On the Nonequilibrium Entropy of Large and Small Systems

Abstract

Thermodynamics makes definite predictions about the thermal behavior of macroscopic systems in and out of equilibrium. Statistical mechanics aims to derive this behavior from the dynamics and statistics of the atoms and molecules making up these systems. A key element in this derivation is the large number of microscopic degrees of freedom of macroscopic systems. Therefore, the extension of thermodynamic concepts, such as entropy, to small (nano) systems raises many questions. Here we shall reexamine various definitions of entropy for nonequilibrium systems, large and small. These include thermodynamic (hydrodynamic), Boltzmann, and Gibbs-Shannon entropies. We shall argue that, despite its common use, the last is not an appropriate physical entropy for such systems, either isolated or in contact with thermal reservoirs: physical entropies should depend on the microstate of the system, not on a subjective probability distribution. To square this point of view with experimental results of Bechhoefer we shall argue that the Gibbs-Shannon entropy of a nano particle in a thermal fluid should be interpreted as the Boltzmann entropy of a dilute gas of Brownian particles in the fluid.

Entropy growth during free expansion of an ideal gas

“…where c is a positive constant [31][32][33]. This property corresponds to equilibrium because the system should stay in (or near) Γ Meq for long times, consistent with the observed stationarity in thermodynamic equilibrium.…”

Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics

“…Boltzmann's approach requires only that macro observables assume their equilibrium values (Goldstein et al, 2010a), whereas a stronger statement is actually true after a long waiting time: that all micro observables assume their equilibrium values . This is true not only for macroscopic systems, but also for small systems (Goldstein et al, 2017a), and has in fact been verified experimentally for a system with as few as 6 degrees of freedom (Kaufman et al, 2016).…”

“…But actually, the H-curve (i.e., the time evolution of entropy) makes complete sense, Boltzmann's vision does work, and von Neumann, Khinchin, and Jaynes were all mistaken, so it is perhaps worth elucidating this point. Many other, deeper discussions can be found in the literature, e.g., qualitative, popular accounts in (Penrose, 1989;Lebowitz and Maes, 2003;Carroll, 2010), overviews in (Goldstein, 2001;Lebowitz, 2008;Goldstein, 2019), more technical and detailed discussions in (Boltzmann, 1898;Ehrenfest and Ehrenfest, 1911;Lanford, 1976;Garrido et al, 2004;Falcioni et al, 2007;Cerino et al, 2016;Goldstein et al, 2017a;Lazarovici, 2018). So we now give a summary of Boltzmann's explanation of the second law.…”

“…Further critiques of subjective entropy can be found in (Callender, 1999;Lebowitz and Maes, 2003;Goldstein et al, 2017a;Goldstein, 2019).…”

“…The term "non-equilibrium" is sometimes understood (Gallavotti, 2003) as referring to so called non-equilibrium steady states (NESS), which concerns, for example, a system S coupled to two infinite reservoirs of different temperature; so S is an open system heated on one side and cooled on another, and it will tend to assume a macroscopically stationary ("steady") state with a temperature gradient, a nonzero heat current, and a positive rate of entropy production (Onsager, 1931;Bergmann and Lebowitz, 1955;Derrida, 2007;Goldstein et al, 2017a). In contrast, in this Section 5 we are considering a closed system (i.e., not interacting with the outside), and "non-equilibrium" refers to any phase point in X mc \ Γ eq .…”