Uncertainty Principle Consequences at Thermal Equilibrium Leonardo A. Pach´on, 1 Johan F. Triana, 1 David Zueco, 2, 3 and Paul Brumer 4 1 Grupo de F´ ısica At´omica y Molecular, Instituto de F´ ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia UdeA; Calle 70 No. 52-21, Medell´ ın, Colombia. 2 Instituto de Ciencia de Materiales de Arag´on y Departamento de F´ ısica de la Materia Condensada, CSIC-Universidad de Zaragoza, Zaragoza E-50012, Spain. 3 Fundaci´on ARAID, Paseo Mar´ ıa Agust´ ın 36, E-50004 Zaragoza, Spain. 4 Chemical Physics Theory Group, Department of Chemistry and Center for Quantum Information and Quantum Control, University of Toronto, Toronto, Canada M5S 3H6 (Dated: January 8, 2014) Contrary to the conventional wisdom that deviations from standard thermodynamics originate from the strong coupling to the bath, it is shown that these deviations are intimately linked to the power spectrum of the thermal bath. Specifically, it is shown that the lower bound of the dispersion of the total energy of the system, imposed by the uncertainty principle, is dominated by the bath power spectrum and therefore, quantum mechanics inhibits the system thermal-equilibrium-state from being described by the canonical Boltzmann’s distribution. This is in sharp contrast to the classical case, for which the thermal equilibrium distribution of a system interacting via central forces with pairwise-self-interacting environment, irrespective of the interaction strength, is shown to be exactly characterized by the canonical Boltzmann distribution. As a consequence of this analysis, we define an effective coupling to the environment that depends on all energy scales in the system and reservoir interaction. Sample computations in regimes predicted by this effective coupling are demonstrated. For example, for the case of strong effective coupling, deviations from standard thermodynamics are present and, for the case of weak effective coupling, quantum features such as stationary entanglement are possible at high temperatures. PACS numbers: 03.65.Yz, 05.70.Ln, 37.10.Jk Introduction.—Thermodynamics was developed before the atomistic description of Nature was formulated. Sta- tistical mechanics was then introduced to understand the laws of thermodynamics in terms of a microscopic de- scription, thus closing the gap between macroscopic and microscopic description. Due to the interest in quantum technologies, there is a major ongoing effort to develop a consistent and well defined extension of thermodynamics to the quantum regime [1–3]. However, the majority of these theories are primarily based on Boltzmann’s origi- nal ideas and are therefore plagued by issues concerning irreversibility, the origin of the second law, the relation be- tween physics and information, the meaning of ergodicity, etc. (see, e.g., Ref. [4]). Despite these issues, it is now well known that, e.g., Onsager’s regression hypothesis fails in the quantum realm [5, 6] and that non-Markovian dynamics are relevant in a variety of fields and applications, from foundations [2, 7], to nuclear physics [8], quantum metrology [9, 10] and biological systems (see, e.g., [11] an references therein). It is also known that the thermal equilibrium state of a quantum system strongly coupled to a thermal bath deviates from the canonical Boltzmann distribution [1, 12– 14], this is also expected to occur in the classical case [15]. Since both are incoherent thermal stationary situations, one would expect that the quantum system is devoid of any coherence and hence, based on the decoherence program [16], that both distributions should coincide. However, one might also suggest that the entanglement between the system and the bath, which has no classical counterpart, could introduce quantum-classical deviations [1]. Furthermore, the fact that extra deviations could be present even if the entanglement between the system and the bath is zero [17] makes the situation even more intriguing. Hence, it seems appropriate to find a situation where the classical and the quantum contributions to the de- viation from the Boltzmann distribution can be clearly isolated and examined. Here we show that irrespective of the interaction strength, there are no deviations from Boltzmann’s distribution when a classical system interacts via central forces with a pairwise-self-interacting environ- ment. Thus, if after quantum-mechanically treating the same case, deviations from the canonical Boltzmann’s distribution are present, then they are purely quantum in nature. As shown below, deviations do appear and, based on completely general arguments, are shown to rely on the uncertainty principle characteristic of quantum mechanics. Therefore, the uncertainty principle not only inhibits the system’s thermal-equilibrium-state from being de- scribed by the canonical Boltzmann distribution, but for each system-bath interaction it also selects which thermal equilibrium states are physically accessible. This latter re- mark, formulated here for the first time in the framework of quantum thermodynamics, constitutes the cornerstone of the theory of pointer states (the states which are ro- bust against the presence of the environment) [16] and arXiv:1401.1418v1 [quant-ph] 7 Jan 2014