^{1, 2}

^{3}

^{3}

^{1}

^{2}

^{3}

We focus attention upon the thermal statistics of the classical analogs of quasi-probabilities (QP) in phase space for the important case of quadratic Hamiltonians. We consider the three more important OPs: Wigner’s,

A quasi-probability distribution is a mathematical construction that resembles a probability distribution but does not necessarily fulfill some of Kolmogorov’s axioms for probabilities [

One usually considers a density operator

There exists a family of different representations, each connected to a different ordering of the creation and destruction operators

In this paper, we wish to apply

We insist that, in this paper, we will regard quasi-probabilities as semiclassical distributions in phase space, analogs of the quantum quasi-probabilistic distributions, and try to ascertain what physical features they are able to describe at such semiclassical level. One has [

The thermodynamics properties associated with coherent states have been the subject of much interest. See, for instance, [

For thermal states, the Gaussian HO-quantum phase spaces distributions are known in the literature for applications in quantum optics.

This paper is organized as follows. Section

Consider a general normalized gaussian distribution in phase space

The information quantifier Fisher’s information measure, specialized for families of shift-invariant distributions, which do not change shape under translations, is [

Fisher measure versus temperature

The logarithmic Boltzmann’s information measure for the the probability distribution (

Left: logarithmic entropies,

Table

Critical temperatures

Frequency ( |
Critical temperatures (°K) | |
---|---|---|

Extremely low frequency (ELF) | 3–30 Hz | 1.4397 |

Super low frequency (SLF) | 30–300 Hz | 1.4397 |

Ultra low frequency (ULF) | 300–3000 Hz | 1.4397 |

Very low frequency (VLF) | 3–30 kHz | 1.4397 |

Low frequency (LF) | 30–300 kHz | 1.4397 |

Medium frequency (MF) | 300 KHz–3 MHz | 1.4397 |

High frequency (HF) | 3–30 MHz | 1.4397 |

Very high frequency (VHF) | 30–300 MHz | 1.4397 |

Ultra high frequency (UHF) | 300 MHz–3 GHz | 1.4397 |

Super high frequency (SHF) | 3–30 GHz | 1.4397 |

Extremely high frequency (EHF) | 30–300 GHz | 1.4397–14.397 |

Tremendously high frequency (THF) | 300 GHz–3000 GHz | 14.397–143.97 |

We see from Table

The statistical complexity

The statistical complexity

Complexities

Another interesting information quantifier is that of the Manfredi-Feix entropy [

Left: linear entropies

The ensuing statistical complexity that uses

In general, the Fano factor is the coefficient of dispersion of the probability distribution

If

For

For

For our Gaussian distribution (

The

We start this section considering the classical Hamiltonian of the harmonic oscillator that reads

Using the definition of the mean value (

Specializing (

Fluctuations versus the temperature

Define now the participation ratio’s analog as [

Participation ratio

We have investigated here the thermal statistics of quasi-probabilities’ analogs

We emphasized the fact that for all of them the semiclassical entropy is a function only of the fluctuation product

it would violate Heisenberg’s-like principle in such a case. The behavior of other information quantifiers reconfirms such an assertion; that is,

Fisher’s measure exceeds its permissible maximum value

the participation ratio becomes

It is also clear then that semiclassical entropy, by itself, in phase space, looks like a kind of “uncertainty” indicator.

We have determined the temperatures for which the statistical complexity becomes maximal as a signature of the well-known transition between classical and nonclassical states of light whose signature is the transition from super-Poissonian to sub-Poissonian distributions [

We have seen that the

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors were supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina. Useful discussions with Professor R. Piasecki of Opole’s University, Poland, are gratefully acknowledged.