Nonequilibrium Statistical Mechanics of Hamiltonian Rotators with Alternated Spins

Abstract

We consider a finite region of a d-dimensional lattice of nonlinear Hamiltonian rotators, where neighbouring rotators have opposite (alternated) spins and are coupled by a small potential of size $\varepsilon ^a,\, a\ge 1/2$. We weakly stochastically perturb the system in such a way that each rotator interacts with its own stochastic thermostat with a force of order $\varepsilon$. Then we introduce action-angle variables for the system of uncoupled rotators ($\varepsilon = 0$) and note that the sum of actions over all nodes is conserved by the purely Hamiltonian dynamics of the system with $\varepsilon >0$. We investigate the limiting (as $\varepsilon \rightarrow 0$) dynamics of actions for solutions of the $\varepsilon$-perturbed system on time intervals of order $\varepsilon ^{-1}$. It turns out that the limiting dynamics is governed by a certain autonomous (stochastic) equation for the vector of actions. This equation has a completely non-Hamiltonian nature. This is a consequence of the fact that the system of rotators with alternated spins do not have resonances of the first order. The $\varepsilon$-perturbed system has a unique stationary measure $\widetilde{\mu }^\varepsilon$ and is mixing. Any limiting point of the family $\{\widetilde{\mu }^\varepsilon \}$ of stationary measures as $\varepsilon \rightarrow 0$ is an invariant measure of the system of uncoupled integrable rotators. There are plenty of such measures. However, it turns out that only one of them describes the limiting dynamics of the $\varepsilon$-perturbed system: we prove that a limiting point of $\{\widetilde{\mu }^\varepsilon \}$ is unique, its projection to the space of actions is the unique stationary measure of the autonomous equation above, which turns out to be mixing, and its projection to the space of angles is the normalized Lebesque measure on the torus $\mathbb {T}^N$. The results and convergences, which concern the behaviour of actions on long time intervals, are uniform in the number $N$ of rotators. Those, concerning the stationary measures, are uniform in $N$ in some natural cases.

Appendices

Appendix 1: The Ito Formula in Complex Coordinates

Let $\{\Omega ,\,{\mathcal {F}},\,{\mathbf {P}}\,;\,{\mathcal {F}}_t\}$ be a filtered probability space and $v(t)=(v_k(t))\in \mathbb {C}^N$ be a complex Ito process on this space of the form

$$\begin{aligned} dv=b\,dt + W\,dB. \end{aligned}$$

Here $b(t)=(b_k(t))\in \mathbb {C}^N$; $B=(\beta ,\overline{\beta })^T,$ $T$ denotes the transposition and $\beta =(\beta _k)\in \mathbb {C}^N$, $\beta _k$ are standard independent complex Brownian motions; the $N\times 2N$-matrix $W$ consists of two blocks $(W_1,W_2)$, so that $W\,dB= W^1\, d\beta + W^2\, d\overline{\beta }$, where $W^{1,2}(t)=(W^{1,2}_{kl}(t))$ are $N\times N$ matrices with complex entires. The processes $b_k(t),\,W^{1,2}_{kl}(t)$ are ${\mathcal {F}}_t$-adapted and assumed to satisfy usual growth conditions, needed to apply the Ito formula. Let

$$\begin{aligned} d^1_{kl}:=\left( \overline{W^1}W^{1T}+ \overline{W^2}W^{2T}\right) _{kl} \quad \text{ and }\quad d_{kl}^2:= \left( {W^2} {W^{1T}} +{W^1} {W^{2T}}\right) _{kl}. \end{aligned}$$

(5.33)

Denote by $(WdB)_k$ the $k$-th element of the vector $WdB$.

