Harmonic Chain with Velocity Flips: Thermalization and Kinetic Theory

Abstract

We consider the detailed structure of correlations in harmonic chains with pinning and a bulk velocity flip noise during the heat relaxation phase which occurs on diffusive time scales, for $t=O(L^2)$ where L is the chain length. It has been shown earlier that for non-degenerate harmonic interactions these systems thermalize, and the dominant part of the correlations is given by local thermal equilibrium determined by a temperature profile which satisfies a linear heat equation. Here we are concerned with two new aspects about the thermalization process: the first order corrections in 1 / L to the local equilibrium correlations and the applicability of kinetic theory to study the relaxation process. Employing previously derived explicit uniform estimates for the temperature profile, we first derive an explicit form for the first order corrections to the particle position-momentum correlations. By suitably revising the definition of the Wigner transform and the kinetic scaling limit we derive a phonon Boltzmann equation whose predictions agree with the explicit computation. Comparing the two results, the corrections can be understood as arising from two different sources: a current-related term and a correction to the position-position correlations related to spatial changes in the phonon eigenbasis.

Appendices

Appendix 1: Computation of q(k) in (3.11)

In this section we show the explicit computation which proves (3.10). We look at (3.9), i.e.

$$\begin{aligned} q(k)=2 \,\gamma \int _0^{\infty } \mathrm{d}t \, \widehat{A}_t(k)^{22} \partial _k \widehat{A}_t(k)^{21}. \end{aligned}$$

(6.1)

Here

$$\begin{aligned}&\widehat{A}_t(k)^{22}=\frac{\mathrm{e}^{-\frac{\gamma \,t}{2}}}{\Omega }\left( -\frac{\gamma }{2}\sinh \Omega \,t +\Omega \cosh \Omega \, t \right) ,\\&\displaystyle \partial _k \widehat{A}_t(k)^{21}= \frac{\mathrm{e}^{-\frac{\gamma \,t}{2}}\Omega '}{\Omega }\left( t\,\cosh \Omega \, t -\frac{\sinh \Omega \,t}{\Omega } \right) . \end{aligned}$$

Therefore the integrand in (6.1), i.e., $2 \,\gamma \, \widehat{A}_t(k)^{22} \partial _k \widehat{A}_t(k)^{21}$ reads

$$\begin{aligned}&2 \,\gamma \frac{\mathrm{e}^{-\gamma t}\Omega '}{\Omega ^2}\left( -\frac{\gamma \, t}{2}\sinh \Omega \, t\, \cosh \Omega \, t +\frac{\gamma }{2\,\Omega } \sinh ^2 \Omega \,t +\Omega \,t \cosh ^2 \Omega \,t-\cosh \Omega \,t\,\sinh \Omega \,t\right) \\&\quad =2 \,\gamma \frac{\Omega '}{\Omega ^2}\left[ -\frac{\gamma \, t}{8}\left( \mathrm{e}^{-t(\gamma -2\,\Omega )}-\mathrm{e}^{-t(\gamma +2\,\Omega )}\right) +\frac{\gamma }{8\,\Omega }\left( \mathrm{e}^{-t(\gamma -2\,\Omega )}+\mathrm{e}^{-t(\gamma +2\,\Omega )}-2\mathrm{e}^{-t\,\gamma }\right) \right] \\&\qquad +2 \,\gamma \frac{\Omega '}{\Omega ^2}\left[ \frac{\Omega \, t}{4}\left( \mathrm{e}^{-t(\gamma -2\,\Omega )}+\mathrm{e}^{-t(\gamma +2\,\Omega )}+2\mathrm{e}^{-t\,\gamma }\right) -\frac{1}{4}\left( \mathrm{e}^{-t(\gamma -2\,\Omega )}-\mathrm{e}^{-t(\gamma +2\,\Omega )}\right) \right] . \end{aligned}$$

