Axiomatic Relation between Thermodynamic and Information-Theoretic Entropies

Weilenmann, Mirjam; Kraemer, Lea; Faist, Philippe; Renner, Renato

doi:10.1103/physrevlett.117.260601

Cited by 58 publications

(88 citation statements)

References 73 publications

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“…In the quantum case, which we consider here for generality, it is a density operator on a Hilbert space H, denoted ρ ∈ Γ = S(H). An adiabatic process, which leaves no trace on the environment except for a change in the relative position of a weight, affects the microscopic degrees of freedom in the following way [25].…”

Section: Adiabatic Processes In the Statistical Approachmentioning

confidence: 99%

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Smooth Entropy in Axiomatic Thermodynamics

Weilenmann

Gabriel

Faist

et al. 2018

Fundamental Theories of Physics

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Thermodynamics can be formulated in either of two approaches, the phenomenological approach, which refers to the macroscopic properties of systems, and the statistical approach, which describes systems in terms of their microscopic constituents. We establish a connection between these two approaches by means of a new axiomatic framework that can take errors and imprecisions into account. This link extends to systems of arbitrary sizes including microscopic systems, for which the treatment of imprecisions is pertinent to any realistic situation. Based on this, we identify the quantities that characterise whether certain thermodynamic processes are possible with entropy measures from information theory. In the error-tolerant case, these entropies are so-called smooth min and max entropies. Our considerations further show that in an appropriate macroscopic limit there is a single entropy measure that characterises which state transformations are possible. In the case of many independent copies of a system (the so-called i.i.d. regime), the relevant quantity is the von Neumann entropy. arXiv:1807.07583v1 [quant-ph]

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Section: Adiabatic Processes In the Statistical Approachmentioning

confidence: 99%

“…This operation can be formally extended to arbitrary α ∈ R ≥0 , which physically involves the consideration of processes on a larger system. For the details of this we refer to [25].…”

Section: Adiabatic Processes In the Statistical Approachmentioning

confidence: 99%

Smooth Entropy in Axiomatic Thermodynamics

Weilenmann

Gabriel

Faist

et al. 2018

Fundamental Theories of Physics

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“…Alternatively, one can start from an axiomatic framework for thermodynamics [19,20]. The importance of using the appropriate form of entropy is highlighted in the recently developed field of stochastic thermodynamics [21][22][23][24][25][26][27][28][29][30], where a central, underlying hypothesis is that Eq.…”

Section: Introductionmentioning

confidence: 99%

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

Gavrilov

Chétrite

Bechhoefer

2017

Proc. Natl. Acad. Sci. U.S.A.

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Stochastic thermodynamics extends classical thermodynamics to small systems in contact with one or more heat baths. It can account for the effects of thermal fluctuations and describe systems far from thermodynamic equilibrium. A basic assumption is that the expression for Shannon entropy is the appropriate description for the entropy of a nonequilibrium system in such a setting. Here, for the first time, we measure experimentally this function. Our system is a micron-scale colloidal particle in water, in a virtual double-well potential created by a feedback trap. We measure the work to erase a fraction of a bit of information and show that it is bounded by the Shannon entropy for a twostate system. Further, by measuring directly the reversibility of slow protocols, we can distinguish unambiguously between protocols that can and cannot reach the expected thermodynamic bounds. INTRODUCTIONBeginning with the foundational work of Clausius, Maxwell, and Boltzmann in the 19th c., the concept of entropy has played a key role in thermodynamics. Yet, despite its importance, entropy is an elusive concept [1][2][3][4][5][6][7][8], with no unique definition; rather, the appropriate definition of entropy depends on the scale, relevant thermodynamic variables, and nature of the system, with ongoing debate existing over the proper definition even for equilibrium cases [9]. Moreover, entropy has not been directly measured but is rather inferred from other quantities, such as the integral of the specific heat divided by temperature. Here, by measuring the work required to erase a fraction of a bit of information, we isolate directly the change in entropy in an open nonequilibrium system, showing that it is compatible with the functional form proposed by Gibbs and Shannon, giving it a physical meaning in this context. Knowing the relevant form of entropy is crucial for efforts to extend thermodynamics to systems out of equilibrium.For a continuous classical system whose state in phase space x is distributed as the probability density function ρ(x), the Gibbs-Shannon entropy is [10,11] where k B is Boltzmann's constant. For quantum systems, von Neumann introduced, in 1927, the corresponding expression in terms of the density matrix [12]. Historically, the system in Eq. 1 has typically been assumed to be in thermal equilibrium. * email: johnb@sfu.caThe physical relevance of Eq. 1 for a nonequilibrium distribution ρ(x) has often been questioned (e.g., [13][14][15][16][17]). One concern is that S is constant on an isolated Hamiltonian system and can change only when evaluated on subsystems, such as those picked out by coarse graining. With many ways to choose subsystems or to coarse grain, is the associated notion of irreversibility intrinsic to the description of the system?In another approach to entropy, advanced in the context of communication and information theory, Shannon [11,18] proved that, up to a multiplicative constant, S is the only possible function satisfying three intuitive axioms. Alternatively, one can start from an...

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“…[23,24,25,26,27,28,29], "What is entropy?" [7,4,8,9,32,34,33,30,15,31], "What is macroscopic?" [35,36,37] have risen to a currently urgent need in the quantum (Q) communities (Q information, Q computing, Q thermal machines, Q fluctuations).…”

Section: Introductionmentioning

confidence: 99%

The fourth law of thermodynamics: steepest entropy ascent

Beretta

2020

Phil. Trans. R. Soc. A.

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When thermodynamics is understood as the science (or art) of constructing effective models of natural phenomena by choosing a minimal level of description capable of capturing the essential features of the physical reality of interest, the scientific community has identified a set of general rules that the model must incorporate if it aspires to be consistent with the body of known experimental evidence. Some of these rules are believed to be so general that we think of them as laws of Nature, such as the great conservation principles, whose ‘greatness’ derives from their generality, as masterfully explained by Feynman in one of his legendary lectures. The second law of thermodynamics is universally contemplated among the great laws of Nature. In this paper, we show that in the past four decades, an enormous body of scientific research devoted to modelling the essential features of non-equilibrium natural phenomena has converged from many different directions and frameworks towards the general recognition (albeit still expressed in different but equivalent forms and language) that another rule is also indispensable and reveals another great law of Nature that we propose to call the ‘fourth law of thermodynamics’. We state it as follows: every non-equilibrium state of a system or local subsystem for which entropy is well defined must be equipped with a metric in state space with respect to which the irreversible component of its time evolution is in the direction of steepest entropy ascent compatible with the conservation constraints. To illustrate the power of the fourth law, we derive (nonlinear) extensions of Onsager reciprocity and fluctuation–dissipation relations to the far-non-equilibrium realm within the framework of the rate-controlled constrained-equilibrium approximation (also known as the quasi-equilibrium approximation). This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.

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Axiomatic Relation between Thermodynamic and Information-Theoretic Entropies

Cited by 58 publications

References 73 publications

Smooth Entropy in Axiomatic Thermodynamics

Smooth Entropy in Axiomatic Thermodynamics

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

The fourth law of thermodynamics: steepest entropy ascent

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