Uncertainty Relations of Statistical Mechanics

Abstract

Recently, we have presented some simple arguments supporting the existence of a certain complementarity between thermodynamic quantities of temperature and energy, an idea suggested by Bohr and Heinsenberg in the early days of Quantum Mechanics. Such a complementarity is expressed as the impossibility to perform an exact simultaneous determination of the system energy and temperature by using an experimental procedure based on the thermal equilibrium with other system regarded as a measuring apparatus (thermometer). In this work, we provide a simple generalization of the last approach with the consideration of a thermodynamic situation with several control parameters.

Energy-temperature uncertainty relation in quantum thermodynamics

“…Appendix A: Proof of 8 We begin by considering an exponential stateρ θ dependent on smooth parameter θ of the formρ(θ) = e − θ /Z, where Z = tr[e − θ ] and θ is some positive hermitian operator. Suppressing the dependence on θ for now, let us denote the spectral decomposition byρ θ = n p n |ψ n ψ n | where the eigenstates satisfy θ |ψ n = λ n |ψ n .…”

Complementarity in Quantum Mechanics and Classical Statistical Mechanics

“…Our discussion does not only demonstrate the existence of complementary relations involving thermodynamic variables (7)(8)(9), but also the existence of a remarkable analogy between the conceptual features of quantum mechanics and classical statistical mechanics. This chapter is organized as follows.…”

Eigenstate-specific temperatures in two-level paramagnetic spin lattices

“…Though temperature is a ubiquitous concept in the physical and biological sciences, its nature and definition have become subjects of fresh debate in the last three decades in two notable contexts: (1) the Feshbach 1 -Kittel 2 -Mandelbrot 3 (FKM) debate regarding the differences between thermodynamic temperatures and the so-called "effective temperatures" 2 or "temperature estimators" 3 for systems in contact with small heat baths 2 and (2) the debate over the thermodynamic legitimacy of negative temperatures in systems with bounded energy spectra, which was recently reignited [4][5][6]31 by the realization of negative temperatures in optical lattices. [28][29][30][31][32] The FKM [1][2][3][34][35][36] and negative temperature [4][5][6]27,31 debates both address issues central to a number of important topics: (1) the thermodynamics of small systems; [1][2][3]34 (2) the thermodynamics of isolated systems; 12,34,35, (3) the thermodynamic uncertainty relation (TUR); [1][2][3][34][35][36] and (4) quantum thermodynamics, 40,60 in which the temperatures of individual quantum eigenstates, hereafter designated eigenstate-specific temperatures (ESTs), apply. a) Author to whom correspondence should be addressed: mmasthay1@ udayton.edu.…”

“…We then detail the adherence of the ESTs to the four laws of thermodynamics (Sec. III D), the relationships of the ESTs to Boltzmann distributions and the TUR, 34,35 the relationships of the ESTs to temperaturedependent system energy levels (TDSELs) 33 (Sec. III E), and the FKM debate (Sec.…”