A Brownian Particle in a Microscopic Periodic Potential

Abstract

We study a model for a massive test particle in a microscopic periodic potential and interacting with a reservoir of light particles. In the regime considered, the fluctuations in the test particle’s momentum resulting from collisions typically outweigh the shifts in momentum generated by the periodic force, so the force is effectively a perturbative contribution. The mathematical starting point is an idealized reduced dynamics for the test particle given by a linear Boltzmann equation. In the limit that the mass ratio of a single reservoir particle to the test particle tends to zero, we show that there is convergence to the Ornstein–Uhlenbeck process under the standard normalizations for the test particle variables. Our analysis is primarily directed towards bounding the perturbative effect of the periodic potential on the particle’s momentum.

Appendix: Exponential Ergodicity

In this section we prove the relaxation of our dynamics to an equilibrium state. Our dynamics is driven by the forward Kolmogorov equation

$$\begin{aligned} \frac{d}{dt}{\Psi }_{t,\lambda }(x,\,p)&= (\mathcal {L}_{\lambda }^{*}{\Psi }_{t,\lambda })(x,\,p)= -p \frac{\partial }{\partial x}{\Psi }_{t,\lambda }(x,p)+\frac{dV}{dx}\big (x\big )\frac{\partial }{\partial p}{\Psi }_{t,\lambda }(x,p) \nonumber \\&+\int \limits _{{\mathbb R}}dp^{\prime }\big ({\mathcal {J}}_{\lambda }(p^{\prime },p){\Psi }_{t,\lambda }(x,p^{\prime })-{\mathcal {J}}_{\lambda }(p,p^{\prime }){\Psi }_{t,\lambda }(x,p) \big ), \end{aligned}$$

(8.1)

which has equilibrium state $\Psi _{\infty ,\lambda }$. The Kolmogorov equation (8.1) determines a transition semigroup $\Phi _{t,\lambda }$, which we take to operate on measures from the right and bounded measurable functions from the left:

$$\begin{aligned} (\mu \,\Phi _{t,\lambda })(ds)=\int \limits _{\Sigma }\mu (ds')\Phi _{t,\lambda }(s',ds)\quad \text {and}\quad (\Phi _{t,\lambda }\,g)(s)=\int \limits _{\Sigma }\Phi _{t,\lambda }(s,ds')g(s'), \end{aligned}$$

for $\mu \in M(\Sigma )$ and $g\in B(\Sigma )$. We identify probability densities with their corresponding measures and denote the total variation norm for finite measures on $\Sigma$ by $\Vert \cdot \Vert _{1}$. The exponential ergodicity that we prove in Theorem 8.1 is not used critically anywhere in our proofs although we need some degree of ergodicity in order to make sense of certain expressions such as the reduced resolvent $\mathfrak {R}_{\lambda }$ of the function $\frac{dV}{dx}$, for instance. A more thorough study of the ergodicity would give some control of the the exponential rate $\alpha (\lambda )$ appearing in Theorem 8.1 for $\lambda \ll 1$. We believe that $\alpha (\lambda )$ can be taken to scale as $\propto \frac{\lambda }{\log (\lambda ^{-1})}$ for small $\lambda$ (i.e. a bit slower than linear in $\lambda$), and if $C$ is replaced on the right side of (8.2) by $C\Vert \Psi \Vert _{w}$ for the weighted norm $\Vert \Psi \Vert _{w}=\int _{\Sigma }|\Psi |(dx\,dp)(1+\lambda ^{\frac{1}{2}}|p|)\,$, then the exponential rate scales as $\propto \lambda$ for $\lambda \ll 1$.

We denote the space of probability measures on $\Sigma$ by $M_{+}^{1}(\Sigma )$.

