Thermodynamic laws in isolated systems Stefan Hilbert,1, ∗ Peter H¨anggi,2, 3 and J¨orn Dunkel4 1Exzellenzcluster Universe, Boltzmannstr. 2, D-85748 Garching, Germany 2Institute of Physics, University of Augsburg, Universit¨atsstraße 1, D-86135 Augsburg, Germany 3Nanosystems Initiative Munich, Schellingstr. 4, D-80799 M¨unchen, Germany 4Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue E17-412, Cambridge, MA 02139-4307, USA (Dated: November 25, 2014) The recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful anal- ysis of the conceptual foundations underlying microcanonical thermostatistics. Here, we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focussing specifically on whether they satisfy or violate the zeroth, first and second law of thermodynamics. Our analysis shows that, for a broad class of systems that includes all standard classical Hamiltonian systems, only the Gibbs volume entropy fulfills all three laws simultaneously. To avoid ambiguities, the discussion is restricted to exact results and analytically tractable examples. PACS numbers: 05.20.-y, 05.30.-d, 05.70.-a Keywords: Classical statistical mechanics – Quantum statistical mechanics – Thermodynamics I. INTRODUCTION system (see Sec. IID). These and other conceptual issues deserve careful and systematic clarification, as they affect Recent advances in experimental and observational the theoretically predicted efficiency bounds of quantum techniques have made it possible to study in detail many- heat engines [2, 13, 14] and the realizability of dark en- particle systems that, in good approximation, are ther- ergy analogues in quantum systems [2]. mally decoupled from their environment. Examples cover The controversy about the existence of negative abso- a wide range of length and energy scales, from isolated lute temperatures revolves around the problem of iden- galactic clusters [1] and nebulae to ultra-cold quantum tifying an entropy definition for isolated systems that is gases [2] and spin systems [3]. The thermostatistical consistent with the laws of thermodynamics [4, 9, 15– description of such isolated systems relies on the mi- 19]. Competing definitions include the ‘surface’ en- crocanonical ensemble (MCE) [4–7]. Conceptually, the tropy, which is often attributed to Boltzmann2, and MCE is the most fundamental statistical equilibrium en- the ‘volume’ entropy derived by Gibbs (Chap. XIV in semble for it only assumes energy conservation, and be- Ref. [4]). Although these and other entropy candidates cause canonical and grand-canonical ensembles can be often yield practically indistinguishable predictions for derived from the MCE (by considering the statistics of the thermodynamic properties of ‘normal’ systems [21], 1 smaller subsystems [6]) but not vice versa. Although such as quasi-ideal gases with macroscopic particle num- these facts are widely accepted, there still exists consider- bers, they can produce substantially different predictions able confusion about the consistent treatment of entropy for mesoscopic systems and ad hoc truncated Hamiltoni- and the role of temperature in the MCE, as evidenced ans with upper energy bounds [9, 17]. A related more by the recent controversy regarding the (non)existence subtle source of confusion is the precise formulation of the of negative absolute temperatures [2, 8–12]. laws of thermodynamics and their interpretation in the The debate has revealed some widespread misconcep- context of isolated systems. Most authors seem to agree tions about the general meaning of temperature in iso- that a consistent thermostatistical formalism should re- lated systems. For example, it is often claimed [10–12] spect the zeroth, first and second law, but often the that knowledge of the microcanonical temperatures suf- laws themselves are stated in a heuristic or ambiguous arXiv:1408.5382v3 [cond-mat.stat-mech] 22 Nov 2014 fices to predict the direction of heat flow. This statement, form [11, 12] that may lead to incorrect conclusions and which is frequently mistaken as being equivalent to the spurious disputes. second law, is true in many situations, but not in gen- Aiming to provide a comprehensive foundation for fu- eral, reflecting the fact that energy, not temperature, is ture discussions, we pursue here a two-step approach: the primary thermodynamic state variable of an isolated Building on the work by Gibbs, Planck and others, we first identify formulations of the zeroth, first and sec- ∗Electronic address: [email protected] 1 This statement is to be understood in a physical sense. Mathe- matically, the microcanonical density operator of many systems 2 According to Sommerfeld [20], this attribution is probably not can be obtained from the canonical density operator via an in- entirely correct historically; see also the historical remarks in verse Laplace transformation. Sec. II C below. 