Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature
The concept of effective temperature (ET) T*(T₀, T ) is used in order to approximately “quantize” the thermodynamic functions of the dynamical object which is in the thermal equilibrium with thermal bath being at constant temperature T (T₀ = E₀/kB, where E₀ is the ground-state energy, kB – Boltz...
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| Цитувати: | Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature / Yu.G. Rudoy, A.D. Sukhanov // Condensed Matter Physics. — 2005. — Т. 8, № 3(43). — С. 507–535. — Бібліогр.: 31 назв. — англ. |
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Rudoy, Yu.G. Sukhanov, A.D. 2017-06-08T11:07:09Z 2017-06-08T11:07:09Z 2005 Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature / Yu.G. Rudoy, A.D. Sukhanov // Condensed Matter Physics. — 2005. — Т. 8, № 3(43). — С. 507–535. — Бібліогр.: 31 назв. — англ. 1607-324X PACS: 05.70.-a, 05.30.-d, 05.40.-a DOI:10.5488/CMP.8.3.507 https://nasplib.isofts.kiev.ua/handle/123456789/119748 The concept of effective temperature (ET) T*(T₀, T ) is used in order to approximately “quantize” the thermodynamic functions of the dynamical object which is in the thermal equilibrium with thermal bath being at constant temperature T (T₀ = E₀/kB, where E₀ is the ground-state energy, kB – Boltzmann constant, is the characteristic “quantum” temperature of the system itself). On these grounds the extensive comparative investigation is carried out for the “standard model” of statistical mechanics – the one-dimensional harmonic oscillator (HO). Three well-known approaches are considered and their thermodynamic consequences thoroughly studied. These are: the exact quantum, or non-classical Planck-Einstein approach, intermediate, or semiclassical Bloch-Wigner approach and, finally, the pure classical, or Maxwell-Boltzmann approach. Концепція ефективної температури (ET) T*(T₀, T) використовується для того, щоб наближено “квантувати” термодинамічні функції динамічного об’єкту, який є в термічній рівновазі з термостатом при постійній температурі T (T₀ = E₀/kB, де E₀ – енергія основного стану, kB – постійна Больцмана, – є характеристичною “квантовою” температурою самої системи). Базуючись на цьому, здійснено екстенсивне порівняння дослідження “стандартної моделі” статистичної механіки – одновимірного гармонійного осцилятора (ГО). Розглянуто три добре відомі підходи та добре вивчено їх термодинамічні наслідки, а саме: точний квантовий або некласичний підхід ПланкаЕйнштейна, проміжний або напівкласичний підхід Блоха-Вігнера і, нарешті, чисто класичний або підхід Максвелла-Больцмана. The authors devote this paper in honour of Prof. Dr. Igor R. Yukhnovsky on the occasion of his 80th birthday. We evaluate very much the fruitful efforts, which Prof. Yukhnovsky has spent to continue and elaborate the ideas of N.N.Bogoliubov’s school in the field of modern statistical mechanics. The present authors are also indebted to the late Prof. Dr. Yu.L.Klimontovich with whom we have had interesting discussions on the theory of open quantum systems. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature До єдиної некласичної фізики: врахування квантових флуктуацій в рівноважній термодинаміці через ефективну температуру Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature |
| spellingShingle |
Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature Rudoy, Yu.G. Sukhanov, A.D. |
| title_short |
Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature |
| title_full |
Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature |
| title_fullStr |
Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature |
| title_full_unstemmed |
Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature |
| title_sort |
towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature |
| author |
Rudoy, Yu.G. Sukhanov, A.D. |
| author_facet |
Rudoy, Yu.G. Sukhanov, A.D. |
| publishDate |
2005 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
До єдиної некласичної фізики: врахування квантових флуктуацій в рівноважній термодинаміці через ефективну температуру |
| description |
The concept of effective temperature (ET) T*(T₀, T ) is used in order to
approximately “quantize” the thermodynamic functions of the dynamical
object which is in the thermal equilibrium with thermal bath being at constant
temperature T (T₀ = E₀/kB, where E₀ is the ground-state energy,
kB – Boltzmann constant, is the characteristic “quantum” temperature of
the system itself). On these grounds the extensive comparative investigation
is carried out for the “standard model” of statistical mechanics – the
one-dimensional harmonic oscillator (HO). Three well-known approaches
are considered and their thermodynamic consequences thoroughly studied.
These are: the exact quantum, or non-classical Planck-Einstein approach,
intermediate, or semiclassical Bloch-Wigner approach and, finally,
the pure classical, or Maxwell-Boltzmann approach.
Концепція ефективної температури (ET) T*(T₀, T) використовується для того, щоб наближено “квантувати” термодинамічні функції
динамічного об’єкту, який є в термічній рівновазі з термостатом при
постійній температурі T (T₀ = E₀/kB, де E₀ – енергія основного
стану, kB – постійна Больцмана, – є характеристичною “квантовою”
температурою самої системи). Базуючись на цьому, здійснено екстенсивне порівняння дослідження “стандартної моделі” статистичної
механіки – одновимірного гармонійного осцилятора (ГО). Розглянуто три добре відомі підходи та добре вивчено їх термодинамічні
наслідки, а саме: точний квантовий або некласичний підхід ПланкаЕйнштейна, проміжний або напівкласичний підхід Блоха-Вігнера і,
нарешті, чисто класичний або підхід Максвелла-Больцмана.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119748 |
| citation_txt |
Towards the unified non-classical physics: account for quantum fluctuations in equilibrium thermodynamics via the effective temperature / Yu.G. Rudoy, A.D. Sukhanov // Condensed Matter Physics. — 2005. — Т. 8, № 3(43). — С. 507–535. — Бібліогр.: 31 назв. — англ. |
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Condensed Matter Physics, 2005, Vol. 8, No. 3(43), pp. 507–535
Towards the unified non-classical
physics: account for quantum
fluctuations in equilibrium
thermodynamics via the effective
temperature
Yu.G.Rudoy∗, A.D.Sukhanov†
People’s Friendship University,
117198, 6 Mikluho-Maklaya Str.,
Moscow, Russia
Received May 13, 2005
The concept of effective temperature (ET) T ∗(T0, T ) is used in order to
approximately “quantize” the thermodynamic functions of the dynamical
object which is in the thermal equilibrium with thermal bath being at con-
stant temperature T (T0 = E0/kB, where E0 is the ground-state energy,
kB – Boltzmann constant, is the characteristic “quantum” temperature of
the system itself). On these grounds the extensive comparative investiga-
tion is carried out for the “standard model” of statistical mechanics – the
one-dimensional harmonic oscillator (HO). Three well-known approaches
are considered and their thermodynamic consequences thoroughly stud-
ied. These are: the exact quantum, or non-classical Planck-Einstein ap-
proach, intermediate, or semiclassical Bloch-Wigner approach and, finally,
the pure classical, or Maxwell-Boltzmann approach.
