Stochastic-process rare event sampling (SPRES) is a
rare-event sampling method in
computer simulation
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
, designed specifically for
non-equilibrium
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of macroscopic quantities (non-equilibrium state variables) that represent an ext ...
calculations, including those for which the rare-event rates are time-dependent (
non-stationary
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
process). To treat systems in which there is time dependence in the dynamics, due either to variation of an external
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
or to
evolution
Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
of the system itself, the scheme for branching paths must be devised so as to achieve sampling which is distributed evenly in time and which takes account of changing
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
es through different regions of the
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
.
Algorithm summary
The SPRES
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
branches simulation paths at fixed time intervals. The process of branching requires that identical paths can be made to diverge from each other, such as by changing the seed in the computer's
random number generator
Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols that cannot be reasonably predicted better than by random chance is generated. This means that the particular out ...
. For systems which would be naturally considered as
deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
, it may be possible to inject an element of randomness, for instance by coupling to a fluctuating
heat bath
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
or by adding random perturbations to account for some elements of the simulation which are not modelled explicitly but which exist in the real system.
The amount of over or under-sampling (the branching density) is decided based on some system-specific 'progress coordinate' which measures progress toward a rare event of interest. The
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of selecting a configuration as the starting point for a new path segment is conditioned jointly by its probability of appearing in an unbiased simulation and by the local flux forwards in the progress coordinate, with a small flux leading adaptively to a larger oversampling.
The method is designed to allow ready observation of rare events with respect to time. An additional benefit relative to methods which mainly split trajectories based on interfaces in the progress coordinate rather than on time is that over most of the progress coordinate space the coordinate only needs to be evaluated at fixed time intervals (rather than continuously) because the exact time-point at which interfaces other than the final interface are reached is no longer of importance.
See also
*
Umbrella sampling
Umbrella sampling is a technique in computational physics and chemistry, used to improve sampling of a system (or different systems) where ergodicity is hindered by the form of the system's energy landscape. It was first suggested by Torrie and ...
Cited references
Stochastic simulation
Computational chemistry
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