## Abstract

We consider a stochastic process which is (a) described by a continuous-time Markov chain on only short time-scales and (b) constrained to conserve a number of hidden quantities on long time-scales. We assume that the transition matrix of the Markov chain is given and the conserved quantities are known to exist, but not explicitly given. To study the stochastic dynamics we propose to use the principle of stationary entropy production. Then the problem can be transformed into a variational problem for a suitably defined “action” and with time-dependent Lagrange multipliers. We show that the stochastic dynamics can be described by a Schrödinger equation, with Lagrange multipliers playing the role of phases, whenever (a) the transition matrix is symmetric or the detailed balance condition is satisfied, (b) the system is not too far from the equilibrium and (c) the number of the conserved quantities is large.

Original language | English (US) |
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Pages (from-to) | 40-53 |

Number of pages | 14 |

Journal | Foundations of Physics |

Volume | 50 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2020 |

### Bibliographical note

Funding Information:The author wishes to acknowledge the hospitality of the Pacific Science Institute where this work began, the University of Niš where the key results were obtained and the Duluth Institute for Advance Study where much of the work in completing the paper was carried out. The work was supported in part by the Foundational Questions Institute (FQXi). Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Funding Information:

The author wishes to acknowledge the hospitality of the Pacific Science Institute where this work began, the University of Ni? where the key results were obtained and the Duluth Institute for Advance Study where much of the work in completing the paper was carried out. The work was supported in part by the Foundational Questions Institute (FQXi).

Publisher Copyright:

© 2019, Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- Emergent phenomena
- Entropy production
- Quantum mechanics