Key words: equilibrium thermodynamics, effective temperature, quantum
fluctuations
PACS: 05.70.-a, 05.30.-d, 05.40.-a
1. Introduction
The problems of quantitative and qualitative differences between classical (C)
and non-classical (NC) physical “worlds”, as well of their interrelations, are ones of
the most stringent nowadays (see, e.g., [1–4]). The important role of non-classical
ideas in modern physics as well as in natural sciences on the whole was stressed in
[5]. Usually one is intended to consider as a non-classical only the QD-“world” of
∗E-mail: rudikar@mail.ru
†E-mail: ogol@oldi.ru
c© Yu.G.Rudoy, A.D.Sukhanov 507
Yu.G.Rudoy, A.D.Sukhanov
quantum dynamics, but in our opinion it is quite reasonable to include therein the
ST-“world” of statistical thermodynamics as well.
The unifying feature of these both “worlds” is the principal need for their prob-
abilistic description and hence for unavoidable use of the concept of fluctuations
and their correlations (the quantum and thermal ones, respectively). Obviously, this
concept is completely alien to “worlds” of classical dynamics (CD) and classical
thermodynamics (CT), which are fully deterministic – at least on the fundamental
level. This means that the concepts of statistical ensemble and distribution functions
may be sometimes used in classical mechanics as well – e.g., in order to account for
the experimental dispersion of the initial conditions for the object’s dynamics1(see,
e.g. [7,8]).
But in principle one can avoid the use of these probabilistic concepts by means
of a required exact specification of the initial values. In other words, the classical
description is the degenerate case of the non-classical one; namely, all the classical
distribution functions may be made to be delta-like (i.e., to have zero spread), where-
as in non-classical case this spread is always strictly positive and regulated by the
fundamental constants which were introduced in physics by Planck and Boltzmann.
Following this line of thought we adopt (as was already formulated in [14,15])
that there exist in general four large-scale divisions in the description of open phys-
ical objects – i.e., interacting with their macroscopic surroundings. These divisions
include two levels (upper and lower)), each containing two versions (left and right).
The vertical “level” division includes (from top to bottom) the dynamic, or “cold”
D-level and the thermodynamic, or “warm” T-level. It is essential that this division
refers not to the physical object itself but rather to its outer surrounding or, more
precisely, to the character of the interaction, or the contact between the object and
surroundings.
The horizontal “version” division includes (from left to right) classical (C), and
non-classical (NC) versions (on the dynamical level the last version is usually called
quantum (Q)). In the scope of non-classical versions it is also accepted to consider
their restricted variants – the so-called semi- (or quasi-) classical versions (SC or
QC). The degree of non-classicality in the proposed scheme is determined by the
degree of fluctuations taken into account, which are called quantum on the D-level
and thermal on the T-level (see for more details in section 2).
It was already noted that on the dynamic level in C-version, the interaction
between the object and the surroundings can be described in a fully deterministic (so
to say, controllable) way, because it is given unambiguously in terms of macroscopic
quantities – external forces or potentials; the results of their effect on the object are
also unambiguous (in the case of strictly specified initial conditions).
On the other hand, in Q-version the dynamic effect of the surroundings becomes
principally uncontrollable due to the minimal value of action equal to the Planck
constant ~. Thus the description in Q-version is always a probabilistic one, and just
1One should mention here the rather delicate case of dynamic chaos, which may require a
probabilistic (stochastic) description even in the scope of classical Newtonian mechanics due to
nonlinearity and ultimate instability caused by small disturbances of initial conditions.
508
Towards the unified non-classical physics
the quantity ~ is a measure of fluctuations (i.e., dispersion) of conjugate quantum
observables. As a mathematical expression of these statements (called sometimes
“quantum kinematics”) serves the Heisenberg indeterminacies relation, later gener-
alized by Schroedinger.
On the thermodynamic level the outer surroundings, or thermal bath is, as a
rule, a macroscopic one, whereas the object may be in general of microscopic origin.
The interaction between the object and the bath in this case is called the thermal
contact and is also (as in quantum dynamics) an uncontrollable kind of action. In
the proposed classification for the T-level we suggest the presence of full thermal
equilibrium (or, possibly, slight deviations from it with small fluctuations) between
the object and thermostat. This state arises after sufficiently long time of ther-
mal contact, which exceeds the characteristic (for the given object) relaxation time
and is characterized by a set of intensive thermodynamic parameters (temperature,
pressure, chemical potential and so on).
The thermal contact is of purely statistical origin and consists in random ex-
changes of some extensive thermodynamic quantities (e.g., inner energy, volume,
particle number and so on) between the object and the bath2. In the simplest (but
seemingly most important) case the thermal equilibrium is characterized by a single
intensive thermodynamic variable – the absolute temperature T of the bath. Note
that due to zeroth law of thermodynamics the temperature of the object also equals
T . In classical version of T-level this equality is understood as an exact one, whereas
in non-classical – only on the average due to fluctuations.
Summing up, the physical object on the T-level in non-classical version can be
described only in the probabilistic terms. Following Einstein, the measure of relevant
thermal fluctuations (i.e., dispersions) of thermodynamically conjugate variables
(e.g., the inner energy and inverse temperature) is proportional to the Boltzmann
constant3 kB.
In the classical version on the T-level no such fluctuations are considered, which
may be formally viewed as the limit kB → 0 (in close analogy with the limit h → 0
for the version on the D-level).
The most interesting class of physical objects is, in our opinion, that of “warm”
microscopic (or “quantum”) objects, being in thermal contact with the macroscopic
surroundings in the form of thermal bath. In this case both quantum and thermal
fluctuations come simultaneously into play: they are due to both types of the un-
controllable effect of the surroundings represented by the fundamental constants h
and kB.
Before we formulate the concrete aim of our investigation let us classify more
precisely the existing physical theories adopting the above sketched classical vs.
non-classical (i.e., “fluctuational”) point of view.
2It is assumed that the thermodynamic state of thermal bath is practically unaffected by this
exchange because the degree of macroscopicity of the thermal bath significantly exceeds that of
the object.