2 ond law that (i) are feasible in the context of isolated Section II C contains brief historical remarks on entropy systems, (ii) permit a natural statistical interpretation naming conventions. Finally, in Sec. II D, we demon- within the MCE, and (iii) provide directly verifiable cri- strate explicitly that, regardless of the adopted entropy teria. In the second step, we analyze whether or not the definition, the microcanonical temperature is, in general, most commonly considered microcanonical entropy defi- not a unique (injective) function of the energy. This fact nitions comply with those laws. In contrast to previous means that knowledge of temperature is, in general, not studies, which considered a narrow range of specific ex- sufficient for predicting the direction of heat flow, im- amples [9–12], the focus here is on exact generic results plying that feasible versions of the second law must be that follow from general functional characteristics of the formulated in terms of entropy and energy (and not tem- density of states (DoS) and, hence, hold true for a broad perature). class of systems. Thereby, we deliberately refrain from imposing thermodynamic limits (TDLs). TDLs provide a useful technical tool for describing phase transitions A. The microcanonical ensemble in terms of formal singularities [22–25] but they are not required on fundamental grounds. A dogmatic restric- We consider strictly isolated3 classical or quantum sys- tion [11] of thermodynamic analysis to infinite systems tems described by a Hamiltonian H(ξ; Z) where ξ de- 4 is not only artificially prohibitive from a practical per- notes the microscopic states and Z = (Z1,...) comprises spective but also mathematically unnecessary: Regard- external control parameters (volume, magnetic fields, less of system size, the thermodynamics laws can be val- etc.). It will be assumed throughout that the microscopic idated based on general properties of the microcanonical dynamics of the system conserves the system energy E, DoS (non-negativity, behavior under convolutions, etc.). that the energy is bounded from below, E 0, and that Therefore, within the qualifications specified in the next the system is in a state in which its thermostatistical≥ sections, the results below apply to finite and infinite sys- properties are described by the microcanonical ensemble tems that may be extensive or non-extensive, including with the microcanonical density operator 5 both long-range and short-range interactions. We first introduce essential notation, specify in de- δ E H(ξ,Z) ρ(ξ E,Z)= − . (1) tail the underlying assumptions, and review the differ- | ω(E,Z) ent microcanonical entropy definitions (Sec. II). The ze- roth, first, and second law are discussed separately in The normalization constant is given by the density of Secs. III, IV and V. Exactly solvable examples that clar- states (DoS) ify practical implications are presented in Sec. VI. Some ω(E,Z) = Tr δ E H(ξ,Z) . (2) of these examples were selected to illustrate explicitly the − behavior of the different entropy and temperature defini- For classical systems, the trace Tr is defined as a phase- tions when two systems are brought into thermal contact. space integral (normalized by symmetry factors and pow- Others serve as counterexamples, showing that certain ers of the Planck constant) and for quantum system by an entropy definitions fail to satisfy basic consistency cri- integral over the basis vectors of the underlying Hilbert teria. Section VII discusses a parameter-free smoothing space. The DoS ω is non-negative, ω(E) 0, and with procedure for systems with discrete spectra. The paper our choice of the ground-state energy, ω≥(E) = 0 for concludes with a discussion of the main results (Sec. VIII) E < 0. and avenues for future study (Sec. IX). The energy derivative of the DoS will be denoted by The main results of our paper can be summarized as follows: Among the considered entropy candidates, only ∂ω(E,Z) ν(E,Z)= . (3) the Gibbs volume entropy, which implies a non-negative ∂E temperature and Carnot efficiencies 1, satisfies all three thermodynamic laws exactly for≤ the vast majority of physical systems (see Table I). 3 We distinguish isolated systems (no heat or matter exchange with environment), closed systems (no matter exchange, but heat ex- change permitted), and open systems (heat and matter exchange II. THE MICROCANONICAL DISTRIBUTION possible). AND ENTROPY DEFINITIONS 4 For classical systems, ξ comprises the canonical coordinates and