h_{A}(0)h_{B}(t)\rangle }{\langle h_{A}\rangle }}}
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69:
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64:
random walk in the path space of the transition trajectories, and thus generate the
598:') at this specific time can be computed with a combination of path sampling and
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Python open source library to perform transition path sampling, Interfaced with
1211:
612:
36:
618:
The rate constant can be viewed as a flux through these interfaces. The rate
83:. Consider the case of a classical many-body system described by coordinates
1190:
Efficient sampling of rare event pathways: from simple models to nucleation
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calculations provided that the interfacial fluxes are time-independent (
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826:) follows from the ratio of the number of paths that reach interface
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1061:
971:
Algorithms for
Chemical Computations, ACS Symposium Series No. 46
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730:{\displaystyle k_{AB}=\Phi _{1,0}\prod _{i=1}^{n-1}P_{A}(i+1|i)}
830: + 1 to the total number of paths in the ensemble.
805:. If the path still starts from A and crosses interface
52:
TPS focuses on the most interesting part of the simulation,
117:
is the length of the path. For a transition from A to B, (
1223:
757:) is the probability for trajectories, coming from state
153:. One of the path times is chosen at random, the momenta
859:
TPS/TIS as normally implemented can be acceptable for
377:) can be rewritten as an average in the path ensemble
178:
In the
Bennett–Chandler procedure, the rate constant k
634:
386:
199:
79:
A powerful and efficient algorithm is the so-called
91:. Molecular dynamics generates a path as a set of (
773:is the flux through the interface closest to
761:and crossing interface i, to reach interface
729:
566:
285:
969:Bennett, C. H. (1977). Christofferson, R. (ed.).
973:. Washington, D.C.: American Chemical Society.
765: + 1. Here interface 0 defines state
582:is an arbitrary time in the plateau region of
769:and interface n defines state B. The factor Φ
8:
1212:C++ source code of an S-PRES wrapper program
1192:(Ph.D. thesis). Universiteit van Amsterdam.
626:transition probabilities between interfaces
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190:is derived from the correlation function
306:is the characteristic function of state
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873:stochastic-process rare-event sampling
7:
793: + 1|i) carry a subscript
611:interfaces but hypersurfaces in the
331:or 0 if not. The time-derivative C'(
323:) is either 1 if the system at time
39:. Standard simulation tools such as
1144:Annual Review of Physical Chemistry
834:relies on the correlation function
1214:, with optional parallelism using
652:
14:
1119:. Vol. 123. pp. 1–84.
1049:The Journal of Chemical Physics
996:The Journal of Chemical Physics
936:The Journal of Chemical Physics
901:The Journal of Chemical Physics
871:methods may be needed, such as
809:, is accepted. The probability
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17:Transition path sampling (TPS)
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606:Transition interface sampling
1117:Advances in Chemical Physics
157:are modified slightly into
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335:) starts at time 0 at the
174:Rate constant computation
855:Time Dependent Processes
182:for the transition from
48:Transition path ensemble
1252:Computational chemistry
337:transition state theory
1224:http://www.pyretis.org
1125:10.1002/0471231509.ch1
731:
693:
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348:and reaches a plateau
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1267:Theoretical chemistry
1110:For a review of TPS:
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369:In the TPS framework
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1179:For a review of TIS
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197:
109:) at discrete times
25:computer simulations
1257:Monte Carlo methods
1184:Moroni, D. (2005).
1156:2002ARPC...53..291B
1071:2010JChPh.133x4101B
1018:2003JChPh.118.7762V
948:1978JChPh..68.2959C
913:1998JChPh.108.9236D
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21:rare-event sampling
1262:Molecular dynamics
893:Dellago, Christoph
727:
564:
387:
283:
41:molecular dynamics
33:chemical reactions
1134:978-0-471-21453-3
1079:10.1063/1.3525099
1026:10.1063/1.1562614
980:978-0-8412-0371-6
897:Bolhuis, Peter G.
600:umbrella sampling
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87:and momenta
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1150:: 291–318.
613:phase space
62:Monte Carlo
1246:Categories
869:rare-event
865:stationary
113:in where
37:nucleation
1062:1001.2456
687:−
669:∏
653:Φ
533:⟩
505:⟨
491:⟩
484:˙
457:⟨
278:⟩
265:⟨
260:⟩
219:⟨
1172:11972010
1095:34154184
1087:21197970
1034:94328349
558:′
525:′
165:, where
149:) is in
66:ensemble
1228:GROMACS
1152:Bibcode
1067:Bibcode
1014:Bibcode
944:Bibcode
909:Bibcode
1232:LAMMPS
1216:OpenMP
1186:"DARE"
1170:
1131:
1093:
1085:
1032:
977:
740:where
310:, and
297:where
1091:S2CID
1057:arXiv
1030:S2CID
1004:arXiv
19:is a
1236:CP2K
1168:PMID
1129:ISBN
1083:PMID
975:ISBN
803:shot
35:and
1194:hdl
1160:doi
1121:doi
1075:doi
1053:133
1022:doi
1000:118
952:doi
917:doi
905:108
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1230:,
1188:.
1166:.
1158:.
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1089:.
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938:.
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163:δp
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31:,
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1218:.
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919::
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