A Curie-Weiss Model with Dissipation

Abstract

We consider stochastic dynamics for a spin system with mean field interaction, in which the interaction potential is subject to noisy and dissipative stochastic evolution. We show that, in the thermodynamic limit and at sufficiently low temperature, the magnetization of the system has a time periodic behavior, despite of the fact that no periodic force is applied.

Appendix A: Proof of Theorem 2.1

For $\mathcal{X}$ a Polish space, denote by $\mathcal{P}(\mathcal {X})$ the space of probability measures on the Borel sets of $\mathcal {X}$; equip $\mathcal{P}(\mathcal{X})$ with the topology of weak convergence, which makes it a Polish space, too. As in Sect. 2, let P ^N be the empirical measure of the N-particle system. We may assume that the $\{-1,1\}\times\mathbb {R}$-valued processes (s _j ,λ _j ) have càdlàg trajectories (i.e., trajectories that are right-continuous with limits from the left). Consequently, P ^N is a probability measure on the Borel sets of $D:= D([0,\infty),\{-1,1\}\times\mathbb{R})$, the space of $\{-1,1\}\times\mathbb{R}$-valued càdlàg functions equipped with the Skorohod topology.

The strategy of proof, here, is to represent both the microscopic and the macroscopic model as solutions of certain stochastic differential equations in order to apply results by [9] on propagation of chaos, which implies convergence of empirical measures.

Let η be Lebesgue measure restricted to the Borel sets on the interval (0,2). Let $((\varOmega,\mathcal{F},\mathbf{P}),(\mathcal {F}_{t})_{t\geq0})$ be a filtered probability space satisfying the usual hypotheses rich enough to carry an independent family $(B_{i},\mathcal{N}_{i})_{i\in\mathbb{N}}$ of one-dimensional $(\mathcal {F}_{t})$-Brownian motions B _i and stationary $(\mathcal {F}_{t})$-Poisson random measures $\mathcal{N}_{i}$ with characteristic measure η. For $N\in\mathbb{N}$, consider the system of Itô-Skorohod equations

$$ \everymath{\displaystyle} \begin{array}{@{}llll} \mathrm {d}\lambda^{N}_{i}(t)= -\alpha\lambda^{N}_{i}(t-)\mathrm {d}{t} + \sigma \mathrm {d}{B}_{i}(t) -\frac{\beta}{N}\sum_{k=1}^{N} \int_{(0,2)} q \bigl(\bigl(s^{N}_{k}(t-), \lambda^{N}_{k}(t-)\bigr),u \bigr) \mathcal{N}_{k}( \mathrm {d}{u},\mathrm {d}{t}), \cr\noalign{\vspace{3pt}} \mathrm {d}{s}^{N}_{i}(t) = \int_{(0,2)} q \bigl( \bigl(s^{N}_{i}(t-),\lambda^{N}_{i}(t-) \bigr),u \bigr) \mathcal{N}_{i}(\mathrm {d}{u},\mathrm {d}{t}), \quad i\in\{1,\ldots,N\}, \end{array} $$

(A.1)

where

$$q\bigl((s,\lambda),u\bigr):= -2h(s)\cdot\mathbf{1}_{(0,1+h(s)\tanh (\lambda ))}(u),\quad(s, \lambda)\in\mathbb{R}^{2},\ u\in(0,2), $$

and h(s):=(−1)∨(s∧1), $s\in\mathbb{R}$. By Theorem 1.2 in [9], existence and uniqueness of solutions hold in the strong sense for the system of Eqs. (A.1) since its coefficients are globally Lipschitz continuous; the jump coefficient, in particular, satisfies the L ¹ Lipschitz assumption of the theorem. Clearly, if s∈{−1,1}, then h(s)=s, h(s)tanh(λ)=tanh(s⋅λ), and

$$\int_{(0,2)} q\bigl((s,\lambda),u\bigr) \eta(\mathrm {d}{u}) = -2 \bigl(s+\tanh(\lambda) \bigr). $$

Thanks to the choice of the jump heights, if (s ^N(0),λ ^N(0)) is such that $s^{N}_{i}(0)\in\{-1,1\}$, then $s^{N}_{i}(t)\in\{-1,1\}$ for all t≥0. Let us fix a sequence of initial conditions $(s^{N}(0),\lambda^{N}(0))_{N\in\mathbb{N}}$ such that s ^N(0)∈{−1,1}^N for all $N\in\mathbb{N}$ and $(\mathbf{P}\circ(s^{N}(0),\lambda^{N}(0))^{-1})_{N\in\mathbb{N}}$ is μ-chaotic for some $\mu\in\mathcal{P}(\{-1,1\}\times\mathbb{R})$. The solution process (s ^N,λ ^N) is then a $\{-1,1\} ^{N}\times\mathbb{R}^{N}$-valued Markov process. Comparing its infinitesimal generator (cf. Eq. (1.2) in [9]) with Eq. (2.3) above shows that (s ^N,λ ^N) is a realization of the N-particle microscopic model. To prove Theorem 2.1 we thus have to prove convergence of the empirical measures $P^{N} := \sum_{i=1}^{N} \delta _{(s^{N}_{i},\lambda^{N}_{i})}$ associated with the solutions of (A.1).

