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Metropolis-Hastings thermal state sampling for numerical simulations of Bose-Einstein condensates.

Grišins P, Mazets IE - Comput Phys Commun (2014)

Bottom Line: We demonstrate the application of the Metropolis-Hastings algorithm to sampling of classical thermal states of one-dimensional Bose-Einstein quasicondensates in the classical fields approximation, both in untrapped and harmonically trapped case.For truncated Wigner simulations the quantum noise can be added with conventional methods (half a quantum of energy in every mode).The advantage of the presented method over the usual analytical and stochastic ones lies in its ability to sample not only from canonical and grand canonical distributions, but also from the generalized Gibbs ensemble, which can help to shed new light on thermodynamics of integrable systems.

View Article: PubMed Central - PubMed

Affiliation: Vienna Center for Quantum Science and Technology, Atominstitut TU Wien, 1020 Vienna, Austria.

ABSTRACT

We demonstrate the application of the Metropolis-Hastings algorithm to sampling of classical thermal states of one-dimensional Bose-Einstein quasicondensates in the classical fields approximation, both in untrapped and harmonically trapped case. The presented algorithm can be easily generalized to higher dimensions and arbitrary trap geometry. For truncated Wigner simulations the quantum noise can be added with conventional methods (half a quantum of energy in every mode). The advantage of the presented method over the usual analytical and stochastic ones lies in its ability to sample not only from canonical and grand canonical distributions, but also from the generalized Gibbs ensemble, which can help to shed new light on thermodynamics of integrable systems.

No MeSH data available.


Related in: MedlinePlus

Influence of the initial state on the rate of convergence to the thermal state. Temperatures during the Metropolis–Hastings ‘evolution’ as a function of the iteration number  in the case of untrapped gas for , averaged over 70 realizations. Thick line: zero-temperature state of the non-interacting gas, cf. Fig. 3(a). Thin line: thermal gas of Bogoliubov quasiparticles with random phases and constant amplitudes (see explanation in the text). Both choices of initial conditions eventually lead to equilibrium, but in case of the ‘Bogoliubov gas’ the convergence is faster, meaning that it is a better ‘initial guess’ for the thermal state. In this particular realization the temperature is rising during the ‘evolution’, but we note that if we had chosen a higher-than-desired initial temperature, then the temperature would be dropping to the desired value. Inset. Temperature of the state, produced by the real-time GPE evolution starting from the achieved thermal state as a function of time. Dots: one single realization, solid line: average over 70 realizations. The stability of the temperature shows that the initial state was indeed the thermal state of the Gross–Pitaevskii Hamiltonian (see the text for further discussion).
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f000020: Influence of the initial state on the rate of convergence to the thermal state. Temperatures during the Metropolis–Hastings ‘evolution’ as a function of the iteration number in the case of untrapped gas for , averaged over 70 realizations. Thick line: zero-temperature state of the non-interacting gas, cf. Fig. 3(a). Thin line: thermal gas of Bogoliubov quasiparticles with random phases and constant amplitudes (see explanation in the text). Both choices of initial conditions eventually lead to equilibrium, but in case of the ‘Bogoliubov gas’ the convergence is faster, meaning that it is a better ‘initial guess’ for the thermal state. In this particular realization the temperature is rising during the ‘evolution’, but we note that if we had chosen a higher-than-desired initial temperature, then the temperature would be dropping to the desired value. Inset. Temperature of the state, produced by the real-time GPE evolution starting from the achieved thermal state as a function of time. Dots: one single realization, solid line: average over 70 realizations. The stability of the temperature shows that the initial state was indeed the thermal state of the Gross–Pitaevskii Hamiltonian (see the text for further discussion).

Mentions: Another independent test whether the achieved state is thermal is the real-time development of the state, as by definition the thermal state should remain thermal during such evolution. To check this criterion we prepared the thermal state of the untrapped gas with Metropolis algorithm and then propagated it in real time with Gross–Pitaevskii equation (there exist efficient algorithms for solving real-time GPE, see e.g.  [20,21]). The results, presented in the inset to Fig. 4, show that indeed the temperature of the state does not change on average, assuring that the initial state was thermal with respect to the Gross–Pitaevskii Hamiltonian.


