Heat Flow in a Periodically Forced, Thermostatted Chain

Abstract

We investigate the properties of a harmonic chain in contact with a thermal bath at one end and subjected, at its other end, to a periodic force. The particles also undergo a random velocity reversal action, which results in a finite heat conductivity of the system. We prove the approach of the system to a time periodic state and compute the heat current, equal to the time averaged work done on the system, in that state. This work approaches a finite positive value as the length of the chain increases. Rescaling space, the strength and/or the period of the force leads to a macroscopic temperature profile corresponding to the stationary solution of a continuum heat equation with Dirichlet-Neumann boundary conditions.

Appendices

Appendix A. The proof of Theorem 1.1

Since the result does not depend on the scaling factor $n^a$, standing by the force ${\mathcal F}(t)$, and the period size $\theta _n$ we assume that $a=0$ and $\theta _n=\theta$. Given a Borel probability measure $\mu$ on ${\mathbb R}^{2(n+1)}$ (see (1.1)) we denote by $\big ({\textbf{q}}_{\mu }(t), {\textbf{p}}_{\mu }(t)\big )$ the solution of (1.3)–(1.4)) such that $\big ({\textbf{q}}_{\mu }(0), {\textbf{p}}_{\mu }(0)\big )$ is distributed according to $\mu$. Denote then by $\big (\overline{{\textbf{q}}}_{\mu }(t), \overline{{\textbf{p}}}_{\mu }(t)\big )$ and $C_{\mu }(t)$ the vector of averages and matrix of the mixed second moments of the solution, correspondingly. They are defined by formulas (2.1) and a $2\times 2$ block matrix

$$\begin{aligned} C _{\mu } (t)=\left[ \begin{array}{ll} C^{(q)}_{\mu } (t)&{}C^{(q,p)}_{\mu } (t)\\ C^{(p,q)}_{\mu } (t) &{}C^{(p)}_{\mu } (t) \end{array} \right] \end{aligned}$$

Each block is an $(n+1)\times (n+1)$ matrix

$$\begin{aligned}&C^{(q)}_{\mu } (t)=[ q_x(t) q_{x'}(t) ]_{x,x'=0,\ldots ,n}\quad C^{(p)}_{\mu } (t)={\mathbb E}[ p_x(t) p_{x'}(t) ]_{x,x'=0,\ldots ,n},\\&C^{(q,p)}_{\mu } (t)={\mathbb E}[ q_x(t) p_{x'}(t)]_{x,x'=0,\ldots ,n} \end{aligned}$$

and $C^{(p,q)}_{\mu } (t)=[C^{(q,p)}_{\mu } (t)]^T$, where the initial state is taken to be $\mu$.

By similar calculation as done in (6.15), their evolution is described by the system of linear differential equations with periodic forcing

$$\begin{aligned} \begin{aligned}&\frac{\textrm{d}}{\textrm{d}t}\left( \begin{array}{c}\overline{\textbf{q}}_{\mu }(t)\\ \overline{\textbf{p}}_{\mu } (t) \end{array}\right) =-A \left( \begin{array}{c}\overline{\textbf{q}}_{\mu } (t)\\ \overline{\textbf{p}}_{\mu } (t) \end{array}\right) + \;{\mathcal F}(t/\theta )\textrm{e}_{p,n+1},\\&\frac{\textrm{d}}{\textrm{d}t}C_{\mu }(t)=-A C_{\mu }(t)- C_{\mu }(t)A^T+\Sigma _2\big (\textbf{c}_{2,\mu }(t)\big )+ {\mathcal F}(t/\theta ) F(t), \end{aligned} \end{aligned}$$

(A.1)

where

$$\begin{aligned} \textbf{c}_{2,\mu }(t)=\left( \begin{array}{c} C^{(p)}_{0,0,\mu }(t) \\ \vdots \\ C^{(p)}_{n,n,\mu }(t) \end{array} \right) \end{aligned}$$

(A.2)

and

$$\begin{aligned} F(t):=\left[ \begin{array}{cc} 0&{} \overline{\textbf{q}}_\mu (t) \otimes \textrm{e}_{p,n+1} \\ &{}\\ \textrm{e}_{p,n+1}\otimes \overline{\textbf{q}}_\mu (t) &{} \ \textrm{e}_{p,n+1}\otimes \overline{\textbf{p}}_\mu (t)+\overline{\textbf{p}}_\mu (t) \otimes \textrm{e}_{p,n+1} \end{array} \right] \end{aligned}$$

(A.3)

Here $\textrm{e}_{p,n+1}$ and $\Sigma _2$ are defined in (2.4) and (6.10) respectively. Suppose that we are given a vector ${\overline{X}}\in {\mathbb R}^{2(n+1)}$ and a symmetric non-negative definite $2(n+1)\times 2(n+1)$ matrix $S\geqslant {\overline{X}}\otimes {\overline{X}}$. Then, equations (A.1) describe the evolution of the first two moments of the solution of (1.3)–(1.4)) whose initial distribution is a random vector with the first two moments given by ${\overline{X}}$ and S, respectively.

1.1 A.1. The existence and uniqueness of the periodic mean and second moment

In the first step we show the existence of a periodic solution of (A.1) that corresponds to the mean and covariance of a certain probability evolution.

Proposition A.1

There exists a unique vector $\overline{\textbf{X}}_\textrm{per}=(\overline{\textbf{q}}_{\textrm{per}}, \overline{\textbf{p}}_{\textrm{per}} ) \in {\mathbb R}^{2(n+1)}$ and a non-negative symmetric matrix $C_\textrm{per}\geqslant \overline{\textbf{X}}_{\textrm{per}}\otimes \overline{\textbf{X}}_{\textrm{per}}$ such that the solution of (A.1) with

$$\begin{aligned} \Big ((\overline{\textbf{q}}(0), \overline{\textbf{p}} (0) ) ,C(0)\Big )=\Big ((\overline{\textbf{q}}_{\textrm{per}}, \overline{\textbf{p}}_{\textrm{per}} ) ,C_{\textrm{per}}\Big ) \end{aligned}$$

satisfies

$$\begin{aligned} \Big ((\overline{\textbf{q}}(0), \overline{\textbf{p}} (0) ) ,C(0)\Big )=\Big ((\overline{\textbf{q}}(\theta ), \overline{\textbf{p}} (\theta ) ) ,C(\theta )\Big ). \end{aligned}$$

(A.4)

In addition, we have

$$\begin{aligned} {C^{(p)}_{x,x}(t)} \geqslant T_-,\quad x=0,\ldots ,n. \end{aligned}$$

(A.5)

The remaining part of this section is devoted to the proof of this results.

