Information and the Second Law of Thermodynamics

Information and the second law of thermodynamics B. Ahmadi,1, 2, ∗ S. Salimi,1, y and A. S. Khorashad1 1Department of Physics, University of Kurdistan, P.O.Box 66177-15175, Sanandaj, Iran 2International Centre for Theory of Quantum Technologies, University of Gdansk, Wita Stwosza 63, 80-308 Gdansk, Poland (Dated: November 14, 2019) The second law of classical thermodynamics, based on the positivity of the entropy production, only holds for deterministic processes. Therefore the Second Law in stochastic quantum thermodynamics may not hold. By making a fundamental connection between thermodynamics and information theory we will introduce a new way of defining the Second Law which holds for both deterministic classical and stochastic quantum thermodynamics. Our work incorporates information well into the Second Law and also provides a thermodynamic operational meaning for negative and positive entropy production. I. INTRODUCTION tic quantum thermodynamics may expose features not present in a classical setting. It will be seen that the stochastic nature of quantum thermodynamics leads to Thermodynamics and information have intricate inter- the results different than that of classical ones. Our aim relations. Soon after establishing the second law of ther- in this work is to introduce a new way of defining the modynamics by Rodulf Clausius, Lord Kelvin and Max Second Law which holds for both deterministic classical Planck [1{3], in his 1867 thought experiment, "Maxwell's and stochastic quantum thermodynamics. In order to do Demon", James Clerk Maxwell attempted to show that this we make a connection between thermodynamics and thermodynamics is not strictly reducible to mechanics information theory in a fundamental way. We will incor- [4{6]. Although Maxwell introduced his demon to ques- porate information into the Second Law, and then show tion the Second Law, established by others, his demon that this relation is fundamental. Based on this relation revealed the relationship between thermodynamics and a generic form for the efficiency of an engine working in information theory for the first time. Clausius et. al an arbitrary cycle will be derived and we will clarify why never considered information playing any role in consti- and how backflow of information can cause a quantum tuting the Second Law. Maxwell illustrated that by us- engine to be more efficient than a Carnot engine. Our re- ing information about the positions and momenta of the sults provide a thermodynamic operational meaning for particles restrictions imposed by the Second Law can be negative entropy production, which until now only had relaxed thus demanding to take into account informa- information-theoretical interpretations; for example, it tion in the Second Law explicitly. In order to do this witnesses the non-Markovianity (or backflow of informa- we must elucidate the physical nature of information so tion) [12]. It will also be shown that a quantum thermo- that the Second Law includes information as a physical dynamic force [13] decodes (encodes) information (not) entity. In 1929 LéoSzilárd[11], inspired by Maxwell's to be used by the system to perform more work than idea, designed an engine working in a cycle, interacting what is expected and consequently the efficiency of the with a single thermal reservoir, which used information system exceeds the Carnot efficiency. (gained by the measurement on the system) to perform work. In classical thermodynamics the Clausius' statement of II. CLASSICAL ENGINES AND ITS the Second Law implies that in an irreversible process LIMITATION the entropy production of a system is always positive arXiv:1809.00611v3 [quant-ph] 13 Nov 2019 which means that information is always lost or encoded Before proceeding with the results let us examine a and never regained or decoded. The second law of classi- classical engine (see Fig. (1)) which gives us the moti- cal thermodynamics only holds for deterministic macro- vation of having a quantum engine more efficient than scopic equilibrium systems [1{3] and is not necessarily Carnot classical engine. For this engine we obtain (see valid in non-equilibrium stochastic microscopic systems Supplementary Note VIII) [7{10]. By equilibrium we mean that both the initial and final states of the system should be equilibrium thermal Tc∆iS ηe − ηC = − ; (1) states. Therefore it is expected that the generalization of ∆Qh the second law of classical thermodynamics to stochas- where ηe = 1 − T3=T1 is the engine efficiency, ηC = 1 − Tc=Th the Carnot efficiency and ∆iS = ∆iS1 + ∆iS2 the entropy production of the total system (the engine plus ∗Electronic address: [email protected] the reservoirs) during a cycle. Based on the second law yElectronic address: [email protected] of classical thermodynamics the entropy production is 2 always positive thus ηe can never exceed ηC . This