Define a function $\bar{b}\!: \mathcal{P}(\mathbb{R}^{2}) \rightarrow\mathbb{R}$ by

$$\bar{b}(\nu):= 2\beta\cdot\int_{\mathbb{R}^{2}} \bigl(h(s) + \tanh( \lambda) \bigr)\nu(\mathrm {d}{s}, \mathrm {d}\lambda). $$

Notice that $\bar{b}$ is Lipschitz continuous with respect to the bounded Lipschitz (or Dudley) metric on $\mathcal{P}(\mathbb{R}^{2})$ as well as with respect to the Wasserstein-1 (or Lipschitz) metric on $\mathcal{P}_{1}(\mathbb{R}^{2})$, the space of probability measures with finite first moments. By Theorem 2.1 in [9], existence and uniqueness of solutions hold in the strong sense for the McKean-Vlasov Itô-Skorohod equation

$$ \everymath{\displaystyle} \begin{array}{@{}rlll} \mathrm {d}\varLambda(t) &=& -\alpha \varLambda(t)dt + \sigma \mathrm {d}{B}_{1}(t) + \bar{b}(P_{t})\mathrm {d}{t}, \cr\noalign{\vspace{3pt}} \mathrm {d}\varSigma(t) &=& \int_{(0,2)} q \bigl(\bigl(\varSigma(t-), \varLambda(t)\bigr),u \bigr) \mathcal{N}_{1}(\mathrm {d}{u},\mathrm {d}{t}), \cr\noalign{\vspace{3pt}} P_{t} &=& \operatorname {law}\bigl(\varSigma(t),\varLambda(t)\bigr). \end{array} $$

(A.2)

Assume that $P_{0} = \mu= \operatorname {law}(\varSigma(0),\varLambda(0))$. Set $P:= \operatorname {law}(\varSigma,\varLambda)$ and observe that $P \in \mathcal {P}(D)$. Comparison of infinitesimal generators yields that the solution (Σ,Λ) of (A.2) is a realization of the nonlinear Markov process given by Eq. (2.11). Moreover, the $\mathcal{P}(\{-1,1\}\times\mathbb{R})$-valued process (P _t ) _t≥0 coincides with the solution of Eq. (2.9) with initial condition P ₀.

For $N\in\mathbb{N}$, consider the system of Itô-Skorohod equations

$$ \everymath{\displaystyle} \begin{array}{@{}rlll} \mathrm {d}\bar{\lambda}^{N}_{i}(t) &=& -\alpha\bar{\lambda}^{N}_{i}(t)\mathrm {d}{t} + \sigma \mathrm {d}{B}_{i}(t) + \bar{b}\bigl(\bar{P}^{N}_{t}\bigr) \mathrm {d}{t}, \cr\noalign{\vspace{3pt}} \mathrm {d}\bar{s}^{N}_{i}(t) &=& \int_{(0,2)} q \bigl(\bigl(\bar{s}^{N}_{i}(t-),\bar{\lambda}^{N}_{i}(t) \bigr),u \bigr) \mathcal{N}_{i}(\mathrm {d}{u},\mathrm {d}{t}),\quad i\in\{1,\ldots ,N\}, \end{array} $$

(A.3)

where $\bar{P}^{N}_{t}:= \sum_{i=1}^{N}\delta_{\bar{(\lambda }^{N}_{i}(t),\bar{s}^{N}_{i}(t))}$ is the empirical measure of the solution at time t. Notice that $\bar{\lambda}^{N}_{i}(t) = \bar {\lambda}^{N}_{i}(t-)$ by continuity of trajectories and that all processes are stochastically continuous. Again by Theorem 1.2 in [9], existence and uniqueness of solutions hold in the strong sense for the system of Eqs. (A.3). If the initial condition $(\bar{s}^{N}(0),\bar{\lambda}^{N}(0))$ for (A.3) is such that $\bar{s}^{N}_{i}(0) \in\{-1,1\}$, then $\bar{s}^{N}_{i}(t)\in\{-1,1\}$ for all t≥0. Fix the initial condition at $(\bar{s}^{N}(0),\bar{\lambda}^{N}(0)):= (s^{N}(0),\lambda^{N}(0))$. Since s ^N(0) takes values in {−1,1}^N and by the continuity of Lebesgue integrals, the system of Eqs. (A.3) can be rewritten as

