Thomson Formulation Research Articles

Fluctuations of work from quantum subensembles: The case against quantum work-fluctuation theorems

We study how Thomson's formulation of the second law of thermodynamics (no work is extracted from an equilibrium ensemble by a cyclic process) emerges in the quantum situation through the averaging over fluctuations of work. The latter concept is carefully defined for an ensemble of quantum systems, the members of which interact with macroscopic sources of work. The approach is based on splitting a mixed quantum ensemble into pure subensembles, which according to quantum mechanics are maximally complete and irreducible. The splitting is done by filtering the outcomes of a measurement process. The approach is corroborated by comparing to relevant experiments in quantum optics. A critical review is given of two other approaches to fluctuations of work proposed in the literature. It is shown that in contrast to those, the present definition (i) is consistent with the physical meaning of the concept of work as mechanical energy lost by the macroscopic sources, or, equivalently, as the average energy acquired by the ensemble; (ii) applies to an arbitrary nonequilibrium state. There is no direct generalization of the classical work-fluctuation theorem to the proper quantum domain. This implies nonclassical scenarios for the emergence of the second law.

Thomson's formulation of the second law for macroscopic and finite work sources

Thomson's formulation of the second law states: no work can be extracted from an equilibrium system through a cyclic process. A simple, general proof is presented for the case of macroscopic sources of work. Next the setup is generalized towards situations, where the corresponding work-source is not macroscopic. It is shown that using such a source one can extract energy from an equilibrium system by means of a cyclic process. However, this extraction is accompanied by an entropy increase of the source, in a manner resembling the Clausius inequality.

Bath-generated work extraction and inversion-free gain in two-level systems

The spin-boson model, often used in NMR and ESR physics, quantum optics and spintronics, is considered in a solvable limit to model a spin one-half particle interacting with a bosonic thermal bath. By applying external pulses to a non-equilibrium initial state of the spin, work can be extracted from the thermalized bath. It occurs on the timescale $\T_2$ inherent to transversal (`quantum') fluctuations. The work (partly) arises from heat given off by the surrounding bath, while the spin entropy remains constant during a pulse. This presents a violation of the Clausius inequality and the Thomson formulation of the second law (cycles cost work) for the two-level system. Starting from a fully disordered state, coherence can be induced by employing the bath. Due to this, a gain from a positive-temperature (inversion-free) two-level system is shown to be possible.

A mathematical theorem as the basis for the second law: Thomson's formulation applied to equilibrium

There are several formulations of the second law, and they may, in principle, have different domains of validity. Here a simple mathematical theorem is proven which serves as the most general basis for the second law, namely the Thomson formulation (“cyclic changes cost energy”), applied to equilibrium. This formulation of the second law is a property akin to particle conservation (normalization of the wave function). It has been strictly proven for a canonical ensemble, and made plausible for a micro-canonical ensemble. As the derivation does not assume time-inversion invariance, it is applicable to situations where persistent currents occur. This clear-cut derivation allows to revive the “no perpetuum mobile in equilibrium” formulation of the second law and to criticize some assumptions which are widespread in literature. The result puts recent results devoted to foundations and limitations of the second law in proper perspective, and structurizes this relatively new field of research.

Violation of the second law of thermodynamics in the quantum microworld

One of the previously reported linear models of open quantum systems (interacting with a single thermal bath but otherwise not aided from outside) endowed with the faculty of spontaneous self-organization challenging standard thermodynamics is reconstructed here. It is then able to produce, in a cyclic manner, a useful (this time mechanical) work at the cost of just thermal energy in the bath whose quanta get properly in-phased. This means perpetuum mobile of the second kind explicitly violating the second law in its Thomson formulation. No approximations can be made responsible for the effect as a special scaling procedure is used that makes the chosen kinetic theory exact. The effect is purely quantum and disappears in the classical limit.

Molecular diffusion and viscous effects on concentration statistics in grid turbulence

Two-particle Lagrangian models in turbulence are used to consider dispersion in decaying isotropic homogeneous turbulence (i.e. approximating grid-generated wind-tunnel flows). Thomson's formulation is used, and his model is extended to incorporate molecular diffusivity and viscosity, i.e. the range of scales modelled includes the dissipation subranges as well as the intertial subrange. New terms are proposed which consistently provide well-mixed models and the impact on Saffman's well-known small-time results is considered. The new model is ideal for comparison with recent concentration-fluctuation measurements in decaying wind-tunnel turbulence and the results are encouraging. In particular, the fluctuation intensities, both along and across the wind tunnel, are well described by the new model. In addition, small source-size effects are far better explained when we include the molecular effects. A surprising results is the persistence in time of both source-geometry and molecular effects upon concentration fluctuations. Both of the effects are negligible at large times for mean concentrations, but persist for significant times for the fluctuations, indicating an important role for small-scale dynamics.