1
|
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.
Collapse
Affiliation(s)
- Edward Bormashenko
- Department of Chemical Engineering, Biotechnology and Materials, Engineering Sciences Faculty, Ariel University, Ariel 407000, Israel
| |
Collapse
|
2
|
Pal PS, Pal A, Park H, Lee JS. Thermodynamic trade-off relation for first passage time in resetting processes. Phys Rev E 2023; 108:044117. [PMID: 37978646 DOI: 10.1103/physreve.108.044117] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2023] [Accepted: 09/14/2023] [Indexed: 11/19/2023]
Abstract
Resetting is a strategy for boosting the speed of a target-searching process. Since its introduction over a decade ago, most studies have been carried out under the assumption that resetting takes place instantaneously. However, due to its irreversible nature, resetting processes incur a thermodynamic cost, which becomes infinite in the case of instantaneous resetting. Here, we take into consideration both the cost and the first passage time (FPT) required for a resetting process, in which the reset or return to the initial location is implemented using a trapping potential over a finite but random time period. An iterative generating function and a counting functional method à la Feynman and Kac are employed to calculate the FPT and the average work for this process. From these results, we obtain an explicit form of the time-cost trade-off relation, which provides the lower bound of the mean FPT for a given work input when the trapping potential is linear. This trade-off relation clearly shows that instantaneous resetting is achievable only when an infinite amount of work is provided. More surprisingly, the trade-off relation derived from the linear potential seems to be valid for a wide range of trapping potentials. In addition, we have also shown that the fixed-time or sharp resetting can further enhance the trade-off relation compared to that of the stochastic resetting.
Collapse
Affiliation(s)
- P S Pal
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
| | - Arnab Pal
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
| | - Hyunggyu Park
- Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Korea
| | - Jae Sung Lee
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
| |
Collapse
|
3
|
Whitelam S. How to train your demon to do fast information erasure without heat production. Phys Rev E 2023; 108:044138. [PMID: 37978603 DOI: 10.1103/physreve.108.044138] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/02/2023] [Accepted: 09/05/2023] [Indexed: 11/19/2023]
Abstract
Time-dependent protocols that perform irreversible logical operations, such as memory erasure, cost work and produce heat, placing bounds on the efficiency of computers. Here we use a prototypical computer model of a physical memory to show that it is possible to learn feedback-control protocols to do fast memory erasure without input of work or production of heat. These protocols, which are enacted by a neural-network "demon," do not violate the second law of thermodynamics because the demon generates more heat than the memory absorbs. The result is a form of nonlocal heat exchange in which one computation is rendered energetically favorable while a compensating one produces heat elsewhere, a tactic that could be used to rationally design the flow of energy within a computer.
Collapse
Affiliation(s)
- Stephen Whitelam
- Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
| |
Collapse
|
4
|
Scandi M, Barker D, Lehmann S, Dick KA, Maisi VF, Perarnau-Llobet M. Minimally Dissipative Information Erasure in a Quantum Dot via Thermodynamic Length. PHYSICAL REVIEW LETTERS 2022; 129:270601. [PMID: 36638287 DOI: 10.1103/physrevlett.129.270601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/2022] [Revised: 11/29/2022] [Accepted: 12/12/2022] [Indexed: 06/17/2023]
Abstract
In this Letter, we explore the use of thermodynamic length to improve the performance of experimental protocols. In particular, we implement Landauer erasure on a driven electron level in a semiconductor quantum dot, and compare the standard protocol in which the energy is increased linearly in time with the one coming from geometric optimization. The latter is obtained by choosing a suitable metric structure, whose geodesics correspond to optimal finite-time thermodynamic protocols in the slow driving regime. We show experimentally that geodesic drivings minimize dissipation for slow protocols, with a bigger improvement as one approaches perfect erasure. Moreover, the geometric approach also leads to smaller dissipation even when the time of the protocol becomes comparable with the equilibration timescale of the system, i.e., away from the slow driving regime. Our results also illustrate, in a single-electron device, a fundamental principle of thermodynamic geometry: optimal finite-time thermodynamic protocols are those with constant dissipation rate along the process.
