The thermal statistics of quasi-probabilities' analogs in phase space

We focus attention upon the thermal statistics of the classical analogs of quasi-probabilities's (QP) in phase space for the important case of quadratic Hamiltonians. We consider the three more important OPs: 1) Wigner's, $P$-, and Husimi's. We show that, for all of them, the ensuing semiclassical entropy is a function {\it only} of the fluctuation product $\Delta x \Delta p$. We ascertain that {\it the semi-classical analog of the $P$-distribution} seems to become un-physical at very low temperatures. The behavior of several other information quantifiers reconfirms such an assertion in manifold ways. We also examine the behavior of the statistical complexity and of thermal quantities like the specific heat.


I. INTRODUCTION
A quasi-probability distribution is a mathematical construction that resembles a probability distribution but does not necessarily fulfill some of the Kolmogorov's axioms for probabilities [1]. Quasi-probabilities exhibit general features of ordinary probabilities. Most importantly, they yield expectation values with respect to the weights of the distribution. However, they disobey the third probability postulate [1], in the sense that regions integrated under them do not represent probabilities of mutually exclusive states. Some quasi-probability distributions exhibit zones of negative probability density. This kind of distributions often arise in the study of quantum mechanics when discussed in a phase space representation, of frequent use in quantum optics, time-frequency analysis, etc.
Most generally, the dynamics of a quantum system is determined by a master equation. We speak of an equation of motion for the density operator (ρ), defined with respect to a complete orthonormal basis. One can show that the density can always be written in a diagonal manner, provided that it is with respect to an overcomplete basis [2]. If this is that of coherent states |α [3] one has [2] ρ = d 2 α π P (α, α * ) |α α|, Here we have d 2 α/π = dxdp/2π , with x and p variables of the phase space. The system evolves as prescribed by the evolution of the quasi-probability distribution function. Coherent states, right eigenstates of the annihilation operator a, serve as the overcomplete basis in such a build-up [2,3]. There exists a family of different representations, each connected to a different ordering of the creation and destruction operatorsâ andâ † . Historically, the first of these is the Wigner quasi-probability distribution W [4], related to symmetric operator ordering. In quantum optics the particle number operator is naturally expressed in normal order and, in the pertinent scenario, the associated representation of the phase space distribution is the Glauber-Sudarshan P one [3]. In addition to W and P , one may find many other quasi-probability distributions emerging in alternative representations of the phase space distribution [5]. A quite popular representation is the Husimi Q one [6][7][8][9], used when operators are in anti-normal order.
In this paper we wish to apply semiclassical information theory tools associated to these P , Q, and W representations (for quadratic Hamiltonians) in order to describe the concomitant thermal semiclassical features (the thermodynamics properties associated to coherent states have been the subject of much interest. See, for instance, Refs. [10] and [11]). It will be seen that useful insights are in this way gained. As stated, we specialize things to the three f −functions associated to a Harmonic Oscillator (HO) of angular frequency ω. In such a scenario the three functions -that we name for sake of convenience f P , f Q , and f W -are simple Gaussians and the treatment becomes entirely analytical, a very convenient feature. The HO is a really important system that yields insights usually having a wide impact. Thus, the HO constitutes much more than a mere simple example. Nowadays, it is of particular interest for the dynamics of bosonic or fermionic atoms contained in magnetic traps [13][14][15] as well as for any system that exhibits an equidistant level spacing in the vicinity of the ground state, like nuclei or Luttinger liquids.
The three important gaussian quantum phase spaces distributions for the HO instance for a thermal states are known in the literature for applications in quantum optics. In this paper we will regard them as semi-classical distributions in phase space -analogs of the quantum quasi-probabilistic distributions-, and try to ascertain what physical features are they able to describe at such semi-classical level. These distributions are [16,17]: with β = 1/k B T , k B the Boltzmann constant, and T the temperature. As stated above, these distributions will be used in the next section as semiclassical statistical weight functions. Since ours is not a quantum approach, the ordering of the HO-creation and destruction operators a and a † plays no role whatsoever. This paper is organized as follows: section II refers to different information quantifiers in a phase space representation for Gaussian distributions. Features of the fluctuations are analyzed in Section III. Also, we discuss the notion of linear entropy. In Section IV we focus attention upon thermodynamic relations and we express them in terms of an effective temperature. Finally, some conclusions are drawn in Section V.

