Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors

Cuneo, Noé; Eckmann, Jean‐Pierre; Poquet, Christophe

doi:10.1088/0951-7715/28/7/2397

Cited by 25 publications

(47 citation statements)

References 18 publications

(46 reference statements)

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“…Our results imply that both effects combine in a way that prevents overheating. See [6,Remark 3.10] for a quantitative discussion of these two effects for a chain of three rotors. See also [17] for a clear exposition of the overheating problem in a related model.…”

Section: Overview Of the Dynamicsmentioning

confidence: 99%

“…It suffices to consider the case j = 2. The variablep 2 is constructed as in [6]. We continue the averaging procedure started above.…”

Section: )mentioning

confidence: 99%

“…The proof of Proposition 3.12, and in particular (3.29), relies on the linear nature of the decoupled dynamics. If we add constant forces τ 1 and τ 4 at the ends of the chain (as in [6]), the method above applies with little modification, and with the replacements…”

Section: Fully Decoupled Dynamics Approximationmentioning

confidence: 99%

“…for some C 9 > 0, where we have used that 6) where the first inequality follows from (2.16) and the second inequality holds because the derivatives of…”

Section: Constructing a Global Lyapunov Functionmentioning

confidence: 99%

“…Now that we have a Lyapunov function (Theorem 1.4), we can prove Theorem 1.3 in the spirit of [6]. In addition to Theorem 1.4, we need a few other ingredients.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

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Non-Equilibrium Steady States for Chains of Four Rotors

Cuneo

Eckmann

2016

Commun. Math. Phys.

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Abstract:We study a chain of four interacting rotors (rotators) connected at both ends to stochastic heat baths at different temperatures. We show that for non-degenerate interaction potentials the system relaxes, at a stretched exponential rate, to a non-equilibrium steady state (NESS). Rotors with high energy tend to decouple from their neighbors due to fast oscillation of the forces. Because of this, the energy of the central two rotors, which interact with the heat baths only through the external rotors, can take a very long time to dissipate. By appropriately averaging the oscillatory forces, we estimate the dissipation rate and construct a Lyapunov function. Compared to the chain of length three (considered previously by C. Poquet and the current authors), the new difficulty with four rotors is the appearance of resonances when both central rotors are fast. We deal with these resonances using the rapid thermalization of the two external rotors.

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Section: Overview Of the Dynamicsmentioning

confidence: 99%

“…It suffices to consider the case j = 2. The variablep 2 is constructed as in [6]. We continue the averaging procedure started above.…”

Section: )mentioning

confidence: 99%

Section: Fully Decoupled Dynamics Approximationmentioning

confidence: 99%

“…for some C 9 > 0, where we have used that 6) where the first inequality follows from (2.16) and the second inequality holds because the derivatives of…”

Section: Constructing a Global Lyapunov Functionmentioning

confidence: 99%

“…Now that we have a Lyapunov function (Theorem 1.4), we can prove Theorem 1.3 in the spirit of [6]. In addition to Theorem 1.4, we need a few other ingredients.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

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Non-Equilibrium Steady States for Chains of Four Rotors

Cuneo

Eckmann

2016

Commun. Math. Phys.

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Nonequilibrium Statistical Mechanics of Weakly Stochastically Perturbed System of Oscillators

Dymov

2015

Ann. Henri Poincaré

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We consider a finite region of a d-dimensional lattice, d ∈ N, of weakly coupled harmonic oscillators. The coupling is provided by a nearest-neighbour potential (harmonic or not) of size ε. Each oscillator weakly interacts by force of order ε with its own stochastic Langevin thermostat of arbitrary positive temperature. We investigate limiting as ε → 0 behaviour of solutions of the system and of the local energy of oscillators on long-time intervals of order ε −1 and in a stationary regime. We show that it is governed by an effective equation which is a dissipative SDE with nondegenerate diffusion. Next we assume that the interaction potential is of size ελ, where λ is another small parameter, independent from ε. Solutions corresponding to this scaling describe small low temperature oscillations. We prove that in a stationary regime, under the limit ε → 0, the main order in λ of the averaged Hamiltonian energy flow is proportional to the gradient of temperature. We show that the coefficient of proportionality, which we call the conductivity, admits a representation through stationary space-time correlations of the energy flow. Most of the results and convergences we obtain are uniform with respect to the number of oscillators in the system.

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Entropic Fluctuations in Thermally Driven Harmonic Networks

2016

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Abstract. We consider a general network of harmonic oscillators driven out of thermal equilibrium by coupling to several heat reservoirs at different temperatures. The action of the reservoirs is implemented by Langevin forces. Assuming the existence and uniqueness of the steady state of the resulting process, we construct a canonical entropy production functional S t which satisfies the Gallavotti-Cohen fluctuation theorem. More precisely, we prove that there exists κ c > 1 2 such that the cumulant generating function of S t has a large-time limit e(α) which is finite on a closed interval [ 1 2 − κ c , 1 2 + κ c ], infinite on its complement and satisfies the Gallavotti-Cohen symmetry e(1 − α) = e(α) for all α ∈ R. Moreover, we show that e(α) is essentially smooth, i.e., that e (α) → ∓∞ as α → 1 2 ∓ κ c . It follows from the Gärtner-Ellis theorem that S t satisfies a global large deviation principle with a rate function I(s) obeying the Gallavotti-Cohen fluctuation relation I(−s) − I(s) = s for all s ∈ R. We also consider perturbations of S t by quadratic boundary terms and prove that they satisfy extended fluctuation relations, i.e., a global large deviation principle with a rate function that typically differs from I(s) outside a finite interval. This applies to various physically relevant functionals and, in particular, to the heat dissipation rate of the network. Our approach relies on the properties of the maximal solution of a one-parameter family of algebraic matrix Riccati equations. It turns out that the limiting cumulant generating functions of S t and its perturbations can be computed in terms of spectral data of a Hamiltonian matrix depending on the harmonic potential of the network and the parameters of the Langevin reservoirs. This approach is well adapted to both analytical and numerical investigations.

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Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors

Cited by 25 publications

References 18 publications

Non-Equilibrium Steady States for Chains of Four Rotors

Non-Equilibrium Steady States for Chains of Four Rotors

Nonequilibrium Statistical Mechanics of Weakly Stochastically Perturbed System of Oscillators

Entropic Fluctuations in Thermally Driven Harmonic Networks

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