Collapse
Affiliation(s)
- Matteo Scandi
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels, Barcelona 08860, Spain
| | - David Barker
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
| | - Sebastian Lehmann
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
| | - Kimberly A Dick
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
- Centre for Analysis and Synthesis, Lund University, Box 124, 22100 Lund, Sweden
| | - Ville F Maisi
- NanoLund and Solid State Physics, Lund University, Box 118, 22100 Lund, Sweden
| | | |
Collapse
|
5
|
Lee JS, Lee S, Kwon H, Park H. Speed Limit for a Highly Irreversible Process and Tight Finite-Time Landauer's Bound. PHYSICAL REVIEW LETTERS 2022; 129:120603. [PMID: 36179191 DOI: 10.1103/physrevlett.129.120603] [Citation(s) in RCA: 9] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2022] [Accepted: 08/25/2022] [Indexed: 06/16/2023]
Abstract
Landauer's bound is the minimum thermodynamic cost for erasing one bit of information. As this bound is achievable only for quasistatic processes, finite-time operation incurs additional energetic costs. We find a tight finite-time Landauer's bound by establishing a general form of the classical speed limit. This tight bound well captures the divergent behavior associated with the additional cost of a highly irreversible process, which scales differently from a nearly irreversible process. We also find an optimal dynamics which saturates the equality of the bound. We demonstrate the validity of this bound via discrete one-bit and coarse-grained bit systems. Our Letter implies that more heat dissipation than expected occurs during high-speed irreversible computation.
Collapse
Affiliation(s)
- Jae Sung Lee
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
| | - Sangyun Lee
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
| | - Hyukjoon Kwon
- School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea
| | - Hyunggyu Park
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea
- Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Korea
| |
Collapse
|
6
|
Ma YH, Chen JF, Sun CP, Dong H. Minimal energy cost to initialize a bit with tolerable error. Phys Rev E 2022; 106:034112. [PMID: 36266886 DOI: 10.1103/physreve.106.034112] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/09/2022] [Accepted: 08/25/2022] [Indexed: 06/16/2023]
Abstract
Landauer's principle imposes a fundamental limit on the energy cost to perfectly initialize a classical bit, which is only reached under the ideal operation with infinitely long time. The question on the cost in the practical operation for a bit has been raised under the constraint by the finiteness of operation time. We discover a raise-up of energy cost by L^{2}(ε)/τ from the Landaeur's limit (k_{B}Tln2) for a finite-time τ initialization of a bit with an error probability ε. The thermodynamic length L(ε) between the states before and after initializing in the parametric space increases monotonously as the error decreases. For example, in the constant dissipation coefficient (γ_{0}) case, the minimal additional cost is 0.997k_{B}T/(γ_{0}τ) for ε=1% and 1.288k_{B}T/(γ_{0}τ) for ε=0.1%. Furthermore, the optimal protocol to reach the bound of minimal energy cost is proposed for the bit initialization realized via a finite-time isothermal process.
Collapse
Affiliation(s)
- Yu-Han Ma
- Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
| | - Jin-Fu Chen
- Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
- Beijing Computational Science Research Center, Beijing 100193, China
- School of Physics, Peking University, Beijing, 100871, China
| | - C P Sun
- Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
- Beijing Computational Science Research Center, Beijing 100193, China
| | - Hui Dong
- Graduate School of China Academy of Engineering Physics, No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
| |
Collapse
|
7
|
Tian Y, Sun P. Information thermodynamics of encoding and encoders. CHAOS (WOODBURY, N.Y.) 2022; 32:063109. [PMID: 35778156 DOI: 10.1063/5.0068115] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/23/2021] [Accepted: 05/09/2022] [Indexed: 06/15/2023]
Abstract
Non-isolated systems have diverse coupling relations with the external environment. These relations generate complex thermodynamics and information transmission between the system and its environment. The framework depicted in the current research attempts to glance at the critical role of the internal orders inside the non-isolated system in shaping the information thermodynamics coupling. We characterize the coupling as a generalized encoding process, where the system acts as an information thermodynamics encoder to encode the external information based on thermodynamics. We formalize the encoding process in the context of the nonequilibrium second law of thermodynamics, revealing an intrinsic difference in information thermodynamics characteristics between information thermodynamics encoders with and without internal correlations. During the information encoding process of an external source Y, specific sub-systems in an encoder X with internal correlations can exceed the information thermodynamics bound on ( X , Y ) and encode more information than system X works as a whole. We computationally verify this theoretical finding in an Ising model with a random external field and a neural data set of the human brain during visual perception and recognition. Our analysis demonstrates that the stronger internal correlation inside these systems implies a higher possibility for specific sub-systems to encode more information than the global one. These findings may suggest a new perspective in studying information thermodynamics in diverse physical and biological systems.