3Einstein [6] attached great importance to this quantity and considered it as a fundamental one
– namely, as a “measure of the thermal stability” of the physical object in the thermal bath.
509
Yu.G.Rudoy, A.D.Sukhanov
2. The “fluctuational” classification of physical theories
On the dynamic, or D-level the relevant fluctuations are the quantum ones of
mechanical observables, usually described by the pairs of canonically conjugated
variables (e.g., coordinates and momenta). Clearly, purely classic dynamic (CD)
theory, which contains no such fluctuations at all, is then the Newton mechanics
(including its relativistic generalization by Einstein) for particles as well as Maxwell
electrodynamics for fields.
On the other hand, fully non-classical theories of D-level are quantum mechanics
and quantum electrodynamics of Heisenberg-Schroedinger-Dirac, because in these
theories are inherent the simultaneous fluctuations (as well as correlations) of the
both canonically conjugate variables. Now, the Bohr-Sommerfeld mechanics permits
the fluctuations of only one variable in each pair and so it is the semiclassical theory
of D-level.
Note that the latter theory differs radically from the Newton-Einstein mechanics,
because it contains (though not quite consistently) the idea of quantization, and so
belongs to the non-classical group of theories of D-level. Clearly, in this group4 the
relevant fundamental constant is the Planck constant h.
Analogously, on the thermodynamic T-level the relevant fluctuations are the
thermal ones which apply to thermodynamic observables, which are also combined
in pairs of conjugated variables – e.g., the inner energy and (inverse) absolute tem-
perature. On this level one may consider as the classical version the Carno-Clausius-
Kelvin thermodynamics, which is applicable to all objects being in thermal equilib-
rium. This theory, like the Newton-Maxwell-Einstein theory on the dynamic D-level,
does not contain any idea of fluctuations at all.
On the other hand, a fully non-classical version of thermodynamic (T-level)
description is at present the Einstein statistical thermodynamics (which is usually
called Einstein’s theory of (quasi)thermodynamic fluctuations), because fluctuations
of the both of the pair of conjugate thermodynamic parameters (e.g., inner energy
and inverse temperature, considered as fully independent quantities) are treated
simultaneously on the same footing.
Not surprising that here one also arrives at some kind of Einstein’s, or thermo-
dynamic “uncertainties relations”. They are formally analogous to the Heisenberg-
Schroedinger dynamic uncertainties relations, but in some respects differ quite no-
ticeably from the latter (for details see [7,8] and references therein). Clearly, in the
group of T-theories the relevant fundamental constant is Boltzmann constant kB.
Finally, as a semiclassical version of the description on the T-level one may
consider the approach which was firstly formulated by Szilard [9] and later on was re-
interpreted and partly modified by Mandelbrot [10], Tisza [11] and others (see, e.g.,
the presentation of some of these results in [12]). In the original Szilard’s formulation
the fluctuations of only one of the pair of conjugated thermodynamic parameters
(as a rule, the extensive one – e.g., the inner energy) was considered.
4For simplicity we consider here only the non-relativistic limit of dynamic description, so the
finitness of the velocity of speed c is irrelevant.
510
Towards the unified non-classical physics
Having taken the classification described above, one may well be concerned
with a description of the physical object, which is, e.g., classical in dynamic re-
spect and non-classical in the thermodynamic one, or vice versa; some other com-
binations are also allowed. Evidently, the more fundamental theories are those
more fully accounting for fluctuations of both kinds – quantum as well as ther-
mal5.
Thus the most desirable goal is to establish the logically consistent fully non-
classical description – in both dynamic and thermodynamic senses. Probably, this
will appear possible on the grounds of the general theory of open quantum systems
(which is now absent) and may, perhaps, deliver the “missing link” even for creating
quantum gravity and cosmology6.
But for more modest practical purposes some intermediate descriptions are some-
times sufficient, and so various approximate procedures might be of use. The pri-
ncipal possibility of such procedures is based on the fact that the physical object
considered as an completely isolated one does not determine unambiguously the
choice of its description. More precisely, depending upon the appropriate physical
conditions this object may stay in classical or non-classical regimes.
The description in classical regime is often simpler than in non-classical, but the
choice between these descriptions is determined not by consideration of simplicity
but by numerical values of relevant dimensionless parameters in the condition of the
given problem.
As a rule, between these two kinds of descriptions there is some connection;
namely, the transition from the non-classical to the classical one is obtained by
some kind of approximate semi- (or quasi-) classical procedure, whereas the reverse
transition is realized by means of some kind of account (more or less exact) for
quantum fluctuations.
On the D-level the parameter of non-classicality is the ratio h/SD, where SD
is the characteristic value of action for the given object in the given physical con-
ditions. If SD significantly exceeds h, the transition from quantum (Q) dynamics
to the classical (C) dynamics may be performed by the approximate quasiclassical
procedure, or Wentzel-Kramers-Brillouin perturbative expansion in powers of small
parameter h/SD.
The reverse transition from C- to Q-regime is also well defined by the quantizati-
on rules of Bohr-Sommerfeld (approximate semiclassical description) and Heisenberg-
Schroedinger-Dirac (exact fully non-classical description).
Analogous classical-non-classical transitions are sometimes possible for the T-
level of description as well; here other parameters of non-classicality are of use. The
most popular of them is the number N of particles in the (macroscopic) physical
object. If N is large enough (∼ 1023), one can perform the transition from the non-
5From this point of view the “Theory Of Everything” (TOE) actively studied now (see, e.g.,
[14]) is not more than the elaboration of only one division of the proposed scheme – namely, the
non-classical D-theory, because it fully ignores thermal fluctuations.
6Note in this connection that the expression for the entropy of the black hole includes all the
four fundamental constants entering the Planck’s universal system of units – namely h, c, G and
especially kB (see, e.g., [15]).
511
Yu.G.Rudoy, A.D.Sukhanov
classical version to the classical one as a perturbative expansion in 1/
√
N , thus
neglecting partially or completely the thermal fluctuations.
Clearly, this is not appropriate already for mesoscopic (and all the more for
nanoscopic) physical objects with N ≈ 106÷102, often used nowadays in the experi-
ment at thermodynamic conditions; the neglect of thermal (and, of course, quantum)
fluctuations is also quite inappropriate near the critical points as well as at ultralow
temperatures [16].
