Quantitative Rates of Convergence to Non-equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators
We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N...
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| Published in | Journal of statistical physics Vol. 181; no. 1; pp. 53 - 94 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.10.2020
Springer Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0022-4715 1572-9613 |
| DOI | 10.1007/s10955-020-02565-5 |
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| Abstract | We study a 1-dimensional chain of
N
weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with
N
dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry–Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein–Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than
N
-
3
. For the purely harmonic chain the order of the rate is in
[
N
-
3
,
N
-
1
]
. This shows that, in this set up, the spectral gap decays at most polynomially with
N
. |
|---|---|
| AbstractList | We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry-Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein-Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than [Formula omitted]. For the purely harmonic chain the order of the rate is in [Formula omitted]. This shows that, in this set up, the spectral gap decays at most polynomially with N. We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry–Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein–Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than N-3. For the purely harmonic chain the order of the rate is in [N-3,N-1]. This shows that, in this set up, the spectral gap decays at most polynomially with N. We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry–Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein–Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than $$N^{-3}$$ N - 3 . For the purely harmonic chain the order of the rate is in $$ [N^{-3},N^{-1}]$$ [ N - 3 , N - 1 ] . This shows that, in this set up, the spectral gap decays at most polynomially with N . We study a 1-dimensional chain of N weakly anharmonic classical oscillators coupled at its ends to heat baths at different temperatures. Each oscillator is subject to pinning potential and it also interacts with its nearest neighbors. In our set up both potentials are homogeneous and bounded (with N dependent bounds) perturbations of the harmonic ones. We show how a generalised version of Bakry–Emery theory can be adapted to this case of a hypoelliptic generator which is inspired by Baudoin (J Funct Anal 273(7):2275-2291, 2017). By that we prove exponential convergence to non-equilibrium steady state in Wasserstein–Kantorovich distance and in relative entropy with quantitative rates. We estimate the constants in the rate by solving a Lyapunov-type matrix equation and we obtain that the exponential rate, for the homogeneous chain, has order bigger than N - 3 . For the purely harmonic chain the order of the rate is in [ N - 3 , N - 1 ] . This shows that, in this set up, the spectral gap decays at most polynomially with N . |
| Audience | Academic |
| Author | Menegaki, Angeliki |
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| CitedBy_id | crossref_primary_10_1016_j_spa_2023_04_006 crossref_primary_10_5802_ahl_129 crossref_primary_10_2140_paa_2021_3_223 crossref_primary_10_1016_j_jfa_2021_109286 crossref_primary_10_1016_j_jfa_2024_110814 |
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| Keywords | Nonequilibrium steady states Exponential convergence Spectral gap Heat bath Hypocoercivity Hypoellipticity Functional inequalities Chain of oscillators |
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| References_xml | – reference: CuneoNPoquetCOn the relaxation rate of short chains of rotors interacting with Langevin thermostatsElectron. Commun. Probab.20172235366685610.1214/17-ECP62 – reference: LetiziaVOllaSNonequilibrium isothermal transformations in a temperature gradient from a microscopic dynamicsAnn. Probab.2017456A39874018372962110.1214/16-AOP1156 – reference: CasherALebowitzJLHeat flow in regular and disordered harmonic chainsJ. Math. Phys.19711217011971JMP....12.1701C10.1063/1.1665794 – reference: EckmannJ-PPilletC-ARey-BelletLEntropy production in nonlinear, thermally driven Hamiltonian systemsJ. Stat. Phys.1999951–23053311999JSP....95..305E170558910.1023/A:1004537730090 – reference: Veselić, K.: Bounds for exponentially stable semigroups. Linear Algebra Appl., 358, 309–333 (2003). Special issue on accurate solution of eigenvalue problems (Hagen, 2000) – reference: KuwadaKDuality on gradient estimates and Wasserstein controlsJ. Funct. Anal.20102581137583774260687110.1016/j.jfa.2010.01.010 – reference: Talay, D.: Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, vol. 8, pp. 163–198 (2002). Inhomogeneous random systems (Cergy-Pontoise, 2001) – reference: CuneoNEckmannJ-PHairerMRey-BelletLNon-equilibrium steady states for networks of oscillatorsElectron. J. Probab.20182328381424910.1214/18-EJP177 – reference: RaquépasRA note on Harris’ ergodic theorem, controllability and perturbations of harmonic networksAnn. Henri Poincaré20192026056292019AnHP...20..605R390884910.1007/s00023-018-0740-0 – reference: GodunovSVKiriljukOPKostinIVSpectral Portraits of Matrices (Russian)1990NovosibirskAN SSSR Siber. Otd – reference: Lepri, S. (eds). Thermal transport in low dimensions, vol. 921 of Lecture Notes in Physics. Springer, Cham (2016). From statistical physics to nanoscale heat transfer – reference: Rey-Bellet, L.: Open Classical Systems. 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| SubjectTerms | Anharmonicity Chains Convergence Mathematical and Computational Physics Oscillators Physical Chemistry Physics Physics and Astronomy Quantum Physics Statistical Physics and Dynamical Systems Steady state Theoretical |
| Title | Quantitative Rates of Convergence to Non-equilibrium Steady State for a Weakly Anharmonic Chain of Oscillators |
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