Collapse
Affiliation(s)
- Yang Tian
- Department of Psychology, Tsinghua Laboratory of Brain and Intelligence, Tsinghua University, Beijing 100084, China
| | - Pei Sun
- Department of Psychology, Tsinghua Laboratory of Brain and Intelligence, Tsinghua University, Beijing 100084, China
| |
Collapse
|
8
|
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.
Collapse
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
| |
Collapse
|
9
|
Remlein B, Seifert U. Optimality of nonconservative driving for finite-time processes with discrete states. Phys Rev E 2021; 103:L050105. [PMID: 34134247 DOI: 10.1103/physreve.103.l050105] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2021] [Accepted: 05/11/2021] [Indexed: 11/07/2022]
Abstract
An optimal finite-time process drives a given initial distribution to a given final one in a given time at the lowest cost as quantified by total entropy production. We prove that for a system with discrete states this optimal process involves nonconservative driving, i.e., a genuine driving affinity, in contrast to the case of a system with continuous states. In a multicyclic network, the optimal driving affinity is bounded by the number of states within each cycle. If the driving affects forward and backwards rates nonsymmetrically, the bound additionally depends on a structural parameter characterizing this asymmetry.
Collapse
Affiliation(s)
- Benedikt Remlein
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| |
Collapse
|
10
|
Dago S, Pereda J, Barros N, Ciliberto S, Bellon L. Information and Thermodynamics: Fast and Precise Approach to Landauer's Bound in an Underdamped Micromechanical Oscillator. PHYSICAL REVIEW LETTERS 2021; 126:170601. [PMID: 33988419 DOI: 10.1103/physrevlett.126.170601] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2021] [Accepted: 04/02/2021] [Indexed: 06/12/2023]
Abstract
The Landauer principle states that at least k_{B}Tln2 of energy is required to erase a 1-bit memory, with k_{B}T the thermal energy of the system. We study the effects of inertia on this bound using as one-bit memory an underdamped micromechanical oscillator confined in a double-well potential created by a feedback loop. The potential barrier is precisely tunable in the few k_{B}T range. We measure, within the stochastic thermodynamic framework, the work and the heat of the erasure protocol. We demonstrate experimentally and theoretically that, in this underdamped system, the Landauer bound is reached with a 1% uncertainty, with protocols as short as 100 ms.
Collapse
Affiliation(s)
- Salambô Dago
- Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
| | - Jorge Pereda
- Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
| | - Nicolas Barros
- Univ Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France
| | - Sergio Ciliberto
- 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
| |
Collapse
|
11
|
Frim AG, Zhong A, Chen SF, Mandal D, DeWeese MR. Engineered swift equilibration for arbitrary geometries. Phys Rev E 2021; 103:L030102. [PMID: 33862711 DOI: 10.1103/physreve.103.l030102] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2020] [Accepted: 02/12/2021] [Indexed: 11/07/2022]
Abstract
Engineered swift equilibration (ESE) is a class of driving protocols that enforce an equilibrium distribution with respect to external control parameters at the beginning and end of rapid state transformations of open, classical nonequilibrium systems. ESE protocols have previously been derived and experimentally realized for Brownian particles in simple, one-dimensional, time-varying trapping potentials; one recent study considered ESE in two-dimensional Euclidean configuration space. Here we extend the ESE framework to generic, overdamped Brownian systems in arbitrary curved configuration space and illustrate our results with specific examples not amenable to previous techniques. Our approach may be used to impose the necessary dynamics to control the full temporal configurational distribution in a wide variety of experimentally realizable settings.
Collapse
Affiliation(s)
- Adam G Frim
- Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
| | - Adrianne Zhong
- Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
| | - Shi-Fan Chen
- Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
| | - Dibyendu Mandal
- Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
| | - Michael R DeWeese
- Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA.,Redwood Center For Theoretical Neuroscience and Helen Wills Neuroscience Institute, University of California, Berkeley, Berkeley, California 94720, USA
| |
Collapse
|
12
|
Falasco G, Esposito M. Dissipation-Time Uncertainty Relation. PHYSICAL REVIEW LETTERS 2020; 125:120604. [PMID: 33016734 DOI: 10.1103/physrevlett.125.120604] [Citation(s) in RCA: 29] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2020] [Accepted: 08/14/2020] [Indexed: 05/10/2023]
Abstract
We show that the entropy production rate bounds the rate at which physical processes can be performed in stochastic systems far from equilibrium. In particular, we prove the fundamental tradeoff ⟨S[over ˙]_{e}⟩T≥k_{B} between the entropy flow ⟨S[over ˙]_{e}⟩ into the reservoirs and the mean time T to complete any process whose time-reversed is exponentially rarer. This dissipation-time uncertainty relation is a novel form of speed limit: the smaller the dissipation, the larger the time to perform a process.