Moreover, the 1/
√
N -expansion does not work at all for such objects as thermal
radiation and black holes (where the very concept of N is not defined), so some
other (more universal) parameters are needed in these cases. By analogy with the
dynamic case, the probable candidate may be the ratio kB/ST, where ST is the
specific entropy, but this kind of expansion has not yet been sufficiently elaborated.
As was pointed above, the fully non-classical description on the T-level, or Ein-
stein’s statistical thermodynamics is formulated as the effective theory (in the sense
of [14]) on the grounds of the so-called Boltzmann principle, which is usually con-
sidered as one of the statements of the second law of thermodynamics (see [29]).
It is important that both descriptions on T-level – Einstein’s non-classical, as well
as the Carno-Kelvin-Clausius’ classical, are applicable to any physical object near
its thermal equilibrium with the thermal bath quite independent of the presence or
absence of any descriptions for this object on the D-level.
The transition from classical to non-classical version on the T-level is much less
investigated compared to the same transition on the D-level. As was pointed above,
some ideas of the appropriate stochastization of classical thermodynamics were out-
lined by Szilard, Mandelbrot and others, see references above), but a consistent
and effective theory of this type (not involving any microscopic D-level, or hidden
variables theories) is absent so far.
At present, the well developed theory which successfully imitates the semiclassi-
cal (in the fluctuational sense!) description on the T-level, is the Maxwell-Boltzmann-
Gibbs’ statistical mechanics [13]. This theory is characterized by the account for
fluctuatuions of only one (as a rule, the extensive one) thermodynamic quantity,
and in order to obtain the fluctuation of the conjugate parameter one should go
over to another Gibbs ensemble.
Moreover, the formulation of statistical mechanics (as it follows from the very
name of this theory) is founded on the use of the concept of microstates (in the
classical or quantum sense – irrelevant in the present aspect) of the physical object.
This concept is quite alien to the effective description on the T-level, which should
employ only the concept of macrostates.
Nevertheless, there exist a direct and physically appealing way to go over from
semiclassical Gibbs approach to a fully non-classical Einstein approach, which is
based on the full stochastization of all the pairs of conjugate thermodynamic quan-
tities (see our previous papers [1,5]).
As was pointed out above, finally we are interested in the full account for both
types of fluctuation – thermal as well as quantum ones, but in the present paper we
are not intended to go so far as to attempt to build the unified fully non-classical
512
Towards the unified non-classical physics
description (both on D- and T-levels). We will confine ourselves to a more modest
task, which will, however, reveal some general tendencies. Namely, we will show
how to build a description which accounts simultaneously for thermal and especially
quantum fluctuations, staying within the Gibbs model for the semiclassical version
of the T-level description.
The degree of the effect of the object’s quantum fluctuations on its thermody-
namic behavior at thermal equilibrium depends largely upon the combination of
internal and external conditions. These conditions allow the object to stay in C- as
well as in Q-regimes during the thermal contact with the thermal bath. It is clear
that the conditions of realization of these regimes on the T-level will significantly
differ from the corresponding conditions for the D-level (see section 4 below).
Once the non-classical (quantum) version of statistical mechanics based on the
Q-regime is achieved, it is also rather simple to obtain its reduced (i.e., semi- or pure
classical) versions for C-regime, when one partially or fully neglects the quantum
fluctuations. This Q → C transition is achieved by means of high-temperature
expansion in powers of 1/T , always accompanied by simultaneous quasiclassical
expansion in powers of h.
These both expansions are, of course, only formal ones; as a matter of fact, the
expansion (like the situation in dynamical case) goes in powers of dimensionless
parameter – in this case the ratio hν/kBT , where ν is some characteristic frequency,
which may be always assigned to the object, T is the absolute temperature of the
thermostat.
Much less trivial task, with which we will be mostly engaged here, is the reverse
C → Q transition, which demands including in some way (as a rule, only approx-
imately) the quantum fluctuations (i.e., the existence of Q-regime), starting from
the classic dynamic description (i.e., from the C-regime) of the given object.
The principal way of performing such a transition was originally sketched in the
works by E. Wigner and F. Bloch early in the 30-ies of the 20th century, presented
by them in the remarkable and up till now often cited papers7 [22] and [23]. These
authors have simultaneously and independently developed rather different theoreti-
cal approaches in the field of quantum statistical mechanics; in particular, in the
articles cited above such fruitful concepts as Wigner function and Bloch equation
were introduced. Loosely speaking, these authors for the first time managed (though
in somewhat different ways) to account for quantum (i.e., dynamical fluctuational)
corrections in the thermodynamic description of the given physical object .
Wigner [22] has built the quasi-classical statistical mechanics for the object in the
arbitrary external field, using the expansion in powers of the small quantity h and
(simultaneously) in powers of the large quantity T . In fact, as was pointed out, this
expansion is in powers of really small dimensionless ratio hν/kBT , where ν is some
characteristic frequency for the object. Wigner found only some first terms of this
expansion (which complexity rises very fast with the number of terms) for the parti-
tion function as well as for distribution functions in coordinate q and momentum p.
7Slightly modified exposition of these results may be found in modern textbooks on statistical
mechanics [18–21]; see also the footnote 8 below.
513
Yu.G.Rudoy, A.D.Sukhanov
Unlike Wigner, Bloch used more general approach, which does not employ any
perturbative ideology (i.e., the expansion in some small parameter). Bloch [23] got
the exact (i.e., valid for all temperatures) expressions for the distribution functions
(d.f.’s) independently in q and p only for the particular case of one-dimensional
harmonic oscillator (1-d HO). Thus Bloch fully took into account the existence of
Q-regime; it is not surprising that in the limit of C-regime the Bloch’s result coincides
with the Wigner’s one for the particular case8 of 1-d HO.
Nevertheless, exact Bloch’s (or, equivalently, refined Wigner’s) result remains to
be only approximate for the partition function (p.f.) in Q-regime because the very
connection between d.f.’s and p.f. in this case is rather nontrivial (more details may
be found in section 8). These results bring about a significant improvement over the
ordinary classical version (accounting for the existence of C-regime only), especially
in the region of low temperatures, where the quantity hν/kBT cannot be considered
as a small one. But as will be shown below, in some respects these results differ
significantly from the exact results for the p.f. of the 1-d HO, obtained by Planck
and Einstein.
Our main aim in the present paper is as follows. We are going to show that most
of the results by Wigner and Bloch may be obtained in a unified – and, which is
even more important – much simpler way using the notion of the generalized, or
effective temperature (ET). We shall consider in detail only the case of 1-d HO,
which possess an exact solution obtained by Planck and Einstein and so permits the
control of accuracy for various approximations.
