Thermodynamics, and in particular work extraction from non-equilibrium states, has been studied in the quantum domain, in the recent years. It introduces radically new insights into quantum statistical and thermal processes. In much these studies, be it classical or quantum, thermal baths are assumed to be considerably large in size compared to systems under consideration. That is why, the baths remains always thermal, with same temperature before and after it interacts with a system. Also, an equilibrated system always shares the same temperature with the bath. Indeed, the assumption large is not fulfilled in every situation. If the baths are finite and small systems, the standard formulation of thermodynamics breaks down. The first problem one would encounter is the inconsistency in the notion of temperature itself. A finite bath could go out of thermal equilibrium, by exchanging energy with a system. Such a situation is relevant for thermodynamics that applies to quantum regime, where system and bath could be small and comparable in size. To incorporate such scenarios, we need to develop a temperature independent thermodynamics, where the bath could be small or large and will not have a special status. Here, we have introduced temperature independent formulation of thermodynamics as an exclusive consequence of (coarse-grained) information conservation. The information is measured in terms of von Neumann entropy.
The formulation is relied on the fact that systems with same entropy can be inter-convertible using entropy preserving operations. Therefore, the states with same entropy forms a constant entropy manifold and there exists a state that possesses minimal amount of energy. This state with minimal energy are known as a completely passive state, which assumes a Boltzmann–Gibb’s canonical form with an intrinsic temperature. The energy of a completely passive state is defined as the bound energy, as this energy cannot be extracted by any entropy preserving operations. For any given state, the free energy is defined as the difference between the internal energy and the bound energy, as this amount of energy can be accessible by means of entropy preserving operations. As shown in [40],two different states possessing identical energy and entropy are thermodynamically equivalent.
Such equivalence enables us to exploit energy-entropy diagram to understand bound, free energies geometrically. With these machinery, we have introduced a completely new definition of heat in terms of bound energy, applicable for arbitrary systems and without any reference to a temperature. We have formulated the laws of thermodynamics accordingly and, as we have seen, they are a consequence of the reversible dynamics of the underlying physical theory.
In particular:
• Zeroth law is a consequence of information conservation.
• First and second laws are a consequence of energy conservation, together with information conservation.
• Third law is a consequence of "strict" information conservation (i.e. microscopic reversibility or unitarity).
There is no third law for processes that only respect "coarse-grained" information conservation. We have applied our formalism to the heat engines that consist of finite bath and demonstrated that the maximum efficiency is in general less, compared to an ideal Carnot’s engine. We have also introduced a resource theoretic framework for intrinsic temperature based thermodynamics. This approach enables us to address the problem of inter-state transformations and work extraction. These results are given a geometric meaning, in terms of the energy-entropy diagram. The information conservation based framework for thermodynamics can be extended to multiple conserved quantities [34]. Analogously, charge-entropy and resource theory can given in this scenario. The extraction of a generalized potential (i.e. linear combinations of charges), becomes analogous to the work extraction (the single charge case).An immediate question arises is that to what extent the formalism can be extended beyond coarse-grained information conservation operations. This is an interesting open question, as in that case, there would be a different notion of bound energy and possibly many more equivalence classes of states. It is also far from clear if energy entropy diagrams would be meaningful there.
