Critical assessment on the minimal work approach in quantum thermodynamics Keyvan Sadri, Afshin Shafiee à Department of Chemistry, Sharif University of Technology, P.O. Box 11365-9516, Tehran, Iran article info Available online 3 July 2009 Keywords: Entropy Work Density matrix abstract According to a well-known approach in quantum thermodynamics, the characteristics of a system in a process can be estimated by calculating work in the process. Here, we are going to show that this approach cannot provide with us a clear depiction of the second law of thermodynamics in some specific (but simple) processes. & 2009 Elsevier B.V. All rights reserved. 1. Introduction In classical thermodynamics, we always try to find state functions (for example, entropy and internal energy) and their changes in some given processes. In quantum thermodynamics, on the other hand, some of these quantities, such as entropy, are not well-defined. The most significant problem here is that we cannot attain thermodynamic limit. As an alternative, in recent years some people have tried to show that this problem can be avoided, if one calculates work in any given process instead of other quantities such as entropy [1–5]. Pathfinders of this view claim that some thermodynamic attitudes emerge when we follow work in a process. For example, authors try to show that the reversible work is minimal in any micro-process for quantum systems to provide a proof for the second law of thermodynamics [1]. In another example, authors compute work in a mixing process to show the origins of the Gibbs paradox [2]. Although the idea seems fascinating, it confronts some problems. Considering one of the recent works as an instance, in Section 2 we show that the proof of minimal work principle (as an alternative statement of the second law) has a limited validity. The other related concept is the notion of relative entropy [1,3]. By relative entropy, authors of Refs. [1,3] mean a new definition of entropy which compares the value of entropy in a process relative to a hypothetical equilibrium situation. In Section 3, we prove that the relative entropy cannot be considered as a coherent and comprehensive concept for illustrating the correct behavior of all thermodynamic processes according to the second law. Subsequently, at the end, in Section 4 we summarize our results. 2. Minimal work principle In a recent article, Nieuwenhuizen and Allahverdyan tried to provide a proof for the minimal work statement of the second law of thermodynamics [1]. When varying the speed of a given process done on an (initially) equilibrium system, the work is minimal for the slowest realization of the process. The main content of their paper is that whenever the rate of accomplishment of a given process for an (initially) equilibrium system depends on time, the work is minimal for the slowest (adiabatic) process. In their proof, it is assumed that the definition of work is given by W ¼ tr Z t f t i rðtÞ dH dt dt ¼ tr½H f rðt f Þ  tr½H i rðt i Þ ð1Þ The initial state of system is assumed to be rðt i Þ¼ X N n¼1 p n jn; t i S/n; t i j ð2Þ where p n ¼ e benðt i Þ P j e be j ðt i Þ : The origin of the definition Eq. (1) comes from the definition of the internal energy U U ¼ / ^ HS ¼ Trðr ^ HÞ ð3Þ According to Eq. (3), we can define change in internal energy as dU ¼ dTrðr ^ HÞ¼ Tr Z t f t i rðt 0 Þ d ^ H dt 0 dt 0 þ Tr Z t f t i ^ Hðt 0 Þ dr dt 0 dt 0 ð4Þ ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/physe Physica E 1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.06.089 à Corresponding author. Tel.: +98 2166165308; fax: +98 2166005718. E-mail address: shafiee@sharif.edu (A. Shafiee). Physica E 42 (2010) 488–490