But it seems to us that the way suggested here possesses much wider (if not
universal) range of applicability if one takes into account the universality of the
concept of ET itself (section 4) and especially its connection with the third law of
thermodynamics (section 5) in its present formulation.
The main advantage of our approach consists in avoiding any dynamic aspects
connected with the transition from Q- to C-regimes (e.g., the intricate enough WKB-
theory). All you need to know about the corrections into thermodynamic quantities
due to quantum fluctuations seems to be already incorporated in the ET; the preli-
minary version of this approach was suggested in [26] (for a fuller discussion see the
next section 4)).
Seemingly, this circumstance reflects partly the universality (in the sense of spe-
cific model independence) of the general thermodynamic laws contained already in
classical version of the T-level, which do not distinguish between the Q- or C-regimes
altogether. These regimes differ only in the specific forms of the dependence of ther-
modynamic quantities upon the ratio hν/kBT ; just this difference is partly taken
into account by the ET. As is the case for any effective (in the sense of [14]) theory,
the relevant quantities are not only calculated but also experimentally measured
according to their operational definitions.
8About half a century later it was revealed [24,25] that after the appropriate (mathematically
sufficiently hard) refining of Wigner’s approach it becomes valid for arbitrary (and not only small)
values of hν/kBT . This refined Wigner’s approach when applied to the 1-d HO leads to the same
exact results for d. f.’s as the Bloch’s one.
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Towards the unified non-classical physics
Contrary to the C-version of thermodynamics, the statistical mechanics is always
based on some version (C- or Q-) of dynamics and so calculated thermodynamic
functions are more or less dependent upon the model of the object. So by refining
the statistical-mechanical description (i.e., going from C- to SC- or even Q-version
for the macroscopic object) one should operate with many dynamic variables in
the high-dimensional space9 – phase space (for C-regime) or Hilbert space (for Q-
regime) and then to average upon these variables.
In this respect the method of ET provides a great economy, because it allows
for any objects during the C → Q transition to deal instead only with few smooth-
behaving thermodynamic quantities in the low-dimensional space of thermodynamic
variables. Of course, the simplification of the mathematical apparatus permits to
better control the physical sense of the obtained results. Moreover, the results of
thus improved descriptions should be in better accordance with the experimental
thermodynamic measurements10.
To avoid confusion we should note in conclusion of this section that despite
its effectiveness, the method of ET (which is described in sections 4, 5) presents
only some kind of extrapolation (from C- to Q-regime in the scope of semiclassical
description on the T-level) and so from the very beginning is an approximate one.
In sections 6–9 we shall give the evaluation of applicability and accuracy of the
ET-method and thus establish more distinct boundaries between the classical and
quantum versions of Gibbs equilibrium statistical mechanics. The latter theory is
briefly sketched in section 3; the description of HO in C-regime (Maxwell-Boltzmann
approach) is given in section 6 and in Q-regime (Planck-Einstein approach) – in secti-
on 7. Sections 8 and 9 are devoted to the semiclassical Bloch and Wigner approaches,
whereas in section 10 some conclusions are formulated.
3. Gibbs statistical mechanics
To describe the physical object on the grounds of Gibbs statistical mechanics (or
SCT-theory according to the nomenclature suggested in section 2) means to calculate
the appropriate thermodynamic functions in terms of the most fundamental quantity
of Gibbs theory – the partition function (p. f.) Z(T ). The starting expressions for
Z(T ) differ for Q- and C-regimes, whereas the recipes (but of course not the results)
for further calculation of thermodynamic functions in both cases are quite identical.
Namely, if the p.f. Z(T ) is known for the object, so the Massieu-Planck function
Φ(T ) and free energy F (T ) are of the form
Φ(T ) = ln Z(T ), F (T ) = −kBTΦ(T ). (1)
Inner energy E(T ) and specific heat capacity CV(T ) at constant volume V have
9One should note that these variables may – and, as a rule, really possess – rather complex and
irregular behavior.
10Note that just this way led Planck [27], who studied as the physical object the thermal radia-
tion, finally to the invention of his famous “quantum” of action h.
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Yu.G.Rudoy, A.D.Sukhanov
correspondingly the forms
E(T ) = kBT 2[dΦ(T )/dT ] = kBT 2[(1/Z(T ))(dZ(T )/dT )], (2)
CV(T ) = dE(T )/dT = 2(E(T )/T ) + kBT 2[d2Φ(T )/dT 2] (3)
and, finally, the entropy S(T ) is given by the relation
S(T ) = −dF (T )/dT = E(T )/T + kBΦ(T ). (4)
Any given physical object on the dynamic level is described by the Hamilton
function H(q, p) (in C-regime) or by Hamiltonian Ĥ(q, p) (in Q-regime); in the
simplest case of one particle in the potential U(q) one obtains:
H(q, p) =
(
p2/2
)
+ U(q), Ĥ(q, p) = −
(
h2/2
) (
d2/dq2
)
+ U(q). (5)
Here H(q, p) is a function, determined uniquely on the phase space11 (q, p) of
coordinates q and momenta p, while H(q, p) is an operator with eigenvalues En > 0
(n = 0, 1, 2, . . . – main quantum number). In terms of these quantities the p.f. Z(T )
is given in the following ways: for C-regime
ZC(T ) =
∫
dqdp(2π~)−s exp [−H(q, p)/kBT ] =
∫
dEg(E) exp [−E/kBT ] , (6)
where g(E) is the phase space structure function, or density of states (s is the
number of degrees of freedom), whereas for Q-regime
ZQ(T ) =
∑
n
gn exp [−En/kBT ] , (7)
where gn is the degeneration factor for n-th energy level En. Strictly speaking, the
expression (6) cannot be considered as a pure classic one, because the ultimate quan-
tum origin of the object is partially taken into account (though in a rather primitive
way) by using the factor (2π~)−s. Following P . and T . Ehrenfest, one introduces the
notion of “cells” in phase space, which partially reflects the existance of uncertainti-
es relation between q and p. Obviously, the “cellular” structure of the phase space is
not more than some kind of quantum “kinematics”, whereas the quantum dynamics
should be (and in fact will be) accounted for more adequately below. Note that due
to the notion of “cells” the Planck constant enters the “classical” expression ZC(T )
from the very beginning in order to obtain it as the limit of ZQ(T ) when (formally)
h → 0. Probably, it would be more correct to substitute the differential element
dp dq of phase space volume in (6) by the product of finite differences ∆p ∆q, so
that the relation ∆p∆q/hs would be an integer number (usually very large). Then
the notation employed in (6) corresponds to the “small h” limit, because only in
11Here and below we use the renormalized coordinates q → q
√
m and momenta p → p/
√
m (m is
the mass of the particle), so that this scale transformation is a canonical one and the phase space
volume element dpdq remains invariant.