1.1.1 A.1.1. The existence of the periodic first moment

Let

$$\begin{aligned} \left( \begin{array}{c}\overline{\textbf{q}}\\ \overline{\textbf{p}} \end{array}\right) := \int _{-\infty }^0 {\mathcal F}(s /\theta ) e^{As}\textrm{e}_{p,n+1}\textrm{d}s. \end{aligned}$$

(A.6)

Thanks to Proposition 2.1 the vector $(\overline{\textbf{q}}, \overline{\textbf{p}} )$ is well defined. One can easily check that the solution of the first equation of (A.1) starting from the vector is given by

$$\begin{aligned} \overline{\textbf{X}}(t)=\left( \begin{array}{c}\overline{\textbf{q}}(t)\\ \overline{\textbf{p}}(t) \end{array}\right) := \int _{-\infty }^t {\mathcal F}(s/\theta ) e^{-A (t-s)}\textrm{e}_{p,n+1}\textrm{d}s. \end{aligned}$$

(A.7)

and is therefore $\theta$-periodic. In fact, thanks to Proposition 2.1 the periodic solution has to be unique. Since the coordinates of $\overline{\textbf{X}}(t)$ satisfy the first equation of (A.1) we conclude that the matrix $\overline{\textbf{X}}_2(t):=\overline{\textbf{X}}(t)\otimes \overline{\textbf{X}}(t)$ satisfies

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\overline{\textbf{X}}_2(t)=-A \overline{\textbf{X}}_2(t)- \overline{\textbf{X}}_2(t) A^T+ {\mathcal F}(t /\theta ) F(t), \end{aligned}$$

and it is given by the formula

$$\begin{aligned} \overline{\textbf{X}}_2(t)= \int _{-\infty }^t{\mathcal F}(s/\theta )e^{-A(t-s)} F(s)e^{-A^T(t-s)}\textrm{d}s ,\quad t\in {\mathbb R}. \end{aligned}$$

(A.8)

1.1.2 A.1.2. The existence of the periodic second moment

Now we are going to establish the existence of a periodic second moment. Suppose that C(t) is a periodic solution of the second equation of (A.1). Using the argument made in the proof of Proposition 6.1 we can conclude that it satisfies the equation

$$\begin{aligned} \begin{aligned} C(t) = \int _{-\infty }^te^{-A(t-s)}\left( \Sigma _2\big (\textbf{c}_{2}(s)\big ) + {\mathcal F}(s/\theta ) F(s)\right) e^{-A^T(t-s)}\textrm{d}s\\ =\int _{-\infty }^te^{-A(t-s)}\Sigma _2\big (\textbf{c}_{2}(s)\big ) e^{-A^T(t-s)}\textrm{d}s +\overline{\textbf{X}}_2(t)\\ = \int _{0}^\infty e^{-As}\Sigma _2\big (\textbf{c}_{2}(t-s)\big ) e^{-A^T s} \textrm{d}s +\overline{\textbf{X}}_2(t)\\ = \sum _{\ell =0}^\infty \int _{0}^\theta e^{-A(s+\ell \theta )} \Sigma _2\big (\textbf{c}_{2}(t-s)\big ) e^{-A^T(s+\ell \theta )} \textrm{d}s +\overline{\textbf{X}}_2(t)\\,\quad t\in {\mathbb R}, \end{aligned} \end{aligned}$$

(A.9)

where the matrix $\Sigma _2$ is defined by (6.10), $\textbf{c}_{2}(s)$ relates to C(s) via (A.2) and F(s) is defined by (A.3), using $(\overline{\textbf{q}}(t), \overline{\textbf{p}}(t) )$ instead of $(\overline{\textbf{q}}_\mu (t), \overline{\textbf{p}}_\mu (t) )$. Conversely, any periodic symmetric matrix valued function C(t) satisfying (A.9) is a periodic solution to the second equation of (A.1).

For $x,x',y=0,\ldots ,n$ define

$$\begin{aligned} {g_{x,x',y}(s):=\sum _{\ell =0}^{+\infty } \Big [e^{-A(s+\ell \theta )}\Big ]_{x+n+1,y+n+1}\Big [e^{-A^T(s+\ell \theta )}\Big ]_{y+n+1,x'+n+1}.} \end{aligned}$$

(A.10)

Consider the following linear mapping: ${\mathcal L}:[C(\mathbb T_\theta )]^{n+1}\rightarrow [C(\mathbb T_\theta )]^{n+1}$, where $\mathbb T_\theta :=\theta \mathbb T$ is the torus of size $\theta$, that assigns to a given vector of $\theta$-periodic functions $\textbf{T}(s)=[T_{0}(s),\ldots ,T_n(s)]$ a vector valued function

$$\begin{aligned} {\mathcal L}{} \textbf{T} :=({\mathfrak {G}}_0\textbf{T},\ldots , {\mathfrak {G}}_n\textbf{T}), \end{aligned}$$

(A.11)

where

$$\begin{aligned} {\mathfrak {G}}_x\textbf{T}(t) = \sum _{y=0}^n \int _0^{\theta }{{\mathfrak {G}}_{x,y}(s)} T_y\left( t-s\right) \textrm{d}s. \end{aligned}$$

(A.12)

Here

$$\begin{aligned} {{\mathfrak {G}}_{x,y}(s)}= 4 \gamma g_{x,x,y}(s),\quad y=0,\ldots ,n. \end{aligned}$$

(A.13)