Proposition 5.7

Let $h:\,\mathbb {C}^N\rightarrow \mathbb {R}$ be a $C^2$-smooth function. Then

$$\begin{aligned} \frac{dh(v(t))}{2}\!=\! \sum \limits _{k}\frac{\partial h}{\partial \overline{v}_k}\cdot b_k\, dt \!+\! \sum \limits _{k,l}\left( \frac{\partial ^2 h}{\partial \overline{v}_k \partial v_l}d^1_{kl} \!+\! \mathrm{Re }\Big ( \frac{\partial ^2 h}{\partial v_k \partial v_l}d^2_{kl} \Big ) \right) \, dt \!+\! \sum \limits _k \frac{\partial h}{\partial \overline{v}_k}\cdot (W d B)_k . \end{aligned}$$

Proof

The result follows from the usual (real) Ito formula. $\square$

Consider the vectors of actions and angles $J=J(v)\in \mathbb {R}^N_{0+}$ and $\psi =\psi (v)\in \mathbb {T}^N$. Using formulas $\partial _{v_k}\psi _k=(2iv_k)^{-1}$ and $\partial _{\overline{v}_k}\psi _k=-(2i\overline{v}_k)^{-1}$, by Proposition 5.7 we get

$$\begin{aligned} dJ_k=(b_k\cdot v_k + d_{kk}^1)\,d t + d M^J_k, \quad d\psi _k=\frac{b_k\cdot (iv_k)- \mathrm{Im }(\overline{v}_kv_k^{-1}d_{kk}^2)}{|v_k|^2}\,d t + d M^\psi _k, \end{aligned}$$

(5.34)

where the martingales $M^J_k(t):=\int \limits _{t_0}^t v_k\cdot ( W d B)_k$ and $M^\psi _k=\int \limits _{t_0}^t\frac{iv_k}{|v_k|^2}\cdot ( W d B)_k$ for some $t_0<t$. By the direct computation we obtain

Proposition 5.8

The diffusion matrices for the $J$- and $\psi$-equations in (5.34) with respect to the real Brownian motion $(\mathrm{Re }\beta _k, \mathrm{Im }\beta _k)$ have the form $S^J=(S^J_{kl})$ and $S^\psi =(S^\psi _{kl})$, where

$$\begin{aligned} S^J_{kl}=\mathrm{Re }(v_k \overline{v}_l d_{kl}^1+ \overline{v}_k \overline{v}_l d_{kl}^2) \quad \text{ and }\quad S^\psi _{kl}=\mathrm{Re }(v_k \overline{v}_l d_{kl}^1 -\overline{v}_k \overline{v}_l d_{kl}^2 )(|v_k||v_l|)^{-2}. \end{aligned}$$

(5.35)

The quadratic variations of $M^J_k$ and $M^\psi _k$ take the form

$$\begin{aligned}{}[M^J_k]_t=\int \limits _{t_0}^t S^J_{kk}\, ds \quad \text{ and } \quad [M^\psi _k]_t=\int \limits _{t_0}^t S^\psi _{kk} \, ds. \end{aligned}$$

(5.36)

Appendix 2: Averaging

Consider a complex coordinates $v=(v_j)\in \mathbb {C}^N$ and the corresponding vectors of actions $J=J(v)$ and angles $\psi =\psi (v)$. Consider a function $P:\mathbb {C}^N\mapsto \mathbb {R}$ and write it in action-angle coordinates, $P(v)=P(J,\psi )$. Its averaging

$$\begin{aligned} \langle P \rangle :=\int \limits _{\mathbb {T}^N} P(J,\psi )\, d\psi \end{aligned}$$

is independent of angles and can be considered as a function $\langle P \rangle (v)$ of $v$, or as a function $\langle P \rangle \big ((|v_j|)_j\big )$ of $(|v_j|)_j$, or as a function $\langle P \rangle (J)$ of $J$.

Proposition 5.9

Let $P\in {\mathcal {L}}_{loc}(\mathbb {C}^N).$ Then

• (i) Its averaging $\langle P \rangle \in {\mathcal {L}}_{loc}(\mathbb {R}^N_{+0})$ with respect to $(|v_j|)$.

• (ii) If $P$ is $C^{2s}$-smooth then $\langle P\rangle$ is $C^{2s}$-smooth with respect to $v$ and $C^{s}$-smooth with respect to $J$.

Proof

(i) Is obvious.

(ii) The first assertion is obvious. To prove the second consider the function $\hat{P}:x\in \mathbb {R}^N\mapsto \mathbb {R}$, $\hat{P}(x):=\langle P\rangle |_{v=x}$. Then $\hat{P}(x)=\langle P\rangle (J)$, where $J_j=x_j^2/2.$ The function $\hat{P}$ is $C^{2s}$-smooth and even in each $x_j$. Any function of finitely many arguments with this property is known to be a $C^s$- smooth function of the square arguments $x_j^2$ (see [37]). $\square$