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1. Thermodynamics of Computations with Absolute Irreversibility, Unidirectional Transitions, and Stochastic Computation Times

   https://doi.org/10.1103/PhysRevX.14.021026  

   Manzano, Gonzalo; Kardeş, Gülce; Roldán, Édgar; Wolpert, David H (May 2024, Physical Review X)

   Developing a thermodynamic theory of computation is a challenging task at the interface of nonequilibrium thermodynamics and computer science. In particular, this task requires dealing with difficulties such as stochastic halting times, unidirectional (possibly deterministic) transitions, and restricted initial conditions, features common in real-world computers. Here, we present a framework which tackles all such difficulties by extending the martingale theory of nonequilibrium thermodynamics to generic nonstationary Markovian processes, including those with broken detailed balance and/or absolute irreversibility. We derive several universal fluctuation relations and second-law-like inequalities that provide both lower and upper bounds on the intrinsic dissipation (mismatch cost) associated with any periodic process—in particular, the periodic processes underlying all current digital computation. Crucially, these bounds apply even if the process has stochastic stopping times, as it does in many computational machines. We illustrate our results with exhaustive numerical simulations of deterministic finite automata processing bit strings, one of the fundamental models of computation from theoretical computer science. We also provide universal equalities and inequalities for the acceptance probability of words of a given length by a deterministic finite automaton in terms of thermodynamic quantities, and outline connections between computer science and stochastic resetting. Our results, while motivated from the computational context, are applicable far more broadly. 

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2. Alexandrov spaces with integral current structure

   https://doi.org/https://dx.doi.org/10.4310/CAG.2021.v29.n1.a4  

   Jaramillo, Maree; Perales, Raquel; Rajan, Priyanka; Searle, Catherine; Siffert, Anna (January 2021, Communications in analysis and geometry)

   Liu, Kefeng (Ed.)

   We endow each closed, orientable Alexandrov space (X, d) with an integral current T of weight equal to 1, ∂T = 0 and set(T) = X, in other words, we prove that (X, d, T) is an integral current space with no boundary. Combining this result with a result of Li and Perales, we show that non-collapsing sequences of these spaces with uniform lower curvature and diameter bounds admit subsequences whose Gromov-Hausdorff and intrinsic flat limits agree. 

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3. Analysis of lower bounds for quantum control times and their relation to the quantum speed limit

   Poggi, Pablo M (April 2020, Anales AFA)

   Limitations to the speed of evolution of quantum systems, typically referred to as quantum speed limits (QSLs), have important consequences for quantum control problems. However, in its standard formulation, is not straightforward to obtain meaningful QSL bounds for time-dependent Hamiltonians with unknown control parameters. In this paper we present a short introductory overview of quantum speed limit for unitary dynamics and its connection to quantum control. We then analyze potential methods for obtaining new bounds on control times inspired by the QSL. We finally extend the work in [Poggi, Lombardo andWisniacki EPL 104 40005 (2013)] by studying the properties and limitations of these new bounds in the context of a driven two-level quantum system. 

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4. Energy Conversion and Entropy Production in Biased Random Walk Processes—From Discrete Modeling to the Continuous Limit

   https://doi.org/10.3390/e25081218  

   Kirchberg, Henning; Nitzan, Abraham (August 2023, Entropy)

   We considered discrete and continuous representations of a thermodynamic process in which a random walker (e.g., a molecular motor on a molecular track) uses periodically pumped energy (work) to pass N sites and move energetically downhill while dissipating heat. Interestingly, we found that, starting from a discrete model, the limit in which the motion becomes continuous in space and time (N→∞) is not unique and depends on what physical observables are assumed to be unchanged in the process. In particular, one may (as usually done) choose to keep the speed and diffusion coefficient fixed during this limiting process, in which case, the entropy production is affected. In addition, we also studied processes in which the entropy production is kept constant as N→∞ at the cost of a modified speed or diffusion coefficient. Furthermore, we also combined this dynamics with work against an opposing force, which made it possible to study the effect of discretization of the process on the thermodynamic efficiency of transferring the power input to the power output. Interestingly, we found that the efficiency was increased in the limit of N→∞. Finally, we investigated the same process when transitions between sites can only happen at finite time intervals and studied the impact of this time discretization on the thermodynamic variables as the continuous limit is approached. 

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5. A Unified Analysis of First-Order Methods for Smooth Games via Integral Quadratic Constraints

   Zhang, Guodong; Bao, Xuchan; Lessard, Laurent; Grosse, Roger (January 2021, Journal of machine learning research)

   The theory of integral quadratic constraints (IQCs) allows the certification of exponential convergence of interconnected systems containing nonlinear or uncertain elements. In this work, we adapt the IQC theory to study first-order methods for smooth and strongly-monotone games and show how to design tailored quadratic constraints to get tight upper bounds of convergence rates. Using this framework, we recover the existing bound for the gradient method~(GD), derive sharper bounds for the proximal point method~(PPM) and optimistic gradient method~(OG), and provide for the first time a global convergence rate for the negative momentum method~(NM) with an iteration complexity O(κ1.5), which matches its known lower bound. In addition, for time-varying systems, we prove that the gradient method with optimal step size achieves the fastest provable worst-case convergence rate with quadratic Lyapunov functions. Finally, we further extend our analysis to stochastic games and study the impact of multiplicative noise on different algorithms. We show that it is impossible for an algorithm with one step of memory to achieve acceleration if it only queries the gradient once per batch (in contrast with the stochastic strongly-convex optimization setting, where such acceleration has been demonstrated). However, we exhibit an algorithm which achieves acceleration with two gradient queries per batch. 

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