Hydrodynamic Limit for a Disordered Quantum Harmonic Chain

Abstract

We study the hydrodynamic limit, in the hyperbolic space-time scaling, for a one-dimensional unpinned chain of quantum harmonic oscillators with random masses. To the best of our knowledge, this is among the first examples where one can prove the hydrodynamic limit for a quantum system rigorously. In fact, we prove that after hyperbolic rescaling of time and space, the distribution of the elongation, momentum, and energy averaged under the proper locally Gibbs state converges to the solution of the Euler equation. Moreover, our result indicates that the temperature profile is frozen in any space-time scale; in particular, the thermal diffusion coefficient vanishes. There are two main phenomena in this chain that enable us to deduce this result. First is the Anderson localization, which decouples the mechanical and thermal energy, providing the closure of the equation for energy. The second phenomenon is similar to some sort of decay of correlation phenomena, which let us circumvent the difficulties arising from the fact that our Gibbs state is not a product state due to the quantum nature of the system.

Appendices

A Properties of the Limiting Function

In this section, we study the properties of $\mathrm {f}^{\mu }_{\beta }$, which is defined in (5.5). We prove a couple of lemmas to facilitate the proof of Theorem 5.4. In particular, we prove the existence of the limit (5.5), this means that $\mathrm {f}^{\mu }_{\beta }$ is well-defined. Moreover, we show that this function is continuous. Furthermore, we treat the equilibrium case and demonstrate that the function $\mathrm {f}^{\mu }(\beta _{eq})$ is well defined, i.e. $\mathrm {f}^{\mu }_{\beta _{eq}}(y)$ does not depend on y. Then we demonstrate the relation (5.36). Finally, we prove a lemma which has been needed in 5.4. All the proofs in this section share the same spirit: We represent in terms of the Taylor series, then cut the series similar to (5.22), and control the expressions depending on the first part of the Taylor series, using the fact that the number of these terms is finite, $\beta$ is continuous and bounded, and the distribution of the masses is compactly supported. Finally, we use the fact that the terms depending on the remainder of the series is small.

Notice that since n is not fixed here, we denote the ensemble average with , in order to emphasize the dependence on n. Before proceeding, recall the definition of $A_p^{\beta }$, and $A_r^{\beta }$ from (3.25), since here n is not fixed and we study matrices with different sizes, we change our notations only in this section and denote these matrices by $A_n^p$, and $A_n^r$, respectively i.e.

$$\begin{aligned} A_n^p:=M_{\beta }^{-\frac{1}{2}}(-\nabla _-\beta ^0 \nabla _+)M_{\beta }^{-\frac{1}{2}}, \quad A_n^r:=(\beta ^o)^{\frac{1}{2}}(-\nabla _+ M_{\beta }^{-1} \nabla _-)(\beta ^o)^{\frac{1}{2}}, \end{aligned}$$

where $M_{\beta }=M\beta ^{-1}$, with $M=\mathrm {diag}(m_1,\dots ,m_n)$, $\beta =\mathrm {diag}(\beta (\frac{1}{n}),\dots ,\beta (\frac{n}{n}))$, and $\beta ^o=\mathrm {diag}(\beta (\frac{1}{n}),\dots ,\beta (\frac{n-1}{n}))$.

Since all these proofs share the same spirit, we set a handful of notation here:

Recall the average expression from (5.18):

(A.1)

Thanks to (5.17), the part corresponding to p can be written as follows:

$$\begin{aligned} \langle x , {\mathfrak {f}}(A_n^p) x\rangle _n= \sum _{k=0}^{\infty } a_k \langle x, (A_n^p-\alpha I_n)^k x \rangle _n. \end{aligned}$$

(A.2)

As we argued in Sect. 5.1, there exists a constant $c_0>0$, such that for every realization of the masses, and $\forall n$, $||A_p^n||_2 \le c_0$. Therefore, using the properties of Taylor series as in (5.25), we observe that $\forall \epsilon >0$, there exists $K_*(\epsilon ) \in {\mathbb {N}}$, such that $\forall n$, $\forall x \in {\mathbb {I}}_n$, and any realization of the masses we have:

$$\begin{aligned} \Big |\sum _{k>K_*(\epsilon )} a_k\langle x, (A_n^p-\alpha I_n)^k x \rangle _n \Big | \le \epsilon . \end{aligned}$$

(A.3)

Notice that we can choose $K_*(\epsilon )$, such that the same bound (A.3) holds when we substitute $A_n^p$ with $A_n^r$. Hence, given $\epsilon >0$ one can define ${\mathfrak {f}}_{\prec }^{\epsilon }(.)$ and ${\mathfrak {f}}_{\succ }^{\epsilon }(.)$ as follows:

$$\begin{aligned} {\mathfrak {f}}_{\prec }^{\epsilon }(A_n^p):= \sum _{k=0}^{K_*(\epsilon )} a_k(A_n^p-\alpha I_{n})^k, \qquad {\mathfrak {f}}_{\succ }^{\epsilon }(A_n^p):= \sum _{k>K_*(\epsilon )}^{\infty } a_k(A_n^p-\alpha I_{n})^k. \end{aligned}$$

(A.4)

In particular, we can rewrite (A.3) in the following way: For any $\epsilon >0$ there exits $K_*(\epsilon )$ such that for any n and $x \in {\mathbb {I}}_n$ we have:

$$\begin{aligned} \begin{aligned}&|\langle x, {\mathfrak {f}}^{\epsilon }_{\succ }(A_n^p) x \rangle _n| \le \epsilon , \quad |{\mathbb {E}}(\langle x, {\mathfrak {f}}^{\epsilon }_{\succ }(A_n^p) x \rangle _n)| \le \epsilon , \\&|\langle x, {\mathfrak {f}}^{\epsilon }_{\succ }(A_n^r) x \rangle _{n-1}| \le \epsilon , \quad |{\mathbb {E}}(\langle x, {\mathfrak {f}}^{\epsilon }_{\succ }(A_n^r) x \rangle _{n-1})| \le \epsilon . \end{aligned} \end{aligned}$$

(A.5)

Fix $(k,n) \in {\mathbb {N}}^2$, denote $A_n^p-\alpha I_{n}$ by ${\tilde{A}}_n^p$, take $x \in {\mathbb {I}}_n$, and consider the following term: $\langle x,({\tilde{A}}_n^p)^k x\rangle$. Here we represent this term in a more appropriate manner, introducing following notations. First, recall the random walk representation of $\langle x,({\tilde{A}}_n^p)^k x\rangle$:

$$\begin{aligned} \begin{aligned} \langle x, ({\tilde{A}}_n^p)^k x \rangle =\sum _{x_1,\dots x_{k-1}=1}^n \langle x,{\tilde{A}}_p^{\beta } x_1 \rangle \langle x_1 ,{\tilde{A}}_p^{\beta } x_2 \rangle \dots \langle x_{k-1} ,{\tilde{A}}_p^{\beta } x \rangle . \end{aligned} \end{aligned}$$

(A.6)

Denote the set of indices with non-zero contribution, in RHS of (A.6) by ${\mathcal {I}}_{n,k}^{x,p}$:

$$\begin{aligned} {\mathcal {I}}_{n,k}^{x,p}:= \{ (x_1,\dots x_{k-1}) \in {\mathbb {I}}_n^{k-1} | \langle x,{\tilde{A}}_n^p x_1 \rangle \langle x_1 ,{\tilde{A}}_n^p x_2 \rangle \dots \langle x_{k-1} ,{\tilde{A}}_n^p x \rangle \ne 0 \}, \end{aligned}$$

(A.7)

where ${\tilde{A}}_n^p=A_n^p-\alpha I_n$. We denote the elements of ${\mathcal {I}}_{n,k}^{x,p}$ by ${\underline{x}}:=(x_1,\dots ,x_{k-1})$. Notice that there is a bijection between ${\mathcal {I}}_{n,k}^{x,p}$ and the set of paths in $[0,k] \times [0,n] \cap {\mathbb {Z}}^2$ from the point (x, 0) to (x, k), consisting of the following vectors: $(-1,1),(0,1),(1,1) \in {\mathbb {Z}}^2$. In fact, this bijection is given as follows: $\forall {\underline{x}} \in {\mathcal {I}}_{n,k}^{x,p}$, assign to ${\underline{x}}$ the path which is given by the following points: (0, x), $(1,x_1)$,$\dots$, $(j,x_j)$, $\dots$,$(k-1,x_{k-1}),(k,x)$.

