Effective Dynamics for a Kinetic Monte–Carlo Model with Slow and Fast Time Scales

Abstract

We consider several multiscale-in-time kinetic Monte Carlo models, in which some variables evolve on a fast time scale, while the others evolve on a slow time scale. In the first two models we consider, a particle evolves in a one-dimensional potential energy landscape which has some small and some large barriers, the latter dividing the state space into metastable regions. In the limit of infinitely large barriers, we identify the effective dynamics between these macro-states, and prove the convergence of the process towards a kinetic Monte Carlo model. We next consider a third model, which consists of a system of two particles. The state of each particle evolves on a fast time-scale while conserving their respective energy. In addition, the particles can exchange energy on a slow time scale. Considering the energy of the first particle, we identify its effective dynamics in the limit of asymptotically small ratio between the characteristic times of the fast and the slow dynamics. For all models, our results are illustrated by representative numerical simulations.

Appendix: Some Useful Results

For convenience, we recall in this appendix some classical results of probability theory that are needed in this article.

Martingales

Several results on martingales are useful in this work. The first one is an existence and uniqueness result for the martingale problem introduced by D.W. Stroock and S.R.S. Varadhan (see e.g. [1] and [26]):

Proposition 3

(Lemma 5.1 of Appendix 1 of [13])

Let (X _t ) _t≥0 be a Markov process and let $(\mathcal{F}_{t} )_{t \geq0}$ be its natural filtration. For any bounded function F, we introduce

$$M^F_t = F (X_t ) - F (X_0 ) - \int^t_0 LF (X_s )ds $$

and

$$N^F_t = \bigl(M^F_t \bigr)^2 - \int^t_0 \bigl(LF^2 (X_s ) - 2F (X_s ) LF (X_s ) \bigr) ds, $$

where L is the generator of the Markov process (X _t ). Then M ^F and N ^F are $\mathcal{F}_{t}$-martingales. In particular, the quadratic variation of M ^F reads

$$\bigl\langle M^F\bigr\rangle _t = \int ^t_0 \bigl(LF^2 (X_s ) - 2F (X_s ) LF (X_s ) \bigr)ds. $$

We recall that for a continuous local martingale M, the process 〈M〉 is defined to be the unique right-continuous and increasing predictable process starting at zero such that M ²−〈M〉 is a local martingale.

The next result is of paramount importance to prove that a process is a jump process, and to identify its generator. We state here this result as a simplified version of [12, Theorem 21.11].

Lemma 4

(Uniqueness result for the martingale problem)

Let F be a countable space, Z _t a stochastic process valued in F and L an operator on bounded functions $\phi:F\rightarrow{\mathbb{R}}$ defined by

$$L\phi(x)=\sum_{x'\in F}L_{x,x'} \bigl( \phi \bigl(x'\bigr)-\phi(x) \bigr), $$

where L _x,x′≥0 for any x,x′∈F and sup _x,x′∈F L _x,x′<∞. If for any bounded function $\phi:F\rightarrow{\mathbb{R}}$, the process

$$M_t^\phi:= \phi (Z_t ) - \phi (Z_0 ) -\int_0^t L\phi (Z_s ) \, ds $$

is a martingale with respect to the natural filtration of (Z _t ) _t≥0, then (Z _t ) _t≥0 is the jump process of initial condition Z ₀ and of generator L.

Convergence of Probability Measures

We now turn to classical results concerning the convergence of probability measures in $D_{\mathbb{R}} [0,\infty )$, which is the space of functions that are right continuous with left limits (the so-called càd-làg functions), defined on [0,∞) and valued in $\mathbb{R}$. Proposition 4 gives an equivalent definition of the Skorohod metric on $D_{\mathbb{R}} [0,\infty )$ (see [7, p. 116–118] for the original definition of the Skorohod metric, that we actually do not use in this work). Theorem 4 states convergence criteria for probability measures on $D_{\mathbb{R}} [0,\infty )$.

Proposition 4

(Proposition 5.3, Chap. 3 of [7])

Let (x _n ) _n≥0 be a sequence in $D_{\mathbb{R}} [0,\infty )$ and $x\in D_{\mathbb{R}} [0,\infty )$. The following assertions are equivalent:

• lim _n→∞ x _n =x in the space $D_{\mathbb{R}} [0,\infty )$ endowed with the Skorohod metric.

• For any T>0, there exists a sequence of strictly increasing, continuous maps (λ _n ) _n≥0 defined on [0,∞) and valued in [0,∞) such that

  $$ \lim_{n \rightarrow\infty} \sup_{0\leq t \leq T} \bigl| \lambda_n (t )-t\bigr|=0 $$

  (45)

  and

  $$ \lim_{n\rightarrow\infty} \sup_{0\leq t \leq T} \bigl|x_n (t )-x \bigl(\lambda_n (t ) \bigr)\bigr|=0. $$

  (46)

Theorem 4

(Aldous’ criterion, Theorem VI.4.5 of [11])

Let (X ⁿ) _n≥1 be a sequence of càd-làg processes, with distributions ${\mathcal{P}}^{n}$. Suppose that

• for any $N\in{\mathbb{N}}$ and ϵ>0, there exists $n_{0}\in{\mathbb{N}}$, n ₀>0, and $K\in{\mathbb{R}}^{+}$ such that, for any n≥n ₀,

  $$ {\mathcal{P}}^n \Bigl( \sup_{t\leq N} \bigl\vert X^n_t\bigr\vert >K \Bigr) \leq\epsilon. $$

  (47)

• for any $N\in{\mathbb{N}}$ and α>0, we have

  $$ \lim_{\theta\rightarrow0} \, \limsup_n \, \sup_{S,T\in\mathfrak{T}_N^n, \;S\leq T\leq S+\theta} {\mathcal{P}}^n \bigl(\bigl\vert X^n_T-X^n_S\bigr\vert \geq \alpha \bigr)=0, $$

  (48)

  where $\mathfrak{T}_{N}^{n}$ is the set of all ${\mathcal{F}}^{n}$ stopping times that are bounded by N.

Then the sequence $(X^{n})_{n\in{\mathbb{N}}}$ is tight.