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<title>Parallel tempering</title>

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<p><b>Parallel tempering</b>, in <a href="Physics" title="Physics">physics</a> and <a href="Statistics" title="Statistics">statistics</a>, is a computer simulation method typically used to find the lowest energy state of a system of many interacting particles. It addresses the problem that at high temperatures, one may have a stable state different from low temperature, whereas simulations at low temperatures 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.

</p><p>More specifically, parallel tempering (also known as <b>replica exchange MCMC sampling</b>), is a <a href="Simulation" title="Simulation">simulation</a> method aimed at improving the dynamic properties of <a href="Monte_Carlo_method" title="Monte Carlo method">Monte Carlo method</a> simulations of physical systems, and of <a href="Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a> (MCMC) sampling methods more generally. The replica exchange method was originally devised by <a href="Robert_Swendsen" title="Robert Swendsen">Robert Swendsen</a> and J. S. Wang,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> then extended by Charles J. Geyer,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and later developed further by <a href="Giorgio_Parisi" title="Giorgio Parisi">Giorgio Parisi</a>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

Koji Hukushima and Koji Nemoto,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

and others.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

Y. Sugita and Y. Okamoto also formulated a <a href="Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a> version of parallel tempering; this is usually known as replica-exchange molecular dynamics or REMD.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>Essentially, one runs <i>N</i> 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.

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<div class="mw-heading mw-heading2"><h2 id="Background">Background</h2></div>

<p>Typically a <a href="Monte_Carlo_simulation" class="mw-redirect" title="Monte Carlo simulation">Monte Carlo simulation</a> using a <a href="Metropolis%E2%80%93Hastings" class="mw-redirect" title="Metropolis–Hastings">Metropolis–Hastings</a> update consists of a single <a href="Stochastic_process" title="Stochastic process">stochastic process</a> that evaluates the <a href="Energy" title="Energy">energy</a> of the system and accepts/rejects updates based on the <a href="Temperature" title="Temperature">temperature</a> <i>T</i>. 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.

</p><p>If we were to run two simulations at temperatures separated by a Δ<i>T</i>, we would find that if Δ<i>T</i> is small enough, then the energy <a href="Histogram" title="Histogram">histograms</a> 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<i>T</i> = 0 the overlap should approach 1.

</p><p>Another way to interpret this overlap is to say that system configurations sampled at temperature <i>T</i><sub>1</sub> are likely to appear during a simulation at <i>T</i><sub>2</sub>. Because the <a href="Markov_chain" title="Markov chain">Markov chain</a> should have no memory of its past, we can create a new update for the system composed of the two systems at <i>T</i><sub>1</sub> and <i>T</i><sub>2</sub>. 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

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<annotation encoding="application/x-tex">{\displaystyle p=\min \left(1,{\frac {\exp \left(-{\frac {E_{j}}{kT_{i}}}-{\frac {E_{i}}{kT_{j}}}\right)}{\exp \left(-{\frac {E_{i}}{kT_{i}}}-{\frac {E_{j}}{kT_{j}}}\right)}}\right)=\min \left(1,e^{(E_{i}-E_{j})\left({\frac {1}{kT_{i}}}-{\frac {1}{kT_{j}}}\right)}\right),}</annotation>

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</math></span><img src="./70ab162a039e036dc0049805f921f59304feef52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; margin-left: -0.089ex; width:64.847ex; height:10.509ex;" alt="{\displaystyle p=\min \left(1,{\frac {\exp \left(-{\frac {E_{j}}{kT_{i}}}-{\frac {E_{i}}{kT_{j}}}\right)}{\exp \left(-{\frac {E_{i}}{kT_{i}}}-{\frac {E_{j}}{kT_{j}}}\right)}}\right)=\min \left(1,e^{(E_{i}-E_{j})\left({\frac {1}{kT_{i}}}-{\frac {1}{kT_{j}}}\right)}\right),}" loading="lazy"></span></dd></dl>

<p>and otherwise the update is rejected. The <a href="Detailed_balance" title="Detailed balance">detailed balance</a> 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.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p><p>This update can be generalized to more than two systems.

</p><p>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.

</p><p>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.

</p><p>The parallel tempering method can be used as a super <a href="Simulated_annealing" title="Simulated annealing">simulated annealing</a> 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.

