A framework for the direct evaluation of large deviations in non-Markovian processes

Abstract

We propose a general framework to simulate stochastic trajectories with arbitrarily long memory dependence and efficiently evaluate large deviation functions associated to time-extensive observables. This extends the "cloning" procedure of Giardiná et al. [Phys. Rev. Lett. 96, 120603 (2006)] to non-Markovian systems. We demonstrate the validity of this method by testing non-Markovian variants of an ion-channel model and the Totally Asymmetric Exclusion Process, recovering results obtainable by other means.

Figures (5)

• Figure 1: Representation of a portion of a trajectory. The time elapsed from the last jump is called age. The conditional probability of having a configuration $x_n$ at an instant $t_n>t_{n-1}$ depends on the age, as well as on events which happened during the history (e.g., the one marked by the red star).
• Figure 2: SCGF of current in ion channel. (a) DTI model with $(k_0,\lambda_0,k_1,\lambda_1)=(0.1,0.01,1,1)$ and $(p_{x_1,x_0,1},p_{x_0,x_1,0},p_{x_0,x_1,-1},p_{x_1,x_0,0})=(0.5,0.6,0.4,0.5)$; the cloning result is consistent with the solution given in Andrieux2008. (b) non-DTI model with Markov representation and inverse scales $(\lambda^L_0,\lambda^R_0,\lambda^L_1,\lambda^R_1)=(10,10,20,20/3)$. The cloning reproduces the leading eigenvalue of the Markovian $s$-modified generator. In both cases $N=10^3$ and $T=10^3$.
• Figure 3: (a) Cloning evaluation of the SCGF for the non-Markovian TASEP, with $(\alpha_0,a,\beta,L)=(0.2,0.1,1,10^3)$, using $T=10^3$. Ensemble size is $N=5\cdot 10^3$ ($N=10^4$) for $s>-2$ ($s<-2$). The markers correspond to the direct evaluation of $e(s)$. Numerical errors are of the order of the symbol size, except for large negative $s$, where finite-ensemble effects still seem to play a role, as documented in Hurtado2009. The red line is obtained as $\int_0^s ( \mathrm{d} e(\sigma)/\mathrm{d}\sigma) \, \mathrm{d} \sigma$, according to the thermodynamic integration of Lecomte2007. (b) Comparison between the Legendre--Fenchel transform of the red line in (a) and the rate function of Harris2015. The dotted line is a numerical artefact due to the finite range of $s$ in (a); the Legendre--Fenchel transform maps the whole linear branch of $e(s)$ to the value at $j^*$ and larger values of $j$ are, in fact, not probed.
• Figure S1: Two graphical representations of the WTD Seq:convenientWTD. The waiting time is equal to the adsorption time of a random walker from the leftmost site to any of the grey sites.
• Figure S2: Graphical representations of the non-DTI ion-channel model with hidden states. The bonds corresponding to biased rates are drawn in thick lines. The modified generators associated with these two models have the same leading eigenvalue. (a) and (b) correspond to the WTD representations of figure \ref{['Sfig:WTD']}.