Obviously, from (A.10), we have ${\mathfrak {G}}_{x,y}(s)\geqslant 0$. Note also that although $g_{x,x',y}(\cdot )$ need not be $\theta$-periodic the functions ${\mathfrak {G}}_x\textbf{T}(t)$, $x=0,\ldots ,n$ are $\theta$-periodic. In addition, if C(t) satisfies (A.9), then

$$\begin{aligned} \textbf{c}_{2}(t) ={\mathcal L}{\mathcal T}(\textbf{c}_{2})(t) +\overline{\textbf{p}}^2(t), \end{aligned}$$

(A.14)

where for a given $\textbf{T}^T=(T_0,\ldots ,T_n)\in {\mathbb R}^{n+1}$

$$\begin{aligned} {\mathcal T} (\textbf{T})= \left( \begin{array}{c} T_-\\ T_1 \\ \vdots \\ T_n \end{array} \right) ,\quad \overline{\textbf{p}}^2(t) = \left( \begin{array}{c} {\overline{p}}_0^2(t)\\ \vdots \\ {\overline{p}}_n^2(t) \end{array} \right) . \end{aligned}$$

Conversely, by finding a solution $\textbf{c}_{2}$ of (A.14) one can define then a $\theta$-periodic function C(t) by the right hand side of (A.9). The entries of the function corresponding to $C_{x,x}$, $x=n+1,\ldots ,2n+1$ coincide with the coordinates of the vector $\textbf{c}_{2}$, by virtue of (A.14). Thus, the function C(t) solves equation (A.9). We have reduced therefore the problem of finding a periodic solution to the second equation of (A.1) to solving equation (A.14).

1.1.3 A.1.3. Solution of (A.14)

Let

$$\begin{aligned} {\mathcal C}_+:=[\textbf{T}=(T_0,T_1,\ldots ,T_n):\,T_x\in C(\mathbb T_\theta )\,\text{ and } T_x\geqslant T_-,\,x=0,\ldots ,n]. \end{aligned}$$

It is a closed subset of $\Big (C(\mathbb T_\theta )\Big )^{n+1}$, equipped with the norm

$$\begin{aligned} |\!\Vert \textbf{T}\Vert \!|:=\max \{ \Vert T_x\Vert _\infty ,\,x=0,\ldots ,n\}. \end{aligned}$$

Consider the mapping

$$\begin{aligned} {\mathfrak {T}}=({\mathfrak {T}}_0,\ldots , {\mathfrak {T}}_n): {\mathcal C}_+\rightarrow \Big (C(\mathbb T_\theta )\Big )^{n+1}, \quad \text{ where } \quad {\mathfrak {T}}{} \textbf{T}:={\mathcal L}{\mathcal T}(\textbf{T})+\overline{\textbf{p}}^2. \end{aligned}$$

(A.15)

Using the notation of (A.12) and (A.14) we have

$$\begin{aligned} {\mathfrak {T}}_x(\textbf{T})(t):=T_-\int _0^{\theta } {\mathfrak {G}}_{x,0}(s)\textrm{d}s+\sum _{x'=1}^n\int _0^\theta {\mathfrak {G}}_{x,x'}(s)T_{x'}\left( t-s\right) \textrm{d}s+{\overline{p}}_x^2(t),\quad x=0,\ldots ,n. \end{aligned}$$

Comparing (6.12) with (A.9), after time averaging over a period, it is easy to identify

$$\begin{aligned} \int _0^{\theta } {\mathfrak {G}}_{x,y}(s)\textrm{d}s = M_{x,y} \end{aligned}$$

(A.16)

defined by (7.2). The matrix $[M_{x,y}]_{x,y=0}^n$ is symmetric, bi-stochastic (as can be easily seen from (7.2)). It also follows immediately that

$$\begin{aligned} 1= \sum _{y=0}^n \int _0^{\theta } {\mathfrak {G}}_{x,y}(s)\textrm{d}s,\quad x=0,1,\ldots ,n \end{aligned}$$

(A.17)

and, as a consequence, ${\mathfrak {T}}\big ({\mathcal C}_+\big )\subset {\mathcal C}_+$. Furthermore, we claim that $M_{x,y}>0$ for all $x,y=0,\ldots ,n$. Indeed, a simple calculation, using (2.3) and (2.5), yields

$$\begin{aligned}&\big [ (\lambda +A)^{-1}\big ]_{x+n+1,y+n+1}= \sum _{j=0}^n\frac{\lambda \psi _j(x)\psi _j(y) }{\lambda ^2+2\gamma \lambda +\mu _j},\nonumber \\&\big [ (\lambda +A)^{-1}\big ]_{x,y+n+1}= \sum _{j=0}^n\frac{ \psi _j(x)\psi _j(y)}{\lambda ^2+2\gamma \lambda +\mu _j}, \nonumber \\&\big [ (\lambda +A)^{-1}\big ]_{x+n+1,y} =-\sum _{j=0}^n\frac{ \mu _{j}\psi _j(x)\psi _j(y) }{\lambda ^2+2\gamma \lambda +\mu _j}, \nonumber \\&\big [ (\lambda +A)^{-1}\big ]_{x,y+n+1}= \sum _{j=0}^n\frac{ \psi _j(x)\psi _j(y)}{\lambda ^2+2\gamma \lambda +\mu _j}. \end{aligned}$$

(A.18)

The poles of the meromorphic functions appearing in (A.18) are given by

$$\begin{aligned} \lambda _{j,\pm }=-\Big (\gamma \pm \sqrt{\gamma ^2-\mu _j}\Big ). \end{aligned}$$

(A.19)

Suppose that $M_{x,y}=0$ for some x, y. From (A.16) we conclude then that

$$\begin{aligned} 0= M_{x,y}=\int _0^{\theta } {\mathfrak {G}}_{x,y}(s)\textrm{d}s =4\gamma \int _0^{+\infty }\Big [e^{-As}\Big ]^2_{x+n+1,y+n+1}\textrm{d}s, \end{aligned}$$

(A.20)

which in turn would implies that $\Big [e^{-As}\Big ]_{x+n+1,y+n+1}\equiv 0$ for all $s\geqslant 0$, thus also

$$\begin{aligned} 0\equiv \big [ (\lambda +A)^{-1}\big ]_{x+n+1,y+n+1}= \sum _{j=0}^n\frac{\lambda \psi _j(x)\psi _j(y) }{\lambda ^2+2\gamma \lambda +\mu _j}. \end{aligned}$$

As a result, we conclude that $\psi _j(x)\psi _j(y)=0$, for all $j=0,\ldots ,n$, which is impossible.

