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Sanders J, Baldovin M, Muratore-Ginanneschi P. Optimal Control of Underdamped Systems: An Analytic Approach. JOURNAL OF STATISTICAL PHYSICS 2024; 191:117. [PMID: 39301104 PMCID: PMC11408580 DOI: 10.1007/s10955-024-03320-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/21/2024] [Accepted: 08/04/2024] [Indexed: 09/22/2024]
Abstract
Optimal control theory deals with finding protocols to steer a system between assigned initial and final states, such that a trajectory-dependent cost function is minimized. The application of optimal control to stochastic systems is an open and challenging research frontier, with a spectrum of applications ranging from stochastic thermodynamics to biophysics and data science. Among these, the design of nanoscale electronic components motivates the study of underdamped dynamics, leading to practical and conceptual difficulties. In this work, we develop analytic techniques to determine protocols steering finite time transitions at a minimum thermodynamic cost for stochastic underdamped dynamics. As cost functions, we consider two paradigmatic thermodynamic indicators. The first is the Kullback-Leibler divergence between the probability measure of the controlled process and that of a reference process. The corresponding optimization problem is the underdamped version of the Schrödinger diffusion problem that has been widely studied in the overdamped regime. The second is the mean entropy production during the transition, corresponding to the second law of modern stochastic thermodynamics. For transitions between Gaussian states, we show that optimal protocols satisfy a Lyapunov equation, a central tool in stability analysis of dynamical systems. For transitions between states described by general Maxwell-Boltzmann distributions, we introduce an infinite-dimensional version of the Poincaré-Lindstedt multiscale perturbation theory around the overdamped limit. This technique fundamentally improves the standard multiscale expansion. Indeed, it enables the explicit computation of momentum cumulants, whose variation in time is a distinctive trait of underdamped dynamics and is directly accessible to experimental observation. Our results allow us to numerically study cost asymmetries in expansion and compression processes and make predictions for inertial corrections to optimal protocols in the Landauer erasure problem at the nanoscale.
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Affiliation(s)
- Julia Sanders
- Department of Mathematics and Statistics, University of Helsinki, 00014 Helsinki, Finland
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Bormashenko E. Landauer Bound in the Context of Minimal Physical Principles: Meaning, Experimental Verification, Controversies and Perspectives. ENTROPY (BASEL, SWITZERLAND) 2024; 26:423. [PMID: 38785672 PMCID: PMC11119825 DOI: 10.3390/e26050423] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2024] [Revised: 04/25/2024] [Accepted: 05/14/2024] [Indexed: 05/25/2024]
Abstract
The physical roots, interpretation, controversies, and precise meaning of the Landauer principle are surveyed. The Landauer principle is a physical principle defining the lower theoretical limit of energy consumption necessary for computation. It states that an irreversible change in information stored in a computer, such as merging two computational paths, dissipates a minimum amount of heat kBTln2 per a bit of information to its surroundings. The Landauer principle is discussed in the context of fundamental physical limiting principles, such as the Abbe diffraction limit, the Margolus-Levitin limit, and the Bekenstein limit. Synthesis of the Landauer bound with the Abbe, Margolus-Levitin, and Bekenstein limits yields the minimal time of computation, which scales as τmin~hkBT. Decreasing the temperature of a thermal bath will decrease the energy consumption of a single computation, but in parallel, it will slow the computation. The Landauer principle bridges John Archibald Wheeler's "it from bit" paradigm and thermodynamics. Experimental verifications of the Landauer principle are surveyed. The interrelation between thermodynamic and logical irreversibility is addressed. Generalization of the Landauer principle to quantum and non-equilibrium systems is addressed. The Landauer principle represents the powerful heuristic principle bridging physics, information theory, and computer engineering.