where the “prime” denotes the derivative with respect to k. Once we integrate with respect to the time variable, we obtain

$$\begin{aligned} q(k)&=2 \,\gamma \frac{\Omega '}{\Omega ^2}\left[ -\frac{\gamma }{8}\left( \frac{1}{(\gamma -2\,\Omega )^2}-\frac{1}{(\gamma +2\,\Omega )^2}\right) +\frac{\gamma }{8\Omega }\left( \frac{1}{(\gamma -2\,\Omega )}+\frac{1}{(\gamma +2\,\Omega )}-\frac{2}{\gamma }\right) \right] \\ {}&\quad +2 \,\gamma \frac{\Omega '}{\Omega ^2}\left[ \frac{\Omega }{4}\left( \frac{1}{(\gamma -2\,\Omega )^2}+\frac{1}{(\gamma +2\,\Omega )^2}+\frac{2}{\gamma ^2}\right) -\frac{1}{4}\left( \frac{1}{(\gamma -2\,\Omega )}-\frac{1}{(\gamma +2\,\Omega )}\right) \right] \\ {}&=2 \,\gamma \frac{\Omega '}{\Omega ^2}\left[ -\frac{\gamma ^2\Omega }{(\gamma ^2-4\,\Omega ^2)^2}+ \frac{\Omega }{\gamma ^2}\left( \frac{\gamma ^4-2\,\Omega ^2\gamma ^2+8\,\Omega ^4}{(\gamma ^2-4\,\Omega ^2)^2}\right) \right] \\&=-\frac{4 \,\Omega ' \Omega }{\gamma \,(\gamma ^2-4\,\Omega ^2)}. \end{aligned}$$

Since $\Omega = (\gamma /2) \sqrt{1 - (2 \omega (k)/\gamma )^2}$ it results that $\Omega ' =-\frac{\omega \,\omega '}{\Omega }.$ By inserting the explicit expression for $\Omega '$ in the previous computation we get

$$\begin{aligned} q(k)= \frac{\partial _k \omega (k)}{\gamma \,\omega (k)}. \end{aligned}$$

Appendix 2: Basic Properties of Lattice Averaging Kernels

In Sect. 4.1, we referred to “lattice averaging kernels” which were understood as convolution sums constructed using the kernel functions

$$\begin{aligned} \varphi (\xi )= \frac{1}{R^d} \sum _{n \in {\mathbb Z}^d} \phi \!\left( \frac{\xi - L n}{R}\right) , \qquad \xi \in {\mathbb R}^d, \end{aligned}$$

(7.1)

for some given $L,R>0$. These kernels are determined via the function $\phi :{\mathbb R}^d\rightarrow {\mathbb R}$ which we assume to satisfy all of the following conditions

• (1) $\phi$ is a Schwartz test function, i.e., $\phi \in \mathscr {S}({\mathbb R}^d)$.

• (2) $\widehat{\phi }$ has a compact support. Let $\rho _\phi >0$ be such that $\widehat{\phi }(p)=0$ whenever $|p|_\infty \ge \rho _\phi$.

• (3) $\phi \ge 0$.

• (4) $\int \!\mathrm{d}y \, \phi (y)=1$.

Since this construction could become useful in phonon models in higher dimensions, we write the results below for arbitrary $d\ge 1$, keeping in mind that in the text they are applied with $d=1$. The main difference comes from the fact that for $d>1$, the max-norm $|y|_\infty := \max _{1\le k\le d} |y_k|$ and the Euclidean norm $|y| := (y_1^2+y_2^2+\cdots + y_d^2)^{1/2}$ no longer give the same numbers. We will mainly need the max-norm for the present lattice systems.

Let us show next that these assumptions guarantee the following properties for $\varphi$:

• (1) (positivity) $\varphi \ge 0$.

• (2) (L-periodicity) $\varphi (\xi +L m)=\varphi (\xi )$ for all $\xi \in {\mathbb R}^d$, $m\in {\mathbb Z}^d$.