II. SEMI-CLASSICAL INFORMATION QUANTIFIERS
The first step in our development is to calculate the entropic quantifiers for these Gaussian distributions. In order to simplify the notation we will consider a general normalized gaussian distribution in phase space whose normalized variance is 1/γ and γ taking values γ P , γ Q and γ W . A.

Shift-invariant Fisher's information measure
The information quantifier Fisher's information measure, specialized for families of shift-invariant distributions, that do not change shape under translations, is [18,19] and, in phase space, adopts the appearance [20] such that considering f (α) given by Eq. (5) we get I = γ, whose specific values are γ P , γ Q , γ W for the three functions f P , f Q , and f W . The behavior of these quantities are displayed in Fig. 1. The solid line is the case P, the dashed one the Wigner one, and the dotted curve is assigned to the Husimi case. Now, it is known that in the present scenario the maximum attainable value for I equals 2 [20]. The P -result violates this restriction at low temperatures, more precisely at with T being expressed in ( ω/k B )−units.

B. Logarithmic entropy S
The logarithmic Boltzmann's information measure for the the probability distribution (5) is so that it acquires the particular values for, respectively, the distributions f P , f Q , and f W . These entropies are plotted in Fig. 2. Details are similar to those of Fig. 1. Notice that S P < 0 for T < ω/(k B ln(1 + e)) ≈ 0.76( ω/k B ) < T crit . Negative classical entropies are well-known. One can cite, as an example, [12].

C. Statistical complexity
The statistical complexity C, according to Lopez-Ruiz, Mancini, and Calvet [21], is a suitable product of two quantifiers, such that C becomes minimal at the extreme situations of perfect order or total randomness. Instead of using the prescription of [21], but without violating its spirit, we will take one of these two quantifiers to be Fisher's measure and the other an entropic form, since it is well known that the two behave in opposite manner [22]. Thus: that vanishes for perfect order or total randomness. For each particular case, we explicitly have for, respectively, the distributions f P , f Q , and f W . The maximum of the statistical complexity occurs when γ = 1 and, the associated temperature values are The statistical complexity C is plotted in Fig. 3.

D. Linear entropy
Another interesting information quantifier is that of the Manfredi-Feix entropy [23], derived from the phase space Tsallis (q = 2) entropy [24]. In quantum information this form is referred to as the linear entropy [25]. It reads Accordingly, we have This is semi-classical result, valid for small γ. The ensuing statistical complexity that uses S l becomes vanishing both for γ = 0 and for γ = 2, the extreme values of the γ−physical range (we showed above that γ cannot exceed 2 without violating uncertainty restrictions). It is easy to see that the derivative of C l with respect to γ vanishes at γ = 1. This is shown in Fig. 4. In particular, Note that in the P -instance the linear entropy becomes negative, once again, for T < T crit . Contrary to what happens for the logarithmic entropy, the linear one can vanish in the W and Q representations.
This fact allows one to conclude that the linear entropy is not as good an indicator of ignorance (with respect to phase space location) as the logarithmic one. Since the former entropy is the first order expansion of the logarithm entering Boltzmann's one, this kind of guarantee of uncertainty's non-violation in phase space provided by the logarithmic entropy should be a second order effect.

III. FLUCTUATIONS
We start this section considering the semi-classical Hamiltonian of the harmonic oscillator that reads where x and p are phase space variables and σ 2 x = /2mω and σ 2 p = mω/2 [26]. Let us further define the semiclassical expectation value of the function A(x, p) as indicating that f (α) is the statistical weight function. Using this general representation, from (24) we immediately find [27] x 2 /2σ 2 with where x f = p f = α f = 0, while γ takes the respective values γ P , γ Q , and γ W . The concomitant variances are ∆x 2 = x 2 f − x 2 f = 2σ 2 p /γ, and ∆p 2 = p 2 f − p 2 f = 2σ 2 p /γ. Hence, for our general gaussian distribution one easily establishes that which shows that γ should be constrained by the restriction if one wishes the inequality to hold.
Specializing (28) for our three quasi-probability distributions yields The restriction (30) applied to the P -result entails that it holds if Thus, the distribution f P seems again to becomes un-physical at temperatures lower than T crit , for which (30) is violated. From (28) we have γ = /U. Accordingly, if we insert this into (9), the logarithmic entropy S can be recast in U−terms via the relation  Fig. 1 .
(also demonstrated in Ref. [28] to hold for the Wehrl entropy) that vanishes for In the P -instance this happens at At this temperature the Heisenberg's-like condition (30) is violated. The W and Q distributions do not allow for such a circumstance. Actually, in the Wigner case, which is exact, the minimum S−value is attained at β = ∞, where The uncertainty restriction (30) seems to impede the phase-space entropy to vanish, a sort of quasi-quantum effect. It is clear then that, in phase space, the logarithmic entropy, by itself, is an uncertainty indicator, in agreement with the work, in other scenarios, of several authors (see, for instance, [29] and references therein).
Define now the participation ratio's analog as [30,31] where J is given by (18). This is an important quantity that measures the number of pure states entering the mixture determined by our general gaussian probability distribution of amplitude γ [30,31]. We again encounter troubles with the P -distribution in this respect. It is immediately realized by looking at Fig. 6 that, for fulfilling the obvious condition m ≥ 1, one needs a temperature T ≥ T crit .

