The Entropy of a Non-Equilibrium Ideal Quantum Gas

Saurav Pandey, Junaid Majeed Bhat, Abhishek Dhar +5 more

Zoran Hadzibabic

Bruno O. Goes

Octavio Obregón, José Luis López, Marco Ortega-Cruz

We explore some important consequences of the quantum ideal Bose gas, the properties of which are described by a non-extensive entropy. We consider in particular two entropies that depend only on the probability. These entropies are defined in the framework of superstatistics, and in this context, such entropies arise when a system is exposed to non-equilibrium conditions, whose general effects can be described by a generalized Boltzmann factor and correspondingly by a generalized probability distribution defining a different statistics. We generalize the usual statistics to their quantum counterparts, and we will focus on the properties of the corresponding generalized quantum ideal Bose gas. The most important consequence of the generalized Bose gas is that the critical temperature predicted for the condensation changes in comparison with the usual quantum Bose gas. Conceptual differences arise when comparing our results with the ones previously reported regarding the q-generalized Bose–Einstein condensation. As the entropies analyzed here only depend on the probability, our results cannot be adjusted by any parameter. Even though these results are close to those of non-extensive statistical mechanics for q ∼ 1 , they differ and cannot be matched for any q.

Luis F. Santos, Victor Hugo M. Ramos, Danilo Cius +2 more

We formulate a canonical quantization of Equilibrium Thermodynamics by applying Dirac's theory of constrained systems. Thermodynamic variables are treated as conjugate pairs of coordinates and momenta, allowing extensive and intensive quantities to be promoted to operators in a Hilbert space. The formalism is applied to the ideal gas, the van der Waals gas, and the photon gas, illustrating both first- and second-class quantization procedures. For the ideal gas, a Schrödinger-like equation emerges in which entropy plays the role of time, and the wave function acquires a phase determined by the internal energy. A pseudo-Hermitian framework restores Hermiticity of the temperature operator and establishes the equivalence among constraint realizations. The approach naturally leads to thermodynamic uncertainty relations and suggests extensions to quantum and topological phase transitions, as well as black-hole and non-equilibrium thermodynamics.

Herbert W. Zimmermann

Abstract The quantum thermodynamics of the ideal gas is treated by means of the particle entropies σ=ka. The non-dimensional entropy numbers a are calculated by the extended, temperature-dependent Schrödinger equation applying periodic boundary conditions. The Boltzmann constant k represents the atomic entropy unit. The quantized particle entropies a are used to calculate the Boltzmann distribution f=exp(α−a), the density of entropy levels D(a), the dimensionless chemical potential α=μ/kT, etc. – The thermodynamics of the ideal gas is treated on the basis of entropy quanta by means of the density matrix formalism. In this way we obtain the total particle entropy A, the thermodynamic entropy S (equation of Sackur-Tetrode), the internal and the Helmholtz free energy E and F, respectively, etc. Moreover, it can also be shown on the basis of entropy quanta that the entropy of a closed system in thermal equilibrium has a maximum value, S=S max. Against this, the entropy S ′ of an arbitrary non-equilibrium state is always smaller, S′ < S max (second law of thermodynamics). – Application of ordinary density matrix theory to a closed system of N particles leads to the result that the entropy S does not change in time, dS/dt=0, regardless whether we consider an equilibrium state, S=S max, or a non-equilibrium state, S=S′. Consequently, S cannot irreversibly change from S′ to S max. However, any irreversible process is accompanied by a positive entropy production P=dS/dt>0. In order to overcome this obvious contradiction, we discuss the entropy evolution in time by means of the commutator equation G =i[ P , t ], which is deduced from very general assumptions. Here P and t are the operators of the entropy production P and the time t. Accordingly, P and t do not commute in general, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties ΔP and Δt, which are expressed by the uncertainty relation of the N-particle system, ΔPΔt≥(1/2)|〈 G 〉|=(γ/2)k. This P − t uncertainty relation easily allows a discussion of the evolution of the entropy S in time t from S′ to S max. Now the irreversible steps are correctly described by the entropy production P=dS/dt > 0, and the thermal equilibrium by P=0, ∆P=0, and thus the lifetime ∆t=∞.