516
Towards the unified non-classical physics
this case ∆p ∆q/hs → dp dq as h → 0. The procedure of “cell quantization” rep-
resents the different ways of determining and counting the number of microstates
of the given object. In C-regime these states form a strictly continuous set, and so
their number is always infinite. In Q-regime the states may form a discontinuous,
or discrete set, denoted usually by some (also discrete) quantum numbers. Clearly,
there may well exist the continuous part of the state (not phase!) space, so the full
number of states may be also infinite (though in many cases countable).
But the most important thing is the following: in Q-regime the uncertainties
relations prevent the description of the phase space states by means of simultaneous
exact specification of both p and q, whereas in C-regime it is quite legitimate. Ob-
viously, in the intermediate, or SC-regime the demands of the quantum uncertainty
principle (and appropriate relations) should be taken into account more carefully
(as compared to simple “cell quantization”).
We propose (section 4) some sort of this procedure, which (in contrast to Bloch
and Wigner) is performed by means of the ET after the Gibbs averaging over the
phase space is fulfilled. Of course, it is a rather difficult task to check the accuracy of
such a procedure for obtaining ZSC(T ) in the general case, when the exact solution
for ZQ(T ) is absent12.
Evidently, the integration upon the continuous variables (q, p) for obtaining
ZC(T ) is, in general, much simpler than the summation upon the discrete ener-
gy eigenvalues En necessary for obtaining ZQ(T ) (separating the problem of finding
these En even for known Ĥ(q, p) ). Clearly, both mathematical procedures (for C-
and Q-regimes respectively) are in general not equivalent to each other. Indeed,
they lead via Z(T) from (6) and (7) to thermodynamic functions (1)–(4) with quite
different dependence upon T ; as a rule, this dependence approximately coincides at
high T , but differs radically at low T (see in this connection any advanced textbook,
e.g., [12–15]).
More precisely, the choice among the expressions (6) and (7) – or, which is the
same, among the type of dynamic, or microscopic regime (C- or Q-) the object
remains in thermal equilibrium – is regulated by the dimensionless temperature
parameter
c = E0/kBT = T0/T, where T0 ≡ E0/kB , (8)
which may be called the degree of object’s “thermal quantumness”.
Here E0 is the lowest possible, or ground-state energy of the object, T0 – the
corresponding quantum dynamic characteristic quantity, having the dimension of
temperature; the quantity kBT has the meaning of some “average” thermal energy,
which approximately relates to one degree of freedom for thermostat (considered as
the classical one).
Obviously, the almost pure C-regime corresponds to the low values of c, when T
is much higher than T0 (T À T0, or in the limiting case T → ∞, c → 0), whereas
the almost pure Q-regime corresponds to the high values of c, when T is much lower
12That is why we employ in this paper as an example just the one-dimensional HO – the simple
enough, but by no means trivial model, fortunately possessing the exact solution ZQ(T ).
517
Yu.G.Rudoy, A.D.Sukhanov
than T0 (T ¿ T0, or in the limiting case T → 0, c → ∞); note, that for T0 = 0
(purely classical object) the C-regime is formally valid for all T > 0.
In other words, to be in the C-regime for a thermodynamic system on the whole
means that the dynamic object should be in thermal contact with a “warm enough”
thermostat, whereas to be in a Q-regime – on the contrary, with a “cold enough”
thermostat.
Consequently, the description of the quantum object in the thermostat depends
not only upon the temperature of the thermostat (as it usually holds for the clas-
sical object), but also upon the property of the quantum object itself (namely, its
characteristic temperature T0). Therefore it is not surprising that both p.f.’s ZC(T )
and ZQ(T ) are in fact the functions of c, while the ZC(T ) is the high-T (low-c) limit
of its quantum counterpart ZQ(T ).
Here naturally does arise the opposite question: if there is given ZC(T ), is it
possible to propose some procedure which could give us some knowledge about the
behavior of its quantum counterpart ZQ(T )? Of course, the question is to describe
not only the high-T (low-c) but also the most interesting low-T (or high-c) limit.
Although the existence of such a procedure is in no way a priori obvious, there
are quite convincing arguments in its favor. These arguments follow unambiguously
from the results of Bloch and Wigner cited and briefly discussed in section 2 (for a
fuller discussion see sections 8 and 9, respectively).
In the context of our paper the most significant thing is that these authors in
fact discovered a tool for carrying out the approximate semiclassical (SC) procedure
that we are interested in. This tool in its very essence proves to be nothing else than
the meant above effective temperature (ET) (see section 4–5); at first glance the
approaches by Wigner and Bloch seem to have little to do with this concept13, but
in section 7–9 we show the opposite.
Being an approximate one, the semiclassical procedure in the case of Wigner
(section 9) supplies us with some additional knowledge only of high-T behavior for
ZQ(T ) (though for arbitrary U(q)), whereas in the case of Bloch (section 8) one
also obtains some knowledge about the much more intriguing low-T region for this
quantity (but only for the HO case).
4. The general concept of effective temperature
The concept of ET (which will be here and below denoted by T ∗) is rather
universal and applicable to any physical object (which are all ultimately of quantum
origin). The object here and below is assumed to be in thermal equilibrium with the
thermostat, possessing the absolute temperature T > 0.
In general T ∗ 6= T (or, more definitely, it appears that always T ∗ > T ), because
the ET depends not only upon the “outer” ordinary, or “classical” temperature T ,
13This notion was explicitly introduced (but not discussed) much later by Landau and Lifshitz [8]
only in connection with Wigner’s result [22]. Unfortunately, there was not established any connec-
tion between the reproduced in [18] results by Wigner and Bloch, as well as their thermodynamic
consequences were not analyzed.
518
Towards the unified non-classical physics
but also upon “inner” characteristic, or “quantum” temperature T0 defined by (8),
so as
T ∗ = T ∗(T0, T ), T ∗
> T, T0 > 0. (9)
Obviously, one can say that the (positive) difference between T ∗ and T is fully
determined by the inherently quantum nature of the object.