• (3) (continuum normalization) $\int _{|\xi |_\infty \le L/2} \!\mathrm{d}\xi \, \varphi (\xi -\xi _0)=1$ for all $\xi _0\in {\mathbb R}^d$.

• (4) (slow variation) To every multi-index $\alpha$ there is a constant $C_\alpha$, which is independent of R and L, such that

  $$\begin{aligned} \left| \partial _\xi ^\alpha \varphi (\xi )\right| \le R^{-|\alpha |} C_\alpha , \quad \text { for all }\xi \in {\mathbb R}^d. \end{aligned}$$

  (7.2)

• (5) (lattice normalization) If $R\ge \rho _\phi$, we have $\sum _{x\in \Lambda _L}\varphi (\xi +x)=1$ for all $\xi \in {\mathbb R}^d$.

• (6) (discrete Fourier transform) If $R\ge 2 \rho _\phi$, we have for all $k\in \Lambda _L^*$, $\xi \in {\mathbb R}^d$

  $$\begin{aligned} \sum _{x\in \Lambda _L} \varphi (\xi -x) \mathrm{e}^{-\mathrm{i}2\pi x\cdot k} = \mathrm{e}^{-\mathrm{i}2\pi \xi \cdot k} \widehat{\phi }(-Rk). \end{aligned}$$

  (7.3)

Hence, the constant L determines the periodicity of the kernel and R the scale of variation, in the sense that each derivative of $\varphi$ will decrease the magnitude by $R^{-1}$.

The items 1 and 2 are obvious consequences of the definition of $\varphi$ and the assumptions on $\phi$. Item 3 is derived by rewriting the sum over integrals as a single integral as follows:

$$\begin{aligned} \int _{|\xi |_\infty \le L} \!\mathrm{d}\xi \, \varphi (\xi -\xi _0)&= \frac{1}{R^d} \sum _{n \in {\mathbb Z}^d} \int _{|\xi |_\infty \le L/2} \!\mathrm{d}\xi \, \phi \!\left( \frac{\xi + L n-\xi _0}{R}\right) \nonumber \\&= \frac{1}{R^d} \int _{{\mathbb R}^d} \!\mathrm{d}y\, \phi \!\left( \frac{y-\xi _0}{R}\right) = 1. \end{aligned}$$

(7.4)

Item 4 follows by taking the derivative inside the sum over n, and then noticing that the result can be bounded by $R^{-|\alpha |}$ times a Riemann sum approximation of the integral $\int \! \mathrm{d}y\, |\partial ^\alpha \phi (y)|$ which is finite since $\phi$ is a Schwartz function.

The lattice normalization condition and Fourier transform in items 5 and 6 need slightly more effort. Applying the definitions of $\varphi$ and of the finite lattice $\Lambda _L$, we obtain for any $k\in \Lambda _L^*$, $\xi \in {\mathbb R}^d$:

$$\begin{aligned} \sum _{x\in \Lambda _L} \varphi (\xi -x) \mathrm{e}^{-\mathrm{i}2\pi x\cdot k} = \sum _{x\in \Lambda _L} \frac{1}{R^d} \sum _{n \in {\mathbb Z}^d} \phi \!\left( \frac{\xi -x - L n}{R}\right) \mathrm{e}^{-\mathrm{i}2\pi (x+L n)\cdot k} = \frac{1}{R^d} \sum _{m \in {\mathbb Z}^d} f(m) \end{aligned}$$

