Abstract
In many real-world situations, there are constraints on the ways in which a physical system can be manipulated. We investigate the entropy production (EP) and extractable work involved in bringing a system from some initial distribution $p$ to some final distribution $p'$, given that the set of master equations available to the driving protocol obeys some constraints. We first derive general bounds on EP and extractable work, as well as a decomposition of the nonequilibrium free energy into an "accessible free energy" (which can be extracted as work, given a set of constraints) and an "inaccessible free energy" (which must be dissipated as EP). In a similar vein, we consider the thermodynamics of information in the presence of constraints, and decompose the information acquired in a measurement into "accessible" and "inaccessible" components. This decomposition allows us to consider the thermodynamic efficiency of different measurements of the same system, given a set of constraints. We use our framework to analyze protocols subject to symmetry, modularity, and coarse-grained constraints, and consider various examples including the Szilard box, the 2D Ising model, and a multi-particle flashing ratchet.
Highlights
For an isothermal process that carries out the transformation p → p0, entropy production (EP) is given by
It can be seen that Eq (5) decomposes the nonequilibrium free energy FE ðpÞ into two terms: an accessible free energy FE 1⁄2φðpÞ, whose decrease over the course of the protocol may be extractable as work, and an inaccessible free energy D1⁄2pkφðpÞ=β, whose decrease over the course of the protocol cannot be turned into work and must be dissipated as EP
By combining Theorem 2 with recent results in stochastic thermodynamics [34,35], we show that the expectation of mp ðxÞ is equal to the difference of expected EPs, hmp ðxÞi 1⁄4 Σðp → p0 Þ − Σ1⁄2φðpÞ → φðp0 Þ, where h·i indicates expectation over trajectories sampled from initial distribution p
Summary
One of the foundational issues in thermodynamics is quantifying how much work is required to transform a system between two thermodynamic states. By Eq (1), the work that can be extracted by bringing this initial distribution to a uniform final distribution is again upper bounded by ðln 2Þ=β It seems that this bound should not be achievable, given the constrained set of available protocols (i.e., one can manipulate the system only by moving the vertical partition left and right). Imagine measuring a bit of information about the location of the particle and using this information to extract work from the system while driving it back to a uniform equilibrium distribution In this case, IðX; MÞ 1⁄4 ln 2 and IðX0 ; MÞ 1⁄4 0; if the system starts and ends with the uniform energy function, Eq (4). This bound depends only on the overall amount of information acquired by the measurement [as quantified by IðX; MÞ] and is completely insensitive to the content of that information [i.e., the particular pattern of correlations quantified by IðX; MÞ]
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