Influence of Non-Markovian Dynamics in Thermal-Equilibrium Uncertainty-Relations

1.4 MB·

by  Leonardo A. Pachon

Uncertainty Principle Consequences at Thermal Equilibrium
Leonardo A. Pach´on,1 Johan F. Triana,1 David Zueco,2,3 and Paul Brumer4
1Grupo de F´ısica At´omica y Molecular, Instituto de F´ısica, Facultad de Ciencias Exactas y Naturales,
Universidad de Antioquia UdeA; Calle 70 No. 52-21, Medell´ın, Colombia.
2Instituto de Ciencia de Materiales de Arag´on y Departamento de F´ısica de la Materia Condensada,
CSIC-Universidad de Zaragoza, Zaragoza E-50012, Spain.
3Fundaci´on ARAID, Paseo Mar´ıa Agust´ın 36, E-50004 Zaragoza, Spain.
4Chemical Physics Theory Group, Department of Chemistry and Center for Quantum Information and Quantum Control,
University of Toronto, Toronto, Canada M5S 3H6
(Dated: January 8, 2014)
Contrary to the conventional wisdom that deviations from standard thermodynamics originate
from the strong coupling to the bath, it is shown that these deviations are intimately linked to the
power spectrum of the thermal bath. Specifically, it is shown that the lower bound of the dispersion
4 of the total energy of the system, imposed by the uncertainty principle, is dominated by the bath
1 power spectrum and therefore, quantum mechanics inhibits the system thermal-equilibrium-state
0 from being described by the canonical Boltzmann’s distribution. This is in sharp contrast to the
2 classical case, for which the thermal equilibrium distribution of a system interacting via central
forceswithpairwise-self-interactingenvironment,irrespectiveoftheinteractionstrength,isshownto
n
be exactly characterized by the canonical Boltzmann distribution. As a consequence of this analysis,
a
we define an effective coupling to the environment that depends on all energy scales in the system
J
and reservoir interaction. Sample computations in regimes predicted by this effective coupling are
7
demonstrated. For example, for the case of strong effective coupling, deviations from standard

thermodynamics are present and, for the case of weak effective coupling, quantum features such as
]
h stationary entanglement are possible at high temperatures.
p
- PACSnumbers: 03.65.Yz,05.70.Ln,37.10.Jk
t
n
a
u Introduction.—Thermodynamics was developed before between the system and the bath, which has no classical
q the atomistic description of Nature was formulated. Sta- counterpart,couldintroducequantum-classicaldeviations
[ tistical mechanics was then introduced to understand the [1]. Furthermore, the fact that extra deviations could be

1 laws of thermodynamics in terms of a microscopic de- present even if the entanglement between the system
v scription, thus closing the gap between macroscopic and and the bath is zero [17] makes the situation even more
8 microscopic description. Due to the interest in quantum intriguing.
1
technologies, there is a major ongoing effort to develop a
4 Hence, it seems appropriate to find a situation where
consistent and well defined extension of thermodynamics
1 the classical and the quantum contributions to the de-
1. to the quantum regime [1–3]. However, the majority of viation from the Boltzmann distribution can be clearly
these theories are primarily based on Boltzmann’s origi-
0 isolated and examined. Here we show that irrespective
nal ideas and are therefore plagued by issues concerning
4 of the interaction strength, there are no deviations from
1 irreversibility,theoriginofthesecondlaw,therelationbe-
Boltzmann’sdistributionwhenaclassicalsysteminteracts
: tween physics and information, the meaning of ergodicity,
v via central forces with a pairwise-self-interacting environ-
etc. (see, e.g., Ref. [4]).
i ment. Thus, if after quantum-mechanically treating the
X
same case, deviations from the canonical Boltzmann’s
r Despite these issues, it is now well known that, e.g.,
a distribution are present, then they are purely quantum
Onsager’sregressionhypothesisfailsinthequantumrealm
in nature. As shown below, deviations do appear and,
[5, 6] and that non-Markovian dynamics are relevant in a
basedoncompletelygeneralarguments, areshowntorely
variety of fields and applications, from foundations [2, 7],
on the uncertainty principle characteristic of quantum
to nuclear physics [8], quantum metrology [9, 10] and
mechanics.
biological systems (see, e.g., [11] an references therein).
