Adaptive Kinetic Monte Carlo#
The adaptive kinetic Monte Carlo (aKMC) method is a method to coarse grain molecular dynamics for rare event systems as described in Henkelman and Jónsson [AK_HJonsson01], Xu and Henkelman [AK_XH08]. A rare event system is one in which the interesting dynamics is governed by short transitions between stable states. The fast vibrational motion within a stable state is considered to be in equilibrium and described statisitcally. A transition between states is assumed to be first order, since it is a rare events, and the rate of the transition is calculated from the harmonic approximation to transition state theory (hTST).
The hTST approximation of a transition rate is calculated from the energy difference between the saddle point along the minimum energy path for the transition and the initial minimum. The vibrational modes at these points are also used to calculate the prefactor. An hTST rate is of the standard Arrhenius form \(R = v \exp (-\Delta E/kT)\) where \(v\) is the product of all positive modes at the minimum divided by those at the saddle, \(\Delta E\) is the energy barrier, and \(kT\) is the thermal enregy.
In order to propogate the dynamics within aKMC, a list of all possible rates leading away from the current stable state to any other state is required. Formally, there are an enourmously large number of such transitions (also called processes) available in a typical atomic system, but in fact only the transitions with the fastest rates with the highest probability of happening are required. The search for processes is then limited to those with rates on the same order as the fastest processes found.
The search for possible processes is the primary task of the aKMC simulations. Each client does a minimum mode (minmode) following search from the miminum of the current state and tries to find a saddle point which connects from the minimum in the current state to an adjacent state [AK_HJonsson99]. A saddle point is connected to a state if a minimization initiated along the negative mode at the saddle converges to the minimum of that state.
Each client is tasked with one or more such searches. It climbs from the minimum to a saddle, and if successful, it minimizes on either side of the saddle to determine the connecting states. The prefactor for the transition is also calculated by finite difference and the hTST rate is calculated. These data are reported back to the server.
Essentially, then, unlike traditional Kinetic Monte Carlo (KMC) methods, adaptive (a), or off-lattice KMC [AK_TMBelandH20] doesn’t require a lattice of predetermined points in configuration space. The method dynamically builds a rate table based on local atomic movements and energy barriers. This allows for more accurate modeling of complex systems where atomic positions are not fixed to a grid. The confidence scheme and thermal accessibility settings in AKMC ensure that transitions between states are based on well-defined criteria, enhancing the precision of simulations.
The server is reponsible for the time evolution of the system by the KMC algorithm. Each process leading to a new state is tabulated in a rate table and one transition is selected stochastically with a probability proportional to its rate. The transition time is drawn from a first-order distribution for the total rate of escape from the state.
A complete tutorial is also provided.
Configuration#
[akmc]
- pydantic model eon.schema.AKMCConfig[source]#
Show JSON schema
{ "title": "AKMCConfig", "type": "object", "properties": { "confidence": { "default": 0.99, "description": "The confidence (out of 1.0) criterion for moving to the next state.", "title": "Confidence", "type": "number" }, "server_side_process_search": { "default": false, "description": "If true, the server does not send out \"process_search\" jobs. Instead, it manages individual dimer, minimization, and Hessian calculations. This option is usually used when a single \"process_search\" job will take a long time (hours or longer).", "title": "Server Side Process Search", "type": "boolean" }, "thermally_accessible_window": { "default": 20.0, "description": "Processes with barriers within this number of kT above the lowest barrier will be used in the rate table and for confidence calculations.", "title": "Thermally Accessible Window", "type": "number" }, "thermally_accessible_buffer": { "default": 0.0, "description": "Processes with barriers of thermally_accessible_window + thermally_accessible_buffer will be stored, in the event that they are thermally accessible later, but are not used in the rate table or for the