Collapse
Affiliation(s)
- Gianmaria Falasco
- Complex Systems and Statistical Mechanics, Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Massimiliano Esposito
- Complex Systems and Statistical Mechanics, Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| |
Collapse
|
13
|
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.
Collapse
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
| |
Collapse
|
14
|
Proesmans K, Ehrich J, Bechhoefer J. Optimal finite-time bit erasure under full control. Phys Rev E 2020; 102:032105. [PMID: 33075986 DOI: 10.1103/physreve.102.032105] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2020] [Accepted: 08/11/2020] [Indexed: 05/25/2023]
Abstract
We study the finite-time erasure of a one-bit memory consisting of a one-dimensional double-well potential, with each well encoding a memory macrostate. We focus on setups that provide full control over the form of the potential-energy landscape and derive protocols that minimize the average work needed to erase the bit over a fixed amount of time. We allow for cases where only some of the information encoded in the bit is erased. For systems required to end up in a local-equilibrium state, we calculate the minimum amount of work needed to erase a bit explicitly, in terms of the equilibrium Boltzmann distribution corresponding to the system's initial potential. The minimum work is inversely proportional to the duration of the protocol. The erasure cost may be further reduced by relaxing the requirement for a local-equilibrium final state and allowing for any final distribution compatible with constraints on the probability to be in each memory macrostate. We also derive upper and lower bounds on the erasure cost.
Collapse
Affiliation(s)
- Karel Proesmans
- Department of Physics, Simon Fraser University, Burnaby, B.C., V5A 1S6, Canada
- Hasselt University, B-3590 Diepenbeek, Belgium
| | - Jannik Ehrich
- Department of Physics, Simon Fraser University, Burnaby, B.C., V5A 1S6, Canada
| | - John Bechhoefer
- Department of Physics, Simon Fraser University, Burnaby, B.C., V5A 1S6, Canada
| |
Collapse
|
15
|
Wolpert DH, Kolchinsky A, Owen JA. A space-time tradeoff for implementing a function with master equation dynamics. Nat Commun 2019; 10:1727. [PMID: 30988296 PMCID: PMC6465315 DOI: 10.1038/s41467-019-09542-x] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2018] [Accepted: 03/15/2019] [Indexed: 11/09/2022] Open
Abstract
Master equations are commonly used to model the dynamics of physical systems, including systems that implement single-valued functions like a computer’s update step. However, many such functions cannot be implemented by any master equation, even approximately, which raises the question of how they can occur in the real world. Here we show how any function over some “visible” states can be implemented with master equation dynamics—if the dynamics exploits additional, “hidden” states at intermediate times. We also show that any master equation implementing a function can be decomposed into a sequence of “hidden” timesteps, demarcated by changes in what state-to-state transitions have nonzero probability. In many real-world situations there is a cost both for more hidden states and for more hidden timesteps. Accordingly, we derive a “space–time” tradeoff between the number of hidden states and the number of hidden timesteps needed to implement any given function. Deterministic maps from initial to final states can always be modelled using the master equation formalism, provided additional “hidden” states are available. Here, the authors demonstrate a tradeoff between the required number of such states and the number of required, suitably defined “hidden time steps”.
Collapse
Affiliation(s)
- David H Wolpert
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501, USA. .,Arizona State University, Tempe, 85281, AZ, USA.