It is also known that the thermal equilibrium state of Therefore, the uncertainty principle not only inhibits
a quantum system strongly coupled to a thermal bath the system’s thermal-equilibrium-state from being de-
deviatesfromthecanonicalBoltzmanndistribution[1,12– scribed by the canonical Boltzmann distribution, but for
14], thisisalsoexpectedtooccurintheclassicalcase[15]. each system-bath interaction it also selects which thermal
Since both are incoherent thermal stationary situations, equilibrium states are physically accessible. This latter re-
one would expect that the quantum system is devoid mark, formulated here for the first time in the framework
of any coherence and hence, based on the decoherence of quantum thermodynamics, constitutes the cornerstone
program [16], that both distributions should coincide. of the theory of pointer states (the states which are ro-
However, one might also suggest that the entanglement bust against the presence of the environment) [16] and
2
couldhavedeepconsequencesforanunderstandingofthe quantum realm, dissipative mechanisms are accompanied
thermalization of quantum systems. by decoherence effects and are bath-nature and coupling-
Classical Thermal-Equilibrium-State.—To provide a particularities sensitive [16, 18] and are then capable of
well-defined situation, consider a classical particle of inducing a variety different thermal states.
mass m with potential energy U (q) and Hamiltonian Quantum Thermal-Equilibrium-State—The quantum
S
H (p,q)= 1 p2+U (q). Now consider a collection of N descriptionfollowsfromEq.(1)bythestandardquantiza-
S 2m S
classical particles interacting via the central force poten- tion procedure. Based on the general description given in
tialUBi,j(qi−qj)andHamiltonianHB(p,q)=(cid:80)Nj [2m1jp2j + [3, 13, 14], one can easily extend the classical definition
(cid:80) Ui,j(q −q )], with which the classical system inter- in Eq. (2) to the quantum regime, namely,
i,j B i j
acts via the central force potential energy V (q −q), so
j j 1 (cid:110) (cid:104) (cid:105) (cid:111)
that the total Hamiltonian is given by ρˆ = tr exp − Hˆ(pˆ,qˆ,pˆ ,ˆq ) β . (4)
S Z B j j
N
(cid:88) TheoperatorcharacterofthevarioustermsinEq.(4)and
H =H (p,q)+H (p,q)+ V (q −q). (1)
S B j j their commutativity relations prevent us from proceeding
j
as we did in the classical case. However, these very same
In classical statistical mechanics, the thermal equilibrium commutativity relations allow the immediate formulation
distribution of the system S is defined by of the following set of inequalities,
1 (cid:90) (cid:89)N [Hˆ ,Vˆ](cid:54)=0⇒∆Hˆ ∆Vˆ ≥ 1|(cid:104)[Hˆ ,Vˆ](cid:105)|, (5a)
ρS(p,q)= Z dpjdqjexp[−H(p,q,pj,qj)β], (2) S S 2 S
j [Vˆ,Hˆ ](cid:54)=0⇒∆Vˆ∆Hˆ ≥ 1|(cid:104)[Vˆ,Hˆ ](cid:105)|, (5b)
B B 2 B
where Z = (cid:82) (cid:81)Ndp dq (cid:82) dpdqexp(−Hβ) denotes the
j j j (cid:113)
partition function of the total system, with β = 1/kBT where Vˆ = (cid:80)NVˆ(ˆq − qˆ) and ∆Oˆ = (cid:104)Oˆ2(cid:105)−(cid:104)Oˆ(cid:105)2
and T being the temperature of the environment. The j j j
denotes the standard deviation of Oˆ, with (cid:104)Oˆ(cid:105)=tr(Oˆρˆ),
integral over p in Eq. (2) trivially cancels out with the
j
ρˆbeingthethermalequilibriumstateofthesystemSand
corresponding contribution in Z. Due to the particular
dependence of V and Ui,j on q , q and q, the integrals the bath B.
j B i j
over {q } can be appropriately manipulated, with the Some implications follow from Eqs. (5). Specifically,
j
net result that they cancel out with the corresponding since |(cid:104)[HˆS,Vˆ](cid:105)| is a measure of the quantum correlations
contribution in Z. Thus, between the system and the bath, it dictates the lower
bound of ∆Hˆ ∆Vˆ. This lower bound is different for
S
ρ (p,q)=Z −1exp[−H (p,q)β], (3) each interaction since each particular form of Vˆ imposes
S S S
a different commutation relation. This last statement
where Z = (cid:82) (cid:81)Ndp dq exp(−H β). Hence, the ther- is precisely what allows, for example, for a connection
S j j j S
mal equilibrium distribution of a bounded particle in betweenthetheoryofpointerstatesandquantumthermo-
contact with a pairwise-self-interacting thermal bath via dynamics. ThegeneralboundsinEqs.(5)predictdifferent
central forces, irrespective of the coupling strength, is ex- thermal equilibrium states for each type of interaction, a
actly given by the canonical Boltzmann distribution. purely quantum effect.
