Validity of the second law in nonextensive quantum thermodynamics

We present an analysis of the foundations of the well known Clausius inequality. It is shown that, strictly speaking, the inequality is not a logical consequence of the Kelvin-Planck formulation of the second law of thermodynamics. Some thought experiments demonstrating the violation of the Clausius inequality are considered. Also, a reformulation of the Landauer's principle in terms of the Clausius inequality is proposed. This version of the inequality may be considered a co...

In a macroscopic (quantum or classical) Hamiltonian system, we prove the second law of thermodynamics in the forms of the minimum work principle and the law of entropy increase, under the assumption that the initial state is described by a general equilibrium distribution. Therefore the second law is a logical necessity once we accept equilibrium statistical mechanics.

The Kullback-Leibler inequality is a way of comparing any two density matrices. A technique to set up the density matrix for a physical system is to use the maximum entropy principle, given the entropy as a functional of the density matrix, subject to known constraints. In conjunction with the master equation for the density matrix, these two ingredients allow us to formulate the second law of thermodynamics in its widest possible setting. Thus problems arising in both quantu...

Deriving the laws of thermodynamics from a microscopic picture is a central quest of statistical mechanics. This tutorial focuses on the derivation of the first and second law for closed and open quantum systems far from equilibrium, where such foundational questions also become practically relevant for emergent nanotechnologies. The derivation is based on a microscopic definition of five essential quantities: internal energy, thermodynamic entropy, work, heat and temperature...

Tsallis' thermostatistics is by now recognized as a new paradigm for statistical mechanical considerations. However, it is still affected by a serious hindrance: the generalization of thermodynamics' zero-th law to a nonextensive scenario is plagued by difficulties. Here we show how to overcome this problem.

We formulate a convenient generalization of the q-expectation value, based on the analogy of the symmetric quantum groups and q-calculus, and show that the q->q^{-1} symmetric nonextensive entropy preserves all of the mathematical structure of thermodynamics just as in the case of non-symmetric Tsallis statistics. Basic properties and analogies with quantum groups are discussed.

In this lecture we briefly review the definition, consequences and applications of an entropy, $S_q$, which generalizes the usual Boltzmann-Gibbs entropy $S_{BG}$ ($S_1=S_{BG}$), basis of the usual statistical mechanics, well known to be applicable whenever ergodicity is satisfied at the microscopic dynamical level. Such entropy $S_q$ is based on the notion of $q$-exponential and presents properties not shared by other available alternative generalizations of $S_{BG}$. The th...

We present a stability analysis of the classical ideal gas in a new theory of nonextensive statistics and use the theory to understand the phenomena of negative specific heat in some self-gravitating systems. The stability analysis is made on the basis of the second variation of Tsallis entropy. It is shown that the system is thermodynamically unstable if the nonextensive parameter is q>5/3, which is exactly equivalent to the condition of appearance of the negative specific h...

We design a heat engine with multi-heat-reservoir, ancillary system and quantum memory. We then derive an inequality related with the second law of thermodynamics, and give a new limitation about the work gain from the engine by analyzing the entropy change and quantum mutual information change during the process. In addition and remarkably, by combination of two independent engines and with the help of the entropic uncertainty relation with quantum memory, we find that the t...

The form invariance of the statement of the maximum entropy principle and the metric structure in quantum density matrix theory, when generalized to nonextensive situations, is shown here to determine the structure of the nonextensive entropies. This limits the range of the nonextensivity parameter to so as to preserve the concavity of the entropies. The Tsallis entropy is thereby found to be appropriately renormalized.