(7.5)

where $f(y):=\phi \!\left( \frac{\xi -y}{R}\right) \mathrm{e}^{-\mathrm{i}2\pi y\cdot k}$ is a Schwartz function. The Fourier transform of f is given by $\widehat{f}(p)=R^d \mathrm{e}^{-\mathrm{i}2\pi \xi \cdot (p+k)} \widehat{\phi }(-R(p+k))$. Therefore, by the Poisson summation formula,

$$\begin{aligned} \sum _{x\in \Lambda _L} \varphi (\xi -x) \mathrm{e}^{-\mathrm{i}2\pi x\cdot k} = \frac{1}{R^d} \sum _{m \in {\mathbb Z}^d} \widehat{f}(m) = \sum _{m \in {\mathbb Z}^d}\mathrm{e}^{-\mathrm{i}2\pi \xi \cdot (m+k)} \widehat{\phi }(-R(m+k)). \end{aligned}$$

(7.6)

If $m\ne 0$, we have $|m+k|_\infty \ge |m|_\infty -|k|_\infty \ge \frac{1}{2}$, and thus $|-R(m+k)|_\infty \ge R/2$. Hence, if $R\ge 2 \rho _\phi$, or $k=0$ and $R\ge \rho _\phi$, all these points lie outside the support of $\widehat{\phi }$, and thus only the “$m=0$” term may contribute to the sum. This yields

$$\begin{aligned} \sum _{x\in \Lambda _L} \varphi (\xi -x) \mathrm{e}^{-\mathrm{i}2\pi x\cdot k} = \mathrm{e}^{-\mathrm{i}2\pi \xi \cdot k} \widehat{\phi }(-Rk). \end{aligned}$$

(7.7)

In particular, if $k=0$, we have $\widehat{\phi }(-Rk)= \widehat{\phi }(0)=\int \!\mathrm{d}y \, \phi (y)=1$, and we obtain $\sum _{x\in \Lambda _L} \varphi (\xi -x)=1$. This completes the proof of both item 5 and item 6.

Appendix 3: Quasi-Stationary Inhomogeneous Solutions

Here we want to show that $\mathcal {P}_t = \mathcal {Q}_t = O(R^{-2})$, $\mathcal {I}_t =O(R^{-1})$ and $\mathcal {H}_t = E_t + O(R^{-2})$ as anticipated in Sect. 4.5. Using the definitions (4.21) including the $\xi$-dependence, as well as the antisymmetry $v(-k)=-v(k)$, from (4.20) we deduce

$$\begin{aligned} \partial _t \mathcal {H}_t(\xi ,k)&= -v(k)\nabla _{\xi } \mathcal {I}_t(k,\xi ) + \bar{\mathscr {C}}[\mathcal {H}_t-\mathcal {P}_t](\xi ,k) + O(R^{-2}) \end{aligned}$$

(8.1)

$$\begin{aligned} \partial _t \mathcal {I}_t(\xi ,k)&= -v(k)\nabla _{\xi } \mathcal {H}_t(k,\xi )-\gamma \mathcal {I}_t(\xi ,k) + O(R^{-2}) \end{aligned}$$

(8.2)

$$\begin{aligned} \partial _t \begin{pmatrix} \mathcal {P}_t(\xi ,k) \\ \mathcal {Q}_t(\xi ,k) \end{pmatrix}&= \mathcal {L}_{v}\begin{pmatrix} \mathcal {P}_t(\xi ,k) \\ \mathcal {Q}_t(\xi ,k) \end{pmatrix} + \begin{pmatrix} \gamma \int _{\Lambda _L^* } \mathrm{d}q \mathcal {P}_t(q) - \bar{\mathscr {C}}[\mathcal {H}_t](\xi ,k) \\ 0 \end{pmatrix} \end{aligned}$$

(8.3)

where

$$\begin{aligned} \mathcal {L}_{v} = \begin{pmatrix} -\gamma &{} 2 \mathrm{i}\omega (k)-v(k)\nabla _{\xi } \\ 2 \mathrm{i}\omega (k)-v(k)\nabla _{\xi } &{} -\gamma \end{pmatrix}. \end{aligned}$$

