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Interactive Statistical Mechanics and Nonlinear Filtering

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Abstract

This paper connects non-equilibrium statistical mechanics and optimal nonlinear filtering. The latter concerns the observation-conditional behaviour of Markov signal processes, and thus provides a tool for investigating statistical mechanics with partial observations. These allow entropy reduction, illustrating Landauer’s Principle in a quantitative way.

The joint process comprising a signal and its nonlinear filter is irreversible in its invariant distribution, which therefore corresponds to a non-equilibrium stationary state of the associated joint system. Macroscopic entropy and energy flows are identified for this state. Since these are driven by observations internal to the system, they do not cause entropy increase, and so the joint system makes statistical mechanical sense in reverse time.

Time reversal yields a dual system in which the signal and filter exchange roles. Despite the structural similarity of the original and dual systems, there is a substantial asymmetry in their complexities. This reveals the direction of time, despite the systems being in stationary states that do not produce entropy.

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Authors and Affiliations

  1. Department of Computing and Electronic Systems, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, UK

    Nigel J. Newton

  2. Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Nigel J. Newton

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  1. Nigel J. Newton
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Correspondence to Nigel J. Newton.

Additional information

This work was partially supported by Leverhulme Trust Research Fellowship 2003/0426, by MURI Grant F49620-02-1-0325 (Complex Adaptive Networks for Co-operative Control), by ARO-MURI Grant DAAD19-00-1-0466 (Data Fusion in Large Arrays of Microsensors (sensor web)) and by NSF-ITR Grant CCR-0325774 Collaborative Research: New Approaches to Experimental Design and Statistical Analysis of Genomic and Structural Biological Data from Multiple Sources.

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Newton, N.J. Interactive Statistical Mechanics and Nonlinear Filtering. J Stat Phys 133, 711–737 (2008). https://doi.org/10.1007/s10955-008-9622-z

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  • DOI: https://doi.org/10.1007/s10955-008-9622-z

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