Let $\tilde{{\mathcal {I}}}_{n,k}^{x,p}$ be the set where every element of ${\mathcal {I}}_{n,k}^{x,p}$ shifted by the vector $(x,\dots ,x) \in ({\mathbb {I}}_n)^{k-1}$, precisely define:

$$\begin{aligned} \begin{aligned} \tilde{{\mathcal {I}}}_{n,k}^{x,p} := \{(x_1-x,\dots ,x_{k-1}-x) | {\underline{x}} \in {\mathcal {I}}_{n,k}^{x,p} \}. \end{aligned} \end{aligned}$$

(A.8)

We denote each element of $\tilde{{\mathcal {I}}}_{n,k}^{x,p}$ by ${\underline{\eta }}=(\eta _1,\dots , \eta _{k-1})$. Notice that ${\underline{\eta }}$ corresponds to a path from (0, 0) to (0, k) in $[0,k]\times [-x,n-x] \cap {\mathbb {Z}}^2$ consisting of the aforementioned vectors. Finally, define $\tilde{{\mathcal {I}}}_k$ as follows:

$$\begin{aligned} \begin{aligned} \tilde{{\mathcal {I}}}_k:= \{ {\underline{\eta }} \in {\mathbb {Z}}^{k-1}| \forall i \in {\mathbb {I}}_{k}, \, |\eta _i-\eta _{i-1}| \le 1,\, \text {with } \eta _0=\eta _k=0\}. \end{aligned} \end{aligned}$$

(A.9)

Observe that each ${\underline{\eta }} \in {\mathcal {I}}_k$, corresponds to a path from (0, 0) to (0, k) in ${\mathbb {Z}}^2$ consisting of $(-1,1),(0,1),(1,1) \in {\mathbb {Z}}^2$.

Having in mind the geometric interpretation of $\tilde{{\mathcal {I}}}_{n,k}^{x,p}$ and $\tilde{{\mathcal {I}}}_k$, one can observe²²:

$$\begin{aligned} \begin{aligned}&\tilde{{\mathcal {I}}}_{n,k}^{x,p} \subset \tilde{{\mathcal {I}}}_k, \quad |\tilde{{\mathcal {I}}}_{n,k}^{x,p}| \subset |\tilde{{\mathcal {I}}}_k| \le 3^k, \\&\tilde{{\mathcal {I}}}_{n,k}^{x,p} = \tilde{{\mathcal {I}}}_k \quad \text {iff} \quad [\frac{k}{2}]\le x \le n-[\frac{k}{2}]. \end{aligned} \end{aligned}$$

(A.10)

By using (A.8), we can rewrite the sum in (A.6), as a sum over the set $\tilde{{\mathcal {I}}}_{n,k}^{x,p}$. Here we introduce a set of notations in order to rewrite each term in (A.6) in a more suitable way.

Fix $k \in {\mathbb {N}}$, and take ${\underline{\eta }} \in \tilde{{\mathcal {I}}_k}$, then consider the set of indices ${\underline{j}}=(j_{-[\frac{k}{2}]},\dots ,j_0,\dots , j_{[\frac{k}{2}]+1})$ $\in {\mathbb {Z}}^{2[\frac{k}{2}]+2}$, where $j_i\ne j_{i'}$ for $i \ne {i'}$. Correspondingly, let ${\underline{m}}^k$ denotes a vector of $2[\frac{k}{2}]+2$ masses indexed by ${\underline{j}}$, i.e. ${\underline{m}}^k=(m_{j_{-[\frac{k}{2}]}},\dots ,m_{j_{[\frac{k}{2}]+1}})$, notice that we extended the set of i.i.d random variables $\{ m_x \}_{x=1}^n$ to the set of i.i.d random variables $\{m_x\}_{x \in {\mathbb {Z}}}$.²³ Moreover, let ${\underline{b}}^k \in [\beta _{min},\beta _{max}]^{2[\frac{k}{2}]+2}$ denotes the following vector: ${\underline{b}}^k=(b_{-[\frac{k}{2}]},\dots ,b_{[\frac{k}{2}]+1})$. We define ${\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k,{\underline{b}}^k)$ as follows:

$$\begin{aligned} \begin{aligned}&{\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k,{\underline{b}}^k)= \theta _{\eta _0\eta _1} \dots \theta _{\eta _i \eta _{i+1}} \dots \theta _{\eta _{k-1}\eta _k}, \\&\theta _{\eta _i\eta _{i+1}} = {\left\{ \begin{array}{ll} b_{\eta _i}(\frac{b_{\eta _i}}{m_{j_{\eta _i}}}+\frac{b_{(\eta _i+1)}}{m_{j_{(\eta _i+1)}}}) -\alpha \quad \text {if} \quad \eta _i=\eta _{i+1}, \\ -\frac{b_{\hat{\eta _i}}}{m_{j_{{\hat{\eta }}_i}}} \sqrt{b_{{\hat{\eta }}_i}b_{({\hat{\eta }}_i+1)}} \quad \text {if} \quad \eta _i \ne \eta _{i+1}, \end{array}\right. } \end{aligned} \end{aligned}$$

(A.11)

where we denoted $\hat{\eta _i}=\min \{ \eta _i,\eta _{i+1} \}$, and $\eta _0=\eta _k=0$. The following properties of ${\mathcal {F}}_{k,{\underline{\eta }}}$ are straightforward, since the distribution of the masses is compactly supported and ${\mathcal {F}}_{k,{\underline{\eta }}}$ is continuous on a compact set:

• ${\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k,{\underline{b}}^k)$ is uniformly continuous in the second component, uniformly in the distribution of the masses. More precisely, $\forall \epsilon >0$, there exists $\delta >0$ such that if $|{\underline{b}}^k_1-{\underline{b}}^k_2|< \delta$,²⁴ then for any realization of the masses, $|{\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k,{\underline{b}}^k_1)-{\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k,{\underline{b}}^k_2)| \le \epsilon$.

• Consider two different set of masses ${\underline{m}}^k_1$ and ${\underline{m}}^k_2$, since masses are i.i.d, we have: ${\mathbb {E}}({\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k_1,{\underline{b}}^k))={\mathbb {E}}({\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k_2,{\underline{b}}^k))$. Notice that we used the assumption $j_i \ne j_{i'}$ for $i \ne i'$.

• From the above properties, it is clear that taking two set of masses ${\underline{m}}^k_1$ and ${\underline{m}}^k_2$ we have: $\forall \epsilon >0$, there exists $\delta >0$ such that if $|{\underline{b}}^k_1-{\underline{b}}^k_2|< \delta$, then $|{\mathbb {E}}({\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k,_1{\underline{b}}^k_1))-{\mathbb {E}}({\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k_2,{\underline{b}}^k_2))|< \epsilon$.

Recall that we fixed k and ${\underline{\eta }} \in \tilde{{\mathcal {I}}}_k$. Let us take $n \in {\mathbb {N}}$ and $x \in {\mathbb {I}}_n$, then we define the vectors ${\underline{m}}^k(x,n)$, ${\underline{b}}^k(x,n)$ as follows:

$$\begin{aligned} \begin{aligned}&{\underline{m}}^k(x,n)_{j_i}= m_{x+i}, \quad -[\frac{k}{2}] \le i \le [\frac{k}{2}]+1, \quad \\&{\underline{b}}^k(x,n)_i= {\left\{ \begin{array}{ll} \beta (\frac{x+i}{n}) \quad \text {if } 0\le x+i \le n, \\ \beta (1) \quad \text {if } n<x+i,\\ \beta (0) \quad \text {if } 0<x+i, \end{array}\right. } \quad -[\frac{k}{2}] \le i \le [\frac{k}{2}]+1. \end{aligned} \end{aligned}$$

(A.12)

Finally, by combining the above notations and definitions, in particular (A.8), (A.11), and (A.12), we end-up with the following identity for $[\frac{k}{2}]+1 \le x \le n- [\frac{k}{2}]+1$:

$$\begin{aligned} \langle x, ({\tilde{A}}^p_n)^k x \rangle _n= \sum _{{\underline{\eta }} \in \tilde{{\mathcal {I}}}_{n,k}^{x,p}} {\mathcal {F}}_{k,{\underline{\eta }}}\big ({\underline{m}}^k(x,n),{\underline{b}}^k(x,n)\big )=\sum _{{\underline{\eta }} \in \tilde{{\mathcal {I}}}_k} {\mathcal {F}}_{k,{\underline{\eta }}}\big ({\underline{m}}^k(x,n),{\underline{b}}^k(x,n)\big ),\nonumber \\ \end{aligned}$$

(A.13)

where we have the second equality thanks to (A.10) and the choice of x. Notice that to check this identity one should compare the definition of ${\mathcal {F}}_{k, {\eta }}$ with the definition of the matrix ${\tilde{A}}_n^p$ in (5.10). Moreover, this identity holds for $x<[\frac{k}{2}]+1$ and $x>n-[\frac{k}{2}]+1$ if one slightly modifies the definition of ${\mathcal {F}}_{k,{\underline{\eta }}}$. However, the current form of this identity is sufficient for our purposes. It is worth mentioning that $\langle x, ({\tilde{A}} ^r_n)^k x \rangle _{n-1}$ can be written in the similar fashion, where one should define $\tilde{{\mathcal {F}}}_{k,{\underline{\eta }}}$ similar to ${\mathcal {F}}_{k,{\underline{\eta }}}$. One can check that $\tilde{{\mathcal {F}}}_{k,{\underline{\eta }}}$ has the three aforementioned properties. Since this task is rather straightforward, we only treat the terms corresponding to p and the terms corresponding to r can be treated similarly.