We shall show that the mapping ${\mathfrak {T}}$ has a unique fixed point in ${\mathcal C}_+$ by proving that the mapping is a contraction in the norm $|\!\Vert \cdot \Vert \!|$. Indeed, for $\textbf{T}_j^T=[T_{j,0},T_{j,1},\ldots ,T_{j,n}]$, $j=1,2$, we have

$$\begin{aligned}&|{\mathfrak {T}}_x(\textbf{T}_1)(t) -{\mathfrak {T}}_x(\textbf{T}_2)(t)|=\Big | \sum _{x'=1}^n\int _0^\theta {\mathfrak {G}}_{x,x'}(s)\Big [T_{1,x'}\left( t-s\right) - T_{2,x'}\left( t-s\right) \Big ]\textrm{d}s\Big |\\&\quad \leqslant \sum _{x'=1}^n\int _0^\theta {\mathfrak {G}}_{x,x'}(s) \Big | T_{1,x'}\left( t-s\right) - T_{2,x'}\left( t-s\right) \Big |\textrm{d}s\\&\quad \leqslant \sum _{x'=1}^n \left( \int _0^\theta {\mathfrak {G}}_{x,x'}(s)\textrm{d}s\right) \Vert T_{1,x'} - T_{2,x'}\Vert _{\infty }\\&\quad \leqslant \left( 1- \int _0^\theta {\mathfrak {G}}_{x,0}(s)\textrm{d}s\right) |\!\Vert \textbf{T}_1-\textbf{T}_2|\!\Vert ,\quad x=0,\ldots ,n. \end{aligned}$$

Therefore

$$\begin{aligned} |\!\Vert {\mathfrak {T}}(\textbf{T}_1)-{\mathfrak {T}}(\textbf{T}_2)\Vert \!| \leqslant \rho |\!\Vert \textbf{T}_1-\textbf{T}_2|\!\Vert , \end{aligned}$$

where

$$\begin{aligned} \rho :=\max \Big [ 1- \int _0^\theta {\mathfrak {G}}_{x,0}(s)\textrm{d}s ,\,x=0,\ldots ,n\Big ]<1. \end{aligned}$$

We have proven that $\Vert {\mathfrak {T}}(\textbf{T}_1) -{\mathfrak {T}} (\textbf{T}_2)\Vert _\infty \leqslant \rho \Vert \textbf{T}_1-\textbf{T}_2\Vert _{\infty }$ and the existence of a unique fixed point follows. This ends the proof of Proposition A.1. $\square$

1.2 A.2. The end of the proof of Theorem 1.1

Suppose now that $\nu$ is a probability law whose first and second moments are $\theta$-periodic, e.g. it could be a Gaussian distribution with the mean and the second moment given by $\textbf{P}_{\textrm{per}}$ and $C_{\textrm{per}}$, respectively. Denote by

$$\begin{aligned} {\mathcal P}_{s,t}F(\textbf{q}, \textbf{p})=\int _{{\mathbb R}^{2(n+1)}}F(\textbf{q}', \textbf{p}'){\mathcal P}_{s,t}(\textbf{q}, \textbf{p}; \textrm{d}\textbf{q}', \textrm{d}\textbf{p}') \end{aligned}$$

the evolution family of transition probability operators corresponding to the dynamics described by (1.3)–(1.4). Consider the event $E:=[N_x({\theta })=0,\,x=1,\ldots ,n].$ We have ${\mathbb P}[E]>0$. Suppose that the dynamics starts at $(\textbf{q}, \textbf{p})$. Then, for any $F\geqslant 0$ we can write

$$\begin{aligned} {\mathcal P}_{0,\theta }F(\textbf{q}, \textbf{p})\geqslant {\mathbb E}\Big [F\big (\textbf{q}(\theta ), \textbf{p}(\theta )\big ),\,E\Big ]={\mathbb P}[E]{\mathcal Q}_{0, \theta }F(\textbf{q}, \textbf{p}), \end{aligned}$$

(A.21)

where ${\mathcal Q}_{s,t}$ is the transition probability operator for the non-homogeneous Ornstein-Uhlenbeck dynamics that corresponds to the generator ${\mathcal {G}}_t^{(g)} = {\mathcal {A}}_t + 2 \gamma S_{-}$, see (1.7) and (1.8). Using the hypoellipticity of ${\mathcal {G}}_t^{(g)}$, see [9, Section A.3], one can prove that there exist strictly positive transition probability density kernels $\rho _{s,t}$ corresponding to ${\mathcal Q}_{s,t}$, see [9, Section A.2] for more details. Thanks to (A.21) we conclude that

$$\begin{aligned} {\mathcal P}_{0, \theta }(\textbf{q}, \textbf{p}; \textrm{d}\textbf{q}', \textrm{d}\textbf{p}') \geqslant c_*\rho _{0,\theta }(\textbf{q}, \textbf{p}; \textbf{q}', \textbf{p}') {\textrm{d}} \textbf{q}' \textrm{d}\textbf{p}', \end{aligned}$$

(A.22)

where $c_*:={\mathbb P}[E]$. Then, $\nu _{0,t}:=\nu {\mathcal P}_{0,t}$ describes the law of $(\textbf{q}(t), \textbf{p}(t) )$ with the prescribed initial data. Thanks to Proposition A.1 we can see that the total energy ${\mathcal H}(t):=\sum _{x=0}^n{\mathcal E}_x(t)$ (see (1.2)) is a Lyapunov function for the above system, since ${\mathbb E}{\mathcal H}(t)$ is $\theta$-periodic. The above implies that the family of laws $\{\nu _{0,t},\,t\geqslant 0\}$ is tight in ${\mathbb R}^{2(n+1)}$. Thus, also the family $\mu _N:=N^{-1}\int _0^{N\theta }\nu _{0,s}\textrm{d}s$ is tight. Suppose that $\mu _\infty$ is its limiting measure, i.e. there exists a sequence $N'\rightarrow +\infty$ such that $\mu _{N'}\rightarrow \mu _\infty$, in the topology of weak convergence. Since ${\mathcal P}_{s,t}$ has the Feller property one can easily conclude that $\mu _\infty {\mathcal P}_{0,\theta }=\mu _\infty$. Hence $\mu _s^P:=\mu _\infty {\mathcal P}_{0,s}$ , $s\in [0,+\infty )$ is a periodic stationary state.

