## HEAT CONDUCTION NETWORKS: DISPOSITION OF HEAT BATHS AND INVARIANT MEASURE (902)

### BibTeX

@MISC{Camanes902heatconduction,

author = {Alain Camanes},

title = {HEAT CONDUCTION NETWORKS: DISPOSITION OF HEAT BATHS AND INVARIANT MEASURE},

year = {902}

}

### OpenURL

### Abstract

Abstract. We consider a model of heat conduction networks consisting of oscillators in contact with heat baths at different temperatures. Our aim is to generalize the results concerning the existence and uniqueness of the stationnary state already obtained when the network is reduced to a chain of particles. Using Lasalle’s principle, we establish a condition on the disposition of the heat baths among the network that ensures the uniqueness of the invariant measure. We will show that this condition is sharp when the oscillators are linear. Moreover, when the interaction between the particles is stronger than the pinning, we prove that this condition implies the existence of the invariant measure. 1. Definitions and Results 1.1. The motivations. We consider an arbitrary graph. At each vertex of this graph, there is a particle interacting with the substrate and with its neighbours. Among these particles, some are linked to heat baths; an Ornstein-Uhlenbeck process models this interaction. Given this graph, we establish conditions on the disposition of the heat baths that entails existence and uniqueness of the invariant measure. When the graph is reduced to a chain, each extremal particle is connected to a heat bath.

### Citations

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Citation Context ...convex polynomials, we expect that the diffusion (1) is Asymptotically Strong Feller at the equilibrium point c0. The Asymptotic Strong Feller property was introduced by M. Hairer and J. Mattingly in =-=[HM06a]-=- to study the stochastic Navier-Stokes equation. We recall briefly the main definitions. Definition A.1 (Totally separating system). An increasing sequence (dn) of pseudo-metrics is a totally separati... |

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Citation Context ... to R n . We recall that the support Suppµ of a measure µ is the set of points z ∈ R n such that for any ball B(z0, ε) of radius ε > 0 centred in z0, µ (B(z0, ε)) > 0. Theorem 8 (Support Theorem, see =-=[SV72]-=-, Section 5). Using the previous notations, Supp Pt0(z0, ·) = St0,z0. 3.2.2. Uniqueness of the invariant measure. Recall that the Hamiltonian H has a unique minimizer c0. By adding a constant term, we... |

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Citation Context ...cribing the system is linear. In the following we will denote (ei) i∈{1,...,n} the canonical base of R n . A natural space occurring in heat conduction networks is the controllability space (see e.g. =-=[Won79]-=-). Let MHEAT CONDUCTION NETWORKS: DISPOSITION OF HEAT BATHS AND INVARIANT MEASURE 3 V U 00 11 00 11 Figure 1.1. Example of a heat conduction network : Damped particles are striped, boundary particles... |

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Citation Context ...potentials but we expect that this property is still true in this setting. The importance of the stability condition is strengthened by the following fact about existence of the invariant measure. In =-=[RBT02]-=-, it is shown that when the pinning potential is weaker than the interaction potential and the network is a chain, there exists an invariant measure. To generalize the method to general networks, we w... |

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Citation Context ...The existence of the invariant measure when the interaction is stronger than the pinning has also been obtained. These results were used in [Car07] to solve some variations of this model developed in =-=[BO05]-=- on the one side and [LS04] on the other. To avoid the particular geometry of the chain, we work with general networks. These heat conduction networks have been introduced in [MNV03] and [RB03]. Let u... |

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Citation Context ...model developed in [BO05] on the one side and [LS04] on the other. To avoid the particular geometry of the chain, we work with general networks. These heat conduction networks have been introduced in =-=[MNV03]-=- and [RB03]. Let us notice that an Ornstein-Uhlenbeck process is the sum of a damping term and an excitation term. To understand the effect of each of these quantities, we will not suppose that the he... |

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Citation Context ...en, if K0 = Q, for any t ≥ 0, Kt = Q and Q is the covariance of the invariant measure. Then, the matrix Q can be written as ∫ ∞ Q = 0 e Mt σσ ⋆ e M⋆ t dt.8 ALAIN CAMANES Moreover, using Lemma 2.3 in =-=[SZ70]-=-, That finishes the proof of Theorem 2. rank(Q) = dim EM,∂V. □ Let us notice that the uniqueness of this solution using Theorem 2.2 in [SZ70] gives an alternative proof of uniqueness Theorem 1. Moreov... |

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Citation Context ...) ≥ µ (Kη) Finally, we have shown that c0 ∈ Suppµ. = P ∗ ∫ Tµ(Kη) = Rn PT (z, Kη) µ(dz) ∫ ≥ PT (z, Kη) µ(dz) > 0. We can now prove uniqueness Theorem 4. BR Proof of Theorem 4. Let us recall (see e.g. =-=[DP06]-=-) that the set of invariant measures is the convex hull of the set of invariant ergodic measures. Moreover, if (Pt) is asymptotically strong Feller at c0, then c0 belongs to at most one invariant ergo... |

