arXiv subjects

M. Baranger

Publications and source records attributed to M. Baranger.

4 recordsLinked to original sources

arXiv2005

Semiclassical Propagation of Wavepackets with Real and Complex Trajectories

We consider a semiclassical approximation for the time evolution of an originally gaussian wave packet in terms of complex trajectories. We also derive additional approximations replacing the complex trajectories by real ones. These yield three different semiclassical formulae involving different real trajectories. One of these formulae is Heller's thawed gaussian approximation. The other approximations are non-gaussian and may involve several trajectories determined by mixed initial-final conditions. These different formulae are tested for the cases of scattering by a hard wall, scattering by an attractive gaussian potential, and bound motion in a quartic oscillator. The formula with complex trajectories gives good results in all cases. The non-gaussian approximations with real trajectories work well in some cases, whereas the thawed gaussian works only in very simple situations.

quant-ph↗

arXiv2001

Semiclassical Approximations in Phase Space with Coherent States

We present a complete derivation of the semiclassical limit of the coherent state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial value representation for the semiclassical propagator, based on an initial gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed gaussian approximation. It is very different from the Herman--Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

quant-ph↗

arXiv2000

Time evolution of thermodynamic entropy for conservative and dissipative chaotic maps

We consider several low--dimensional chaotic maps started in far-from-equilibrium initial conditions and we study the process of relaxation to equilibrium. In the case of conservative maps the Boltzmann-Gibbs entropy S(t) increases linearly in time with a slope equal to the Kolmogorov-Sinai entropy rate. The same result is obtained also for a simple case of dissipative system, the logistic map, when considered in the chaotic regime. A very interesting results is found at the chaos threshold. In this case, the usual Boltzmann-Gibbs is not appropriate and in order to have a linear increase, as for the chaotic case, we need to use the generalized q-dependent Tsallis entropy $S_q(t)$ with a particular value of a q different from 1 (when q=1 the generalized entropy reduces to the Boltzmann-Gibbs). The entropic index q appears to be characteristic of the dynamical system.

cond-mat.stat-mech↗

arXiv1999

The rate of entropy increase at the edge of chaos

Under certain conditions, the rate of increase of the statistical entropy of a simple, fully chaotic, conservative system is known to be given by a single number, characteristic of this system, the Kolmogorov-Sinai entropy rate. This connection is here generalized to a simple dissipative system, the logistic map, and especially to the chaos threshold of the latter, the edge of chaos. It is found that, in the edge-of-chaos case, the usual Boltzmann-Gibbs-Shannon entropy is not appropriate. Instead, the non-extensive entropy $S_q\equiv \frac{1-\sum_{i=1}^W p_i^q}{q-1}$, must be used. The latter contains a parameter q, the entropic index which must be given a special value $q^*\ne 1$ (for q=1 one recovers the usual entropy) characteristic of the edge-of-chaos under consideration. The same q^* enters also in the description of the sensitivity to initial conditions, as well as in that of the multifractal spectrum of the attractor.

cond-mat.stat-mech↗