Thanks to (A.4) and (A.13), we can establish the existence of the following limit: , for every $y \in [0,1]$. We prove this fact by showing that the sequence is a Cauchy sequence.

Lemma A.1

Recall the assumption on the distribution of the masses, where $\mu (x)$ is smooth and supported on $[m_{min},m_{max}]$, $0< m_{min}< m_{max} < \infty$. We have $\forall y \in [0,1]$, and $\forall \epsilon >0$, there exists $N_0$, such that $\forall n,l>N_0$, we have .

Proof

Take $\epsilon >0$, and recall (A.1) then we have:

(A.14)

Since $\beta$ is continuous, and $0<\beta _{min} \le \beta (y)\le \beta _{max}$, we have $\beta _{[ny]}=\beta (\frac{[ny]}{n})$ and $\beta (\frac{[ly]}{l})$ are sufficiently close. In addition, ${\mathfrak {f}}(A^r_n)$ ,${\mathfrak {f}}(A^p_n)$ are uniformly bounded in n. Hence, it is sufficient to show

$$\begin{aligned} \begin{aligned}&\big |{\mathbb {E}}\big (\langle [ny],{\mathfrak {f}}(A_n^r) [ny] \rangle _{n-1} + \langle [ny], {\mathfrak {f}}(A_n^p) [ny] \rangle _n + \epsilon ^{[ny]}_n\big )\\&\quad - {\mathbb {E}}\big (\langle [ly],{\mathfrak {f}}(A_l^r) [ly] \rangle _{l-1} + \langle [ly], {\mathfrak {f}}(A^p_l) [ly] \rangle _l + \epsilon ^{[ly]}_l\big )\big |<\epsilon , \end{aligned} \end{aligned}$$

(A.15)

for $n,l>N_0$ with $N_0$ large enough. But the terms $|\epsilon ^{[ny]}_n|$ and $|\epsilon ^{[ly]}|$ are bounded by $\frac{{\mathcal {C}}}{n}$ and $\frac{{\mathcal {C}}}{l}$, respectively. Therefore, for $N_0$ large enough, they will be small, and it is enough to show:

$$\begin{aligned} \big |{\mathbb {E}} \big ( \langle [ny],{\mathfrak {f}}(A_n^r) [ny] \rangle _{n-1} + \langle [ny], {\mathfrak {f}}(A_n^p) [ny] \rangle _n \big ) - {\mathbb {E}} \big ( \langle [ly],{\mathfrak {f}}(A_l^r) [ly] \rangle _{l-1} + \langle [ly], {\mathfrak {f}}(A^p_l) [ly] \rangle _l \big )\big |<\epsilon , \end{aligned}$$

for proper $N_0$. Actually, we prove that there exists $N_0$, such that for $n,l>N_0$,

$$\begin{aligned} \big |{\mathbb {E}}\big (\langle [ny],{\mathfrak {f}}(A_n^p) [ny] \rangle _{n}\big ) - {\mathbb {E}}\big (\langle [ly],{\mathfrak {f}}(A_n^p) [ly] \rangle _{l}\big ) |<\epsilon . \end{aligned}$$

(A.16)

The term $|{\mathbb {E}}( \langle [ny], {\mathfrak {f}}(A_n^r) [ny] \rangle _{n-1} - {\mathbb {E}}( \langle [ly], {\mathfrak {f}}(A^r_l) [ly] \rangle _{l-1}|$ can be treated exactly the same way. In order to demonstrate (A.16), recall the definition of ${\mathfrak {f}}_{\prec }^{\frac{\epsilon }{4}}(A_n^p)$ and ${\mathfrak {f}}_{\succ }^{\frac{\epsilon }{4}}(A_n^r)$ from the expression (A.4), and recall $K_*(\frac{\epsilon }{4})$, which is given in this definition. Taking advantage of (A.5), we get

$$\begin{aligned} \Big |{\mathbb {E}}\Big (\langle [ny] , {\mathfrak {f}}_{\succ }^{\frac{\epsilon }{4}}(A_n^p) [ny] \rangle _n\Big )-{\mathbb {E}}\Big (\langle [ly] , {\mathfrak {f}}_{\succ }^{\frac{\epsilon }{4}}(A_l^p) [ly] \rangle _l\Big )\Big | \le \frac{\epsilon }{2}. \end{aligned}$$

By using the fact that ${\mathfrak {f}}(A_n^p)={\mathfrak {f}}_{\prec }^{\frac{\epsilon }{4}}(A_n^p)+{\mathfrak {f}}_{\succ }^{\frac{\epsilon }{4}}(A_n^p)$, and ${\mathfrak {f}}(A_l^p)={\mathfrak {f}}_{\prec }^{\frac{\epsilon }{4}}(A_l^p)+{\mathfrak {f}}_{\succ }^{\frac{\epsilon }{4}}(A_l^p)$, it is sufficient to prove that for $n,l>N_0$:

$$\begin{aligned} \Big |{\mathbb {E}}\Big (\langle [ny] , {\mathfrak {f}}_{\prec }^{\frac{\epsilon }{4}}(A_n^p) [ny] \rangle \Big )-{\mathbb {E}}\Big (\langle [ly] , {\mathfrak {f}}_{\prec }^{\frac{\epsilon }{4}}(A_l^p) [ly] \rangle \Big )\Big | \le \frac{\epsilon }{2}. \end{aligned}$$

(A.17)

Since $K_*(\frac{\epsilon }{4})$ is independent of n and l, it is enough to prove that $\forall {\tilde{\epsilon }}>0$, there exist $N_0$ such that $\forall k \in \{0,\dots ,K_*(\frac{\epsilon }{4})\}$ and $\forall n,l>N_0$:

$$\begin{aligned} \big |{\mathbb {E}}\big (\langle [ny] , (A_n^p-\alpha I_{n})^k [ny] \rangle \big )-{\mathbb {E}}\big (\langle [ly] , (A_l^p-\alpha I_{l})^k [ly] \rangle \big )\big | <{\tilde{\epsilon }}. \end{aligned}$$

(A.18)

Then, taking ${\tilde{\epsilon }}=\frac{\epsilon }{2cK_*(\frac{\epsilon }{4})}$, where c is the bound on $|a_0|,\dots ,|a_{K_*(\frac{\epsilon }{4})}|$, completes the proof. We can obtain (A.18) by using (A.13) and properties of ${\mathcal {F}}_{k,{\underline{\eta }}}$ as follows: Let us assume $y \in (0,1)$,²⁵ fix $k \in \{1, \dots K_*(\frac{\epsilon }{4}) \}$, and take $N'$ such that for $n,l>N'$ we have: $K_*(\frac{\epsilon }{4})<[ny]<n-K_*{\frac{\epsilon }{4}}$ and $K_*(\frac{\epsilon }{4})<[ly]<l-K_*(\frac{\epsilon }{4})$. By this choice we can use (A.13) and observe:

$$\begin{aligned} \begin{aligned}&\big |{\mathbb {E}}\big (\langle [ny] , (A_n^p-\alpha I_{n})^k [ny] \rangle _n\big )-{\mathbb {E}}\big (\langle [ly] , (A_l^p-\alpha I_{l})^k [ly] \rangle _l\big )\big | \\&\quad = \bigg |\sum _{{\underline{\eta }} \in \tilde{{\mathcal {I}}}_k} \bigg ( {\mathbb {E}} \Big ( {\mathcal {F}}_{k,{\underline{\eta }}}\big ({\underline{m}}^k([ny],n),{\underline{b}}^k([ny],n)\big ) \Big ) - {\mathbb {E}}\Big ( {\mathcal {F}}_{k,{\underline{\eta }}}\big ({\underline{m}}^k([ly],l),{\underline{b}}^k([ly],l)\big ) \Big ) \bigg ) \bigg |. \end{aligned}\nonumber \\ \end{aligned}$$

(A.19)