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<div class="mw-heading mw-heading2"><h2 id="Implementations">Implementations</h2></div>

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<ul><li><a href="Abalone_(molecular_mechanics)" title="Abalone (molecular mechanics)">Abalone</a></li>

<li>ACEMD</li>

<li><a href="AMBER" title="AMBER">AMBER</a></li>

<li><a href="CHARMM" title="CHARMM">CHARMM</a></li>

<li><a href="Desmond_(software)" title="Desmond (software)">Desmond</a></li>

<li><a href="GROMACS" title="GROMACS">GROMACS</a></li>

<li>GENESIS</li>

<li><a href="LAMMPS" title="LAMMPS">LAMMPS</a></li>

<li>RASPA-2.0</li>

<ul><li><a href="Bennett_acceptance_ratio" title="Bennett acceptance ratio">Bennett acceptance ratio</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Swendsen RH and Wang JS (1986) <a rel="nofollow" class="external text" href="https://www.researchgate.net/profile/Robert_Swendsen/publication/13255490_Replica_Monte_Carlo_Simulation_of_Spin-Glasses/links/0046352309b5f54715000000.pdf">Replica Monte Carlo simulation of spin glasses</a> Physical Review Letters 57 : 2607–2609</span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">C. J. Geyer, (1991) in <i>Computing Science and Statistics</i>, Proceedings of the 23rd Symposium on the Interface, American Statistical Association, New York, p. 156.</span>

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</style><cite id="CITEREFMarinariParisi1992" class="citation journal cs1">Marinari, E; Parisi, G (1992-07-15). <a rel="nofollow" class="external text" href="https://iopscience.iop.org/article/10.1209/0295-5075/19/6/002">"Simulated Tempering: A New Monte Carlo Scheme"</a>. <i>Europhysics Letters</i>. <b>19</b> (6): <span class="nowrap">451–</span>458. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-lat/9205018">hep-lat/9205018</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992EL.....19..451M">1992EL.....19..451M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1209%2F0295-5075%2F19%2F6%2F002">10.1209/0295-5075/19/6/002</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0295-5075">0295-5075</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:250781561">250781561</a>.</cite></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFHukushima,_KojiNemoto,_Koji1996" class="citation journal cs1">Hukushima, Koji & Nemoto, Koji (1996). "Exchange Monte Carlo method and application to spin glass simulations". <i>J. Phys. Soc. Jpn</i>. <b>65</b> (6): <span class="nowrap">1604–</span>1608. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/9512035">cond-mat/9512035</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996JPSJ...65.1604H">1996JPSJ...65.1604H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1143%2FJPSJ.65.1604">10.1143/JPSJ.65.1604</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15032087">15032087</a>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFMarco_FalcioniMichael_W._Deem1999" class="citation journal cs1">Marco Falcioni & Michael W. Deem (1999). "A Biased Monte Carlo Scheme for Zeolite Structure Solution". <i>J. Chem. Phys</i>. <b>110</b> (3): 1754. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/9809085">cond-mat/9809085</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999JChPh.110.1754F">1999JChPh.110.1754F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.477812">10.1063/1.477812</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13963102">13963102</a>.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">David J. Earl and Michael W. Deem (2005) <a rel="nofollow" class="external text" href="http://www.rsc.org/Publishing/Journals/CP/article.asp?doi=b509983h">"Parallel tempering: Theory, applications, and new perspectives"</a>, <i>Phys. Chem. Chem. Phys.</i>, 7, 3910</span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFY._SugitaY._Okamoto1999" class="citation journal cs1">Y. Sugita & Y. Okamoto (1999). "Replica-exchange molecular dynamics method for protein folding". <i>Chemical Physics Letters</i>. <b>314</b> (<span class="nowrap">1–</span>2): <span class="nowrap">141–</span>151. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999CPL...314..141S">1999CPL...314..141S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0009-2614%2899%2901123-9">10.1016/S0009-2614(99)01123-9</a>.</cite></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFRadford_M._Neal1996" class="citation journal cs1">Radford M. Neal (1996). "Sampling from multimodal distributions using tempered transitions". <i>Statistics and Computing</i>. <b>6</b> (4): <span class="nowrap">353–</span>366. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00143556">10.1007/BF00143556</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11106113">11106113</a>.</cite></span>

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