Thisresultissurprisingbecause, inthestrongcoupling For example, since [HˆS,Vˆ] = 0 implies a pure de-
regime, there is no apparent physical reason why the cohering interaction, which can be treated here in the
equilibrium thermodynamic properties of a system are in- framework of fluctuations without dissipation [19], the
dependent, for a wide class of systems, of both the nature equilibrium state is an incoherent mixture of system’s
of the bath to which it is coupled and of the functional eigenstates and is expected to be well characterized by
formoftheobservablesthatmediatetheinteraction. The the canonical Boltzmann distribution [20]. In this case
physical picture that emerges from this result is that in [HˆS,Vˆ] = 0, so that ∆HˆS∆Vˆ ≥ 0, meaning that the
thelongtimeregime,anydissipativemechanismisequally commutativityrelationhereresultsintheminimumlower
effective in taking the system to thermal equilibrium. In boundon∆HˆS∆Vˆ. Notethatthesamelowestlimitisob-
other words, dissipative dynamics can contract the classi- tained if, as in the classical case, the thermal equilibrium
cal phase-space volume with no fundamental restriction state of the system ρˆS is formally the canonical Boltz-
and therefore, the resultant equilibrium state is indepen- mann distribution ρˆcSan. Specifically, if ρˆ= ρˆcSan⊗trSρˆ,
dent of the dissipative coupling and the rate at which then |(cid:104)[Hˆ ,Vˆ](cid:105)| = tr([Hˆ ,Vˆ]ρˆ) = tr([ρˆ,Hˆ ]Vˆ) = 0 since
S S S
equilibrium [Eq. (3)] is reached. This suggests that the [ρˆcan,Hˆ ] = 0, giving ∆Hˆ ∆Vˆ ≥ 0. This is just a con-
S S S
concept of intrinsic and extensive thermodynamic vari- sequence of the fact that the Boltzmann distribution is
ables [4] can be extended, in some cases, to the strong the zero-order-in-the-coupling thermal equilibrium state
coupling regime. By contrast, and as shown below, in the and therefore, disregards quantum correlations between
3
the system and the bath. Below, it shown that the bath Markovian dynamics [23, 26]. Since at fixed T, this non-
spectrum is also related to the lower bound and thus, Markovian character can be modified by the functional
Eq. (5) will ALSO allow for a clear connection to other form of the spectral density [27], Eq. (6) makes clear
fundamental features such as the failure of the Onsager’s that the equilibrium system properties depend on the
regression hypothesis in the quantum regime [5, 6]. non-Markovian character. This means that the quantum
Influence of the Spectrum of the Bath: Non-Markovian equilibrium statistical properties of a system experienc-
Character at Thermal Equilibrium—Although the set of ing Markovian dynamics (flat spectrum) are expected to
inequalities (5) are fairly general, it is not possible to in- differ from those of the same system experiencing non-
fer the role that standard quantities such as the spectral Markovian dynamics (non-flat spectrum), which is in
density or the spectrum of the bath play in establishing sharp contrast to the classical case (see Eq. 3). This can
the thermodynamic bounds above. To provide a concrete be clearly understood in terms of the different thermody-
expressionforthelowerboundinEq.(5a),theinteraction namic lower bounds resulting from either Markovian or
between the bath particles is set to zero, i.e., Ui,j = 0 non-Markovian interactions [see Eq. (6)].