Thanks to the third property of ${\mathcal {F}}_{k,{\underline{\eta }}}$, $\forall$ $\epsilon '>0$, there exists $\delta _{{\underline{\eta }}}(\epsilon ')$ such that for $|{\underline{b}}^k([ly],l)-{\underline{b}}^k([ny],n)| \le \delta _{{\underline{\eta }}}(\epsilon ')$, we have

$$\begin{aligned} \Big |{\mathbb {E}} \Big ( {\mathcal {F}}_{k,{\underline{\eta }}}\big ({\underline{m}}^k([ny],n),{\underline{b}}^k([ny],n)\big ) \Big ) - {\mathbb {E}}\Big ( {\mathcal {F}}_{k,{\underline{\eta }}}\big ({\underline{m}}^k([ly],l),{\underline{b}}^k([ly],l)\big ) \Big )\Big | \le \epsilon '. \end{aligned}$$

On the other hand, since $\beta (.)$ is continuous, there exist $N^k_{{\underline{\eta }}}$ such that for $n,l>N^k_{{\underline{\eta }}}$ and for all $-[\frac{k}{2}]-1<i<[\frac{k}{2}]+1$, we have

$$\begin{aligned} |\beta (\frac{[ny]+i}{n})-\beta (\frac{[ly]+i}{l})| \le \frac{\delta _{{\underline{\eta }}}(\frac{{\tilde{\epsilon }}}{3^{K_*(\frac{\epsilon }{4})}})}{\sqrt{k+3}}. \end{aligned}$$

Hence, thanks to the definition of ${\underline{b}}^k$ (A.12), for $n,l > N_{{\underline{\eta }}}^k$ we get $|{\underline{b}}^k([ly],l)-{\underline{b}}^k([ny],n)| \le \delta _{{\underline{\eta }}}(\frac{{\tilde{\epsilon }}}{3^{K_*(\frac{\epsilon }{4})}})$. Consequently, if we take $N_k= \max _{ \{ {\underline{\eta }} \in \tilde{{\mathcal {I}}}_k \} } \{ N_{{\underline{\eta }}}^k , N' \}$, $\forall n,l > N_k$, we have $\forall {\underline{\eta }} \in \tilde{{\mathcal {I}}}_k$:

$$\begin{aligned} \Big |{\mathbb {E}} \Big ( {\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k([ny],n),{\underline{b}}^k([ny],n)) \Big ) - {\mathbb {E}}\Big ( {\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k([ly],l),{\underline{b}}^k([ly],l)) \Big )\Big | \le \frac{{\tilde{\epsilon }}}{3^{K_*(\frac{\epsilon }{4})}}. \end{aligned}$$

Combining the latter with the estimate $|\tilde{{\mathcal {I}}}_k| \le 3^k \le 3^{K_*(\frac{\epsilon }{4})}$, we get (A.18). Finally, taking $N_0 = \max _{\{ k \in {\mathbb {I}}_{K_*(\frac{\epsilon }{4})}\}} \{N_k, N'\}$ finishes the proof.

Notice that in order to deal with the term $|{\mathbb {E}}( \langle [ny], {\mathfrak {f}}(A_n^r) [ny] \rangle _{n-1} - {\mathbb {E}}( \langle [ly], {\mathfrak {f}}(A^r_l) [ly] \rangle _{l-1}|$ one should properly modify the definition of ${\mathcal {F}}_{{\underline{\eta }},k}$ and $\tilde{{\mathcal {I}}}^{x,p}_{n,k}$. In particular, in this case ${\mathcal {F}}_{{\underline{\eta }},k}$ is given by:

$$\begin{aligned} \begin{aligned}&\tilde{{\mathcal {F}}}_{n,{\underline{\eta }}}({\underline{m}}^k,{\underline{b}}^k)= \theta _{\eta _0\eta _1} \dots \theta _{\eta _i \eta _{i+1}} \dots \theta _{\eta _{k-1}\eta _k}, \\&\theta _{\eta _i,\eta _{i+1}} = {\left\{ \begin{array}{ll} {\underline{b}}^k_{\eta _i}(\frac{{\underline{b}}^k_{\eta _i}}{m_{\eta _i+k}}+\frac{{\underline{b}}_{\eta _i+1}}{(m_{\eta _i+1)}} -\alpha \quad \text {if} \quad \eta _i=\eta _{i+1}, \\ -\frac{{\underline{b}}^k_{hat{\eta _i}+1}}{m_{{\hat{\eta }}_i}} \sqrt{{\underline{b}}^k_{{\hat{\eta }}_i}{\underline{b}}^k_{{\hat{\eta }}_i+1}} \quad \text {if} \quad \eta _i \ne \eta _{i+1}. \end{array}\right. } \end{aligned} \end{aligned}$$

(A.20)

Since this function satisfies the same properties, the rest of the proof is exactly similar to the previous case.$\square$

As an obvious consequence of Lemma A.1 we have:

Corollary A.1.1

$\forall y \in [0,1]$, the limit exists, and the function $\mathrm {f}^{\mu }_{\beta }$ is well-defined.

Moreover, following the proof of Lemma A.1, we can deduce the following corollary as well:

Corollary A.1.2

In thermal equilibrium i.e. when for $\beta _{eq} \in (0,\infty )$, $\beta (y)=\beta _{eq}$ is constant in y, we have:

$$\begin{aligned} \forall y,y' \in (0,1), \quad \quad \mathrm {f}^{\mu }_{\beta }(y)= \mathrm {f}^{\mu }_{\beta }(y'). \end{aligned}$$

(A.21)

In particular, the function $\mathrm {f}^{\mu }(.)$ in (5.35) is well defined.

Proof

In order to proof (A.21) it is enough to show that

We omit the subscript of $\rho$ since it is clear that we are in thermal equilibrium. Take $\epsilon >0$, first recall the expression of (A.1), then rewrite ${\mathfrak {f}}(.)={\mathfrak {f}}_{\prec }^{\frac{\epsilon }{4}}(.)+{\mathfrak {f}}_{\succ }^{\frac{\epsilon }{4}}(.)$ as it has been defined in (A.4) and observe:

(A.22)

where we bounded the terms involving ${\mathfrak {f}}_{\succ }^{\frac{\epsilon }{4}}$ by $\frac{\epsilon }{2}$, thanks to (A.5). Moreover, $|\epsilon _{n}^{[ny]}|+|\epsilon _n^{[ny']}|$ is bounded by $\frac{C}{n}$; therefore, it has been bounded by $\frac{\epsilon }{2}$, by taking $n>N_1$, for proper $N_1$. Lastly, recall $K_*(\frac{\epsilon }{4})$ from (A.4), and choose $N_2$ such that for $n>N_2$,

$$\begin{aligned} {[}\frac{K_*(\frac{\epsilon }{4})}{2}]+1<[ny]<n-[\frac{K_*(\frac{\epsilon }{4})}{2}]-1, \quad [\frac{K_*(\frac{\epsilon }{4})}{2}]+1<[ny']<n-[\frac{K_*(\frac{\epsilon }{4})}{2}]-1. \end{aligned}$$

Thanks to this choice, and by using (A.13), for any $k \in \{ 1,\dots , K_*(\frac{\epsilon }{4}) \}$ we have:

$$\begin{aligned} \begin{aligned}&|{\mathbb {E}}(\langle [ny] , (A_n^p-\alpha I_{n})^k [ny] \rangle _n)-{\mathbb {E}}(\langle [ny'] , (A_n^p-\alpha I_{n})^k [ny'] \rangle _n)| \\&\quad = \bigg |\sum _{{\underline{\eta }} \in \tilde{{\mathcal {I}}}_k} \bigg ( {\mathbb {E}} \Big ( {\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k([ny],n),{\underline{b}}^k([ny],n)) \Big ) - {\mathbb {E}}\Big ( {\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k([ny'],n),{\underline{b}}^k([ny'],n)) \Big ) \bigg ) \bigg |=0, \end{aligned}\nonumber \\ \end{aligned}$$

(A.23)

where, first, we used the fact that in thermal equilibrium we have ${\underline{b}}^k([ny],n)={\underline{b}}^k([ny'],n)$, then we took advantage of the second property of ${\mathcal {F}}_{k,{\underline{\eta }}}$. Therefore, by using the definition of ${\mathfrak {f}}_{\prec }^{\frac{\epsilon }{4}}$, the term in second line of (A.22) is zero, for $n>N_2$ (the part corresponding to $A_n^r$ is completely analogous). Hence taking $n> \max \{ N_1,N_2 \}$ gives us the desirable result. $\square$

Proposition A.1.1

The function $\mathrm {f}^{\mu }_{\beta }(y): (0,1) \rightarrow {\mathbb {R}}$ is continuous.