Theorem 8.1

Let $\Phi _{t,\lambda }$ be the transition semigroup corresponding to the Kolmogorov equation (8.1). There exist $\alpha (\lambda ), C>0$ such that for all $\Psi \in M_{+}^{1}(\Sigma )$ and $t\in {\mathbb R}^+$,

$$\begin{aligned} \big \Vert \Psi \Phi _{t,\lambda }-\Psi _{\infty ,\lambda }\big \Vert _{1}\le Ce^{-t\alpha (\lambda )}. \end{aligned}$$

(8.2)

Proof

Since the parameter $\lambda >0$ is fixed in this proof, we will not attach it as a subscript or superscript to mathematical objects that depend upon it as we did in earlier sections. It is sufficient to work with the resolvent chain rather than the original process and show that there are $C,\alpha >0$

$$\begin{aligned} Ce^{-n\alpha }&\ge \big \Vert \Psi \mathcal {T}^{n}-\Psi _{\infty } \big \Vert _{1} \nonumber \\&= \sup _{\Vert g\Vert _{\infty }\le 1} \Big |\int \limits _{\Sigma }(\Psi \mathcal {T}^{n})(ds)g(s) -\Psi _{\infty }(g) \Big |, \end{aligned}$$

(8.3)

where $\mathcal {T}:B(\Sigma )\rightarrow B(\Sigma )$ is the transition operator for the resolvent chain and has the form

$$\begin{aligned} \mathcal {T}=\int \limits _{0}^{\infty }dre^{-r}\Phi _{r}. \end{aligned}$$

Let $\mu _{L}\ge 0$ be defined as the Lebesgue measure of the region $\{ H(s)\le L\}\subset \Sigma$ for $L>0$. There exist $L>0$ and $0<\epsilon <1$ such that the following statements hold:

• (i). For all $s,s'\in \Sigma$ with $H(s),H(s')\le L$,

  $$\begin{aligned} \mathcal {T}(s,ds')>\epsilon \,\frac{ ds'}{ \mu _{L}}. \end{aligned}$$

• (ii). For all $s\in \Sigma$ with $H(s)> L$,

  $$\begin{aligned} \int \limits _{H(s')\le L}\mathcal {T}(s,ds') \ge \epsilon . \end{aligned}$$

Given any $L>0$, we can find an $\epsilon$ such that statement (i) is true by the argument in the proof of Part (1) of Proposition 2.3 (in which the cutoff was $l=1+2\sup _{x}V(x)$ rather than an arbitrary fixed $L$, but this does not change the argument). Statement (ii) follows from the extreme momentum-contractive nature of the jump rates:

$$\begin{aligned} \mathcal {J}(p',p)=\frac{1+\lambda }{64}|p'-p|e^{-\frac{1}{2}(\frac{1-\lambda }{2}p'-\frac{1+\lambda }{2}p)^{2}}. \end{aligned}$$

A test particle with momentum $|p'|\gg 1$ will suffer a collision after a waiting time on the order of $\frac{1}{|p'|}$, and the resulting momentum of the test particle will be concentrated around the contracted value $p'\frac{1-\lambda }{1+\lambda }$. Consequently, the test particle descends from arbitrarily high energy to the low energy region $\{ H(s)\le L\}\subset \Sigma$ in finite time. The statements (i) and (ii) imply that the probability of jumping into the region $\{ s \in \Sigma \,\big |\,H(s)\le L \}$ is at least $\epsilon >0$ from any starting point. The statements (i) and (ii) and the fact that $\Psi _{\infty ,\lambda }$ is a stationary state for the dynamics are sufficient to prove the exponential ergodicity without further reference to the dynamics at hand. The proof of (ii) is below, and we proceed next with the remainder of the proof.