is, in thermodynamics, i.e., distance from the equilibrium state fact, equivalent to the relation Tc ≤ T3 ≤ T1 ≤ Th which affects the minimal work that can be extracted. Another always holds for classical engines. Now the question is: interesting point is that, as in equilibrium thermodynam- does this relation also always hold for quantum engines? ics, based on our partitioning the amount of the internal If it does not, therefore quantum engines may be more energy which is encoded not to be used by the system 1 efficient than that of Carnot. In the following we will to do work equals ∆ S. This is notable because Eq. investigate this question and see how it can be violated β i in the quantum realm hence demanding a new form of (4) indicates the fact that regardless of the equilibrium the Second Law. or non-equilibrium thermodynamics the amount of the 1 internal energy which is encoded equals ∆ S. In other β i 1 words the relation ∆W = ∆ S as a link between irr β i thermodynamics and information theory is fundamental. This relation will serve as an important building block in the rest of this investigation. We must note that ∆Wrev and ∆Wirr in Eq. (2) are not done by the system during the process. The work which is done by the system is ∆W . j∆Wrevj is the maximal amount of the internal energy which is supposed to be spent by the system as work if there was no irreversibility during the process and pos- FIG. 1: (Color online) A classical engine working between itive ∆W is the amount of the internal energy which two reservoirs at temperatures T > T . Irreversibility occurs irr h c is not allowed to be spent by the system as work due to between the engine and the reservoirs not in the interior of the engine. irreversibility (loss of information). This explains why, regardless of the equilibrium or non-equilibrium thermodynamics, Eq. (4) has the same form in both stochastic and deterministic thermodynamics and thus is fundamental. Hence the encoded internal energy which is not III. REVERSIBLE AND IRREVERSIBLE WORK supposed to be used as work is always directly related to the entropy production (or information). In other words, The work done by a thermodynamic system, in the we can say weak coupling limit, can always be appropriately parti- tioned into two parts: reversible work and irreversible ∆iS = β×(internal enery not to be spent as work): (5) work (see Supplementary NoteIX), We can go further and define a non-equilibrium free en- ∆W = ∆Wrev + ∆Wirr; (2) ergy for a generic statistical state ρ of a quantum system in contact with a thermal bath as in which 1 F (ρ, H) ≡ E − TS = trfρHg − TS(ρ); (6) ∆W ≡ ∆I + ∆F β; (3) rev β and we find and 1 1 ∆iS = ∆Wirr = ∆W − ∆F: (7) ∆W ≡ ∆ S: (4) β irr β i The associated non-equilibrium free energy is analogous It is seen that the total work can always be parti- to its equilibrium counterpart in non-equilibrium pro- tioned into two parts, the reversible part and the irre- cesses. As can be seen from Eq. (7) the minimal work, versible part. This partitioning seems plausible since on average, necessary to drive the system from one ar- whenever the process is reversible all the work is re- bitrary state to another is the difference, ∆F , between versible and there exists no irreversible work as expected. the non-equilibrium free energy in each state. The ex- The reversible work defined in Eq. (3) is different from cess work with respect to this minimum is the dissipated the definition of reversible work in the literature, i.e., or irreversible work, ∆Wirr. If the entropy production β 1 ∆Wrev ≡ ∆F by the term ∆I. Classical thermody- is positive ∆iS ≥ 0 then the generalized minimal work β formulation (the generalized second law) for an isother- namics is the equilibrium thermodynamics and the minimal process with given initial and final non-equilibrium mal work can be extracted only in equilibrium processes, distributions is obtained as hence the minimal work equals the equilibrium work, i.e., ∆W = ∆F β. The term ∆I in Eq. (3) explains the 1 min ∆W ≥ ∆F β + ∆I: (8) fact that quantum thermodynamics is a non-equilibrium β 3 The generalized minimal work formulation of thermo- thermodynamics, this is also true as long as the process is dynamics for non-equilibrium distributions gives an im- Markovian. But if the process is non-Markovian ∆iS can portant relation between two major concepts in physics, be negative [12, 15], i.e., some of the internal energy can energy and information. In the following we will show be decoded to be used by the system as work and conse- that in non-equilibrium quantum thermodynamics the in- quently the efficiency can exceed the Carnot efficiency. ternal energy can also be decoded (negative irreversible (b) Consider a quantum engine operating in a cycle be- work) to be used by the system to perform more work tween two heat reservoirs at temperatures Th and Tc with than what is typically expected.

Load more