Suppose that $\mu (\textrm{d}\textbf{q}, \textrm{d}\textbf{p})=f (\textbf{q}, \textbf{p}) \textrm{d}\textbf{q} \textrm{d}\textbf{p}$, where f is a $C^\infty$ smooth probability density. One can show, using the regularity theory of stochastic differential equations, that $\mu {\mathcal P}_{0,\theta }$ is absolutely continuous w.r.t. the Lebesgue measure and its density is also $C^\infty$ smooth, see e.g. [6, Corollary III.3.4, p. 303]. This allows us to conclude further that $\mu {\mathcal P}_{0,\theta }$ is absolutely continuous, provided that $\mu$ is absolutely continuous. We shall denote by ${\mathcal P}_{0,\theta }$ the corresponding operator induced on $L^1({\mathbb R}^{2(n+1)})$. The operator ${\mathcal Q}_{0,\theta }$ corresponding to the Gaussian dynamics transforms $\mu _\infty$ into an absolutely continuous measure. Thanks to (A.22) we conclude that

$$\begin{aligned} \mu _\infty (\textrm{d}\textbf{q}', \textrm{d}\textbf{p}')=\mu _\infty {\mathcal P}_{0,\theta }(\textrm{d}\textbf{q}', \textrm{d}\textbf{p}')\geqslant c_*\mu _\infty {\mathcal Q}_{0,\theta }( \textbf{q}', \textbf{p}')\textrm{d}\textbf{q}' \textrm{d}\textbf{p}'. \end{aligned}$$

(A.23)

Therefore the singular part of $\mu _\infty$ is of at most mass $1-c_*$. Since ${\mathcal P}_{0,\theta }$ transforms the space of abolutely continuous measures into itself, both the singular and absolutely continuous parts of $\mu _\infty$, after normalization, become invariant under ${\mathcal P}_{0,\theta }$. Iterating this procedure we conclude, after m steps, that the singular part can be of at most mass $(1-c_*)^m$, which eventually leads to the conclusion that the measure $\mu _\infty$ is absolutely continuous. The respective density is positive, due to (A.22). This ends the proof of Theorem 1.1. $\square$

Appendix B. Green Functions Convergence

Recall that $G_{\omega _0}$ and $G^n_{\omega _0}$ are the Green’s functions corresponding to $\omega _0^2-\Delta$ and $\omega _0^2-\Delta _{\textrm{N}}$, where $\Delta$ is the free lattice laplacian on ${\mathbb {Z}}$ and $\Delta _{\textrm{N}}$ is the Neumann discrete laplacian on $\{0,1,\dots , n\}$, see Sects. 2.3 and 2.4, respectively.

1.1 B.1. Estimantes on oscillating sums

Define $\chi _n(x)$ as the $n+1$-periodic extension of $\chi _n(x):=(1+x)\wedge (n+2-x)$, $x\in [0,n+1]$. Suppose that $\Phi :{\mathbb R}^2\rightarrow {\mathbb {C}}$ is a $\theta ,\theta '$-periodic function in each variable respectively. Denote

$$\begin{aligned} H_{x,x'}^{(n)} =\frac{1}{(n+1)^2}\sum _{j,j'=0}^n\Phi \left( \frac{\theta j}{n+1}, \frac{\theta ' j'}{n+1}\right) \exp \left\{ \frac{ 2i\pi j x}{n+1}\right\} \exp \left\{ \frac{ 2i\pi j'x'}{n+1}\right\} \end{aligned}$$

for $x,x'\in {\mathbb Z}$.

Lemma B.1

Suppose that $\Phi$ is of $C^k$-class for some $k\geqslant 1$. Then, there exists C such that

$$\begin{aligned} \Big |H_{x,x'}^{(n)}\Big |\leqslant \frac{C}{\chi ^\ell _n(x) \chi ^{\ell '}_n(x')},\quad x,x'\in {\mathbb Z},\,n\geqslant 1,\,\ell ,\ell '\geqslant 0,\,\ell +\ell '\leqslant k. \end{aligned}$$

(B.1)

Proof

To simplify the notation we suppose that $\theta =\theta '=1$. Summation by parts yields

$$\begin{aligned} H_{x,x'}^{(n)}&=\frac{1}{(n+1)^2}\sum _{j,j'=0}^n\Phi \left( \frac{ j}{n+1}, \frac{ j'}{n+1}\right) \frac{\exp \left\{ \frac{ 2 i\pi x}{n+1}\right\} -1}{\exp \left\{ \frac{ 2 i\pi x}{n+1}\right\} -1}\exp \left\{ \frac{ 2 i\pi jx}{n+1}\right\} \exp \left\{ \frac{ 2i\pi j'x'}{n+1}\right\} \\&=\frac{1}{(n+1)^2 [\exp \left\{ \frac{ i\pi x}{n+1}\right\} -1]} \sum _{j,j'=0}^n\Big [\Phi \left( \frac{ j-1}{n+1}, \frac{ j'}{n+1}\right) - \Phi \left( \frac{ j}{n+1}, \frac{ j'}{n+1}\right) \Big ] \\&\quad \times \exp \left\{ \frac{ 2 i\pi jx}{n+1}\right\} \exp \left\{ \frac{ 2 i\pi j'x}{n+1}\right\} \end{aligned}$$

Since $\Phi$ is of $C^1$ class

$$\begin{aligned} \Big |\Phi \left( \frac{ j-1}{n+1}, \frac{ j'}{n+1}\right) - \Phi \left( \frac{ j}{n+1}, \frac{ j'}{n+1}\right) \Big |\leqslant \frac{C}{n+1} \end{aligned}$$

for some constant $C>0$. In addition, there exists $c>0$ such that

$$\begin{aligned} \left| \exp \left\{ \frac{ 2 i\pi x}{n+1}\right\} -1\right| \geqslant \frac{c\chi _n(x) }{n+1} \end{aligned}$$

for $x,x'\in {\mathbb Z},\,n\geqslant 1$. Thus, there exists $C>0$ such that

$$\begin{aligned} |H_{x,x'}^{(n)}|\leqslant \frac{C}{\chi _n(x)},\quad x,x'\in {\mathbb Z}. \end{aligned}$$

Iterating this argument in the regularity degree k of $\Phi$ we conclude (B.1). $\square$

1.2 B.2. Application

An application concerns the approximation of the Green’s function $G_{\omega _0}$ by $G_{\omega _0}^n$ along the diagonal.

