/* Irreversible local Markov chains with rapid convergence towards equilibrium */
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| ==Irreversible local Markov chains with rapid convergence towards equilibrium== | ==Irreversible local Markov chains with rapid convergence towards equilibrium== | ||
| - | [[Image:Figure1_Kapfer_Krauth_2017a.jpg|left|600px|border|Mixing time scales for local Markov chains in 1d]] Monte Carlo algorithms, generally satisfy the detailed balance condition, which prescribes that in the limit of infinite times, the ''probability flow'' from a configuration '''a''' to a configuration '''b''' equals the flow from '''b''' to '''a'''. This may seem terrible abstract, but it simply means that if, in a room full of air molecules, each molecule moves to the left and to the right with the same probability (and sometimes does not move at all, because there is already another particle where it wants to go), the density of air will be more or less uniform. In [[Kapfer_Krauth_2017a|a recent paper with Sebastian Kapfer]], we systematically studied irreversible local Markov chain, that is, Monte Carlo algorithms which only satisfy the global balance condition, but not the detailed balance (in the example of the air-filled room, this corresponds to algorithms where the molecules are much more likely to move in one direction than the other, but where the asymptotic density is still uniform). We considered the case of hard-sphere gases in one spatial dimension with periodic boundary conditions and, to our greatest surprise, came up with Markov chains such as the 'forward Metropolis algorithm' or the 'lifted forward Metropolis algorithm', or even the 'lifted forward Metropolis algorithm with restart' that mix much faster than the usual methods, although they reach exactly the same steady state in the limit of infinite times. We even made contact with the vast research literature on the TASEP (totally asymmetric simple exclusion process), a discrete variant of our Markov chains. We are all the more excited that the algorithms studied are but special versions of the [[Bernard_Krauth_Wilson_2009|event-chain algorithm]], that we used a lot during the last years. | + | [[Image:Figure1_Kapfer_Krauth_2017a.jpg|left|600px|border|Mixing time scales for local Markov chains in 1d]] Monte Carlo algorithms, generally satisfy the detailed balance condition, which prescribes that in the limit of infinite times, the ''probability flow'' from a configuration '''a''' to a configuration '''b''' equals the flow from '''b''' to '''a'''. This may seem terrible abstract, but it simply means that if, in a room full of air molecules, each molecule moves to the left and to the right with the same probability (and sometimes does not move at all, because there is already another particle where it wants to go), the density of air will be more or less uniform. In [[Kapfer_Krauth_2017a|a recent paper with Sebastian Kapfer, to appear in Physical Review Letters]], we systematically studied irreversible local Markov chain, that is, Monte Carlo algorithms which only satisfy the global balance condition, but not the detailed balance (in the example of the air-filled room, this corresponds to algorithms where the molecules are much more likely to move in one direction than the other, but where the asymptotic density is still uniform). We considered the case of hard-sphere gases in one spatial dimension with periodic boundary conditions and, to our greatest surprise, came up with Markov chains such as the 'forward Metropolis algorithm' or the 'lifted forward Metropolis algorithm', or even the 'lifted forward Metropolis algorithm with restart' that mix much faster than the usual methods, although they reach exactly the same steady state in the limit of infinite times. We even made contact with the vast research literature on the TASEP (totally asymmetric simple exclusion process), a discrete variant of our Markov chains. We are all the more excited that the algorithms studied are but special versions of the [[Bernard_Krauth_Wilson_2009|event-chain algorithm]], that we used a lot during the last years. |
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