By (i) we have the minorization condition $\mathcal {T}(s,ds')\ge h(s)\nu (ds')$ for

$$\begin{aligned} h(s)= \epsilon \, \chi \big (H(s)\le L \big ) \quad \text {and} \quad \nu (ds)= \frac{ds}{ \mu _{L}}\chi \big (H(s)\le L \big ). \end{aligned}$$

We can thus apply standard Nummelin splitting [28] to define a chain $\tilde{\sigma }_{n}\in \tilde{\Sigma }=\Sigma \times \{0,1\}$ with an atom set $\Sigma \times 1$ using the pair $h, \nu$ above. Let $\tilde{n}_{m}$, $m\ge 1$ be the sequence of return times to the atom. The times $\tilde{n}_{m}$ form a delayed renewal chain in which the delay distribution is $\tilde{\mathbb {P}}_{\tilde{\Psi }}[\tilde{n}_{1}= n]$ and the jumps have distribution $\tilde{\mathbb {P}}_{\tilde{\nu }}[\tilde{n}_{1}= n]$; recall from (2.2) that $\tilde{\mu }$ refers to the splitting of a measure $\mu$ on $\Sigma$.

We denote the first component of the chain $\tilde{\sigma }_{n}\in \tilde{\Sigma }$ by $\sigma _{n}$. We will treat the difference between $\Psi \mathcal {T}^{n}$ and $\Psi _{\infty }$ in the norm $\Vert \cdot \Vert _{1}$ through the formula (8.3). For $g\in B(\Sigma )$,

$$\begin{aligned}&\int \limits _{\Sigma }(\Psi \mathcal {T}^{n})(ds)g(s) =\mathbb {E}_{\Psi }\big [g(\sigma _{n}) \big ]=\tilde{\mathbb {E}}_{\tilde{\Psi }}\big [g(\sigma _{n})\big ]\nonumber \\&\quad = \tilde{\mathbb {E}}_{\tilde{\Psi }}\big [g(\sigma _{n})\chi (\tilde{n}_{1}\ge n) \big ]+ \tilde{\mathbb {E}}_{\tilde{\Psi }}\Big [\sum _{m=1}^{n}\sum _{r=1}^{\infty } \chi (m-1=\tilde{n}_{r}) g(\sigma _{n})\chi (\tilde{n}_{r+1}\ge n) \Big ] \nonumber \\&\quad = \tilde{\mathbb {E}}_{\tilde{\Psi }}\big [g(\sigma _{n})\chi (\tilde{n}_{1}\ge n) \big ]+\sum _{m=1}^{n} F_{\Psi }(m-1)\, \tilde{\mathbb {E}}_{\tilde{\nu }}\big [g(\sigma _{n-m})\chi (\tilde{n}_{1}\ge n-m) \big ], \end{aligned}$$

(8.4)

where $F_{\Psi }:\mathbb {N}\rightarrow {\mathbb R}^{+}$ is the renewal function

$$\begin{aligned} F_{\Psi }(m)= \sum _{r=1}^{\infty }\tilde{\mathbb {P}}_{\tilde{\Psi }}\big [\tilde{n}_{r}=m\big ], \end{aligned}$$

and the last equality in (8.4) uses the strong Markov property. We will delay the demonstration of the following two statements until the end of the proof.

• (I). There is a $c>0$ such that for all $m\in \mathbb {N}$ and probability measures $\Psi \in M_{+}^{1}(\Sigma )$,

  $$\begin{aligned} \tilde{\mathbb {P}}_{\tilde{\Psi }}\big [\tilde{n}_{1}\ge m \big ] \le c e^{-\epsilon \,m}. \end{aligned}$$

• (II). There is a $c>0$ such that for all $m\in \mathbb {N}$ and probability measures $\Psi \in M_{+}^{1}(\Sigma )$,

  $$\begin{aligned} \Vert F_{\Psi }(m)- \Psi _{\infty }(h) \Vert _{\infty }\le ce^{-\epsilon \,m}. \end{aligned}$$