Lemma B.2

We have

$$\begin{aligned} G_{\omega _0}^n(y,y)=G_{\omega _0}(0)+{\widetilde{H}}^{(n)}(y) +O\Big (\frac{1}{n}\Big ),\quad y=0,\ldots ,n,\,n\geqslant 1. \end{aligned}$$

(B.2)

Here for some constant $C>0$ we have

$$\begin{aligned} |{\widetilde{H}}^{(n)}(y)|\leqslant \frac{C}{\chi ^2(y)} ,\quad y=0,\ldots ,n,\,n\geqslant 1. \end{aligned}$$

(B.3)

Proof

Using the definition of the Green’s function (2.13) (with $\ell =0$) and formulas (2.14) we obtain

$$\begin{aligned} G_{\omega _0}^n(y,y) =\frac{ 1}{n+1}\sum _{j=0}^n \Xi \left( \frac{ j}{n+1}\right) \left[ 1+\cos \left( \frac{\pi j(2y+1)}{n+1}\right) \right] +O\Big (\frac{1}{n}\Big ), \end{aligned}$$

where

$$\begin{aligned} \Xi \left( u \right) =\left\{ 4\sin ^2\left( \frac{\pi u}{2}\right) +\omega _0^2 \right\} ^{-1}. \end{aligned}$$

As a result we write $G_{\omega _0}^n(y,y)$ in the form (B.2), with

$$\begin{aligned} {\widetilde{H}}_y^{(n)}=\frac{ 1}{2(n+1)}\sum _{j=-n-1}^n\cos \left( \frac{\pi j(2y+1)}{n+1}\right) \Xi \left( \frac{ j}{n+1}\right) . \end{aligned}$$

Estimate (B.3) is then a consequence of Lemma B.1. $\square$

Appendix C. Proofs of Lemmas 9.2, 9.3 and 9.5

1.1 C.1. Proof of Lemma 9.2

For $m\in {\mathbb Z}$ and $g=(g_0,\ldots ,g_n)\in {\mathbb R}^{n+1}$ define

$$\begin{aligned} S(m):=\int _0^{+\infty }e^{-2\pi i ms/\theta }e^{-As}\Sigma (g) e^{-A^Ts} \textrm{d}s= \left[ \begin{array}{cc} S^{q}(m)&{} S^{qp}(m)\\ &{}\\ S^{pq}(m)&{} S^{p}(m) \end{array} \right] , \end{aligned}$$

where $\Sigma _2$ is defined in (6.10). Note that

$$\begin{aligned} \sum _{y=0}^nM_{x,y}(m)g_y=S^{p}_{x,x}(m). \end{aligned}$$

(C.1)

Arguing similarly as in the proof of (6.16) we get

$$\begin{aligned} A S(m)+ S(m) A^T +2i\alpha _mS(m)=\Sigma _2(g), \end{aligned}$$

(C.2)

where $\alpha _m:=\pi m/\theta$. Denote

$$\begin{aligned} \widetilde{ S}^{q,p}_{j,j'}(m)=\sum _{x,x'=0}^nS^{q,p}_{x,x'}(m)\psi _j(x) \psi _{j'}(x') \end{aligned}$$

Following the same manipulations as those leading to (6.20) we obtain

$$\begin{aligned}&{\widetilde{S}}^{p}_{j,j'} (m)=\frac{1}{2} \widetilde{S}^{q}_{j,j'}(m) \Big [\mu _{j'}+\mu _j + 2(\gamma +i\alpha _m) i\alpha _m \Big ] , \nonumber \\&{\widetilde{S}}^{q}_{j,j'}(m)\Big [\mu _{j'}-\mu _j -4 i\alpha _m (\gamma +i\alpha _m)\Big ]+4(\gamma +i\alpha _m) {\widetilde{S}}^{qp}_{j,j'} (m) =0,\nonumber \\&{\widetilde{S}}^{qp}_{j,j'}(m)(\mu _{j}-\mu _{j'} )+ 2 i\alpha _m\widetilde{S}^{q}_{j,j'}(m)\mu _{j'} +2(2\gamma +i\alpha _m){\widetilde{S}}^{p}_{j,j'}(m)= {\widetilde{F}}_{j,j'} \end{aligned}$$

(C.3)

and ${\widetilde{F}}_{j,j'}=\sum _{x=0}^n\psi _j(x)\psi _{j'}(x)g_x.$ Solving the above system using the procedure used to deal with (6.20) we obtain

$$\begin{aligned} {\widetilde{S}}^{p}_{j,j'}(m)=\sum _{y}\Theta _m(\mu _j,\mu _{j'})\psi _j(y)g_y\psi _{j'}(y), \end{aligned}$$

with

$$\begin{aligned}&\Theta _m(c,c') :=\frac{2\gamma }{2\gamma +i\alpha _m} \Bigg \{\Big [ (2\gamma +i\alpha _m)\Big (c+c' +2 (\gamma +i\alpha _m)i\alpha _m\Big ) \Big ]^{-1} \nonumber \\&\quad \quad \quad \quad \quad \quad \times \Big [\frac{1}{4}(\gamma +i\alpha _m)^{-1}(c-c')^2 + i\alpha _m(c+c') \Big ] +1 \Bigg \}^{-1} . \end{aligned}$$

(C.4)

