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# Direct evaluation of dynamical large-deviation rate functions using a variational ansatz

## Abstract

We describe a simple form of importance sampling designed to bound and compute large-deviation rate functions for time-extensive dynamical observables in continuous-time Markov chains. We start with a model, defined by a set of rates, and a time-extensive dynamical observable. We construct a reference model, a variational ansatz for the behavior of the original model conditioned on atypical values of the observable. Direct simulation of the reference model provides an upper bound on the large-deviation rate function associated with the original model, an estimate of the tightness of the bound, and, if the ansatz is chosen well, the exact rate function. The exact rare behavior of the original model does not need to be known in advance. We use this method to calculate rate functions for currents and counting observables in a set of network- and lattice models taken from the literature. Straightforward ansätze yield bounds that are tighter than bounds obtained from Level 2.5 of large deviations via approximations that involve uniform scalings of rates. Here, we show how to correct these bounds in order to recover the rate functions exactly. Our approach is complementary to more specialized methods and offers a physically transparent framework for approximating and calculatingmore »

- Authors:

- California Inst. of Technology (CalTech), Pasadena, CA (United States)
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1601210

- Grant/Contract Number:
- [AC02-05CH11231]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review E

- Additional Journal Information:
- [ Journal Volume: 100; Journal Issue: 5]; Journal ID: ISSN 2470-0045

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY

### Citation Formats

```
Jacobson, Daniel, and Whitelam, Stephen. Direct evaluation of dynamical large-deviation rate functions using a variational ansatz. United States: N. p., 2019.
Web. doi:10.1103/physreve.100.052139.
```

```
Jacobson, Daniel, & Whitelam, Stephen. Direct evaluation of dynamical large-deviation rate functions using a variational ansatz. United States. doi:10.1103/physreve.100.052139.
```

```
Jacobson, Daniel, and Whitelam, Stephen. Mon .
"Direct evaluation of dynamical large-deviation rate functions using a variational ansatz". United States. doi:10.1103/physreve.100.052139.
```

```
@article{osti_1601210,
```

title = {Direct evaluation of dynamical large-deviation rate functions using a variational ansatz},

author = {Jacobson, Daniel and Whitelam, Stephen},

abstractNote = {We describe a simple form of importance sampling designed to bound and compute large-deviation rate functions for time-extensive dynamical observables in continuous-time Markov chains. We start with a model, defined by a set of rates, and a time-extensive dynamical observable. We construct a reference model, a variational ansatz for the behavior of the original model conditioned on atypical values of the observable. Direct simulation of the reference model provides an upper bound on the large-deviation rate function associated with the original model, an estimate of the tightness of the bound, and, if the ansatz is chosen well, the exact rate function. The exact rare behavior of the original model does not need to be known in advance. We use this method to calculate rate functions for currents and counting observables in a set of network- and lattice models taken from the literature. Straightforward ansätze yield bounds that are tighter than bounds obtained from Level 2.5 of large deviations via approximations that involve uniform scalings of rates. Here, we show how to correct these bounds in order to recover the rate functions exactly. Our approach is complementary to more specialized methods and offers a physically transparent framework for approximating and calculating the likelihood of dynamical large deviations.},

doi = {10.1103/physreve.100.052139},

journal = {Physical Review E},

number = [5],

volume = [100],

place = {United States},

year = {2019},

month = {11}

}

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