Refined Second Law for Markov Processes

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Probability density evolution during erasure: overdamped vs underdamped dynamics
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Optimal overdamped erasure protocol
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Optimal transition \( [0.9,0.05,0.05]\to [0.05,0.05,0.9] \) for a jump process
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Evolution of mean position and momentum for minimum entropy production transitions between Gaussian states for underdamped dynamics.
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Erasure protocol between equilibria in the valley method description
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Consequences of speed constraints on the protocol
$$ E\,\geq\,\mathrm{K_{B}}\,T\,\ln 2 $$ In a series of distinct collaborations, we proved that thermodynamic transitions in non-equilibrium thermodynamics described by Markov processes (overdamped, underdamped with arbitrary or kinetic plus potential Hamiltonian, pure jump ) always occur with average entropy production bounded away from zero. Optimal control problems are notoriously sensitive to the class of admissible protocols. In all the cases above, optimal protocol are smooth and non-equilibrium: the current velocity at the end of the control horizon is non-vanishing. We also succeeded to develop a systematic analytic theory to compute underdamped corrections to overdamped Schrödinger bridges.
It is also possible to impose constraints (via valley method or via an upper bound ) on the protocols' acceleration. This is in my view a realistic way to inquire speed way to address speed limits and shortcuts to equilibration of therodynamic transitions.