Metropolis-Hastings thermal state sampling for numerical simulations of Bose-Einstein condensates.

Grišins P, Mazets IE - Comput Phys Commun (2014)

Influence of the initial state on the rate of convergence to the thermal state. Temperatures during the Metropolis–Hastings ‘evolution’ as a function of the iteration number  in the case of untrapped gas for , averaged over 70 realizations. Thick line: zero-temperature state of the non-interacting gas, cf. Fig. 3(a). Thin line: thermal gas of Bogoliubov quasiparticles with random phases and constant amplitudes (see explanation in the text). Both choices of initial conditions eventually lead to equilibrium, but in case of the ‘Bogoliubov gas’ the convergence is faster, meaning that it is a better ‘initial guess’ for the thermal state. In this particular realization the temperature is rising during the ‘evolution’, but we note that if we had chosen a higher-than-desired initial temperature, then the temperature would be dropping to the desired value. Inset. Temperature of the state, produced by the real-time GPE evolution starting from the achieved thermal state as a function of time. Dots: one single realization, solid line: average over 70 realizations. The stability of the temperature shows that the initial state was indeed the thermal state of the Gross–Pitaevskii Hamiltonian (see the text for further discussion).
© Copyright Policy - CC BY
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4376078&req=5

f000020: Influence of the initial state on the rate of convergence to the thermal state. Temperatures during the Metropolis–Hastings ‘evolution’ as a function of the iteration number in the case of untrapped gas for , averaged over 70 realizations. Thick line: zero-temperature state of the non-interacting gas, cf. Fig. 3(a). Thin line: thermal gas of Bogoliubov quasiparticles with random phases and constant amplitudes (see explanation in the text). Both choices of initial conditions eventually lead to equilibrium, but in case of the ‘Bogoliubov gas’ the convergence is faster, meaning that it is a better ‘initial guess’ for the thermal state. In this particular realization the temperature is rising during the ‘evolution’, but we note that if we had chosen a higher-than-desired initial temperature, then the temperature would be dropping to the desired value. Inset. Temperature of the state, produced by the real-time GPE evolution starting from the achieved thermal state as a function of time. Dots: one single realization, solid line: average over 70 realizations. The stability of the temperature shows that the initial state was indeed the thermal state of the Gross–Pitaevskii Hamiltonian (see the text for further discussion).
Mentions: Another independent test whether the achieved state is thermal is the real-time development of the state, as by definition the thermal state should remain thermal during such evolution. To check this criterion we prepared the thermal state of the untrapped gas with Metropolis algorithm and then propagated it in real time with Gross–Pitaevskii equation (there exist efficient algorithms for solving real-time GPE, see e.g.  [20,21]). The results, presented in the inset to Fig. 4, show that indeed the temperature of the state does not change on average, assuring that the initial state was thermal with respect to the Gross–Pitaevskii Hamiltonian.

Bottom Line: We demonstrate the application of the Metropolis-Hastings algorithm to sampling of classical thermal states of one-dimensional Bose-Einstein quasicondensates in the classical fields approximation, both in untrapped and harmonically trapped case.For truncated Wigner simulations the quantum noise can be added with conventional methods (half a quantum of energy in every mode).The advantage of the presented method over the usual analytical and stochastic ones lies in its ability to sample not only from canonical and grand canonical distributions, but also from the generalized Gibbs ensemble, which can help to shed new light on thermodynamics of integrable systems.

View Article: PubMed Central - PubMed

Affiliation: Vienna Center for Quantum Science and Technology, Atominstitut TU Wien, 1020 Vienna, Austria.

ABSTRACT

We demonstrate the application of the Metropolis-Hastings algorithm to sampling of classical thermal states of one-dimensional Bose-Einstein quasicondensates in the classical fields approximation, both in untrapped and harmonically trapped case. The presented algorithm can be easily generalized to higher dimensions and arbitrary trap geometry. For truncated Wigner simulations the quantum noise can be added with conventional methods (half a quantum of energy in every mode). The advantage of the presented method over the usual analytical and stochastic ones lies in its ability to sample not only from canonical and grand canonical distributions, but also from the generalized Gibbs ensemble, which can help to shed new light on thermodynamics of integrable systems.

No MeSH data available.


Related in: MedlinePlus