Note that $\Theta _0(c,c')=\Theta (c,c')$ defined in (6.24). From (C.1) we conclude that

$$\begin{aligned} M_{x,y}(m)= \sum _{j,j'=0}^n\Theta _m(\mu _j,\mu _{j'}) \psi _j(x)\psi _{j'}(x) \psi _j(y)\psi _{j'}(y) \end{aligned}$$

(C.5)

As in (7.5), for any sequence $(f_x)\in {\mathbb {C}}^{n+1}$ we can write

$$\begin{aligned} \sum _{x,y=0}^n(\delta _{x,y}-M_{x,y}(m))f_y^\star f_x = \sum _{j,j'=0}^n \left( 1- \Theta _m(\mu _j,\mu _{j'})\right) \left| \sum _{x=0}^n \psi _j(x)f_x\psi _{j'}(x)\right| ^2. \end{aligned}$$

We have

$$\begin{aligned}&1- \Theta _m(c,c')\\&\quad = \Bigg \{i\alpha _m+\Big [c+c' +2 (\gamma +i\alpha _m) i\alpha _m\Big ]^{-1} \Big [\frac{1}{4}(\gamma +i\alpha _m)^{-1}(c-c')^2 + i\alpha _m (c+c') \Big ]\Bigg \}\\&\quad \quad \times \Bigg \{\Big [ c+c' +2 (\gamma +i\alpha _m) i\alpha _m \Big ]^{-1} \Big [\frac{1}{4}\Big (\gamma +i\alpha _m\Big )^{-1}(c-c')^2 + i\alpha _m (c+c') \Big ] +(2\gamma +i\alpha _m) \Bigg \}^{-1}. \end{aligned}$$

Thus $\lim _{m\rightarrow +\infty }\Big (1- \Theta _m(c,c')\Big )=1.$ On the other hand, if $m\not =0$, an easy calculation shows that $1- \Theta _m(c,c')=0$ implies that $(c-c')^2=8\alpha ^2_m(\alpha ^2_m+\gamma ^2)$ and $\quad c+c'=2\alpha ^2_m$, where $\alpha _m=\pi m/\theta$. But this would clearly lead to a contradiction, as then we would have $|c-c'|>\sqrt{2}(c+c')$, which is clearly impossible (remember that $c,c'>0$). Hence, there exists ${\mathfrak {C}}_*>0$ such that

$$\begin{aligned} |1- \Theta _m(c,c')|\geqslant {\mathfrak {C}}_*,\quad |m|\geqslant 1,\,c,c'\in [0,\omega _0^2+4]. \end{aligned}$$

This ends the proof of (9.9). $\square$

1.2 C.2. Proof of Lemma  9.3

Using (C.5) we obtain

$$\begin{aligned} \sum _{x=0}^n|M_{x,0}(m)|^2= \sum _{j_1,\ldots ,j_4 =0}^n\Theta _m(\mu _{j_1},\mu _{j_2}) \Theta _m(\mu _{j_3},\mu _{j_4}) \prod _{k=1}^4\psi _{j_k}(0) \sum _{x=0}^n\prod _{k=1}^4\psi _{j_k}(x) . \end{aligned}$$

(C.6)

Applying elementary trigonometric identities we conclude that

$$\begin{aligned}&\sum _{x=0}^n\prod _{k=1}^4\psi _{j_k}(x) =\frac{1}{(n+1)^2} \left\{ \prod _{k=1}^4(2-\delta _{0,j_k})\right\} ^{1/2} \sum _{x=0}^n\prod ^4_{k=1}\cos \left( \frac{\pi j_k(2x+1)}{2(n+1)}\right) \\&\quad =\frac{1}{2^5(n+1)} \left\{ \prod _{k=1}^4(2-\delta _{0,j_k})\right\} ^{1/2} \sum _{\iota _1,\ldots ,\iota _4\in \{-1,1\}}\cos \left( \frac{\pi }{2(n+1)}\sum _{k=1}^4\iota _kj_k\right) 1_{2(n+1){\mathbb Z}}\Big (\sum _{k=1}^4\iota _kj_k\Big ) \end{aligned}$$

Therefore we can write

$$\begin{aligned}&\sum _{x=0}^n|M_{x,0}(m)|^2= \sum _{\iota ,\iota _1\in \{-1,1\}}\sum _{\iota ',\iota _1'\in \{-1,1\}}\int _0^1\textrm{d}u\int _0^1\textrm{d}u' \int _0^1\textrm{d}u_1\int _0^1\textrm{d}u'_1 {\mathfrak {V}}_{m}(u,u'){\mathfrak {V}}_{m}^\star (u_1,u'_1)\end{aligned}$$

(C.7)

$$\begin{aligned}&\quad \times \exp \left\{ \textrm{i}\pi \Big ( \iota u+\iota 'u'+\iota _1 u_1+\iota '_1u'_1\Big )\right\} \sum _{q\in {\mathbb Z}}\delta _q\Big (\iota u+\iota 'u'+\iota _1 u_1+\iota '_1u'_1\Big )+O_m\left( \frac{1}{n}\right) , \end{aligned}$$

(C.8)

where $O_m\left( \frac{1}{n}\right) \leqslant \frac{C}{n}$ for some constant $C>0$, independent of n and m, and

$$\begin{aligned} {\mathfrak {V}}_{m}(u,u'):= \Theta _m\Big (\omega _0^2+4\sin ^2\left( \frac{\pi u}{2}\right) , \omega _0^2+4\sin ^2\left( \frac{\pi u'}{2}\right) \Big )\cos \left( \frac{\pi u}{2}\right) \cos \left( \frac{\pi u'}{2}\right) . \end{aligned}$$