Proof

Fix $\epsilon >0$, and observe

$$\begin{aligned} |\mathrm {f}_{\beta }^{\mu }(y)- \mathrm {f}_{\beta }^{\mu }(y')| \le |\mathrm {f}_{\beta }^{\mu }(y)- \mathrm {f}_{\beta ,n}^{\mu }(y)| +|\mathrm {f}_{\beta ,n}^{\mu }(y)- \mathrm {f}_{\beta ,n}^{\mu }(y')|+ |\mathrm {f}_{\beta ,n}^{\mu }(y')- \mathrm {f}_{\beta }^{\mu }(y')| ,\nonumber \\ \end{aligned}$$

(A.24)

where $\mathrm {f}^{\mu }_{\beta ,n}(y):={\mathbb {E}}(\langle {\tilde{e}}_{[ny]} \rangle _{\rho ^n})$. Since $\lim _{n \rightarrow \infty } \mathrm {f}_{\beta ,n}^{\mu }(y)=\mathrm {f}_{\beta }^{\mu }(y)$, if we take $n>N_1(\epsilon )$, then

$$\begin{aligned} |\mathrm {f}_{\beta }^{\mu }(y)- \mathrm {f}_{\beta ,n}^{\mu }(y)| + |\mathrm {f}_{\beta ,n}^{\mu }(y')- \mathrm {f}_{\beta }^{\mu }(y')| \le \frac{\epsilon }{2}. \end{aligned}$$

Moreover, we claim that there exist $N_2(\epsilon )$, such that for $n>N_2(\epsilon )$, there exists $\delta$ such that for $|y-y'| < \delta$, $|\mathrm {f}_{\beta ,n}^{\mu }(y)- \mathrm {f}_{\beta ,n}^{\mu }(y')| < \frac{\epsilon }{2}$. Proving this claim completes the proof, since we can take $n > \max \{N_1(\epsilon ), N_2(\epsilon ) \}$ and observe that for $|y-y'|< \delta$, we have $|\mathrm {f}_{\beta }^{\mu }(y)-\mathrm {f}_{\beta }^{\mu }(y')|<\epsilon$.

However, the proof of this statement follows the same lines of Lemma A.1 and Corollary (A.1.2). Similar to (A.22), we divide ${\mathfrak {f}}(.)={\mathfrak {f}}_{\prec }^{\frac{\epsilon }{4}}(.)+{\mathfrak {f}}_{\succ }^{\frac{\epsilon }{4}}(.)$, and take $N_3(\epsilon )$ such that for $n>N_2(\epsilon )$ we have

$$\begin{aligned} {[}\frac{K_*(\frac{\epsilon }{4})}{2}]+1<[ny]<n-[\frac{K_*(\frac{\epsilon }{4})}{2}]-1 , \quad [\frac{K_*(\frac{\epsilon }{4})}{2}]+1<[ny']<n-[\frac{K_*(\frac{\epsilon }{4})}{2}]-1. \end{aligned}$$

Therefore, we have:

(A.25)

where we can find $N_4$ and $\delta _0$ such that the last line will be bounded by $\frac{3 \epsilon }{4}$, for $n>N_4$ and $|y-y'|<\delta _0$. Let us take $N_2(\epsilon )= \max \{N_3(\epsilon ),N_4 \}$. Recall the definition of ${\mathfrak {f}}_{\prec }^{\frac{\epsilon }{4}}(.)$ (A.4) as a Taylor sum up to $K_*(\frac{\epsilon }{4})$ terms. By using the choice of $N_2(\epsilon )$ rewrite each term of this sum as a sum over the paths ${\underline{\eta }} \in \tilde{{\mathcal {I}}}_k$ as in (A.13). The rest of the proof boils down to demonstrating the fact that for $n>N_2(\epsilon )$, for all $1\le k \le K_*(\frac{\epsilon }{4})$, and for all ${\underline{\eta }} \in \tilde{{\mathcal {I}}}_k$, we have: $\forall {\hat{\epsilon }}>0$ there exist $\delta _{k,{\underline{\eta }}}({\hat{\epsilon }})>0$, such that if $|y-y'| < \delta _{k,{\underline{\eta }}}({\hat{\epsilon }})$ then

$$\begin{aligned} \Big |{\mathbb {E}} \Big ( {\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k([ny],n),{\underline{b}}^k([ny],n)) \Big )-{\mathbb {E}} \Big ( {\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k([ny'],n),{\underline{b}}^k([ny'],n)) \Big )\Big | < {\hat{\epsilon }}.\nonumber \\ \end{aligned}$$

(A.26)

However, recalling the definition of ${\underline{b}}^k([ny],n)$ from (A.12), since $\beta (.)$ is continuous, one can observe that if $|y-y'|< {\tilde{\delta }}$, $|{\underline{b}}^k([ny],n)-{\underline{b}}^k([ny'],n)|$ is sufficiently small for a proper choice of ${\tilde{\delta }}$. Therefore, by using the third property of ${\mathcal {F}}_{k,\eta }$, we obtain the desired $\delta _{k,{\underline{\eta }}}({\hat{\epsilon }})$. Finally, taking

$$\begin{aligned} {\hat{\epsilon }}=\frac{\epsilon }{3^{K_*(\frac{\epsilon }{4})}K_*(\frac{\epsilon }{4})4c}, \quad \delta = \min _{0<k \le K_*(\frac{\epsilon }{4})} \{\delta _0, \min _{{\underline{\eta }} \in \tilde{{\mathcal {I}}}_k} \{ \delta _{k,{\underline{\eta }}} \} \} \end{aligned}$$

finishes the proof. Notice that here c is the uniform bound on Taylor coefficients $|a_0|,\dots |a_{K_*({\frac{\epsilon }{4}}})|$. Moreover, the choice of ${\hat{\epsilon }}$ is justified by the bound $|\tilde{{\mathcal {I}}}_k| \le 3^k$. Furthermore, as usual the part corresponding to r can be treated exactly in the same way. $\square$

The next proposition proves (5.36), and illustrates the fact that $\mathrm {f}_{\beta }^{\mu }(y)$ is in fact a function of inverse temperature at point y i.e. $\beta (y)$. Since the proof of this proposition is similar to the previous lemma and proposition, we just sketch the proof and only highlight the differences:

Proposition A.1.2

Let $\beta \in C^0([0,1])$ satisfying the assumptions stated in the definition (2.9). Recall the definition of $\mathrm {f}^{\mu }_{\beta }:[0,1] \rightarrow {\mathbb {R}}$ from (5.5), and $\mathrm {f}^{\mu }:(0,\infty ) \rightarrow {\mathbb {R}}$ from (5.35), then we have $\forall y \in (0,1)$:

$$\begin{aligned} \mathrm {f}^{\mu }_{\beta }(y)=\mathrm {f}^{\mu }(\beta (y)). \end{aligned}$$

(A.27)

Proof

Fix $y \in (0,1)$ and recall that we denote the average in Gibbs state in thermal equilibrium at inverse temperature $\beta _{eq}$, with . Since in thermal equilibrium, we have translation invariance in the limit thanks to (A.21) in Corollary A.1.2, it is enough to prove that $\forall \epsilon >0$, there exist $N(\epsilon )$, such that for $n>N(\epsilon )$:

(A.28)

Let us denote the matrices corresponding to thermal averages at temperature profile $\beta (.)$ (3.25) by $A_n^{p,\beta (.)}$ and $A_n^{r,\beta (.)}$, only for the sake of this proposition. Similarly, denote the same matrices in thermal equilibrium at temperature $\beta (y)$ by $A_n^{p,\beta (y)}$, $A_n^{r,\beta (y)}$. The proof goes as follows: we rewrite (A.28) in terms of ${\mathfrak {f}}(A_n^{p,\beta (.)})$, ${\mathfrak {f}}(A_n^{r,\beta (.)})$, ${\mathfrak {f}}(A_n^{p,\beta (y)})$, and ${\mathfrak {f}}(A_n^{r,\beta (y)})$ thanks to (A.1), up to a vanishing error. Then we decompose ${\mathfrak {f}}(.)={\mathfrak {f}}_{\prec }^{\frac{\epsilon }{4}}(.)+{\mathfrak {f}}_{\succ }^{\frac{\epsilon }{4}}(.)$ and we bound all the terms corresponding to ${\mathfrak {f}}_{\succ }^{\frac{\epsilon }{4}}(.)$ by $\frac{\epsilon }{2}$, thanks to (A.5). Notice that the definition of ${\mathfrak {f}}$, ${\mathfrak {f}}_{\succ }^{\epsilon }$ and ${\mathfrak {f}}_{\prec }^{\epsilon }$ depends on the matrices through the constants $\alpha$ and $K_*(\epsilon )$. Here, we take the definition which is given by matrices $A_n^{p,\beta (.)}$ and $A_n^{r,\beta (.)}$. Then it is straightforward to check that ${\mathfrak {f}}(A_n^{r,\beta (y)})$ and ${\mathfrak {f}}(A_n^{p,\beta (y)})$, are well defined and they satisfy the same bounds as in (A.5), since $\beta _{\min }\le \beta (y) \le \beta _{max}$, and the same uniform bound $c_0$ in Lemma 5.1 holds for $||A_n^{p,\beta (y)}||$ and $||A_n^{r,\beta (y)}||$. The terms involving ${\mathfrak {f}}^{\frac{\epsilon }{4}}_{\prec }$ can be treated similar to Proposition A.1.1, we sketch the terms related to p, the other ones is similar. First, we choose $N_0(\epsilon )$ such that for $n> N_0(\epsilon )$, $K_*(\frac{\epsilon }{4}) \le [ny] \le n-K_*(\frac{\epsilon }{4})$, then we expand each term of the sum appearing in ${\mathfrak {f}}^{\frac{\epsilon }{4}}_{\prec }(.)$ by using the random walk representation in (A.13), similar to (A.26), it is enough to show that for any ${\hat{\epsilon }}>0$, $1 \le k \le K_*(\frac{\epsilon }{4})$ and ${\underline{\eta }} \in \tilde{{\mathcal {I}}}_k$ there exists N such that for $n>N$, we have:

$$\begin{aligned} \Big |{\mathbb {E}} \Big ( {\mathcal {F}}_{k, {\underline{\eta }}}({\underline{m}}^k([ny],n),{\underline{b}}^k([ny],n)) \Big )-{\mathbb {E}} \Big ({\mathcal {F}}_{k, {\underline{\eta }}}({\underline{m}}^k([ny],n),\tilde{{\underline{b}}}^k([ny],n)) \Big )\Big | \le {\hat{\epsilon }},\nonumber \\ \end{aligned}$$

(A.29)

where $\tilde{{\underline{b}}}^k([ny],n)$ is defined analogous to ${\underline{b}}^k([ny],n)$ (A.12) for a constant profile of temperature at inverse temperature $\beta (y)$ i.e. $\tilde{{\underline{b}}^k}([ny],n)_i=\beta (y)$ for $-[\frac{k}{2}] \le i \le [\frac{k}{2}]+1$. However, existence of N such that for $n>N$ (A.29) holds is evident from the third property of ${\mathcal {F}}_{k,{\underline{\eta }}}$ and the fact that $\beta (.)$ is continuous. $\square$

Recall the definition of , and the sequence $n_k$ in the proof of 5.4, where $n_k \rightarrow \infty$ with $\frac{n_{k+1}}{n_k} \rightarrow 1$. The following lemma was an essential part of the proof:

Lemma A.2

Fix $\epsilon >0$, then for every realization of the masses, there exists $N_*$ such that, for every $n_k>N_*$ and every n with $n_k \le n < n_{k+1}$, we have: $\forall x \in {\mathbb {I}}_{n_k}$, with $n_*<x<n_k-n_*$, $|Y^n_x -Y^{n_k}_x| \le \epsilon$. Here, $n_*$ is a constant only depending on $\epsilon$.

Proof

Fix $\epsilon >0$, and recall the definition of $Y_x^n$, we have

(A.30)

The second term in (A.30) can be treated by using Lemma A.1, and continuity of g (note that we used the choice of n and $n_k$, where $\frac{n}{n_k} \rightarrow 1$, as well). Hence, there exits $N_1$, such that for $n_k>N_1$, we have:

(A.31)

where $c_1$ is the bound on |g|, and $c_2$ is the bound on .

Similarly, for the first term in (A.30) , it is enough to deal with . Thanks to the expression of in (A.1), for proper $N_2$, we have for $n>N_2$:

(A.32)

where we chose $N_2$ such that $|\epsilon ^x_n|+|\epsilon ^x_{n_k}| \le \frac{\epsilon }{6}$. From now on, let us show $\frac{\epsilon }{c_1C_0}$ by $\epsilon$. Now it is enough to find $N_3$, such that for $n_k>N_3$,

$$\begin{aligned} \big |\langle x, {\mathfrak {f}}(A^p_n) x \rangle _{n}-\langle x, {\mathfrak {f}}(A^p_{n_k}) x \rangle _{n_k}\big | \le \frac{\epsilon }{6}. \end{aligned}$$

Let us decompose ${\mathfrak {f}}(.)={\mathfrak {f}}^{\frac{\epsilon }{12}}_{\prec }(.)+{\mathfrak {f}}^{\frac{\epsilon }{12}}_{\succ }(.)$ as in (A.4), recall $K_*(\frac{\epsilon }{12})$ from (A.4) and let $n_*=K_*(\frac{\epsilon }{12})$, notice that $n_*$ only depends on $\epsilon$. Therefore, thanks to (A.5) we have:

$$\begin{aligned} \big |\langle x, {\mathfrak {f}}(A^p_n) x \rangle _{n}-\langle x, {\mathfrak {f}}(A^p_{n_k}) x \rangle _{n_k}\big | \le \big |\langle x, {\mathfrak {f}}_{\prec }^{\frac{\epsilon }{12}}(A^p_n) x \rangle _{n}-\langle x, {\mathfrak {f}}_{\prec }^{\frac{\epsilon }{12}}(A^p_{n_k}) x \rangle _{n_k}\big | + \frac{\epsilon }{12}.\nonumber \\ \end{aligned}$$

(A.33)

Now, it is sufficient to show that

$$\begin{aligned} \big |\langle x, {\mathfrak {f}}_{\prec }^{\frac{\epsilon }{12}}(A^p_n) x \rangle _{n}-\langle x, {\mathfrak {f}}_{\prec }^{\frac{\epsilon }{12}}(A^p_{n_k}) x \rangle _{n_k}\big | \le \frac{\epsilon }{12}, \end{aligned}$$

(A.34)

for $n_k$ sufficiently large. In order to control this term, recall the definition of ${\underline{b}}^k$ from (A.12) and notice that for any $\delta >0$, there exists $N_4(\delta )$ such that for $n_k>N_4(\delta )$, $|{\underline{b}}^k(x,n)-{\underline{b}}^k(x,n_k)| < \delta$, since $\beta$ is continuous, $n_k\le n < n_{k+1}$, and $\frac{n_{k+1}}{n_k} \rightarrow 1$. By using the first property of ${\mathcal {F}}_{k,{\underline{\eta }}}$, we choose $\delta >0$ such that

$$\begin{aligned} |{\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k(x,n),{\underline{b}}^k(x,n))-{\mathcal {F}}_{k,{\underline{\eta }}}({\underline{m}}^k(x,n_k),{\underline{b}}^k(x,n_k))| \le \frac{\epsilon }{12c_1n_*3^{n_*}}, \end{aligned}$$

where $c_1$ is the uniform bound on coefficients $a_0,\dots ,a_{n_*}$ in (A.4).

Taking $n>\max \{N_1,N_2,N_3,N_4(\delta ), n_* \}$, and combining this last estimate with the expression of ${\mathfrak {f}}_{\prec }^{\frac{\epsilon }{12}}$ in (A.4) as well as the random walk representation relation (A.13), gives us (A.34) for any $n_*<x<n-n_*$. The term corresponding to r can be bounded by $\frac{\epsilon }{6}$ similarly. $\square$

B Alternative Coordinates

Describing our system in the r coordinates with $\sum _{x=1}^n p_x=0$ means that our system is described by an observer at center of mass; hence, macroscopically the center of mass should be fixed as well, i.e., $\int _0^1 {\bar{p}}(y)dy=0$. Equivalently, this system can be modeled in terms of q coordinates, where one should be careful about the center of mass. Since it is more fashionable to treat quantum harmonic chains in terms of p and q coordinates, in this section, we introduce our model in terms of these coordinates and rewrite the main transformations in terms of q coordinates, in order to illustrate the link between our setup and the conventional one. Define on the Hilbert space $L^2({\mathbb {R}}^n)$, the following Hamiltonian operator:

$$\begin{aligned} H_n=\frac{1}{2}\sum _{x=1}^n (\frac{p_x^2}{m_x} + (q_{x+1}-q_{x})^2), \end{aligned}$$

(B.1)

where for each $x \in {\mathbb {I}}_n$, $q_x$ denotes the position operator of the particle x, i.e. denoting the space variable by $\pmb {\zeta } \in {\mathbb {R}}^n$, for any , , and $p_x$ denotes the corresponding momentum operator $p_x=-i \partial \setminus \partial _{\pmb {\zeta }_x}$. Moreover, we assume free boundary conditions: $q_0=q_1$ and $q_n=q_{n+1}$. Here $m_x$ are positive i.i.d random variables.

