# Effective fluctuation and response theory

I launched a new paper in the deep sky of the arXiv:

M. Polettini, M. Esposito, Effective fluctuation and response theory, arXiv:1803.03552

It’s by far my most ambitious work so far (and I don’t think I will embark in a similar enterprise anytime soon…). 41 pages, double column, tiny fond. arXiv did not even accept the abstract, because it was at least four times longer than conventional…

I also happen to be at the Deutsche Physikalische Gesellschaft, where in a few minutes I will be faced with the daunting task of presenting all this material in 10 minutes flat, last speaker right before the lunch break. Even given the small time available, still I stubbornly chose the slowest tool for presentations: the blackboard. I find using the blackboard forces me to communicate the exact amount of information that can be digested, and to avoid certain common mistakes in presentations, like overloading presentations with symbols and formulas. But it’s already difficult to communicate this stuff at the blackboard in 40′, what’s going to happen in 10′? Here an unreadable synthesis of the talk (which will be accompanied by figures and formulas):

1) A systems’ perspective on thermodynamics

• We are interested in the interaction of a system with an environment.
• The environment is resolved into several baths.
• (Time-extensive) currents flow through the system, to and from baths.
• Thermo-dynamics: affinities-currents.

2) Thermodynamics based on Markov processes

• The internal dynamics of the system is a Markov process.
• On a discrete space (=> graph-theoretic techniques).
• A single realization of the process (path) is described by waitings and jumps.
• Currents are linear combinations of the number of transitions along a path

DISCLAIMER ————————————————————————————————–

From now on: currents are single-edge. Generalization is possible but lots of ifs.

———————————————————————————————————————–

3) “Complete” results

• “Complete” edge currents cover all cycles. Otherwise: “marginal”.
• A list of results standard for complete currents: the Fluctuation Relation, the Integral Fluctuation Relation, the 2nd Law of Thermodynamics, the characterization of equilibrium, the (Symmetrized) Fluctuation-Dissipation Relation at equilibrium, the Reciprocal Relations
• Can be resumed in the inference diagram FR => IFR => 2nd, FR => RR, IFR => S-FDR

4) Marginal results

• What if we take a marginal subset of currents? Do there exist effective affinities?
• A list of new results standard for complete currents: the Integral Fluctuation Relation, the 2nd Law of Thermodynamics, a characterization of stalling, the S-FDR
• Can be resumed in the inference diagram IFR => 2nd, IFR => S-FDR
• Operational definition of the effective affinities
• We can formulate a new auxiliary dynamics that reinstates the FR and the RR [see slides]

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