We claim that there exists $C>0$, independent of n and m, such that ${\mathfrak {V}}_{m}(u,u')\leqslant C$ for all $u,u'\in [0,\omega _0^2+4]$. Indeed, as can be seen directly from (C.4), we have $\lim _{m\rightarrow +\infty }\Theta _m(c,c')=0$ uniformly in $c,c'\in [0,\omega _0^2+4]$. On the other hand the function ${\mathbb R}\times [0,\omega _0^2+4]^2\ni (m,c,c')\rightarrow \Theta _m(c,c')$ is bounded on compact set. If otherwise, this would imply that there exist $(m,c,c')\in {\mathbb R}\times [0,\omega _0^2+4]^2$ such that

$$\begin{aligned}&0=\Big [ (2\gamma +i\alpha _m)\Big (c+c' +2 (\gamma +i\alpha _m) i\alpha _m\Big ) \Big ]^{-1} \\&\quad \times \Big [\frac{1}{4}(\gamma +i\alpha _m)^{-1}(c-c')^2 + i\alpha _m (c+c') \Big ] +1. \end{aligned}$$

An easy calculation gives $c+c'= 2\alpha ^2_m$ and $(c-c')^2=8(\alpha ^2_m+\gamma ^2) \alpha ^2_m,$ where $\alpha _m=\pi m/\theta$. This leads to a contradiction, as then $|c-c'|>\sqrt{2}(c+c')$ (but both $c,c'>0$). Thus the conclusion of the lemma follows. $\square$

1.3 C.3. Proof of Lemma  9.4

From (A.18) we obtain

$$\begin{aligned} {[}e^{-At}]_{x+n+1,x'+n+1}= \sum _{j=0}^nE_j(t)\psi _j(x)\psi _j(x'), \end{aligned}$$

where (cf (6.18))

$$\begin{aligned} E_j(t):=\frac{1}{2\sqrt{\gamma ^2-\mu _j}}\Big [ -\lambda _{j,+} \exp \left\{ \lambda _{j,+}t\right\} + \lambda _{j,-} \exp \left\{ \lambda _{j,-}t\right\} \Big ],\quad \text{ if } \mu _j\not = \gamma ^2. \end{aligned}$$

In the case $\mu _j= \gamma ^2$ (then $\lambda _{j,\pm }=\gamma$, cf (A.19)) we have $E_j(t):=(1-\gamma t)e^{-\gamma t} .$ Using (A.6) we obtain therefore

$$\begin{aligned} {\overline{p}}_x(t)&= \frac{1}{n^{1/2}} \sum _{j=0}^n\int _{0}^{+\infty } {\mathcal F}((t-s)/\theta ) [e^{-A s}]_{x+n+1,2n+1} \textrm{d}s \nonumber \\&=\frac{1}{n^{1/2}} \sum _{j=0}^n\psi _j(x)\psi _j(n) \int _{0}^{+\infty } {\mathcal F}((t-s)/\theta )E_j(s)\textrm{d}s. \end{aligned}$$

(C.9)

From (C.9) we conclude that there exists ${\mathfrak {p}}_*>0$ such that

$$\begin{aligned} \sup _{t\in {\mathbb R},x=0,\ldots ,n}|{\overline{p}}_x(t)|\leqslant \frac{{\mathfrak {p}}_*}{n^{1/2}},\quad n=1,2,\ldots . \end{aligned}$$

(C.10)

Estimate (9.11) is then a straightforward consequence of (C.10) and (4.11). $\square$

1.4 C.4. Proof of Lemma  9.5

Multiplying both sides of (9.4) by $V_x(t)$ and averaging over time we get

$$\begin{aligned} \langle \langle V_x^2\rangle \rangle&= \sum _{x'=1}^n \int _0^\theta {\mathfrak {g}}_{x,x'}(s) \langle \langle V_x(\cdot )V_{x'}(\cdot -s)\rangle \rangle \textrm{d}s+ \langle \langle V_x v_x\rangle \rangle \nonumber \\&\leqslant \sum _{x'=1}^n M_{x,x'} \langle \langle V_x^2\rangle \rangle ^{1/2}\langle \langle V_{x'}^2\rangle \rangle ^{1/2}+ \langle \langle V_x v_x\rangle \rangle . \end{aligned}$$

(C.11)

Summing up over x we obtain

$$\begin{aligned}&\sum _{x,x'=0}^n \Big (\delta _{x,x'} -M_{x,x'}\Big ) \langle \langle V_x^2\rangle \rangle ^{1/2}\langle \langle V_{x'}^2\rangle \rangle ^{1/2}+\langle \langle V_0^2\rangle \rangle ^{1/2}\sum _{x=0}^n M_{x,0} \langle \langle V_{x}^2\rangle \rangle ^{1/2}\\&\quad \leqslant \sum _{x=0}^n \langle \langle V_x v_x\rangle \rangle . \end{aligned}$$

Using (7.4) and the Cauchy-Schwarz inequality we obtain in particular that

$$\begin{aligned} M_{0,0}\langle \langle V_0^2\rangle \rangle \leqslant \left\{ \sum _{x=0}^n \langle \langle V_x^2 \rangle \rangle \right\} ^{1/2}\left\{ \sum _{x=0}^n \langle \langle v_x^2 \rangle \rangle \right\} ^{1/2}\leqslant \frac{C}{n}\left\{ \sum _{x=0}^n \langle \langle V_x \rangle \rangle ^2\right\} ^{1/2} \end{aligned}$$

for some $C>0$ independent of n. The last estimate follows from (9.11). To finish the proof note that from (7.2) we have

$$\begin{aligned} M_{0,0}&=\frac{4}{(n+1)^2}\sum _{j,j'=0}^n(1-\delta _{0,j})(1-\delta _{0,j'})\Theta (\mu _j,\mu _{j'})\cos ^2\left( \frac{\pi j}{2(n+1)}\right) \cos ^2\left( \frac{\pi j'}{2(n+1)}\right) \\&\approx 4\gamma ^2\int _0^1\int _0^1 \frac{\Big [\omega _0^2+2\sin ^2(\pi u/2)+2\sin ^2(\pi u'/2)\Big ]\cos ^2(\pi u/2)\cos ^2(\pi u'/2)\textrm{d}u\textrm{d}u'}{\gamma ^2 \Big (\omega _0^2+2\sin ^2(\pi u/2)+2\sin ^2(\pi u'/2) \Big )+ \Big (\sin ^2(\pi u/2)-\sin ^2(\pi u'/2))\Big )^2}, \end{aligned}$$

where the equality holds up to a term of order O(1/n). $\square$