With (8.4) we can write the difference between $\int _{\Sigma }(\Psi \mathcal {T}^{n})(ds)g(s)$ and $\Psi _{\infty }(g)$ as

$$\begin{aligned}&\int \limits _{\Sigma }(\Psi \mathcal {T}^{n})(ds)g(s)-\Psi _{\infty }(g)= \tilde{\mathbb {E}}_{\tilde{\Psi }}\big [g(\sigma _{n})\chi (\tilde{n}_{1}\ge n) \big ]\nonumber \\&\quad +\sum _{m=1}^{\lfloor \frac{n}{2} \rfloor }F_{\Psi }(m-1)\, \tilde{\mathbb {E}}_{\tilde{\nu }}\big [g(\sigma _{n-m})\chi (\tilde{n}_{1}\ge n-m) \big ]\nonumber \\&\quad + \sum _{m=\lfloor \frac{n}{2} \rfloor +1}^{n}\big (F_{\Psi }(m-1)-\Psi _{\infty ,\lambda }(h) \big )\, \tilde{\mathbb {E}}_{\tilde{\nu }}\big [g(\sigma _{n-m})\chi (\tilde{n}_{1}\ge n-m) \big ]\nonumber \\&\quad +\Psi _{\infty }(h) \Big (\sum _{m=\lfloor \frac{n}{2} \rfloor +1}^{n}\tilde{\mathbb {E}}_{\tilde{\nu }}\big [g(\sigma _{n-m})\chi (\tilde{n}_{1}\ge n-m) \big ] -\frac{ \Psi _{\infty }(g) }{ \Psi _{\infty }(h)} \Big ). \end{aligned}$$

(8.5)

Notice that the bottom line of (8.5) can be rewritten as $-\Psi _{\infty }(h) \sum _{m=\lfloor \frac{n}{2} \rfloor }^{\infty }\tilde{\mathbb {E}}_{\tilde{\nu }}\big [g(\sigma _{m})\chi (\tilde{n}_{1}\ge m) \big ]$ since by Part (2) of Proposition 2.4 we have the second equality below:

$$\begin{aligned} \sum _{m=0}^{\infty } \tilde{\mathbb {E}}_{\tilde{\nu }}\big [g(\sigma _{m})\chi (\tilde{n}_{1}\ge m) \big ] = \tilde{\mathbb {E}}_{\tilde{\nu }}\Big [\sum _{m=0}^{\tilde{n}_{1}}g(\sigma _{m}) \Big ]=\frac{ \Psi _{\infty }(g) }{ \Psi _{\infty }(h)}. \end{aligned}$$

(8.6)

By applying the triangle inequality to (8.5), statements (I) and (II), and that $F_{\Psi }^{(\lambda )}(m)\le 1$, we have the inequality

$$\begin{aligned} \Big |\int \limits _{\Sigma }(\Psi \mathcal {T}^{n})(ds)g(s)-\Psi _{\infty }(g)\Big |\le c\Vert g\Vert _{\infty } \Big ((1-\epsilon )^{n}+\frac{e^{-\epsilon \lfloor \frac{n}{2} \rfloor }}{\epsilon }+ \frac{e^{-\epsilon \lfloor \frac{n}{2} \rfloor }}{\epsilon }+\frac{e^{-\epsilon \lfloor \frac{n}{2} \rfloor }}{ \epsilon }\Psi _{\infty }(h) \Big ). \end{aligned}$$

The above is $\Vert g\Vert _{\infty }$ multiplied by $O (e^{-\frac{n\epsilon }{2}})$, so we have proven (8.3) for $\alpha =\frac{\epsilon }{2}$ when assuming statements (ii), (I), and (II).