| | | | - Jeremy A Owen
- Physics of Living Systems Group, Department of Physics, Massachusetts Institute of Technology, 400 Tech Square, Cambridge, MA, 02139, USA
| |
Collapse
|
16
|
Thermodynamics of Majority-Logic Decoding in Information Erasure. ENTROPY 2019; 21:e21030284. [PMID: 33266999 PMCID: PMC7514764 DOI: 10.3390/e21030284] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/10/2019] [Revised: 02/25/2019] [Accepted: 03/11/2019] [Indexed: 01/31/2023]
Abstract
We investigate the performance of majority-logic decoding in both reversible and finite-time information erasure processes performed on macroscopic bits that contain N microscopic binary units. While we show that for reversible erasure protocols single-unit transformations are more efficient than majority-logic decoding, the latter is found to offer several benefits for finite-time erasure processes: Both the minimal erasure duration for a given erasure and the minimal erasure error for a given erasure duration are reduced, if compared to a single unit. Remarkably, the majority-logic decoding is also more efficient in both the small-erasure error and fast-erasure region. These benefits are also preserved under the optimal erasure protocol that minimizes the dissipated heat. Our work therefore shows that majority-logic decoding can lift the precision-speed-efficiency trade-off in information erasure processes.
Collapse
|
17
|
Strasberg P, Esposito M. Non-Markovianity and negative entropy production rates. Phys Rev E 2019; 99:012120. [PMID: 30780330 DOI: 10.1103/physreve.99.012120] [Citation(s) in RCA: 48] [Impact Index Per Article: 9.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2018] [Indexed: 11/07/2022]
Abstract
Entropy production plays a fundamental role in nonequilibrium thermodynamics to quantify the irreversibility of open systems. Its positivity can be ensured for a wide class of setups, but the entropy production rate can become negative sometimes. This is often taken as an indicator of non-Markovianity. We make this link precise by showing under which conditions a negative entropy production rate implies non-Markovianity and when it does not. For a system coupled to a single heat bath, this can be established within a unified language for two setups: (i) the dynamics resulting from a coarse-grained description of a Markovian master equation and (ii) the classical Hamiltonian dynamics of a system coupled to a bath. The quantum version of the latter result is shown not to hold despite the fact that the integrated thermodynamic description is formally equivalent to the classical case. The instantaneous fixed point of a non-Markovian dynamics plays an important role in our study. Our key contribution is to provide a consistent theoretical framework to study the finite-time thermodynamics of a large class of dynamics with a precise link to its non-Markovianity.
Collapse
Affiliation(s)
- Philipp Strasberg
- Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Massimiliano Esposito
- Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| |
Collapse
|
18
|
Chu D, Spinney RE. A thermodynamically consistent model of finite-state machines. Interface Focus 2018; 8:20180037. [PMID: 30443334 DOI: 10.1098/rsfs.2018.0037] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 09/14/2018] [Indexed: 01/26/2023] Open
Abstract
Finite-state machines (FSMs) are a theoretically and practically important model of computation. We propose a general, thermodynamically consistent model of FSMs and characterize the resource requirements of these machines. We model FSMs as time-inhomogeneous Markov chains. The computation is driven by instantaneous manipulations of the energy levels of the states. We calculate the entropy production of the machine, its error probability, and the time required to complete one update step. We find that a sequence of generalized bit-setting operations is sufficient to implement any FSM.
Collapse
Affiliation(s)
- Dominique Chu
- School of Computing, University of Kent, Canterbury CT2 7NF, UK
| | - Richard E Spinney
- Centre for Complex Systems, The University of Sydney, Sydney, New South Wales 2006, Australia
| |
Collapse
|
19
|
Boyd AB, Mandal D, Crutchfield JP. Correlation-powered information engines and the thermodynamics of self-correction. Phys Rev E 2017; 95:012152. [PMID: 28208508 DOI: 10.1103/physreve.95.012152] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2016] [Indexed: 05/23/2023]
Abstract
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 thermodynamic mechanism based on nonergodicity that underlies error correction as it operates to support resilient engineered and biological systems.
Collapse
Affiliation(s)
- Alexander B Boyd
- Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, California 95616, USA
| | - Dibyendu Mandal
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - James P Crutchfield
- Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, California 95616, USA
| |
Collapse
|
20
|
|
21
|
Zulkowski PR, DeWeese MR. Optimal protocols for slowly driven quantum systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:032113. [PMID: 26465432 DOI: 10.1103/physreve.92.032113] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2015] [Indexed: 06/05/2023]
Abstract
The design of efficient quantum information processing will rely on optimal nonequilibrium transitions of driven quantum systems. Building on a recently developed geometric framework for computing optimal protocols for classical systems driven in finite time, we construct a general framework for optimizing the average information entropy for driven quantum systems. Geodesics on the parameter manifold endowed with a positive semidefinite metric correspond to protocols that minimize the average information entropy production in finite time. We use this framework to explicitly compute the optimal entropy production for a simple two-state quantum system coupled to a heat bath of bosonic oscillators, which has applications to quantum annealing.