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Affiliation(s)
- Edward Bormashenko
- Department of Chemical Engineering, Biotechnology and Materials, Engineering Sciences Faculty, Ariel University, Ariel 407000, Israel
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Seet I, Ouldridge TE, Doye JPK. Simulation of reversible molecular mechanical logic gates and circuits. Phys Rev E 2023; 107:024134. [PMID: 36932514 DOI: 10.1103/physreve.107.024134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2022] [Accepted: 10/21/2022] [Indexed: 06/18/2023]
Abstract
Landauer's principle places a fundamental lower limit on the work required to perform a logically irreversible operation. Logically reversible gates provide a way to avoid these work costs and also simplify the task of making the computation as a whole thermodynamically reversible. The inherent reversibility of mechanical logic gates would make them good candidates for the design of practical logically reversible computing systems if not for the relatively large size and mass of such systems. In this paper we outline the design and simulation of reversible molecular mechanical logic gates that come close to the limits of thermodynamic reversibility even under the effects of thermal noise, and outline associated circuit components from which arbitrary combinatorial reversible circuits can be constructed and simulated. We demonstrate that isolated components can be operated in a thermodynamically reversible manner, and explore the complexities of combining components to implement more complex computations. Finally, we demonstrate a method to construct arbitrarily large reversible combinatorial circuits using multiple external controls and signal boosters with a working half-adder circuit.
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Affiliation(s)
- Ian Seet
- Physical and Theoretical Chemistry Laboratory, Department of Chemistry, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom
| | - Thomas E Ouldridge
- Department of Bioengineering, Imperial College London, Royal School of Mines, Exhibition Road, London SW7 2AZ, United Kingdom
| | - Jonathan P K Doye
- Physical and Theoretical Chemistry Laboratory, Department of Chemistry, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom
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Dago S, Bellon L. Dynamics of Information Erasure and Extension of Landauer's Bound to Fast Processes. PHYSICAL REVIEW LETTERS 2022; 128:070604. [PMID: 35244423 DOI: 10.1103/physrevlett.128.070604] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2021] [Accepted: 01/31/2022] [Indexed: 06/14/2023]
Abstract
Using a double-well potential as a physical memory, we study with experiments and numerical simulations the energy exchanges during erasure processes, and model quantitatively the cost of fast operation. Within the stochastic thermodynamics framework we find the origins of the overhead to Landauer's bound required for fast operations: in the overdamped regime this term mainly comes from the dissipation, while in the underdamped regime it stems from the heating of the memory. Indeed, the system is thermalized with its environment at all times during quasistatic protocols, but for fast ones, the inefficient heat transfer to the thermostat is delayed with respect to the work influx, resulting in a transient temperature rise. The warming, quantitatively described by a comprehensive statistical physics description of the erasure process, is noticeable on both the kinetic and potential energy: they no longer comply with equipartition. The mean work and heat to erase the information therefore increase accordingly. They are both bounded by an effective Landauer's limit k_{B}T_{eff}ln2, where T_{eff} is a weighted average of the actual temperature of the memory during the process.
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Affiliation(s)
- Salambô Dago
- Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
| | - Ludovic Bellon
- Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
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Boyd AB, Patra A, Jarzynski C, Crutchfield JP. Shortcuts to Thermodynamic Computing: The Cost of Fast and Faithful Information Processing. JOURNAL OF STATISTICAL PHYSICS 2022; 187:17. [PMID: 35400756 PMCID: PMC8960662 DOI: 10.1007/s10955-022-02871-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2021] [Accepted: 12/17/2021] [Indexed: 05/10/2023]
Abstract
Landauer's Principle states that the energy cost of information processing must exceed the product of the temperature, Boltzmann's constant, and the change in Shannon entropy of the information-bearing degrees of freedom. However, this lower bound is achievable only for quasistatic, near-equilibrium computations-that is, only over infinite time. In practice, information processing takes place in finite time, resulting in dissipation and potentially unreliable logical outcomes. For overdamped Langevin dynamics, we show that counterdiabatic potentials can be crafted to guide systems rapidly and accurately along desired computational paths, providing shortcuts that allow for the precise design of finite-time computations. Such shortcuts require additional work, beyond Landauer's bound, that is irretrievably dissipated into the environment. We show that this dissipated work is proportional to the computation rate as well as the square of the information-storing system's length scale. As a paradigmatic example, we design shortcuts to create, erase, and transfer a bit of information metastably stored in a double-well potential. Though dissipated work generally increases with operation fidelity, we show that it is possible to compute with perfect fidelity in finite time with finite work. We also show that the robustness of information storage affects an operation's energetic cost-specifically, the dissipated work scales as the information lifetime of the bistable system. Our analysis exposes a rich and nuanced relationship between work, speed, size of the information-bearing degrees of freedom, storage robustness, and the difference between initial and final informational statistics.
