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Register a new userZeroth law of thermodynamics for thermalized open quantum systems having integrals of motion
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Added by
Alexander Lisyansky
Publication date
2018
fields
Physics
and research's language is
English
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We study the evolution of an open quantum system described by a dynamical semigroup having the Lindblad superoperator as a generator. This generator may have an eigenfunction with a unity eigenvalue, referred to as a constant of motion (COM). An open quantum system has a unique stationary state if and only if it has no COMs. A system with multiple stationary states has a basis of COMs; any COM of the system is a linear combination of the basis COMs. The basis divides the space of system states into subspaces. Each subspace has its own stationary state, and any stationary state of the system is a linear combination of these states. Usually, neither the basis of COMs nor even the number of COMs is known. We demonstrate that finding the stationary state of the system does not require looking for the COMs. Instead, one can construct a set of invariant subspaces. If the system evolution begins from one of these subspaces, the system will remain in it, arriving at a stationary state independent of evolution in other subspaces. We suggest a direct way of finding the invariant subspaces by studying the evolution of the system. We show that the sets of invariant subspaces and subspaces generated by the basis of COMs are equivalent. A stationary state of the system is a weighted sum of stationary states in each invariant subspace; the weighted factors are determined by the initial state of the system.
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Thermalization of isolated quantum systems has been studied intensively in recent years and significant progresses have been achieved. Here, we study thermalization of small quantum systems that interact with large chaotic environments under the consideration of Schr{o}dinger evolution of composite systems, from the perspective of the zeroth law of thermodynamics. Namely, we consider a small quantum system that is brought into contact with a large environmental system; after they have relaxed, they are separated and their temperatures are studied. Our question is under what conditions the small system may have a detectable temperature that is identical with the environmental temperature. This should be a necessary condition for the small quantum system to be thermalized and to have a well-defined temperature. By using a two-level probe quantum system that plays the role of a thermometer, we find that the zeroth law is applicable to quantum chaotic systems, but not to integrable systems.
We consider open quantum systems consisting of a finite system of independent fermions with arbitrary Hamiltonian coupled to one or more equilibrium fermion reservoirs (which need not be in equilibrium with each other). A strong form of the third law of thermodynamics, $S(T) rightarrow 0$ as $Trightarrow 0$, is proven for fully open quantum systems in thermal equilibrium with their environment, defined as systems where all states are broadened due to environmental coupling. For generic open quantum systems, it is shown that $S(T)rightarrow gln 2$ as $Trightarrow 0$, where $g$ is the number of localized states lying exactly at the chemical potential of the reservoir. For driven open quantum systems in a nonequilibrium steady state, it is shown that the local entropy $S({bf x}; T) rightarrow 0$ as $T({bf x})rightarrow 0$, except for cases of measure zero arising due to localized states, where $T({bf x})$ is the temperature measured by a local thermometer.
We show that systems with negative specific heat can violate the zeroth law of thermodynamics. By both numerical simulations and by using exact expressions for free energy and microcanonical entropy it is shown that if two systems with the same intensive parameters but with negative specific heat are thermally coupled, they undergo a process in which the total entropy increases irreversibly. The final equilibrium is such that two phases appear, that is, the subsystems have different magnetizations and internal energies at temperatures which are equal in both systems, but that can be different from the initial temperature.
The phenomenon described by our title should surprise no one. What may be surprising though is how easy it is to produce a quantum system with this feature; moreover, that system is one that is often used for the purpose of showing how systems equilibrate. The violation can be variously manifested. In our detailed example, bringing a detuned 2-level system into contact with a monochromatic reservoir does not cause it to relax to the reservoir temperature; rather, the system acquires the reservoirs level-occupation-ratio.
We study the twirling semigroups of (super)operators, namely, certain quantum dynamical semigroups that are associated, in a natural way, with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group. The link connecting this class of semigroups of operators with (classical) Brownian motion is clarified. It turns out that every twirling semigroup associated with a finite-dimensional representation is a random unitary semigroup, and, conversely, every random unitary semigroup arises as a twirling semigroup. Using standard tools of the theory of convolution semigroups of measures and of convex analysis, we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite-dimensional unitary representation of a Lie group.
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