A. Fano factor's analog F
The Fano factor [20,32] is the coefficient of dispersion of the gaussian probability distribution f , defined as If one sets x = |α| 2 one has the Fano-analog and, after to resolve the pertinent mean values according to (25), the Fano factor becomes that, for a Gaussian distribution, links the Fano factor to the distribution's width and to the Fisher's measure I. Now, if one builds up a Poisson distribution (F = 1) in the variable |α| 2 , one sees that the pertinent Fano factor becomes unity [16,33]. We remind the reader of two situations: 1. for F < 1, sub-Poissonian processes occur, while 2. for F > 1, corresponds to a super-Poissonian process.
The first processes are of a quantum nature and can nor take place in a classical environment. Thus, with reference to the critical temperature defined in (8) we have to deal with The f Q −case reaches the super to sub-Poissonian transition only at T = 0, while the other two cases reach it at finite temperatures. Table I lists a set of critical temperatures T crit for typical radio waves.

A. Thermodynamic quantities
The mean energy of the hamiltonian H(x, p) is written in the fashion where f (α) is the statistical weight function while γ takes the respective values γ P , γ Q , and γ W explained in Introduction. The free energy and the specific heat, respectively, read Additionally, the thermodynamic entropy S ′ is where we have added the Boltzmann constant k B . The specific heat adopts the same value in all three cases, i.e., Thus, it is clear that, at T = 0 we have U = ω/2 for the Wigner case, the minimum for the energy. This condition is clearly violated in the P -instance for T < T crit .

B. Effective temperature
The mean energy can be viewed as a function of thermodynamic entropy S ′ given by (49). Accordingly, we can write the associated, fundamental equation as U = U (S ′ ). Thus, the differential of U is where we have considered the volume V to be constant. Combining (46) with the thermodynamic entropy (49) we get Thus, after effecting the pertinent replacements one has and Accordingly, we find which suggests introducing an effective temperature T ef f . Using T ef f we obtain a unified picture that encompasses the three distributions f P , f Q , and f W , in a single thermodynamic description. We have such that Note that in the three instances, T ef f = ∞ for T = ∞. However, if T = 0, T ef f = 0 only in the f P -case. It equals 1/2 in the Wigner instance and equals 1 in the Husimi case, as depicted in the accompanying figure.
From (46) and (58) we can rewrite the mean energy in terms of effective temperature.
that corresponds to the classical mean energy of a harmonic oscillator of temperature T ef f , with k B T ef f /2 contributions for each of the two pertinent degrees of freedom. Similarly, the thermodynamic entropy is recast as and the Helmholtz free energy is given by The effective specific heat is defined as that using (59) becomes which is precisely the specific heat for the classical harmonic oscillator which is independent of the temperature. This becomes the Dulong and Petit's rule at the classical limit. In view of (58) and (61) the analog partition function Z * is given by and, according to Eqs. (9), (46), and (64) we find with Thus, one reobtains all the thermal results pertaining to a classical HO at the temperature T ef f . The statistical complexity in terms of T ef f becomes Keeping in mind T ef f 's definition, it is easy to see that the maximum for the complexity C ′ is attained when This implies, according to Eq. (57) that the maximum of the Fisher measure es I max = 1. At the complexity-peak, thermodynamic quantities take the values a remarkable simplicity! Note that the whole thermal description becomes now of a classical character. All the quantum effects are contained in the relationship between T ef f and T .

V. CONCLUSIONS
We have investigated here the thermal statistics of quasi-probabilities-analogs f (α) in phase space for the important case of quadratic Hamiltonians, focusing attention on the three more important instances, i.e., those of Wigner, P -, and Husimi distributions.