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. Conclusion 65Appendix A Calculating the Prefactor by the Determinantal Method 68Appendix B Proof of eq. (3.27) 73Appendix C Cancelation of first order terms in S + I 75 IntroductionSemiclassical approximations in phase space using coherent states have been discussed extensively for several decades. This attractive topic, a favorite of many theoretical physicists and chemists, turns out to be very difficult. In this contribution to its literature, we shall attempt to sort out and clarify the web of contradictions and inconsistencies that have characterized the recent state of the field. We shall do so for the simplest possible case, one-dimensional coordinate space, i. e. two-dimensional phase space. This is the case where it is relatively easy to check the semiclassical approximations. We have done work in higher dimensions as well, but we do not include it here, as it would only obscure the basic relationships and further lengthen the paper. The conclusions we have reached are stated in section 7, and the reader who is already familiar with the subject may jump to them now to get an overall view. Because the pitfalls are numerous, however, we shall follow a slower approach, a historical one in this introduction, and then a systematic and detailed one in the body of the paper.The study of semiclassical methods has two basic motivations. First, it provides approximations to quantum mechanical quantities in terms of classical ingredients. These approximations should be very good if the typical classical actions are much larger than Planck's constant. Interestingly, they are often fairly good even at very low quantum numbers. Second, semiclassical methods also help in understanding the quantum mechanical processes themselves, providing a more intuitive description. This description includes quantum mechanical interference, since both amplitudes and phases can be calculated semiclassically.The semiclassical approximation for the evolution operator, or propagator, in the coordinate representation has been known for more than 70 ...

In recent years, strikingly consistent patterns of biodiversity have been identified over space, time, organism type and geographical region. A neutral theory (assuming no environmental selection or organismal interactions) has been shown to predict many patterns of ecological biodiversity. This theory is based on a mechanism by which new species arise similarly to point mutations in a population without sexual reproduction. Here we report the simulation of populations with sexual reproduction, mutation and dispersal. We found simulated time dependence of speciation rates, species-area relationships and species abundance distributions consistent with the behaviours found in nature. From our results, we predict steady speciation rates, more species in one-dimensional environments than two-dimensional environments, three scaling regimes of species-area relationships and lognormal distributions of species abundance with an excess of rare species and a tail that may be approximated by Fisher's logarithmic series. These are consistent with dependences reported for, among others, global birds and flowering plants, marine invertebrate fossils, ray-finned fishes, British birds and moths, North American songbirds, mammal fossils from Kansas and Panamanian shrubs. Quantitative comparisons of specific cases are remarkably successful. Our biodiversity results provide additional evidence that species diversity arises without specific physical barriers. This is similar to heavy traffic flows, where traffic jams can form even without accidents or barriers.

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We elucidate the connection between the Kolmogorov-Sinai entropy rate k and the time evolution of the physical or statistical entropy S. For a large family of chaotic conservative dynamical systems including the simplest ones, the evolution of S͑t͒ for far-from-equilibrium processes includes a stage during which S is a simple linear function of time whose slope is k. We present numerical confirmation of this connection for a number of chaotic symplectic maps, ranging from the simplest two-dimensional ones to a four-dimensional and strongly nonlinear map. [S0031-9007(98)08099-5] PACS numbers: 05.45.Ac, 05.70.LnThis paper tries to clarify the connection between the Kolmogorov-Sinai (KS) entropy and the physical entropy for a chaotic conservative dynamical system. This connection is obviously very important if one is to understand the impact on thermodynamics and statistical mechanics of the large amount of work done by mathematicians on the behavior of chaotic systems [1]. To start with the KS entropy, it is not really an entropy but an entropy per unit time, or an "entropy rate." It is a single number k, which is a property solely of the chaotic dynamical system considered. As for the physical entropy S͑t͒, the entropy of the second law of thermodynamics, it is a function of time, and this function depends not only on the particular dynamical system, but also on the choice of an initial probability distribution for the state of that system. Though it is clear that the original definition of k [2] was meant to provide a connection with S͑t͒, the precise connection does not seem to be well known nowadays, and the few statements found in the textbooks are often vague [3].The simplest connection one might guess would be the following: the KS entropy rate would be the maximum possible absolute value of the rate of variation of the physical entropy, i.e., jdS͞dtj # k. But this is wrong, because a counterexample can easily be found [4,5]. The actual connection is less direct and, in many cases, it requires that S͑t͒ be averaged over many histories (or trajectories), so as to give equal weights to initial distributions from all regions of phase space. Then, assuming these initial distributions to be very far from equilibrium, the variation with time of the physical entropy goes through three successive, roughly separated stages. In the first stage, S͑t͒ is heavily dependent on the details of the dynamical system and of the initial distribution; no generic statement can be made; dS͞dt can be positive or negative, large or small; and, in particular, it can be larger than k. In the second stage, S͑t͒ is a linear increasing function whose slope is k. In the third stage, S͑t͒ tends asymptotically toward the constant value which characterizes equilibrium, for which the distribution is uniform in the available part of phase space. It may happen, however, that the simple and generic stage 2 is absent, with stages 1 and 3 merging into each other. This is true when the initial distribution is not sufficiently different from the equ...