As is well known, in general E0 is non-zero (positive) quantity (E0 > 0), and so
is T0 > 0; this is due to the fluctuations of purely quantum origin. But the precise
value of E0 significantly depends upon the physical nature of the relevant object’s
degree of freedom.
For example, for the translational motion of the quantum object in the macro-
scopic (large enough) box one has E0 ≈ 0, so that T0 ≈ 0 and for all T holds:
T ∗(0, T ) ≈ T. (10)
But for most internal degrees of freedom of the single quantum object one has
E0 > 0, T0 > 0, so in the low-temperature limit T → 0 it holds
T ∗(T0, 0) = T0. (11)
Thus, ET should interpolate between the “external” temperature T of the ther-
mostat and the “internal” temperature T0 of the object itself. Generally speaking,
one may expect, that for different classes of physical objects the functional form of
(9) is also different, although the very concept of ET seems to be a universal one.
Further, on rather general grounds caused by the very structure of the expressions
(6) and (7) (see also section 7) all the thermodynamic functions (1)–(4) do depend
not upon T0 and T apart, but upon their ratio c. So one should also anticipate the
self-similar, or “scaling” character of the ET dependence (9) upon T0 and T , namely,
T ∗(T0, T ) = T ∗(c), c = T0/T. (12)
This is much more physical, because the high- and low-T limits may be achieved
now not only in formal limits for the temperature itself (T → ∞ and T → 0), but in
essentially less stringent ones for the dimensionless temperature parameter (c → 0
means only T À T0 and c → ∞ – accordingly T ¿ T0).
Loosely speaking, in the first limit the system almost fully “loses” its quantum
features (though T may be finite, but high enough to largely exceed T0)
T ∗(c) → TC∗ ≡ T (c → 0). (13)
In the opposite limit the system, on the contrary, becomes almost fully “quantum
degenerate” (though T may strongly differ from zero, but at the same time being
much less than T0), so that
T ∗(c) → T ∗
Q ≡ T0(c → ∞). (14)
For the future purposes it is convenient to introduce besides c from (8) two
additional dimensionless parameters, or relative effective temperatures, namely
T ∗(c)/T = f(c), T ∗(c)/T0 = ϕ(c), ϕ(c) = f(c)/c, (15)
519
Yu.G.Rudoy, A.D.Sukhanov
so that for all values of c
T ∗(c) = Tf(c) = T0ϕ(c); f(c), ϕ(c) > 1. (16)
It is instructive to calculate also the derivatives of ET on T . Denoting here and
below the derivative on the variable c by prime, we have
f ′(c) = ϕ(c) − cψ(c), ψ(c) ≡ ϕ2(c) − 1 > 0, (17)
while according to the definition (8)
dc(T )/dT = −c/T = −c2/T0 6 0, Tdc/dT = −c 6 0; (18)
finally one obtains:
dT ∗(c)/dT = f(c) + f ′(c)T [dc(T )/dT ] = c2ψ(c) > 0. (19)
Further, noting that ψ′(c) = −2ϕ(c)ψ(c) < 0, one easily finds:
d2T ∗(c)/dT 2 = (2/T0) c3ψ(c)[f(c) − 1], (20)
d3T ∗(c)/dT 3 =
(
2/T 2
0
)
c4ψ(c)
[
3(f(c) − 1)2 − c2
]
. (21)
Note, that both equations (19) and (20) are strictly positive for all values of c.
However the latter equation (21) is alternating in sign and has a zero at c0 =√
3[f(c0) − 1]; so the equation (20) possesses maximum at c0, whereas the equati-
on (21) – the inflection point. Let us remark in conclusion that in order to make not
only qualitative but also quantitative predictions one should, of course, obtain some
explicit expression for ET, which is defined in general form in (16).
5. Effective temperature and the third law of thermodynamics
The physical meaning of the term “effective temperature” (ET) and its connecti-
on with the third law of thermodynamics can be seen most natural when one applies
ET to the concept of object’s inner energy E(T ). To start with, we recall one of the
most prominent results of the classical statistical mechanics (see, e.g., [12–15]) –
namely, the virial theorem14 (or, as it is sometimes called, “equipartition” of the
inner energy in the degrees of freedom):
EC(T ) = sαkBT, CC
V = sαkB = const. (22)
The first of these expressions states the presence of equal portions αkBT of the
inner energy per each of the s object’s degrees of freedom; here α = 1/2+1/l, where l
is the homogeneity index of the potential energy function U(q); in particular, for the
14This theorem may be easily obtained from the Maxwell-Boltzmann classical version of Gibbs
approach (see section 6) on the grounds of classical p.f. (6) and the definitions (2) and (3).
520
Towards the unified non-classical physics
HO l = 2, so α = 1 (later on we consider only the case s = 1). So for one-dimensional
CHO one obtains
EC(c) = E0(1/c), CC
V = kB . (23)
The second expressions in (22) and (23) follow immediately from the first ones and
confirm the independence of specific heat upon the temperature T .
The latter statement is, unfortunately, in an obvious contradiction with the third
law of thermodynamics, or Nernst theorem, which maintains:
Cv(T ) → 0 (as T → 0 or c → ∞). (24)
This contradiction shows that the “equipartition” theorem is significantly violated
at low T -region (the experiment also confirms this fact).
The resolution of this difficulty on the phenomenological level is well known: one
usually hears about the “effective” (i.e., depending upon the thermostat temperature
T ) number s(T ) of object’s degrees of freedom. It is assumed that in the low-T (high-
c) region they are “frozen out”, so that s(T ) > 0 at T > 0, and so does the specific
heat Cv(T ).
Physically such an explanation is qualitatively quite acceptable, because it elim-
inates the contradiction with the third law of thermodynamics, which is inherent to
CC
V from the equations (22) or (23). Indeed, this simply means that at low enough
values of T the system’s degrees of freedom are almost not activated. This in turn
signifies (as is most clearly demonstrated by (24)), that the system cannot effectively
exchange the energy in the form of heat with the thermostat.
But it seems much more natural and convincing to let the number s of object’s
degrees of freedom be constant (as it really is) and introduce instead of s(T ) into
the first expressions in equations (22) and (23) for the inner energy in place of T
the ET defined in (15). Then one obtains15
EQ(T ) = EC[T ∗(c)] = EC(T )f(c) (25)
and, taking into account equation (4), also
CQ
V (T ) = dEQ(T )/dT = CC
V [dT ∗(c)/dT ] = CC
Vc2ψ(c). (26)
Equation (25) obviously violates the “theorem” of equipartition (which is of
course valid only classically), because the ET, defined by equation (15), depends
not only upon T but also upon T0 and thus is different for every degree of freedom,
which may be of various physical nature and possesses different values of T0.