The Hamiltonian (B.1) can be diagonalized with a linear transformation and written as sum of free bosons:

$$\begin{aligned} H_n = {\hat{p}}_0^2+\sum _{k=1}^{n-1}\omega _{k}({\hat{b}}_k^* {\hat{b}}_k+\frac{1}{2}), \end{aligned}$$

(B.2)

where $b_k$ and $b_k^*$ are bosonic annihilation and creation operators with commutation relations $[{\hat{b}}_k^*,{\hat{b}}_{k'}]=\delta _{k,k'}, [{\hat{b}}_k^*,{\hat{b}}_{k'}^*]=[{\hat{b}}_k,{\hat{b}}_{k'}]=0$. Here ${\hat{p}}_o=\sum _{x=1}^n p_x$ is the total momentum operator. Since $[{\hat{p}}_0,H_n]=0$, after a straightforward analysis, the expression (B.2) indicates that $H_n$ has a purely continuous spectrum (cf. Remark 3.3 of [36], and Remark 3.6 of [35] for more details). In order to avoid technical difficulties concerning continuous spectrum of $H_n$, we alternatively define our model in r coordinates. Notice that, one can observe that the Heisenberg dynamics generated by (B.2) on $L^2({\mathbb {R}}^n)$ is similar to (3.22) up to ${\hat{p}}_0=(\sum _{x=1}^n m_x)^{-1} p_x$ operator which is constant in time.

Another technical problem arising in this description, concerns the density operator $\rho ^n_{{\bar{p}},{\bar{r}},\beta }=\exp (-{\tilde{H}}_{\beta }^n)$ (2.9). The pseudo-Hamiltonian ${\tilde{H}}_{\beta }^n$ can be written as

$$\begin{aligned} {\tilde{H}}_{\beta }^n= \frac{\tilde{{\mathfrak {p}}}_0^2}{2}+ \sum _{k=1}^{n-1} \gamma _k(\tilde{{\mathfrak {b}}}_k^*\tilde{{\mathfrak {b}}}_k +\frac{1}{2} ), \end{aligned}$$

(B.3)

where $\tilde{{\mathfrak {b}}}_k^*$ and $\tilde{{\mathfrak {b}}}_k$ are another set of bosonic operators (3.35), (B.13), and

$$\begin{aligned} \tilde{{\mathfrak {p}}}_0=(\sum _{x=1}^n\frac{m_x}{\beta _x})^{-\frac{1}{2}} \big (\sum _{x=1}^n p_x -\Pi _0\big )= (\sum _{x=1}^n\frac{m_x}{\beta _x})^{-\frac{1}{2}} ({\hat{p}}_o-\Pi _0). \end{aligned}$$

(B.4)

Thanks to the definition of $\tilde{{\mathfrak {p}}}_0$, one can observe that similar to $H_n$, ${\tilde{H}}_{\beta }^n$ has a continuous spectrum as an operator on $L^2({\mathbb {R}}^n)$ in this coordinates. Moreover, $\rho ^n_{{\bar{p}},{\bar{r}},\beta }$ is not trace-class in this description, anymore. Although $\rho ^n_{{\bar{p}},{\bar{r}},\beta }$ seems to be a natural choice for our locally Gibbs state corresponding to ${\bar{r}}, {\bar{p}}, \beta$, we need to slightly modify it, in order to circumvent the above mentioned technicalities. Recall that the total momentum ${\hat{p}}_o= \sum _{x=1}^n p_x$ is conserved by the dynamics. Ideally, one would be tempted to fix the total momentum apriori to the value prescribed by the macroscopic profile i.e.,

$$\begin{aligned} \Pi _0 = \sum _{x=1}^n {\bar{p}}(\frac{x}{n})\frac{m_x}{{\bar{m}}}. \end{aligned}$$

(B.5)

However, this is not convenient for technical reasons, instead we can modify our initial locally Gibbs state as follows: Let us denote $L^{2}({\mathbb {R}}^n)$ by ${{\mathcal {H}}}_n$, only in this section. Inspiring from diagonalization (3.31), we decompose ${{\mathcal {H}}}_n$ as follows. Let $\tilde{\pmb {\zeta }}_0:= (\sum _{x=1}^n \frac{m_x}{\beta _x})^{-\frac{1}{2}} \sum _{x=1}^n \frac{m_x}{\beta _x} \pmb {\zeta }_x$, and $V_{\tilde{\pmb {\zeta }}_0}^{\perp } \subset {{\mathbb {R}}}^n$, be the orthogonal complement of $\mathrm {Span}(\tilde{\pmb {\zeta }}_0)$. Denote the Lebesgue measure on $V_{\hat{\pmb {\zeta }}_0}^{\perp } \subset {{\mathbb {R}}}^n$, by $d\nu _{n-1}$; then, we have: ${{\mathcal {H}}}_n= L^2({{\mathbb {R}}},d\tilde{\pmb {\zeta }}_0) \otimes L^2 (V_{\tilde{\pmb {\zeta }}_0}^{\perp }, d\nu _{n-1}) \equiv {{\mathcal {H}}}_o \otimes {{\mathcal {H}}}_{n-1}^-$. Again appealing to (3.31), observe that ${\tilde{H}}_{\beta }^{n}=\frac{1}{2}\tilde{{\mathfrak {p}}}_0^2+ H_{\beta }^{n,-}$, where

$$\begin{aligned} \tilde{{\mathfrak {p}}}_0=(\sum _{x=1}^n\frac{m_x}{\beta _x})^{-\frac{1}{2}} \big (\sum _{x=1}^n p_x -\Pi _0\big )= (\sum _{x=1}^n\frac{m_x}{\beta _x})^{-\frac{1}{2}}({\hat{p}}_o-\Pi _0), \end{aligned}$$

(B.6)

only acts on ${{\mathcal {H}}}_o$ and

$$\begin{aligned} H_{\beta }^{n,-}:= {\tilde{H}}_{\beta }^{n}-\frac{1}{2}\tilde{{\mathfrak {p}}}_0^2, \end{aligned}$$

(B.7)

only acts on ${{\mathcal {H}}}_{n-1}^-$.

Take such that , , ; this means the total momentum has the following average and uncertainty: and

, (cf. Remark B.2). Now, we define the locally Gibbs state with “fixed total momentum” as:

(B.8)

where denotes the projection operator into the subspace spanned by the pure state . Notice that $H_{\beta }^{n,-}$ can be mapped into $H_{\beta }^n$ (2.10) by a unitary transformation. Consequently, one can check that $H_{\beta }^{n,-}$ has a discrete spectrum with non-negative eigenvalues (with a process similar to Sect. 3). We can express $H_{\beta }^{n,-}$ in terms of sum of free bosonic operators, and obtain its spectrum explicitly similar to what we did in Sect. 3. Hence, using spectral theorem, one can observe that $\rho ^n_{{\bar{p}},{\bar{r}},\beta }$ is well defined and trace-class.

As before, one can observe that , , and are well defined, this suggests that (B.8) is an appropriate modification of (2.9) in this coordinate.

In order to avoid the aforementioned difficulties, we describe our model in terms of elongation operators. One can argue that elongation operators are “physically” relevant, since in the classical counterpart of our system the elongation is the “real” physical variable, rather than the position of the particles. Let us highlight the relation between these models by a couple of remarks:

Remark B.1

Notice that same result of Theorem 2.1 holds for this system, as well. In fact, initially the previous description can be mapped into this new description via a unitary transformation (cf. Remark 3.3 in [36]). The proof in this new coordinate is basically identical, except for some considerations concerning center of mass, which makes the proof even simpler. For example ${\mathscr {E}}_x^n$ in (3.41) does not appear anymore thanks to the proper choice of .

In the previous description (2.2), by definition we have $\sum _{x=1}^n p_x=0$. This is because we begin the description of the system by quantizing the classical description corresponding to the observer in center of mass. Consequently, we should take $\int _0^1 {\bar{p}}(y) dy=0$. In this new coordinates, since we describe the center of mass separately by , we can take any ${\bar{p}} \in C^1([0,1])$ with non zero average.

Remark B.2

Physically the initial state (B.8), means that initially we prepare our system in the lab such that our system’s center of mass is known and is given by the wave function (pure state) , where its above mentioned averages (momentum and kinetic energy contribution) is prescribed by the macroscopic profile of momentum and temperature. Furthermore, in this state other degrees of freedom are subjected to thermal and mechanical fluctuations prescribed by the macroscopic profiles (mixed state). Mathematically, this state is more convenient. We should emphasize that the assumption is not crucial. Our result holds as long as we replace $(\sum _{x=1}^n \frac{m_x}{\beta _x})$ with any constant of order n. One can replace the pure state with any mixed state $\rho _o$ acting on ${\mathcal {H}}_o$, with , such that

and obtain the similar result.