(ii). We will return to a consideration of the process $S_{t}$ underlying the resolvent chain $\sigma _{n}$. Let $\tau$ be a mean one exponential and $S_{0}=s\in \Sigma$. For $H(s)> L$ let $\varsigma$ be the hitting time that $H(S_{\varsigma })\le L$. Since the Hamiltonian evolution preserves energy, we can give a lower bound for our quantity of interest through

$$\begin{aligned} \int \limits _{H(s')\le L}\mathcal {T}(s,ds')&\ge \mathbb {P}_{s}\big [\varsigma \le \tau \text { and no collisions occur over the time interval} (\varsigma ,\tau ] \big ] \nonumber \\&\ge \Big (\inf _{H(x',p')\le L}\frac{1}{1+\mathcal {E}(p')} \Big ) \mathbb {P}_{s}\big [\varsigma \le \tau \big ]. \end{aligned}$$

(8.7)

The second inequality uses that the collisions occur with Poisson rate $\mathcal {E}(S_{t})$ and that $\tau -\varsigma$ is a mean one exponential in the event that $\varsigma \le \tau$ and conditioned on the information up to time $\varsigma$. The escape rates $\mathcal {E}(S_{t})$ are uniformly bounded over any compact region, and it is thus sufficient for us to give a lower bound for $\mathbb {P}_{s}\big [\varsigma \le \tau \big ]$. For any $T>0$ that we choose,

$$\begin{aligned} \inf _{H(s)>L} \mathbb {P}_{s}\big [\varsigma \le \tau \big ]&= \inf _{H(s)>L} \int \limits _{0}^{\infty }dt e^{-t}\mathbb {P}_{s}\big [\varsigma \le t \big ] \nonumber \\&\ge e^{-T}- e^{-T}\sup _{H(s)>L} \mathbb {P}_{s}\big [\varsigma > T \big ]. \end{aligned}$$

If we show $\sup _{H(s)>L} \mathbb {P}_{s}\big [\varsigma > T \big ]$ is small for large $T$ (or even merely bounded away from one), then combining this result with (8.7) completes the proof of statement (ii).

Define the function $w:\Sigma \rightarrow [\frac{1}{2},1]$ such that

$$\begin{aligned} w(s)=1-\frac{1}{2}\frac{ 1}{ 1+H^{\frac{1}{2}}(s)}. \end{aligned}$$

There exists $L$ large enough so that for some $\delta >0$

$$\begin{aligned} \sup _{ H(x,p)>L} \int \limits _{{\mathbb R}}dp'\mathcal {J}(p,p')\Big (w(x,p') -w(x,p) \Big )\le -\delta . \end{aligned}$$

(8.8)

The inequality (8.8) follows from the momentum-contractive nature of the jump rates $\mathcal {J}(p,p')$ described below the statement of (ii), and we will not go through its proof. The significance of the inequality (8.8) is that $t\,\delta +w(S_{t})$ behaves as a supermartingale over the time interval $[0,\varsigma ]$ since for $s=(X_{t},P_{t})$

$$\begin{aligned} \frac{d}{dr}\Big |_{r=0}\mathbb {E}_{s}\big [r\delta + w(X_{r}, P_{r})\big ]=\delta + \int \limits _{{\mathbb R}}dp'\mathcal {J}(P_{t},p')\Big (w(X_{t},p') -w(X_{t},P_{t}) \Big )\le 0. \end{aligned}$$

The equality above follows since the function $w$ is invariant of the Hamiltonian evolution. We have the following sequence of inequalities:

$$\begin{aligned} \sup _{H(s)>L}\mathbb {P}_{s}\big [\varsigma >T \big ]&\le \Big (\sup _{H(s)>L}\mathbb {P}_{s}\big [\varsigma > \delta ^{-1} \big ] \Big )^{\lfloor \delta T \rfloor }\le \Big (\sup _{H(s)>L} \delta \, \mathbb {E}_{s}\big [\varsigma \wedge \delta ^{-1} \big ] \Big )^{\lfloor \delta T \rfloor } \\&\le \Big (\sup _{H(s)>L}\mathbb {E}_{s}\big [w(s)-w(S_{ \varsigma \wedge \frac{1}{\delta }}) \big ] \Big )^{\lfloor \delta T \rfloor }\le 2^{-\lfloor \delta T \rfloor }. \end{aligned}$$

For the first inequality, the event $\varsigma >T$ requires that the time $\varsigma$ failed to occur over disjoint time intervals $(\frac{n}{\delta } ,\frac{n+1}{\delta }]$ for $0 \le n\le \lfloor \delta T \rfloor -1$. The second inequality is Chebyshev’s, and the third is by the optional stopping theorem since $w(s)-w(S_{t})- \delta \,t$ is a submartingale over the time interval $t\in [0, \varsigma \wedge \frac{1}{ \delta }]$. The last inequality follows from the constraint $\frac{1}{2}\le w(s)\le 1$. We can choose $T$ to make the right side arbitrarily small.