Collapse
Affiliation(s)
- Patrick R Zulkowski
- Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA Department of Mathematics, Berkeley City College, Berkeley, California 94704, USA and Redwood Center for Theoretical Neuroscience, University of California, Berkeley, Berkeley, California 94720, USA
| | - Michael R DeWeese
- Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA Redwood Center for Theoretical Neuroscience, University of California, Berkeley, Berkeley, California 94720, USA and Helen Wills Neuroscience Institute, University of California, Berkeley, Berkeley, California 94720, USA
| |
Collapse
|
22
|
Zulkowski PR, DeWeese MR. Optimal control of overdamped systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:032117. [PMID: 26465436 DOI: 10.1103/physreve.92.032117] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2015] [Indexed: 05/25/2023]
Abstract
Nonequilibrium physics encompasses a broad range of natural and synthetic small-scale systems. Optimizing transitions of such systems will be crucial for the development of nanoscale technologies and may reveal the physical principles underlying biological processes at the molecular level. Recent work has demonstrated that when a thermodynamic system is driven away from equilibrium then the space of controllable parameters has a Riemannian geometry induced by a generalized inverse diffusion tensor. We derive a simple, compact expression for the inverse diffusion tensor that depends solely on equilibrium information for a broad class of potentials. We use this formula to compute the minimal dissipation for two model systems relevant to small-scale information processing and biological molecular motors. In the first model, we optimally erase a single classical bit of information modeled by an overdamped particle in a smooth double-well potential. In the second model, we find the minimal dissipation of a simple molecular motor model coupled to an optical trap. In both models, we find that the minimal dissipation for the optimal protocol of duration τ is proportional to 1/τ, as expected, though the dissipation for the erasure model takes a different form than what we found previously for a similar system.
Collapse
Affiliation(s)
- Patrick R Zulkowski
- Department of Physics, University of California, Berkeley, California 94720, USA; Department of Mathematics, Berkeley City College, Berkeley, California 94704, USA; and Redwood Center for Theoretical Neuroscience, University of California, Berkeley, California 94720, USA
| | - Michael R DeWeese
- Department of Physics, University of California, Berkeley, California 94720, USA; Redwood Center for Theoretical Neuroscience, University of California, Berkeley, California 94720, USA; and Helen Wills Neuroscience Institute, University of California, Berkeley, California 94720, USA
| |
Collapse
|
23
|
Acconcia TV, Bonança MVS. Degenerate optimal paths in thermally isolated systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042141. [PMID: 25974472 DOI: 10.1103/physreve.91.042141] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2015] [Indexed: 06/04/2023]
Abstract
We present an analysis of the work performed on a system of interest that is kept thermally isolated during the switching of a control parameter. We show that there exists, for a certain class of systems, a finite-time family of switching protocols for which the work is equal to the quasistatic value. These optimal paths are obtained within linear response for systems initially prepared in a canonical distribution. According to our approach, such protocols are composed of a linear part plus a function which is odd with respect to time reversal. For systems with one degree of freedom, we claim that these optimal paths may also lead to the conservation of the corresponding adiabatic invariant. This points to an interesting connection between work and the conservation of the volume enclosed by the energy shell. To illustrate our findings, we solve analytically the harmonic oscillator and present numerical results for certain anharmonic examples.
Collapse
Affiliation(s)
- Thiago V Acconcia
- Instituto de Física 'Gleb Wataghin', Universidade Estadual de Campinas, 13083-859 Campinas, São Paulo, Brazil
| | - Marcus V S Bonança
- Instituto de Física 'Gleb Wataghin', Universidade Estadual de Campinas, 13083-859 Campinas, São Paulo, Brazil
| |
Collapse
|
24
|
Barato AC, Seifert U. Stochastic thermodynamics with information reservoirs. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042150. [PMID: 25375481 DOI: 10.1103/physreve.90.042150] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/05/2014] [Indexed: 05/10/2023]
Abstract
We generalize stochastic thermodynamics to include information reservoirs. Such information reservoirs, which can be modeled as a sequence of bits, modify the second law. For example, work extraction from a system in contact with a single heat bath becomes possible if the system also interacts with an information reservoir. We obtain an inequality, and the corresponding fluctuation theorem, generalizing the standard entropy production of stochastic thermodynamics. From this inequality we can derive an information processing entropy production, which gives the second law in the presence of information reservoirs. We also develop a systematic linear response theory for information processing machines. For a unicyclic machine powered by an information reservoir, the efficiency at maximum power can deviate from the standard value of 1/2. For the case where energy is consumed to erase the tape, the efficiency at maximum erasure rate is found to be 1/2.