Notice that the aforementioned constants are random and they have been defined for a realization of the masses. However, in the thermodynamic limit $\frac{\Pi _0}{n} \rightarrow \int _0^1 {\bar{p}}(y)dy,$ and $\frac{1}{n} \sum _{x=1}^n \frac{m_x}{\beta _x} \rightarrow \int _{0}^1 \frac{{\bar{m}}}{\beta (y)}dy,$ almost surely, thanks to law of large numbers, which further justifies our choice.

Finally, one can construct such easily, via an inverse Fourier transform of a Gaussian function.

We show our main transformations in terms of q coordinates: Rewrite the Hamiltonian in q coordinate as:

$$\begin{aligned} H_n=\frac{1}{2}(\langle p, M^{-1} p \rangle _n + \langle \nabla _+ q , \nabla _+ q \rangle _{n-1}))= \frac{1}{2}\langle p, M^{-1} p \rangle _n + \langle q,-\Delta q \rangle _n). \end{aligned}$$

The proper transformation of q in order to diagonalize the Hamiltonian is

$$\begin{aligned} {\hat{q}}_k= \langle \varphi ^k, M^{\frac{1}{2}}q \rangle _n = \sum _{x=1}^n \sqrt{m_x}\varphi ^k_x q_x, \end{aligned}$$

where ${\hat{p}}_k$ is defined as before. We can find the following relation between ${\hat{q}}_k$ and ${\hat{r}}_k$:

$$\begin{aligned} \begin{aligned} {\hat{r}}_k = \omega _k{\hat{q}}_k. \end{aligned} \end{aligned}$$

(B.9)

Canonical commutation relation (CCR) in terms of q reads:

$$\begin{aligned} {[}{\hat{q}}_k,{\hat{p}}_{k'}]=i\delta _{k,k'}, \qquad {[}{\hat{q}}_k,{\hat{q}}_{k'}]=[{\hat{p}}_k,{\hat{p}}_{k'}]=0. \end{aligned}$$

(B.10)

The inverse is given by

$$\begin{aligned} q=M^{-\frac{1}{2}}O{\hat{q}}, \qquad q_x= \frac{1}{\sqrt{m_x}}\sum _{k=0}^{n-1} \varphi _x^k {\hat{q}}_k, \end{aligned}$$

and the Hamiltonian reads

$$\begin{aligned} \begin{aligned} H_n = \frac{{\hat{p}}_0}{2}+\frac{1}{2}\sum _{k=1}^{n-1}( {\hat{p}}_k^2+\omega _k^2 {\hat{q}}_k^2), \end{aligned} \end{aligned}$$

(B.11)

The bosonic operators have the following form in terms of ${\hat{q}}$ coordinates:

$$\begin{aligned} {\hat{b}}_k=\frac{1}{\sqrt{2}}(\sqrt{\omega _k}{\hat{q}}_k+i\frac{1}{\sqrt{\omega _k}}{\hat{p}}_k), \quad {\hat{b}}_k^*= \frac{1}{\sqrt{2}} (\sqrt{\omega _k}{\hat{q}}_k-i\frac{1}{\sqrt{\omega _k}}{\hat{p}}_k). \end{aligned}$$

In order to expose the aforementioned link further, we introduce the coordinate $\tilde{{\mathfrak {q}}}_k$ similar to ${\hat{q}}_k$:

Define ${\bar{q}}_x$ as follows: First, construct ${\bar{q}}_x$, for $x \in {\mathbb {I}}_n$, from ${\bar{r}}_x$, by defining ${\bar{q}}_1={\bar{q}}_0 = c$ (c is an arbitrary constant, corresponding to the macroscopic position of the first particle) and let ${\bar{q}}_x= \sum _{y=1}^{x-1} {\bar{r}}_y + {\bar{q}}_1$. Then, we have ${\tilde{q}}_x= q_x-{\bar{q}}_x$, which gives us:

$$\begin{aligned} {\tilde{H}}_{\beta }^n=\frac{1}{2} (\langle {\tilde{p}},M_{\beta }^{-1} {\tilde{p}} \rangle _n + \langle \nabla _+ {\tilde{q}}, \beta ^o \nabla _+ {\tilde{q}} \rangle _{n-1} ) = \frac{1}{2} (\langle {\tilde{p}},M_{\beta }^{-1} {\tilde{p}} \rangle _n + \langle {\tilde{q}},-\nabla _- \beta ^o \nabla _+ {\tilde{q}} \rangle _n ). \end{aligned}$$

Therefore, $\tilde{{\mathfrak {q}}}_k$ is defined as:

$$\begin{aligned} \tilde{{\mathfrak {q}}}= O_{\beta }^{\dagger } M_{\beta }^{\frac{1}{2}}{\tilde{q}}, \qquad \tilde{{\mathfrak {q}}}_k=\langle \psi ^k,M_{\beta }^{\frac{1}{2}} {\tilde{q}} \rangle _n = \sum _{x=1}^{n} \sqrt{\frac{m_x}{\beta _x}} \psi ^k_x {\tilde{q}}_x. \end{aligned}$$

Moreover, it is illuminating to know the relation between $\tilde{{\mathfrak {r}}}_k$ and $\tilde{{\mathfrak {q}}}_k$:

$$\begin{aligned} \tilde{{\mathfrak {r}}}_k=\gamma _k \tilde{{\mathfrak {q}}}_k. \end{aligned}$$

The inverse relation for $\tilde{{\mathfrak {q}}}_k$ is given by:

$$\begin{aligned} {\tilde{q}}=M_{\beta }^{-\frac{1}{2}} O_{\beta }\tilde{{\mathfrak {q}}}, \qquad {\tilde{q}}_x= \sqrt{\frac{\beta _x}{m_x}}\sum _{k=0}^{n-1} \psi ^k_x\tilde{{\mathfrak {q}}}_k. \end{aligned}$$

Finally, the pseudo-Hamiltonian is diagonalized as follows:

$$\begin{aligned} \begin{aligned} {\tilde{H}}_{\beta }^n=\frac{\tilde{{\mathfrak {p}}}}{2}+\frac{1}{2}\sum _{k=1}^{n-1} (\tilde{{\mathfrak {p}}}_k^2+\gamma _k^2\tilde{{\mathfrak {q}}}_k^2), \end{aligned} \end{aligned}$$

(B.12)

Commutation relation is given by

$$\begin{aligned} {[}\tilde{{\mathfrak {q}}}_k,\tilde{{\mathfrak {q}}}_{k'}]=[\tilde{{\mathfrak {p}}}_k,\tilde{{\mathfrak {p}}}_{k'}]=0, \qquad [\tilde{{\mathfrak {q}}}_k,\tilde{{\mathfrak {p}}}_{k'}] =i\delta _{k,k'}, \quad \forall k \in {\mathbb {I}}_{n-1}^0, \end{aligned}$$

and the bosonic operators are given by:

$$\begin{aligned}&\tilde{{\mathfrak {b}}}_k=\frac{1}{\sqrt{2}}(\sqrt{\gamma _k} \tilde{{\mathfrak {q}}}_k +i \frac{1}{\sqrt{\gamma _k}}\tilde{{\mathfrak {p}}}_k), \quad \quad \tilde{{\mathfrak {b}}}_k^*=\frac{1}{\sqrt{2}}(\sqrt{\gamma _k} \tilde{{\mathfrak {q}}}_k -i \frac{1}{\sqrt{\gamma _k}}\tilde{{\mathfrak {p}}}_k). \end{aligned}$$

(B.13)

C Average of Bosonic Operators

Recall the average of bosonic operators (3.38) from Lemma 3.2. One can compute these averages and proof the lemma as follows (Notice that since $E_0$ is a constant, we can simply omit it from our computation and it does not change the desired averages) :

Proof

First we compute , using spectral theorem, we expand the trace in the basis of eigenvalues of $H_{\beta }^n$ i.e. . Then, the following computation gives us the result: (we abbreviate $\sum _{k=1}^{n-1} \theta _k \gamma _k$ by ${\bar{\theta }}.{\bar{\gamma }}$, where ${\bar{\gamma }}$ stands for the vector ${\bar{\gamma }}:=(\gamma _1,\dots ,\gamma _{n-1})$)²⁶

(C.1)

Now from (3.32), (3.37), observe that

Therefore, since are orthonormal, we have

Moreover, if $k\ne k'$, we have

Since are orthonormal, if we expand in the basis of $H_{\beta }^n$, we deduce the first set of equalities in (3.38). Furthermore, for $k \ne k'$, we deduce as well.

On the other hand, if $k=k'$, the same relations (5.21), (3.32)imply:

thus, we have . Consequently, we can compute :

(C.2)

Since we have the commutator relation $[\tilde{{\mathfrak {b}}}_k ,\tilde{{\mathfrak {b}}}_k^*]=1$, we obtain the last equality: . $\square$