• (I). The second equality below follows from an inductive argument using the fact that the probability of the event $\tilde{\sigma }_{n}\in \Sigma \times 1$ is $h(\sigma _{n})$ when conditioned on the information up to time $n-1$ and the value $\sigma _{n}$. This property is visible from the form of the transition operator $\tilde{\mathcal {T}}$ given in Sect. 2.

  $$\begin{aligned} \tilde{\mathbb {P}}_{\tilde{\Psi }}\big [\tilde{n}_{1}\ge m \big ]&= \tilde{\mathbb {E}}_{\tilde{\Psi }}\big [\chi (\tilde{n}_{1}\ne 0)\cdots \chi (\tilde{n}_{1}\ne m-1) \big ]\\&= \tilde{\mathbb {E}}_{\tilde{\Psi }}\big [\big (1-h(\sigma _{0})\big )\cdots \big (1-h(\sigma _{m-1})\big )\big ]\\&= \mathbb {E}_{\Psi }\big [\big (1- h(\sigma _{0}) \big )\cdots \big (1-h(\sigma _{m-2})\big )\mathbb {E}\big [\big (1- h(\sigma _{m-1})\big )\,\big |\,\sigma _{0},\ldots ,\sigma _{m-2} \big ]\big ] \\&\le (1-\epsilon ) \mathbb {E}_{\Psi }\big [\big (1- h(\sigma _{0})\big )\cdots \big (1-h(\sigma _{m-2})\big )\big ]. \end{aligned}$$

  The third equality identifies the expectation as from the original statistics, and the inequality uses that the probability $\sigma _{m-1}$ jumps from $\sigma _{m-2}$ to the support of $h$ is $\ge (1-\epsilon )$ by (i) and (ii). Applying this inductively, we get the desired bound.

• (II). By the renewal theorem, $F_{\Psi }(n)$ converges as $n\rightarrow \infty$ to the inverse of the expectation of the renewal increments $\tilde{n}_{m+1}-\tilde{n}_{m}$. The expectation of the increments $\tilde{n}_{m+1}-\tilde{n}_{m}$, $m\ge 1$ is given by

  $$\begin{aligned} \tilde{\mathbb {E}}_{\tilde{\Psi }}\big [\tilde{n}_{m+1}-\tilde{n}_{m} \big ]= 1+\tilde{\mathbb {E}}_{\tilde{\nu }}\big [\tilde{n}_{1} \big ]= \tilde{\mathbb {E}}_{\tilde{\nu }}\Big [\sum _{n=0}^{\tilde{n}_{1}}1 \Big ]= \frac{ 1}{ \Psi _{\infty }(h)} \end{aligned}$$

  since the distribution for the split chain following a return to the atom is $\tilde{\nu }$ by Part (1) of Proposition 2.1, and where the second equality is by Part (1) of Proposition 2.4. The tails of the delay distribution $\tilde{\mathbb {P}}_{\tilde{\Psi }}\big [\tilde{n}_{1}>n \big ]$ and the renewal jump distribution $\tilde{\mathbb {P}}_{\tilde{\nu }}\big [\tilde{n}_{1}>n \big ]$ decay with order $O (e^{-\epsilon n})$ for large $n$ by (I). It follows that the renewal function $F_{\Psi }(n)$ converges exponentially with rate $\epsilon$ to the value $\Psi _{\infty }(h)$.

$\square$