Parallel tempering in
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 ...
and
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 ...
, is a computer simulation method typically used to find the lowest free energy state of a system of many interacting particles at low temperature. That is, the one expected to be observed in reality. It addresses the problem that at high temperature one may have a stable state different from low temperature, whereas simulations at low temperature may become "stuck" in a metastable state. It does this by using the fact that the high temperature simulation may visit states typical of both stable and metastable low temperature states.
More specifically, parallel tempering (also known as replica exchange MCMC sampling), is a
simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...
method aimed at improving the dynamic properties of
Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...
simulations of physical systems, and of
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 ...
(MCMC) sampling methods more generally. The replica exchange method was originally devised by Swendsen and Wang then extended by Geyer and later developed, among others, by
Hukushima and
Nemoto,
Giorgio Parisi
Giorgio Parisi (born 4 August 1948) is an Italian theoretical physicist, whose research has focused on quantum field theory, statistical mechanics and complex systems. His best known contributions are the QCD evolution equations for parton densit ...
,
Sugita and Okamoto formulated a
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
version of parallel tempering: this is usually known as replica-exchange molecular dynamics or REMD.
Essentially, one runs ''N'' copies of the system, randomly initialized, at different temperatures. Then, based on the Metropolis criterion one exchanges configurations at different temperatures. The idea of this method
is to make configurations at high temperatures available to the simulations at low temperatures and vice versa.
This results in a very robust ensemble which is able to sample both low and high energy configurations.
In this way, thermodynamical properties such as the specific heat, which is in general not well computed in the canonical ensemble, can be computed with great precision.
Background
Typically a
Monte Carlo simulation
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
using a
Metropolis–Hastings update consists of a single
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
that evaluates the
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
of the system and accepts/rejects updates based on the
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
''T''. At high temperatures updates that change the energy of the system are comparatively more probable. When the system is highly correlated, updates are rejected and the simulation is said to suffer from critical slowing down.
If we were to run two simulations at temperatures separated by a Δ''T'', we would find that if Δ''T'' is small enough, then the energy
histogram
A histogram is an approximate representation of the distribution of numerical data. The term was first introduced by Karl Pearson. To construct a histogram, the first step is to " bin" (or "bucket") the range of values—that is, divide the ent ...
s obtained by collecting the values of the energies over a set of Monte Carlo steps N will create two distributions that will somewhat overlap. The overlap can be defined by the area of the histograms that falls over the same interval of energy values, normalized by the total number of samples. For Δ''T'' = 0 the overlap should approach 1.
Another way to interpret this overlap is to say that system configurations sampled at temperature ''T''
1 are likely to appear during a simulation at ''T''
2. Because the
Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
should have no memory of its past, we can create a new update for the system composed of the two systems at ''T''
1 and ''T''
2. At a given Monte Carlo step we can update the global system by swapping the configuration of the two systems, or alternatively trading the two temperatures. The update is accepted according to the Metropolis–Hastings criterion with probability
:
and otherwise the update is rejected. The
detailed balance The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at equilibrium, each elementary process is in equilibrium with its reve ...
condition has to be satisfied by ensuring that the reverse update has to be equally likely, all else being equal. This can be ensured by appropriately choosing regular Monte Carlo updates or parallel tempering updates with probabilities that are independent of the configurations of the two systems or of the Monte Carlo step.
This update can be generalized to more than two systems.
By a careful choice of temperatures and number of systems one can achieve an improvement in the mixing properties of a set of Monte Carlo simulations that exceeds the extra computational cost of running parallel simulations.
Other considerations to be made: increasing the number of different temperatures can have a detrimental effect, as one can think of the 'lateral' movement of a given system across temperatures as a diffusion process.
Set up is important as there must be a practical histogram overlap to achieve a reasonable probability of lateral moves.
The parallel tempering method can be used as a super
simulated annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. It ...
that does not need restart, since a system at high temperature can feed new local optimizers to a system at low temperature, allowing tunneling between metastable states and improving convergence to a global optimum.
Implementations
See also
*
Bennett acceptance ratio
The Bennett acceptance ratio method (BAR) is an algorithm for estimating the difference in free energy between two systems (usually the systems will be simulated on the computer).
It was suggested by Charles H. Bennett in 1976.
Preliminaries
Tak ...
References
{{DEFAULTSORT:Parallel Tempering
Markov chain Monte Carlo
Molecular dynamics
Heuristics
Statistical mechanics
Stochastic optimization