$$ \everymath{\displaystyle} \begin{array}{@{}rlll} \mathrm {d}\bar{\lambda}^{N}_{i}(t) &=& -\alpha\bar{\lambda}^{N}_{i}(t-)\mathrm {d}{t} + \sigma \mathrm {d}{B}_{i}(t) -\frac{\beta}{N}\sum_{k=1}^{N} \int_{(0,2)} q \bigl(\bigl(\bar{s}^{N}_{k}(t-), \bar{\lambda}^{N}_{k}(t-)\bigr),u \bigr)\eta(\mathrm {d}{u})\mathrm {d}{t}, \cr\noalign{\vspace{3pt}} \mathrm {d}\bar{s}^{N}_{i}(t) &=& \int_{(0,2)} q \bigl(\bigl(\bar{s}^{N}_{i}(t-),\bar{\lambda}^{N}_{i}(t-) \bigr),u \bigr) \mathcal{N}_{i}(\mathrm {d}{u},\mathrm {d}{t}),\quad i\in\{1,\ldots ,N\}. \end{array} $$

(A.4)

Set $\bar{P}^{N}\doteq\sum_{i=1}^{N} \delta_{(\bar{s}^{N},\bar {\lambda}^{N})}$. By Theorem 4.1 in [9], the sequence $(\operatorname {law}(\bar{s}^{N},\bar{\lambda}^{N}))_{N\in\mathbb{N}}$ is P-chaotic. This implies, by the Tanaka-Sznitman theorem (for instance, Theorem 3.2 in [8]), that the sequence $(\bar{P}^{N})_{N\in\mathbb{N}}$ of $\mathcal{P}(D)$-valued random variables converges in distribution to the probability measure P. In order to establish convergence of $(P^{N})_{N\in\mathbb{N}}$ to P, it is therefore enough to show that

$$\hat{d}_{bL} \bigl(\operatorname {law}\bigl(P^{N}\bigr),\operatorname {law} \bigl(\bar{P}^{N}\bigr) \bigr) \stackrel{N\to\infty}{\longrightarrow} 0, $$

where $\hat{d}_{bL}$ is the bounded Lipschitz metric on $\mathcal {P}(\mathcal{P}(D))$. By definition of $\hat{d}_{bL}$ and since both P ^N and $\bar{P}^{N}$ are empirical measures for processes defined on the same stochastic basis, we have

$$\hat{d}_{bL} \bigl(\operatorname {law}\bigl(P^{N}\bigr),\operatorname {law} \bigl(\bar{P}^{N}\bigr) \bigr) \leq \mathbf{E}\bigl[ d_{bL} \bigl(P^{N},\bar{P}^{N} \bigr) \bigr] \leq\frac{1}{N} \sum_{i=1}^{N} \mathbf{E}\bigl[ d_{Sko} \bigl( \bigl(s^{N}_{i},\lambda^{N}_{i} \bigr), \bigl(\bar{s}^{N}_{i},\bar{\lambda }^{N}_{i}\bigr) \bigr) \bigr], $$

where d _bL is the bounded Lipschitz metric on $\mathcal{P}(D)$ and d _Sko the Skorohod metric on D. For $i\in\mathbb{N}$, let $\tilde{\mathcal{N}}_{i}$ be the compensated Poisson random measure associated with $\mathcal{N}_{i}$, that is, $\tilde{\mathcal {N}}_{i}(\mathrm {d}{u},\mathrm {d}{t}) = \mathcal{N}_{i}(\mathrm {d}{u},\mathrm {d}{t}) - \eta (du)dt$. Then for T>0, i∈{1,…,N}, $N\in\mathbb{N}$,

Since

it follows that

An application of Gronwall’s lemma yields, for every T>0,

$$\frac{1}{N}\sum_{i=1}^{N}\mathbf{E}\Bigl[\,\sup_{t\in[0,T]} \bigl(\bigl|s^{N}_{i}(t) - \bar{s}^{N}_{i}(t)\bigr| + \bigl|\lambda^{N}_{i}(t) - \bar{\lambda}^{N}_{i}(t)\bigr| \bigr) \Bigr] \stackrel{N\to \infty}{\longrightarrow} 0, $$

which implies the desired convergence.