B
and consider the second order picture of the system-bath To make a connection with previous studies, the failure
central force interaction, i.e., Vˆ ≈ (cid:80)N 1m ω2(qˆ −qˆ)2, Onsager’s regression hypothesis in quantum mechanics
j 2 j j j
whichyieldstothewell-knownUllersma-Caldeira-Leggett [5, 6] is discussed next. In doing so, note that the hypoth-
model [21, 22]. After expanding Vˆ, it is possible to rede- esisthatknowingallmeanvaluessufficestodeterminethe
fineHˆ inEq.(1)asHˆ =Hˆ(cid:48)+Hˆ(cid:48) +Vˆ withVˆ =Bˆ⊗Sˆ. quantum dynamics of the correlation functions is valid
S B SB SB
HereBˆ =(cid:80)Nm ω2qˆ andSˆ=qˆ,whichactintheHilbert only under Markovian dynamics [5, 6] and when corre-
j j j j
space of the bath and the system, respectively. The com- lations between the bath and the system are negligible
mutator [Hˆ ,Vˆ], calculated to second order in Vˆ β, is at equilibrium (in general, at any time) [28]. Based on
S SB
then given by the fact that formal Markovian dynamics can only be
achieved for flat spectra (bare Ohmic spectral density
 (cid:126)β  with (cid:126)β →0 [27]), these two conditions can be seen as a
 (cid:90) 
|(cid:104)[Hˆ ,Vˆ](cid:105)|∝tr [Hˆ ,Sˆ]e−HˆSβ dσSˆ(−iσ)K(σ) , single one when formulated in terms of Eq. (6). Specif-
S S S
  ically, Markovian dynamics imply |(cid:104)[Hˆ ,Vˆ](cid:105)| → 0 and,
0 S
(6) hence, the system-bath correlations vanish. This implies
where (cid:126)K(σ) = (cid:104)Bˆ(−iσ)Bˆ(0)(cid:105) denotes the two-time that Onsager’s regression hypothesis, as well as the Boltz-
B
correlation of the bath operators given by [23] K(σ) = mann distribution, pertains exclusively to the classical
π−1(cid:82) dωJ(ω)cosh(cid:0)1(cid:126)βω−iσ(cid:1)/sinh(cid:0)1(cid:126)βω(cid:1), J(ω) = realm.
π(cid:80)∞ 1m ω3δ(ω−ω2) being the spectr2al density of the To provide some insight into the magnitude and con-
j 2 j j j
bath. Note that as long as second order perturba- sequences of the fundamental limit derived above and,
tion theory is valid, Eq. (6) holds for any Sˆ and Bˆ in particular, of the role of the spectral density, an effec-
and can be straightforwardly generalized to the case of tive coupling to the bath is introduced below and sam-
Vˆ =(cid:80) Bˆ ⊗Sˆ . ple aspects of two complementary regimes are analyzed.
SB α α α
The main feature of the quantum thermodynamic Specifically, (i) effective strong coupling characterized by
bound in Eq. (6) is the presence of the power spectrum deviations from standard thermodynamics, and (ii) an
of the bath I(ω,T)=(cid:126)J(ω)coth(1(cid:126)βω), which for bare effective weak coupling that is shown below to allow for
2
Ohmic dissipation, J(ω) = mγω, at high temperatures, the survival of entanglement between two oscillators in
(cid:126)β →0, is the flat I(ω,T)≈2mγk T. Note that in this thermal equilibrium at high temperatures.
B
high temperature limit, the upper limit of the integral in Effective Coupling to the Bath.—For the Ullersma-
Eq. (6) vanishes, leading to the vanishing of the commu- Caldeira-Leggett model, a standard calculation [23], after
tator, evenif[Hˆ ,Sˆ](cid:54)=0. Asimilarseries-expansionanal- removing a local contribution in the correlation func-
S
ysis leads to the conclusion that the thermal equilibrium tion, yields K(τ) = 2(cid:126)mβ ddτ (cid:80)∞l=1γ˜(|νl|)sin(|νl|τ), where
state ρˆ formally approaches the canonical Boltzmann ν =lΩ, with Ω=2π/(cid:126)β, are the Matsubara frequencies
S l
distribution only when (cid:126)β → 0. In other words, in the and γ˜(z) defines an effective coupling to the bath. Note
high temperature limit the quantum correlations between that γ˜(|ν |) contains all the information about the cor-
l
the bath and the system disappear and the thermal equi- relations of the bath operators and therefore defines the
librium state is described by the canonical distribution, influenceofthebathonthesystematthermalequlibrium.
