On the convergence to the non-equilibrium steady state of a Langevin dynamics with widely separated time scales and different temperatures
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| Format: | Preprint |
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2021
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| _version_ | 1866916149752496128 |
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| author | Alberici, Diego Macris, Nicolas Mingione, Emanuele |
| author_facet | Alberici, Diego Macris, Nicolas Mingione, Emanuele |
| contents | We study the solution of the two-temperatures Fokker-Planck equation and rigorously analyse its convergence towards an explicit non-equilibrium stationary measure for long time and two widely separated time scales. The exponential rates of convergence are estimated assuming the validity of logarithmic Sobolev inequalities for the conditional and marginal distributions of the limit measure. We show that these estimates are sharp in the exactly solvable case of a quadratic potential. We discuss a few examples where the logarithmic Sobolev inequalities are satisfied through simple, though not optimal, criteria. In particular we consider a spin-glass model with slowly varying external magnetic fields whose non-equilibrium measure corresponds to Guerra's hierarchical construction appearing in Talagrand's proof of the Parisi formula. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_11356 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the convergence to the non-equilibrium steady state of a Langevin dynamics with widely separated time scales and different temperatures Alberici, Diego Macris, Nicolas Mingione, Emanuele Mathematical Physics Disordered Systems and Neural Networks Analysis of PDEs Probability We study the solution of the two-temperatures Fokker-Planck equation and rigorously analyse its convergence towards an explicit non-equilibrium stationary measure for long time and two widely separated time scales. The exponential rates of convergence are estimated assuming the validity of logarithmic Sobolev inequalities for the conditional and marginal distributions of the limit measure. We show that these estimates are sharp in the exactly solvable case of a quadratic potential. We discuss a few examples where the logarithmic Sobolev inequalities are satisfied through simple, though not optimal, criteria. In particular we consider a spin-glass model with slowly varying external magnetic fields whose non-equilibrium measure corresponds to Guerra's hierarchical construction appearing in Talagrand's proof of the Parisi formula. |
| title | On the convergence to the non-equilibrium steady state of a Langevin dynamics with widely separated time scales and different temperatures |
| topic | Mathematical Physics Disordered Systems and Neural Networks Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2112.11356 |