In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, multicanonical ensemble (also called multicanonical sampling or flat histogram) is a
Markov chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain ...
sampling technique that uses the
Metropolis–Hastings algorithm
In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. This seque ...
to compute
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s where the integrand has a rough landscape with multiple
local minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
. It samples states according to the inverse of the
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
,
[ which has to be known a priori or be computed using other techniques like the ]Wang and Landau algorithm The Wang and Landau algorithm, proposed by Fugao Wang and David P. Landau, is a Monte Carlo method designed to estimate the density of states of a system. The method performs a non-Markovian random walk to build the density of states by quickly vi ...
.[ Multicanonical sampling is an important technique for ]spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
systems like the Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
or spin glass
In condensed matter physics, a spin glass is a magnetic state characterized by randomness, besides cooperative behavior in freezing of spins at a temperature called 'freezing temperature' ''Tf''. In ferromagnetic solids, component atoms' magne ...
es.[
]
Motivation
In systems with a large number of degrees of freedom, like spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
systems, Monte Carlo integration
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at ...
is required. In this integration, importance sampling
Importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally att ...
and in particular the Metropolis algorithm
A metropolis () is a large city or conurbation which is a significant economic, political, and cultural center for a country or region, and an important hub for regional or international connections, commerce, and communications.
A big ci ...
, is a very important technique.[ However, the Metropolis algorithm samples states according to where beta is the inverse of the temperature. This means that an ]energy barrier
In chemistry and physics, activation energy is the minimum amount of energy that must be provided for compounds to result in a chemical reaction. The activation energy (''E''a) of a reaction is measured in joules per mole (J/mol), kilojoules p ...
of on the energy spectrum is exponentially difficult to overcome.[ Systems with multiple local energy minima like the ]Potts model
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenome ...
become hard to sample as the algorithm gets stuck in the system's local minima.[ This motivates other approaches, namely, other sampling distributions.
]
Overview
Multicanonical ensemble uses the Metropolis–Hastings algorithm with a sampling distribution given by the inverse of the density of states of the system, contrary to the sampling distribution of the Metropolis algorithm.[ With this choice, on average, the number of states sampled at each energy is constant, i.e. it is a simulation with a "flat histogram" on energy. This leads to an algorithm for which the energy barriers are no longer difficult to overcome. Another advantage over the Metropolis algorithm is that the sampling is independent of the temperature of the system, which means that one simulation allows the estimation of thermodynamical variables for all temperatures (thus the name "multicanonical": several temperatures). This is a great improvement in the study of first order ]phase transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
s.[
The biggest problem in performing a multicanonical ensemble is that the density of states has to be known ''a priori''.][ One important contribution to multicanonical sampling was the ]Wang and Landau algorithm The Wang and Landau algorithm, proposed by Fugao Wang and David P. Landau, is a Monte Carlo method designed to estimate the density of states of a system. The method performs a non-Markovian random walk to build the density of states by quickly vi ...
, which asymptotically converges to a multicanonical ensemble while calculating the density of states during the convergence.[
The multicanonical ensemble is not restricted to physical systems. It can be employed on abstract systems which have a cost function ''F''. By using the density of states with respect to F, the method becomes general for computing higher-dimensional integrals or finding local minima.][
]
Motivation
Consider a system and its phase-space characterized by a configuration in and a "cost" function ''F'' from the system's phase-space to a one-dimensional space :