Recall that $\mathcal {H}_t, \mathcal {I}_t, \mathcal {P}_t$ and $\mathcal {Q}_t$ are L-periodic in $\xi$. To solve (8.1), (8.2) and (8.3) we look at the Fourier coefficients of those observables:

$$\begin{aligned} \partial _t \widehat{\mathcal {H}}_t(n,k)&= -2 \mathrm{i}\pi n L^{-1}v(k) \widehat{ \mathcal {I}}_t(n,k) + \bar{\mathscr {C}}[\widehat{\mathcal {H}}_t-\widehat{\mathcal {P}}_t](n,k) + O(R^{-2}) \end{aligned}$$

(8.4)

$$\begin{aligned} \partial _t \widehat{\mathcal {I}}_t(n,k)&= -2 \mathrm{i}\pi n L^{-1}v(k) \widehat{ \mathcal {H}}_t(n, k)-\gamma \widehat{\mathcal {I}}_t(n,k) + O(R^{-2}) \end{aligned}$$

(8.5)

$$\begin{aligned} \partial _t \begin{pmatrix} \widehat{\mathcal {P}}_t(n,k) \\ \widehat{\mathcal {Q}}_t(n,k) \end{pmatrix}&= \widehat{\mathcal {L}}_{v}\begin{pmatrix} \widehat{\mathcal {P}}_t(n,k) \\ \widehat{\mathcal {Q}}_t(n,k) \end{pmatrix} + \begin{pmatrix} \gamma \int _{\Lambda _L^* } \mathrm{d}q \widehat{\mathcal {P}}_t(n,q) - \bar{\mathscr {C}}[\widehat{\mathcal {H}}_t](n,k) \\ 0 \end{pmatrix}, \end{aligned}$$

(8.6)

where

$$\begin{aligned} \widehat{\mathcal {L}}_{v} = \begin{pmatrix} -\gamma &{} 2 \mathrm{i}( \omega (k)-\pi n L^{-1}v(k)) \\ 2 \mathrm{i}( \omega (k)-\pi nL^{-1} v(k)) &{} -\gamma \end{pmatrix} \end{aligned}$$

and $\widehat{\mathcal {H}}_t(n,k) = L^{-1}\int _{0}^L \mathrm{d}\xi \, \mathrm{e}^{-2 \pi \mathrm{i}L^{-1} n \cdot \xi } \mathcal {H}_t(\xi ,k)$ with $n \in {\mathbb Z}$ and analogously for $\widehat{\mathcal {I}}_t,\widehat{\mathcal {P}}_t$ and $\widehat{\mathcal {Q}}_t$.

Assuming that the time derivative yields a contribution order $O(R^{-2})$ we have

$$\begin{aligned} \widehat{\mathcal {I}}_t(n,k)&= -\frac{2 \mathrm{i}\pi n L^{-1} v(k)}{\gamma } \widehat{ \mathcal {H}}_t(n, k) + O(R^{-2}), \end{aligned}$$

(8.7)

which implies that (8.4) becomes

$$\begin{aligned} \bar{\mathscr {C}}[\widehat{\mathcal {H}}_t](n,k)&= \bar{\mathscr {C}}[\widehat{\mathcal {P}}_t](n,k) + \gamma ^{-1} (2\pi n L^{-1} v(k))^2 \widehat{ \mathcal {H}}_t(n,k) + O(R^{-2}). \end{aligned}$$

(8.8)

Moreover, for (8.6) we have

$$\begin{aligned} \begin{pmatrix} \widehat{\mathcal {P}}_t(n,k) \\ \widehat{\mathcal {Q}}_t(n,k) \end{pmatrix}&= -\widehat{\mathcal {L}}_{v}^{-1} \begin{pmatrix} \gamma \int _{\Lambda _L^* } \mathrm{d}q \widehat{\mathcal {P}}_t(n,q) - \bar{\mathscr {C}}[\widehat{\mathcal {H}}_t](n,k) \\ 0 \end{pmatrix} + O(R^{-2}), \end{aligned}$$

