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Heat release by controlled continuous-time Markov jump processes
Statistical Mechanics · Submitted 19 Mar 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Thus, our results may be interpreted as a refined version of the second law of thermodynamics.
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The abstract, sentence by sentence
We derive the equations governing the protocols minimizing the heat released by a continuous-time Markov jump process on a one-dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon. In particular, we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Bellman-Jacobi-type. As the state-space mesh tends to zero, these equations converge to those satisfied by the diffusion process minimizing the heat released in the Langevin formulation of the same problem. We also show that in full analogy with the continuum case, heat minimization is equivalent to entropy production minimization. Thus, our results may be interpreted as a refined version of the second law of thermodynamics.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We derive the equations governing the protocols minimizing the heat released by a continuous-time Markov jump process on a one-dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon. | contribution 0.93 | 0.96 | 0.34 | asserted 1.00 |
| s1 | In particular, we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Bellman-Jacobi-type. | result 0.82 | 0.94 | 0.12 | asserted 1.00 |
| s2 | As the state-space mesh tends to zero, these equations converge to those satisfied by the diffusion process minimizing the heat released in the Langevin formulation of the same problem. | result 0.97 | 0.85 | 0.11 | asserted 1.00 |
| s3 | We also show that in full analogy with the continuum case, heat minimization is equivalent to entropy production minimization. | result 0.91 | 0.90 | 0.29 | asserted 1.00 |
| s4 | Thus, our results may be interpreted as a refined version of the second law of thermodynamics. | interpretation 1.00 | 0.57 | 0.36 | hedged possibility 0.99 |
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The paper
Original abstract
We derive the equations governing the protocols minimizing the heat released by a continuous-time Markov jump process on a one-dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon. In particular, we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Bellman-Jacobi-type. As the state-space mesh tends to zero, these equations converge to those satisfied by the diffusion process minimizing the heat released in the Langevin formulation of the same problem. We also show that in full analogy with the continuum case, heat minimization is equivalent to entropy production minimization. Thus, our results may be interpreted as a refined version of the second law of thermodynamics.
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How the literature uses it
5 arXiv papers cite this in 7 sentences
Papers citing it · 5 sentences without a confident label
Uses its method or tool 1 paper
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We parameterize the transition rates as [ref] [equation] with constant, symmetric part $κ_ij= κ_ji$, which sets the characteristic time-scale for the transition from state $i$ to $j$, and time-dependent, antisymmetric $A_ij(t) = - A_ji(t)$.
Optimality of non-conservative driving for finite-time processes with discrete states · Introduction · 0.80
Cites it as background 1 paper
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Considerable effort has been put into finding a theoretical solution to optimal control problems for physically relevant Markov models of the dynamics: overdamped [ref] , discrete configuration space jump [ref] , and underdamped [ref] processes.
Minimal work protocols for inertial particles in non-harmonic traps · Introduction · 0.98
What it cites · 45 sentences without a confident label
Uses the method or tool of 1 paper
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Under these hypotheses, Girsanov formula (see e.g. [ref] or formula (2.5) of [ref] ) provides us with two equivalent representations of the Radon-Nikodym derivative of the measure $𝒫_ξ$ of $ξ$ with respect to that $𝒫_η$ of $η$.
On and beyond entropy production: the case of Markov jump processes · Time-reversal of the probability measure · 0.94
Compares with 2 papers
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Our results are the counter-part for Markov jump processes of the entropy bounds derived in [ref] for the Langevin dynamics.
Refined Second Law of Thermodynamics for fast random processes · Conclusions · 0.90
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For the two state dynamics we show that our optimal protocol corresponds to an entropy production lower than the one generated by optimizing the Glauber jump process as in [ref] .
Finite time thermodynamics for a single level quantum dot · Introduction · 1.00
Cites as support 1 paper
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This is indeed the case by virtue of a result of [ref] .
On and beyond entropy production: the case of Markov jump processes · Heat release as a “cost functional” · 1.00
Cites as background 10 papers
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Fluctuation relations [ref] impose general constraints on these probability distributions which can be and have been extensively tested experimentally see e.g. [ref] .
A nonequilibrium equality for free energy differences · Introduction · 0.83
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Fluctuation relations [ref] impose general constraints on these probability distributions which can be and have been extensively tested experimentally see e.g. [ref] .
Path ensembles averages in systems driven far-from-equilibrium · Introduction · 0.83
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Fluctuation relations [ref] impose general constraints on these probability distributions which can be and have been extensively tested experimentally see e.g. [ref] .
A Gallavotti-Cohen Type Symmetry in the Large Deviation Functional for Stochastic Dynamics · Introduction · 0.83
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