irrespective of the coupling strength or the functional For the subsequent discussion we adopt the most
form of the spectral density J(ω). It is clear that these commonly used spectral density, the regularized Drude
results cast doubt on arguments in favor of canonical model with a high frequency cutoff ω , J(ω) =
D
typicality in the quantum regime [24, 25] at other that m γωω2/(cid:0)ω2+ω2(cid:1), where γ is the standard strength
0 D D
high temperatures. coupling constant to the thermal bath and ω dictates
D
Forout-of-equilibriumquantumdynamics,thelowtem- the degree of non-Markovian dynamics. For this par-
perature condition, finite (cid:126)β, is associated with non- ticular case, the effective coupling is given by γ˜(|ν |) =
l
4
FIG.1. log(Z/Z )foraharmonicoscillatorasafunctionoftheratiosk T/(cid:126)ω andω /ω . Wecomparethepartitionfunction
can B 0 D 0
forγ =0.1ω (a),γ =0.05ω (b),γ =0.01ω (c)andγ =0.005ω (d). Weobservethat(i)thestrongerthedampingrateγ,the
0 0 0 0
stronger the deviation from the canonical partition function, (ii) the larger the cutoff frequency, the larger the deviation from
the canonical partition function and (iii) in the high temperature regime Ω/ω (cid:29)1, regardless the damping rate, no deviations
D
are obtained.
γ/(1 + |l|Ω/ω ). Below we analyze the effective weak from the discussion above. For the von Neumann entropy
D
coupling, Ω/ω (cid:29) 1, and the effective strong coupling, S =tr [ρˆ ln(ρˆ )],thebehavioroftheratiolog(S/S )is
D S S S can
Ω/ω (cid:28)1, regimes. essentially the same as the one described for the partition
D
Strong Effective Coupling Regime—To quantify the function ratio in Fig. 1, and is not shown here.
consequencesofnon-flatspectrainthisregime,consideras Since γ˜(z)= 1 (cid:82)∞ dωJ(ω) 2z , the use of different
thesystemaharmonicoscillatorofmassm0andfrequency spectral densitiems0ch0angπes tωheωf2u+nzc2tional form of the ef-
ω0coupledtoathermalbath[12,13]. Inparticular,weare fective coupling and therefore, of the thermal equilibrium
interested in quantifying: (i) the generation of squeezing properties. Hence, as long as (cid:126)β remains finite, different
in the thermal equilibrium state, (ii) the deviation from spectral densities lead to different thermal equilibrium
the canonical partition function Zcan =trSeHˆSβ and (iii) states.
the deviation from the canonical von-Neumann entropy Weak Effective Coupling Regime—As an example
Scan =trS[ρˆcanln(ρˆcan)]. in the weak effective coupling regime, consider the
For this case the momentum and position variances survival of entanglement at thermal equilibrium be-
are given by [12, 13] (cid:104)p2(cid:105) = m2ω2(cid:104)q2(cid:105) + ∆, where tween two identical harmonic oscillators with masses
0 0
(cid:104)q2(cid:105) = (cid:104)qc2l(cid:105)+ m20β (cid:80)∞n=1(cid:2)ω02+νn2 +γ˜(|νn|)|νn|(cid:3)−1 and mco0nsatnandtfrceq.uTenhceieHs aωm0illtinoenaiarlnyicsoguipvleendbwyitHhˆ c=ouHpˆlin+g
the squeezing parameter ∆=−2m γβ−1∂lnZ(cid:48)/∂γ with 0 S
Z(cid:48) = (cid:126)β1ω (cid:81)∞n=1|νn|(cid:2)ω02+νn2 +γ˜(|ν0n|)|νn|(cid:3)−1. We recall (cid:80)Nj,α,2(cid:104)2pˆm2j,αj + mj2ωj2 (qˆj,α−qˆα)2(cid:105), with α = {1,2} and
that for this model, the classical theory predicts (cid:104)p2(cid:105)= Hˆ = 1 (pˆ2+pˆ2)+1m ω2(qˆ2+qˆ2)−c q q . The intro-
m2ω2(cid:104)q2(cid:105) and (cid:104)q2(cid:105)=k T/m ω2, so that ∆ =0. cl duSction2mo0f in1depe2nden2t b0at0hs1for e2ach os0ci1lla2tor ensures
0 0 cl cl B 0 0 cl
For the effective weak coupling regime Ω/ω (cid:29) 1, that no deviations from Boltzmann’s distribution are
D
disgarding terms of the order ω /ω and γ/ω gives present in the classical case. This can be verified directly
0 D D
π(cid:126)γmω /6Ω [12]. Thus ∆ vanishes at high tempera- from the multi-particle-system generalization of Eq. (1).
