# Foundations of Quantum Thermodynamics and Statistical Mechanics

In a completely different direction, I worked on the foundations of statistical mechanics and thermodynamics. The main problem facing the foundation of statistical mechanics is that, in principle, all the features of the theory should be derived from the basic equations of motion – Newton or Schrodinger’s laws. This has been hitherto impossible; instead many basic features were simply postulated and the rest of the theory was then derived from the postulates. Furthermore, subjective lack of knowledge needed to be invoked. My aim was to prove the postulates from the basic laws. That every physical system when left undisturbed eventually reaches thermal equilibrium is one of the most fundamental facts of nature. As a first step towards showing this, in [16], in collaboration with Short and Winter, I showed that given a sufficiently small subsystem of a large closed system almost every individual state of the system is such that the subsystem is approximately in a canonical state. As this is a property of individual states, ensemble or time-averaging are not required and hence the “equal a-priori probability” postulate of statistical mechanics is redundant. Building upon the concepts formulated in [16], in collaboration with Linden, Short and Winter [17], I proved that, with virtual generality, (i.e. for almost all Hamiltonians and almost all initial states, including non-typical ones), reaching equilibrium is a universal property of quantum systems: in a large closed system, almost any subsystem will reach an equilibrium state and remain close to it for almost all times. We have thus proven, from first principles, the postulate of equilibration.

In quantum thermodynamics I raised the question of whether or not there are limitations on the size of thermal machines – could thermal machines be built with only a small number of quantum states? And if they could, would they reach maximal (Carnot) efficiency, or there is a trade-off between size and efficiency. We first presented the smallest possible refrigerator, consisting only of two qubits (two-state systems, such as spin ½ particles) and proved that this refrigerator can reach maximal efficiency. Following from this work we introduced [18] the notion of virtual temperatures, which effectively gives a new approach to thermodynamics. More recently using this approach we showed that the laws of thermodynamics that were originally found to apply to large ensembles of particles apply also to individual quantum ones; in particular, that one can define a notion of “free energy” for individual particles, very similar to the ordinary free energy, and that the work one can extract from one particle is equal to the change of its free energy [20].