Memoryless Thermodynamics? A Reply

We reply to arXiv:1508.00203 `Comment on "Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets" (arXiv:1507.01537v2)'.Comment: 4 pages; http://csc.ucdavis.edu/~cmg/compmech/pubs/MerhavReply.ht

Correlation-powered Information Engines and the Thermodynamics of Self-Correction

Information engines can use structured environments as a resource to generate work by randomizing ordered inputs and leveraging the increased Shannon entropy to transfer energy from a thermal reservoir to a work reservoir. We give a broadly applicable expression for the work production of an information engine, generally modeled as a memoryful channel that communicates inputs to outputs as it interacts with an evolving environment. The expression establishes that an information engine must have more than one memory state in order to leverage input environment correlations. To emphasize this functioning, we designed an information engine powered solely by temporal correlations and not by statistical biases, as employed by previous engines. Key to this is the engine's ability to synchronize---the engine automatically returns to a desired dynamical phase when thrown into an unwanted, dissipative phase by corruptions in the input---that is, by unanticipated environmental fluctuations. This self-correcting mechanism is robust up to a critical level of corruption, beyond which the system fails to act as an engine. We give explicit analytical expressions for both work and critical corruption level and summarize engine performance via a thermodynamic-function phase diagram over engine control parameters. The results reveal a new thermodynamic mechanism based on nonergodicity that underlies error correction as it operates to support resilient engineered and biological systems.Comment: 22 pages, 13 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/tos.ht

Identifying Functional Thermodynamics in Autonomous Maxwellian Ratchets

We introduce a family of Maxwellian Demons for which correlations among information bearing degrees of freedom can be calculated exactly and in compact analytical form. This allows one to precisely determine Demon functional thermodynamic operating regimes, when previous methods either misclassify or simply fail due to approximations they invoke. This reveals that these Demons are more functional than previous candidates. They too behave either as engines, lifting a mass against gravity by extracting energy from a single heat reservoir, or as Landauer erasers, consuming external work to remove information from a sequence of binary symbols by decreasing their individual uncertainty. Going beyond these, our Demon exhibits a new functionality that erases bits not by simply decreasing individual-symbol uncertainty, but by increasing inter-bit correlations (that is, by adding temporal order) while increasing single-symbol uncertainty. In all cases, but especially in the new erasure regime, exactly accounting for informational correlations leads to tight bounds on Demon performance, expressed as a refined Second Law of Thermodynamics that relies on the Kolmogorov-Sinai entropy for dynamical processes and not on changes purely in system configurational entropy, as previously employed. We rigorously derive the refined Second Law under minimal assumptions and so it applies quite broadly---for Demons with and without memory and input sequences that are correlated or not. We note that general Maxwellian Demons readily violate previously proposed, alternative such bounds, while the current bound still holds.Comment: 13 pages, 9 figures, http://csc.ucdavis.edu/~cmg/compmech/pubs/mrd.ht

Above and Beyond the Landauer Bound: Thermodynamics of Modularity

Information processing typically occurs via the composition of modular units, such as universal logic gates. The benefit of modular information processing, in contrast to globally integrated information processing, is that complex global computations are more easily and flexibly implemented via a series of simpler, localized information processing operations which only control and change local degrees of freedom. We show that, despite these benefits, there are unavoidable thermodynamic costs to modularity---costs that arise directly from the operation of localized processing and that go beyond Landauer's dissipation bound for erasing information. Integrated computations can achieve Landauer's bound, however, when they globally coordinate the control of all of an information reservoir's degrees of freedom. Unfortunately, global correlations among the information-bearing degrees of freedom are easily lost by modular implementations. This is costly since such correlations are a thermodynamic fuel. We quantify the minimum irretrievable dissipation of modular computations in terms of the difference between the change in global nonequilibrium free energy, which captures these global correlations, and the local (marginal) change in nonequilibrium free energy, which bounds modular work production. This modularity dissipation is proportional to the amount of additional work required to perform the computational task modularly. It has immediate consequences for physically embedded transducers, known as information ratchets. We show how to circumvent modularity dissipation by designing internal ratchet states that capture the global correlations and patterns in the ratchet's information reservoir. Designed in this way, information ratchets match the optimum thermodynamic efficiency of globally integrated computations.Comment: 17 pages, 9 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/idolip.ht

PT-Symmetric, Quasi-Exactly Solvable matrix Hamiltonians

Matrix quasi exactly solvable operators are considered and new conditions are determined to test whether a matrix differential operator possesses one or several finite dimensional invariant vector spaces. New examples of $2\times 2$-matrix quasi exactly solvable operators are constructed with the emphasis set on PT-symmetric Hamiltonians.Comment: 14 pages, 1 figure, one equation corrected, results adde

On sums and reciprocal sum of generalized fibonacci numbers

The purpose of this report is to analyze the properties of Fibonacci numbers modulo a Lucas numbers. Any Fibonacci number, except the first two, is the sum of the two immediately preceding Fibonacci numbers and closely related to Fibonacci numbers are Lucas number. Fibonacci numbers are used in the application of computer algorithms. They can be used to compress audio files and generate code. The most recently Fibonacci number have been used to symbolize mathematical relationship in the Davinci code as well as in the TV shows fringe, criminal minds. In this report, some generalized identities of Holliday and Komatsu have been studied results of Liu and Zhao obtained by applying the floor function to the reciprocal of infinite sums of reciprocal generalized Fibonacci numbers and the infinite sums of reciprocal generalized Fibonacci sums have been extended

Parity Nonconservation in Odd-isotopes of Single Trapped Atomic Ions

We have estimated the size of the light-shifts due to parity nonconservation (PNC) interactions in different isotopes of Ba+ and Ra+ ions based on the work of Fortson [Phys. Rev. Lett. 70, 2383 (1993)]. We have used the nuclear spin independent (NSI) amplitudes calculated earlier by us [Phys. Rev. Lett. 96, 163003 (2006); Phys. Rev. A 78, 050501(R) (2008)] and we have employed the third order many-body perturbation theory (MBPT(3)) in this work to estimate the nuclear spin dependent (NSD) amplitudes in these ions. Ra+ is found to be more favourable than Ba+ for measuring both the NSI and NSD PNC observables.Comment: 5 pages, 1 tabl