D
tures, and the classical unsqueezed state is recovered. At equilibrium, the entanglement between the two har-
However, for the strong coupling regime Ω/ω (cid:28) 1, monic oscillators can survive only when k T/(cid:126)ω (cid:28) 1
D B 0
∆ ≈ (cid:126)γmln(2πω /Ω) [12], meaning that the deviation (see, e.g., Ref. [29]). However, this limit only applies in
D
from the canonical state translates into squeezing of the the Markovian regime and γ → 0. Thus, based on the
equilibrium state. This feature may be of relevance to- discussion above, and supported by the recent observa-
ward the generation of non-classical states, e.g., in nano- tion that non-Markovian dynamics assists entanglement
mechanical resonators. in the longtime limit [9], we expect that this limit needs
Deviations from the canonical result are also evident in to be refined in order to account for the non-Markovian
the partition function Z. Figure 1 shows the logarithmic character of the interaction and the finite value of γ.
of the ratio of Z to the canonical partition function Z Figure 2 shows the logarithmic negativity for different
can
as a function of the dimensionless parameters k T/(cid:126)ω values of the damping constant γ as a function of the
B 0
and ω /ω for (from left to right) γ =0.1ω , γ =0.05ω , dimensionless ratios k T/(cid:126)ω and ω /ω . As expected,
D 0 0 0 B 0 D 0
γ =0.01ω and γ =0.005ω . Deviations are observed at (i) the more coupled the oscillators are, the higher the
0 0
low temperatures and for high cutoff frequencies (i.e., in temperatures and the damping rate at which entangle-
theeffectivestrongcouplingregime). Intheoppositelimit, ment can survive at equilibrium (not shown), and (ii) the
regardless of the coupling parameter γ, both calculated smaller the damping rate (the more isolated the system
partition functions show the same behavior, as expected is), the higher the temperature at which entanglement
5
FIG. 2. Logarithmic Negativity in the presence of non-Markovian interactions for c =0.1m ω2 with γ =0.1ω (a), γ =0.05ω
0 0 0 0 0
(b), γ = 0.01ω (c) and γ = 0.005ω (d) as a function of the dimensionless parameters k T/(cid:126)ω and ω /ω . We observe
0 0 B 0 D 0
that (i) the stronger the coupling between the oscillators, c , the higher the temperature at which the system can support
0
entanglement at equilibrium (not shown); (ii) the higher the damping rate γ, the stronger the influence of the non-Markovian
interaction depicted by the ratio ω /ω ; and (iii) the smaller the ratio ω /ω (a more non-Markovian interaction), the higher
D 0 D 0
the temperature at which the system can support entanglement at equilibrium.
canbemaintained. Thenewfeaturehereisthatthemore pleasure. This work was supported by the Comit´e para
non-Markovian the interaction, the higher the tempera- el Desarrollo de la Investigacio´n –CODI– of Universidad
ture and the damping rate at which entanglement can be de Antioquia, Colombia under contract number E01651
maintained at equilibrium. and under the Estrategia de Sostenibilidad 2013-2014, by
Discussion.—It has been shown that the fundamental the Departamento Administrativo de Ciencia, Tecnolog´ıa
bound imposed by the Heisenberg uncertainty principle e Innovaci´on –COLCIENCIAS– of Colombia under the
[Eq. (6)] prevents quantum systems to relax the Gibbs grant number 111556934912, by NSERC and by the US
state dictated by the system Hamiltonian only. The AirForceOfficeofScientificResearchundercontractnum-
Gibbs state is only recovered in the classical-high T limit ber FA9550-13-1-0005, by the Spanish MINECO project
((cid:126)β → 0). The implications at low-T for quantum ther- FIS2011-25167, the Gobierno de Aragon (FENOL group)
modynamics are so crucial as the failure for the Onsager and the EU (PROMISCE).
hypothesisorthedifficultyonthedefinitionsofthespecific
heat [30] and the temperature definition [31]. Specifically,
thehightemperatureregimedefinedby(cid:126)β →0,modified
by an appropriate effective coupling, emerges as the main
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