Microcanonical entropy for classical systems

The entropy definition in the microcanonical ensemble is revisited. We propose a novel definition for the microcanonical entropy that resolve the debate on the correct definition of the microcanonical entropy. In particular we show that this entropy definition fixes the problem inherent the exact extensivity of the caloric equation. Furthermore, this entropy reproduces results which are in agreement with the ones predicted with standard Boltzmann entropy when applied to macroscopic systems. On the contrary, the predictions obtained with the standard Boltzmann entropy and with the entropy we propose, are different for small system sizes. Thus, we conclude that the Boltzmann entropy provides a correct description for macroscopic systems whereas extremely small systems should be better described with the entropy that we propose here.Comment: 5 pages, 2 figure

Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

We consider a generic classical many particle system described by an autonomous Hamiltonian $H(x^{_1},...,x^{_{N+2}})$ which, in addition, has a conserved quantity $V(x^{_1},...,x^{_{N+2}})=v$, so that the Poisson bracket $\{H,V \}$ vanishes. We derive in detail the microcanonical expressions for entropy and temperature. We show that both of these quantities depend on multidimensional integrals over submanifolds given by the intersection of the constant energy hypersurfaces with those defined by $V(x^{_1},...,x^{_{N+2}})=v$. We show that temperature and higher order derivatives of entropy are microcanonical observable that, under the hypothesis of ergodicity, can be calculated as time averages of suitable functions. We derive the explicit expression of the function that gives the temperature.Comment: 4 pages, preprin

Entanglement estimation in non-optimal qubit states

In the last years, a relationship has been established between the quantum Fisher information (QFI) and quantum entanglement. In the case of two-qubit systems, all pure entangled states can be made useful for sub-shot-noise interferometry while their QFI meets a necessary and sufficient condition [1]. In M-qubit systems, the QFI provides just a sufficient condition in the task of detecting the degree of entanglement of a generic state [2]. In our work, we show analytically that, for a large class of one-parameter non-optimal two-qubit states, the maximally entangled states are associated with stationary points of the QFI, as a function of such parameter. We show, via numerical simulations, that this scenario is maintained for the generalisation of this class of states to a generic M-qubit system. Furthermore, we suggest a scheme for an interferometer able to detect the entanglement in a large class of two-spin states.Comment: 7 pages, 7 figure

Nonclassical dynamics of Bose condensates in an optical lattice in the superfluid regime

A condensate in an optical lattice, prepared in the ground state of the superfluid regime, is stimulated first by suddenly increasing the optical lattice amplitude and then, after a waiting time, by abruptly decreasing this amplitude to its initial value. Thus the system is first taken to the Mott regime and then back to the initial superfluid regime. We show that, as a consequence of this nonadiabatic process, the system falls into a configuration far from equilibrium whose superfluid order parameter is described in terms of a particular superposition of Glauber coherent states that we derive. We also show that the classical equations of motion describing the time evolution of this system are inequivalent to the standard discrete nonlinear Schreodinger equations. By numerically integrating such equations with several initial conditions, we show that the system loses coherence, becoming insulating.Comment: 5 pages, 4 figure

Newton's cradle analogue with Bose-Einstein condensates

We propose a possible experimental realization of a quantum analogue of Newton's cradle using a configuration which starts from a Bose-Einstein condensate. The system consists of atoms with two internal states trapped in a one dimensional tube with a longitudinal optical lattice and maintained in a strong Tonks-Girardeau regime at maximal filling. In each site the wave function is a superposition of the two atomic states and a disturbance of the wave function propagates along the chain in analogy with the propagation of momentum in the classical Newton's cradle. The quantum travelling signal is generally deteriorated by dispersion, which is large for a uniform chain and is known to be zero for a suitably engineered chain, but the latter is hardly realizable in practice. Starting from these opposite situations we show how the coherent behaviour can be enhanced with minimal experimental effort.Comment: To appear in Journal of Physics

Topology and Phase Transitions: towards a proper mathematical definition of finite N transitions

A new point of view about the deep origin of thermodynamic phase transitions is sketched. The main idea is to link the appearance of phase transitions to some major topology change of suitable submanifolds of phase space instead of linking them to non-analyticity, as is usual in the Yang-Lee and in the Dobrushin-Ruelle-Sinai theories. In the new framework a new possibility appears to properly define a mathematical counterpart of phase transitions also at finite number of degrees of freedom. This is of prospective interest to the study of those systems that challenge the conventional approaches, as is the case of phase transitions in nuclear clusters.Comment: 6 pages, 1 figur

A microcanonical entropy correcting finite-size effects in small systems

In a recent paper [Franzosi, Physica A {\bf 494}, 302 (2018)], we have suggested to use of the surface entropy, namely the logarithm of the area of a hypersurface of constant energy in the phase space, as an expression for the thermodynamic microcanonical entropy, in place of the standard definition usually known as Boltzmann entropy. In the present manuscript, we have tested the surface entropy on the Fermi-Pasta-Ulam model for which we have computed the caloric equations that derive from both the Boltzmann entropy and the surface entropy. The results achieved clearly show that in the case of the Boltzmann entropy there is a strong dependence of the caloric equation from the system size, whereas in the case of the surface entropy there is no such dependence. We infer that the issues that one encounters when the Boltzmann entropy is used in the statistical description of small systems could be a clue of a deeper defect of this entropy that derives from its basic definition. Furthermore, we show that the surface entropy is well founded from a mathematical point of view, and we show that it is the only admissible entropy definition, for an isolated and finite system with a given energy, which is consistent with the postulate of equal a-priori probability

On the apparent failure of the topological theory of phase transitions

The topological theory of phase transitions has its strong point in two theorems proving that, for a wide class of physical systems, phase transitions necessarily stem from topological changes of some submanifolds of configuration space. It has been recently argued that the $2D$ lattice $\phi^4$-model provides a counterexample that falsifies this theory. It is here shown that this is not the case: the phase transition of this model stems from an asymptotic ($N\to\infty$) change of topology of the energy level sets, in spite of the absence of critical points of the potential in correspondence of the transition energy.Comment: 5 pages, 4 figure

On the dispute between Boltzmann and Gibbs entropy

Very recently, the validity of the concept of negative temperature has been challenged by several authors since they consider Boltzmann's entropy (that allows negative temperatures) inconsistent from a mathematical and statistical point of view, whereas they consider Gibbs' entropy (that does not admit negative temperatures) the correct definition for microcanonical entropy. In the present paper we prove that for systems with equivalence of the statistical ensembles Boltzmann entropy is the correct microcanonical entropy. Analytical results on two systems supporting negative temperatures, confirm the scenario we propose. In addition, we corroborate our proof by numeric simulations on an explicit lattice system showing that negative temperature equilibrium states are accessible and obey standard statistical mechanics prevision.Comment: To appear in Annals of Physic