Thermodynamic laws in isolated systems

Abstract

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 analysis of the conceptual foundations underlying microcanonical thermostatistics. Here we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focusing specifically on whether they satisfy or violate the zeroth, first, and second laws 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.

Merits and qualms of work fluctuations in classical fluctuation theorems

“…Deep insights into work fluctuations are also relevant to designs of energy devices. Our general results would not be possible were the system under consideration not chaotic (systems with only one degree of freedom can be exceptions [23] because they are typically ergodic). This study shows that, when suppression of work fluctuations is of interest, the underlying chaotic dynamics affords general predictions and hence may be of expedient value in designing work protocols.…”

“…Thus, Ω(E; λ) in Eq. (1) is a thermodynamic adiabatic invariant [23,24]. The remarkable result that work fluctuations in chaotic systems vanish identically Figure 1.…”

“…Those early theorems showed that the ratio of two work distributions for forward and backward protocols, independent of the timevariation of λ(t), equals the ratio of two DoS, namely ω(E 0 + W )/ω(E 0 ). Now given that Ω(E; λ) is an adiabatic invariant, i.e., Ω(E τ ; λ τ ) = Ω(E 0 ; λ 0 ), this ratio reduces to that of corresponding two thermodynamic Gibbs temperatures k B T G = Ω(·)/ω(·) [23,24] …”

“…For initial states prepared as a microcanonical distribution, work fluctuations vanish identically in the (mechanical) adiabatic limit. For initial states prepared with a canonical distribution at the inverse Boltzmann temperature β [23,24], the variance in e −βW as an exponential form of work W , namely, Var(e −βW ) ≡ e −2βW − e −βW 2 , is minimized by adiabatic work protocols. This second result indicates that if Var(e −βW ) diverges for an adiabatic protocol, then it diverges for any nonadiabatic work protocol sharing the same initial and final Hamiltonians.…”

Thermodynamics, Statistical Mechanics and Entropy

“…However, because small effects of the order of 1/N, where N is the number of particles, have played a significant role in References [1][2][3][4][5][6][7][8][9][10][11], my discussion will also take small effects seriously.…”