confidence calculations. Processes with barriers higher than the sum of these two values will be discarded.", "title": "Thermally Accessible Buffer", "type": "number" }, "max_kmc_steps": { "default": 0, "description": "The maximum number of KMC transitions in a row. In MPI or continuous mode, EON will exit after performing this many KMC steps. If this is set to 0, EON will run forever.", "title": "Max Kmc Steps", "type": "integer" }, "confidence_scheme": { "default": "new", "description": "The scheme used for confidence calculation.", "enum": [ "old", "new", "sampling", "dynamics" ], "title": "Confidence Scheme", "type": "string" }, "confidence_correction": { "default": false, "description": "If true, applies a correction to the confidence calculation.", "title": "Confidence Correction", "type": "boolean" }, "max_rate": { "default": 0.0, "description": "Sets the maximum allowable rate for a process. If a calculated rate exceeds this value, it is capped to prevent exceedingly high rates from dominating the simulation. This helps in maintaining numerical stability and realistic dynamics in the simulation.", "title": "Max Rate", "type": "number" }, "eq_rate": { "default": 0.0, "description": "Sets the equilibrium rate. If the forward and reverse rates exceed this value, they are adjusted to maintain equilibrium conditions. This prevents exceedingly high rates from dominating the simulation and ensures that both forward and reverse processes are considered symmetrically.", "title": "Eq Rate", "type": "number" } } }
- Config:
use_attribute_docstrings: bool = True
- Fields:
- field confidence_correction: bool = False#
If true, applies a correction to the confidence calculation.
- field confidence_scheme: Literal['old', 'new', 'sampling', 'dynamics'] = 'new'#
Options: -
old: Traditional confidence scheme based on the number of unique processes (\(Nf\)) and the number of searches (\(Ns\)). -new: Enhanced confidence scheme using a mathematical function to account for the ratio of \(Nf\) to \(Ns\). -sampling: Confidence based on sampling probabilities derived from the repeat counts of processes. -dynamics: Confidence derived from molecular dynamics, considering time extrapolation and rates at different temperatures.The scheme used for confidence calculation.
- field eq_rate: float = 0.0#
Sets the equilibrium rate. If the forward and reverse rates exceed this value, they are adjusted to maintain equilibrium conditions. This prevents exceedingly high rates from dominating the simulation and ensures that both forward and reverse processes are considered symmetrically.
- field max_kmc_steps: int = 0#
The maximum number of KMC transitions in a row. In MPI or continuous mode, EON will exit after performing this many KMC steps. If this is set to 0, EON will run forever.
- field max_rate: float = 0.0#
Sets the maximum allowable rate for a process. If a calculated rate exceeds this value, it is capped to prevent exceedingly high rates from dominating the simulation. This helps in maintaining numerical stability and realistic dynamics in the simulation.
- field server_side_process_search: bool = False#
If true, the server does not send out “process_search” jobs. Instead, it manages individual dimer, minimization, and Hessian calculations. This option is usually used when a single “process_search” job will take a long time (hours or longer).
- field thermally_accessible_buffer: float = 0.0#
Processes with barriers of thermally_accessible_window + thermally_accessible_buffer will be stored, in the event that they are thermally accessible later, but are not used in the rate table or for the confidence calculations. Processes with barriers higher than the sum of these two values will be discarded.
References#
Graeme Henkelman and Hannes Jónsson. A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives. The Journal of Chemical Physics, 111(15):7010–7022, October 1999. doi:10.1063/1.480097.
Graeme Henkelman and Hannes Jónsson. Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table. The Journal of Chemical Physics, 115(21):9657–9666, November 2001. doi:10.1063/1.1415500.
Mickaël Trochet, Normand Mousseau, Laurent Karim Béland, and Graeme Henkelman. Off-Lattice Kinetic Monte Carlo Methods. In Wanda Andreoni and Sidney Yip, editors, Handbook of Materials Modeling: Methods: Theory and Modeling, pages 715–743. Springer International Publishing, Cham, 2020. doi:10.1007/978-3-319-44677-6_29.
Lijun Xu and Graeme Henkelman. Adaptive kinetic Monte Carlo for first-principles accelerated dynamics. The Journal of Chemical Physics, 129(11):114104, September 2008. doi:10.1063/1.2976010.