(8.9)

where

$$\begin{aligned} \widehat{\mathcal {L}}_{v}^{-1} =- \frac{1}{\gamma ^2 + 4(\omega (k)-\pi n L^{-1} v(k))^2} \begin{pmatrix} \gamma &{} 2 \mathrm{i}(\omega (k)-\pi n L^{-1} v(k)) \\ 2 \mathrm{i}(\omega (k)-\pi n L^{-1} v(k)) &{} \gamma \end{pmatrix}. \end{aligned}$$

Combining (8.8) and (8.9) we get

$$\begin{aligned} \widehat{\mathcal {P}}_t(n,k) = -\frac{( \pi n L^{-1} v(k))^2}{(\omega (k)-\pi nL^{-1} v(k))^2} \widehat{\mathcal {H}}_t(n,k). \end{aligned}$$

(8.10)

By the definition of the test function $\varphi$ given in (4.4), $\widehat{\mathcal {H}}_t(n,k)$ is concentrated on values of n such that n / L is of order $O(R^{-1})$. In fact, from the definition of $\mathcal {H}_t(\xi , k)$ we get the explicit form of $\widehat{\mathcal {H}}_t(n,k)$:

$$\begin{aligned} \widehat{\mathcal {H}}_t(n,k) = \frac{1}{L} \int _0^L \mathrm{d}\xi \, \mathrm{e}^{-2 \pi \mathrm{i}L^{-1}n \cdot \xi } \sum _{x \in \Lambda _L} \varphi (\xi -x) V_t(x,k) \end{aligned}$$

(8.11)

where

$$\begin{aligned} V_t(x,k)= \sum _{y \in \Lambda _L} \mathrm{e}^{-2 \pi \mathrm{i}y \cdot k} {\mathbb E}[\psi _t(x,-1) \psi _t(x+y, +1)+\psi _t(x,+1) \psi _t(x+y, -1)]. \end{aligned}$$

Thanks to (7.7), (8.11) becomes

$$\begin{aligned} \widehat{\mathcal {H}}_t(n,k) =&\frac{1}{L} \int _{\Lambda _L^*} \mathrm{d}k' \, \widehat{V}_t(k',k) \widehat{\phi }(R k') \int _0^L \mathrm{d}\xi \, \mathrm{e}^{-2 \pi \mathrm{i}\xi \cdot (n L^{-1} -k')} \nonumber \\ =&\widehat{V}_t(n L^{-1},k) \widehat{\phi }(R n L^{-1}) \end{aligned}$$

(8.12)

where $\widehat{V}_t(k',k):= \sum _{x \in \Lambda _L^*} \mathrm{e}^{-2 \pi \mathrm{i}x \cdot k'} V_t(x,k)$ and we used the fact that

$$\begin{aligned} \frac{1}{L} \int _0^L \mathrm{d}\xi \, \mathrm{e}^{-2 \pi \xi \cdot (n L^{-1} -k')} = \mathbb {1}(k' = nL^{-1})\, \ \ \text{ for } \text{ any } k' \in \Lambda _L^*. \end{aligned}$$

Therefore, $nL^{-1} \in \Lambda _L^*$ and, since $\widehat{\phi }$ has compact support [see assumption (2) in Appendix 2], we get that $\widehat{\phi }(RnL^{-1})=0$ whenever $|nL^{-1}| \le \rho _{\phi } R^{-1}$, from which the claim follows.

The fact that $\widehat{ \mathcal {H}}_t(n,k)$ vanishes for $|nL^{-1}| \ge O(R^{-1})$ indicates that ${\mathcal {P}}_t(\xi ,k)=O(R^{-2})$. Then clearly ${\mathcal {Q}}_t(\xi ,k)=O(R^{-2})$ and $\bar{\mathscr {C}}[{\mathcal {H}}_t](\xi ,k)=O(R^{-2})$, thus implying also $\mathcal {H}_t = E_t + O(R^{-2})$.