Collapse
Affiliation(s)
- Andre C Barato
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| |
Collapse
|
25
|
Sandberg H, Delvenne JC, Newton NJ, Mitter SK. Maximum work extraction and implementation costs for nonequilibrium Maxwell's demons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042119. [PMID: 25375450 DOI: 10.1103/physreve.90.042119] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/17/2014] [Indexed: 06/04/2023]
Abstract
We determine the maximum amount of work extractable in finite time by a demon performing continuous measurements on a quadratic Hamiltonian system subjected to thermal fluctuations, in terms of the information extracted from the system. The maximum work demon is found to apply a high-gain continuous feedback involving a Kalman-Bucy estimate of the system state and operates in nonequilibrium. A simple and concrete electrical implementation of the feedback protocol is proposed, which allows for analytic expressions of the flows of energy, entropy, and information inside the demon. This let us show that any implementation of the demon must necessarily include an external power source, which we prove both from classical thermodynamics arguments and from a version of Landauer's memory erasure argument extended to nonequilibrium linear systems.
Collapse
Affiliation(s)
- Henrik Sandberg
- Department of Automatic Control, KTH Royal Institute of Technology, Stockholm, Sweden
| | | | - Nigel J Newton
- School of Computer Science and Electronic Engineering, University of Essex, Colchester, United Kingdom
| | - Sanjoy K Mitter
- Laboratory for Information and Decision Systems, MIT, Cambridge, Massachusetts, USA
| |
Collapse
|
26
|
Zulkowski PR, DeWeese MR. Optimal finite-time erasure of a classical bit. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052140. [PMID: 25353772 DOI: 10.1103/physreve.89.052140] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2013] [Indexed: 05/25/2023]
Abstract
Information erasure inevitably leads to the generation of heat. Minimizing this dissipation will be crucial for developing small-scale information processing systems, but little is known about the optimal procedures required. We have obtained closed-form expressions for maximally efficient erasure cycles for deletion of a classical bit of information stored by the position of a particle diffusing in a double-well potential. We find that the extra heat generated beyond the Landauer bound is proportional to the square of the Hellinger distance between the initial and final states divided by the cycle duration, which quantifies how far out of equilibrium the system is driven. Finally, we demonstrate close agreement between the exact optimal cycle and the protocol found using a linear response framework.
Collapse
Affiliation(s)
- Patrick R Zulkowski
- Department of Physics, University of California, Berkeley, California 94720, USA and Redwood Center for Theoretical Neuroscience, University of California, Berkeley, California 94720, USA
| | - Michael R DeWeese
- Department of Physics, University of California, Berkeley, California 94720, USA; Redwood Center for Theoretical Neuroscience, University of California, Berkeley, California 94720, USA; and Helen Wills Neuroscience Institute, University of California, Berkeley, California 94720, USA
| |
Collapse
|
27
|
Horowitz JM, Jacobs K. Quantum effects improve the energy efficiency of feedback control. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:042134. [PMID: 24827219 DOI: 10.1103/physreve.89.042134] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/2013] [Indexed: 06/03/2023]
Abstract
The laws of thermodynamics apply equally well to quantum systems as to classical systems, and because of this, quantum effects do not change the fundamental thermodynamic efficiency of isothermal refrigerators or engines. We show that, despite this fact, quantum mechanics permits measurement-based feedback control protocols that are more thermodynamically efficient than their classical counterparts. As part of our analysis, we perform a detailed accounting of the thermodynamics of unitary feedback control and elucidate the sources of inefficiency in measurement-based and coherent feedback.
Collapse
Affiliation(s)
- Jordan M Horowitz
- Department of Physics, University of Massachusetts at Boston, Boston, Massachusetts 02125, USA
| | - Kurt Jacobs
- Department of Physics, University of Massachusetts at Boston, Boston, Massachusetts 02125, USA and Hearne Institute for Theoretical Physics, Louisiana State University, Baton Rouge, Louisiana 70803, USA
| |
Collapse
|