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Citation Context ...ess is the sum of a damping term and an excitation term. To understand the effect of each of these quantities, we will not suppose that the heat baths have non-negative temperatures. A recent work of =-=[BLLO08]-=- uses this kind of result to prove the existence of a self-consistent temperature profile. We will see that their results are closely related to the geometry of the network they consider. First we int... |

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Citation Context ...re C(µ, ν) is the set of couplings of (µ, ν). The functional W defines the Wasserstein distance on the set P1 of probability measures such that for one z0 ∈ X, W(δz0, µ) < +∞. Let us recall (see e.g. =-=[Bol07]-=-) that (P1, W) is a complete space. In the following, we denote d(µ, ν) = W(µ, ν). Proof of the first part of Theorem 1. We first prove a contraction inequality for Dirac measures. Indeed, using a tri... |

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Citation Context ...ant measure when the interaction is stronger than the pinning has also been obtained. These results were used in [Car07] to solve some variations of this model developed in [BO05] on the one side and =-=[LS04]-=- on the other. To avoid the particular geometry of the chain, we work with general networks. These heat conduction networks have been introduced in [MNV03] and [RB03]. Let us notice that an Ornstein-U... |

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Citation Context ...particle entails the behaviour of its neighbour and so on...The existence of the invariant measure when the interaction is stronger than the pinning has also been obtained. These results were used in =-=[Car07]-=- to solve some variations of this model developed in [BO05] on the one side and [LS04] on the other. To avoid the particular geometry of the chain, we work with general networks. These heat conduction... |

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Citation Context ...e using this method that the convergence speed to the invariant measure is exponential. Indeed, this rate was obtained via compactness properties that we could not generalize. However, the results of =-=[HM07]-=- are still valid : when the interaction potential is quadratic and the pinning potential is at least of degree 4, 0 is in the essential spectrum of the extension of the generator L to the space L 2 (e... |

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Citation Context ...ped in [BO05] on the one side and [LS04] on the other. To avoid the particular geometry of the chain, we work with general networks. These heat conduction networks have been introduced in [MNV03] and =-=[RB03]-=-. Let us notice that an Ornstein-Uhlenbeck process is the sum of a damping term and an excitation term. To understand the effect of each of these quantities, we will not suppose that the heat baths ha... |

3 |
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Citation Context ...the pinning, then there exists a unique invariant measure. Section 2 is devoted to the proofs of Theorems 1 and 2. In Section 3, we will briefly prove Theorem 3 using results obtained by M. Hairer in =-=[Hai05b]-=-. Then, we will present Lasalle’s principle and the associated stability condition. Finally, we will present the proof of the existence of the invariant measure when interaction is stronger than pinni... |

2 | Strange heat flux in (an)harmonic networks
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Citation Context ...i.e. dim EM,D ̸= n, there exists a quantity invariant with respect to the Hamiltonian flow and the invariant measure is not unique. Let us notice that this uniqueness condition was already known (see =-=[EZ04]-=-). We give here a new proof using completeness and provide an explicit quantity invariant by the Hamiltonian flow. The shape of the support of the invariant measure is then described by the position o... |

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1 | On the controllability of conservative systems
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Citation Context ...es are excited, i.e. D = ∂V. Hörmander’s condition will be defined precisely in Section 3.1. The following theorem will be a straightforward consequence of the weak controllability result obtained in =-=[Hai05a]-=-. Theorem 3. If D = ∂V and Hörmander’s condition is satisfied, there exists at most one invariant measure. Finally, in the more general setting where ∂V ⊂ D, we will provide a condition (see Section 3... |

1 |
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Citation Context ... to Section 3.1 we do not suppose that all the damped particles are excited, ∂V ⊂ D. This method is based on the contraction properties of the noise-free dynamics used in the lectures of J. Mattingly =-=[Mat07]-=- to prove uniqueness of the invariant measure for Stochastic Navier-Stokes equations. Our aim is to prove that the contraction point is in the support of every invariant ergodic measure. Then, since e... |

1 |
Nonlinear systems, volume 10 of Interdisciplinary Applied Mathematics
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(Show Context)
Citation Context ...finition 3.1 (Invariant set). A set A ⊂ R n is called invariant if all the trajectories starting from A stay in A, i.e. for any z0 ∈ A, for all t ≥ t0, z z0 t ∈ A. Theorem 7 (Lasalle’s principle, see =-=[Sas99]-=-, Proposition 5.22). Suppose there exists a function H : R n → R+ of class C 1 satisfying the following conditions: for all a > 0, (1) Ωa = {z; H(z) ≤ a} is bounded, (2) ˙ H ∣ ∣Ωa ≤ 0. We denote S = {... |