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Affiliation(s)
- Alexander B. Boyd
- Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616 USA
| | - Ayoti Patra
- Department of Physics, University of Maryland, College Park, MD 20742 USA
| | - Christopher Jarzynski
- Department of Physics, University of Maryland, College Park, MD 20742 USA
- Department of Chemistry and Biochemistry, University of Maryland, College Park, MD 20742 USA
- Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742 USA
| | - James P. Crutchfield
- Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616 USA
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Zhen YZ, Egloff D, Modi K, Dahlsten O. Universal Bound on Energy Cost of Bit Reset in Finite Time. PHYSICAL REVIEW LETTERS 2021; 127:190602. [PMID: 34797137 DOI: 10.1103/physrevlett.127.190602] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2021] [Accepted: 09/28/2021] [Indexed: 06/13/2023]
Abstract
We consider how the energy cost of bit reset scales with the time duration of the protocol. Bit reset necessarily takes place in finite time, where there is an extra penalty on top of the quasistatic work cost derived by Landauer. This extra energy is dissipated as heat in the computer, inducing a fundamental limit on the speed of irreversible computers. We formulate a hardware-independent expression for this limit in the framework of stochastic processes. We derive a closed-form lower bound on the work penalty as a function of the time taken for the protocol and bit reset error. It holds for discrete as well as continuous systems, assuming only that the master equation respects detailed balance.
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Affiliation(s)
- Yi-Zheng Zhen
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
| | - Dario Egloff
- Institute of Theoretical Physics, Technische Universität Dresden, D-01062 Dresden, Germany
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
| | - Kavan Modi
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - Oscar Dahlsten
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
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Proesmans K, Ehrich J, Bechhoefer J. Finite-Time Landauer Principle. PHYSICAL REVIEW LETTERS 2020; 125:100602. [PMID: 32955336 DOI: 10.1103/physrevlett.125.100602] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2020] [Accepted: 08/14/2020] [Indexed: 06/11/2023]
Abstract
We study the thermodynamic cost associated with the erasure of one bit of information over a finite amount of time. We present a general framework for minimizing the average work required when full control of a system's microstates is possible. In addition to exact numerical results, we find simple bounds proportional to the variance of the microscopic distribution associated with the state of the bit. In the short-time limit, we get a closed expression for the minimum average amount of work needed to erase a bit. The average work associated with the optimal protocol can be up to a factor of 4 smaller relative to protocols constrained to end in local equilibrium. Assessing prior experimental and numerical results based on heuristic protocols, we find that our bounds often dissipate an order of magnitude less energy.
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Affiliation(s)
- Karel Proesmans
- Department of Physics, Simon Fraser University, Burnaby, British Columbia,V5A 1S6, Canada
- Hasselt University, B-3590 Diepenbeek, Belgium
| | - Jannik Ehrich
- Department of Physics, Simon Fraser University, Burnaby, British Columbia,V5A 1S6, Canada
| | - John Bechhoefer
- Department of Physics, Simon Fraser University, Burnaby, British Columbia,V5A 1S6, Canada
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