The equation (26) should fulfill the condition (24) and, consequently, the function
ψ(c) should go to zero by c → ∞ not slowly than 1/c3. Note once more, that the
demand (24) of Nernst theorem signifies in fact that the heat exchange between the
quantum system and the thermostat is strongly suppressed at low T .
By the latter argument not only the third law of thermodynamics but also the
zeroth law (in slightly modified form) may be fully explained. At high enough values
15Note that the equations (25) and (26) are the exact ones in the case of QHO for the particular
choice of f(c) (see section 7), and so we take them as heuristically valid also in the general case.
521
Yu.G.Rudoy, A.D.Sukhanov
of T (low c) any system behaves as almost classical one (as if E0 ≡ 0), and so due
to equation (13) T ∗ ≡ T . As the thermostat temperature T lowers (accordingly, c
rises) the value of T ∗ differs from T more and more.
Clearly, owing to Nernst theorem (24) the object becomes less and less capable
of exchanging heat with the thermostat. In other words, in the limit T → 0 (c → ∞)
the object in fact ceases to be in thermal contact with the thermostat and gradually
goes over to the adiabatically isolated system. So it is not surprising that T ∗ becomes
finally quite independent of T and equals T0. Note that for almost “classical” object
(i.e., object with T0 ≈ 0) the approximate equality T ∗ ≈ T holds practically in the
whole interval of T .
6. Maxwell-Boltzmann classical approach
One of the most important notions in statistical mechanics is the notion of dis-
tribution function (d.f.) – as a rule, in coordinates q and momenta p (sometimes also
in energy E and, maybe, other dynamic variables). For the separable Hamiltonian
of the form (5) these d.f.s for the system in C-regime are the familiar one Maxwell
d.f.
PC(p; T ) =
[
ZC
p (T )
]−1
exp
[
−(1/kBT )(p2/2)
]
(27)
and Boltzmann d.f.
PC(q; T ) = [ZC
q (T )]−1 exp[−(1/kBT )U(q)]. (28)
Of course, for the object in C-regime it is quite legitimate to speak about the
simultaneous or joint d.f. in the system’s phase space, i.e. about the Gibbs d.f.
PC(p, q; T ) = PC(p; T )PC(q; T ) =
[
ZC(T )
]−1
exp [−(1/kBT )H(p, q)] . (29)
Due to statistical independence of the distribution in p and q in this case (or, which
is just the same, the absence of any correlations among p and q) the joint d.f. (29)
is given by the product of d.f.s (27) and (28). In other words, the d.f. (29) possesses
a physical meaning since from the classical point of view every phase space point
(p, q) is always exactly defined.
The normalization factor in (29) is obviously equal to the product of the corre-
sponding factors in (27) and (28),
ZC(T ) = ZC
p (T )ZC
q (T ) (30)
and thus coincides with the expression (6) for the p.f. of any object (described in
terms of H(p, q)) in C-regime.
For the particular case of HO (see footnote 13) one has
U(q) = (1/2)ω2q2, (31)
so both d.f.s (27) and (28) get the Gaussian form
PC(p; T ) = [ZC
p (T )]−1 exp[−αp(T )p2], (32)
PC(q; T ) = [ZC
q (T )]−1 exp[−αq(T )q2]. (33)
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Towards the unified non-classical physics
Using twice the Poisson integrals, one obtains the partial p.f.s in the form
ZC
p,q(T ) = (2π~)−1/2[π/αp,q(T )]1/2, (34)
where
αp(T ) = 1/2kBT, αq(T ) = ω2αp(T ) = ω2/2kBT. (35)
Then it follows from (30), (34) and (35) that
ZC(T ) = kBT/~ω, or ZC(c) = 1/2c (36)
and hence, according to (2)–(4), the equation (23) as well as the expression for the
entropy
(1/kB)SC(c) = σ0 − ln c, σ0 = 1 − ln 2, 0 < σ0 < 1, (37)
are easily obtained. Evidently, SC(c) → −∞, i.e. diverges logarithmically as c → ∞,
and so the quantity SC(c) (as well as CC
V) is in strong contradiction with another
demand of the Nernst theorem: the value of S(T ) as T → 0 should be a finite non-
negative one (presumably zero).
7. Planck-Einstein non-classical approach
As mentioned above, the use of HO as a test model is justified, because the study
of this model in various approximations (both theoretically and experimentally) has
a rather long history and is well familiar. Moreover, due to Planck [27] and Einstein
[29] we possess the exact solution ZQ(T ) for the one-dimensional QHO with s = 1.
Perhaps, Wien was the first who proved the self-similar, or “scaling” form of
spectral energy d.f. (and hence of all thermodynamic functions) in the theory of
thermal radiation 16. According to Wien [27], the expression (8) for c should be of
the form c = α(υ/T ) (α = const), and it was Planck [27] who pretty soon showed
that α = h/kB.
Further on, using mainly the thermodynamic arguments Planck [27] deduced an
explicit functional form for the Kirchhoff-Wien’s spectral energy d.f. and the inner
energy of HO, whereas Einstein [29] calculated the p.f. for HO. In fact, Planck for the
first time “quantized” the energy of HO, but only later he managed [28] to account
for the zero-point energy E0 = 1/2 · ~ω, where ω is the characteristic frequency of
HO. Obviously, for QHO the characteristic “quantum” temperature is of the form
T0 = E0/kB = ~ω/2kB (QHO), (38)
where ω is the characteristic frequency of HO.
It follows from [27,29] that according to the definitions (16) and (25) for T ∗(c)
it is natural to define the Planck-Einstein ET by the following functions:
ϕ(c) = cth c, f(c) = c cth c, (39)
16Long before Kirchhoff [31] stressed that this function should be a universal one, i.e. should
not depend upon the physical nature of particular radiating body or thermal bath. Note that both
Kirchhoff and Wien used pure thermodynamic arguments following the traditions of Carno.
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Yu.G.Rudoy, A.D.Sukhanov
with the well known asymptotic expansions
ϕ(c) =
{
(1/c) [1 + (1/3)c2 − (1/45)c3 + · · ·] , (c → 0),
1 + 2e−2c (1 + e−2c + · · ·) , (c → ∞)
(40)
and, accordingly,
f(c) =
{
1 + (1/3)c2 + O(c3), (c → 0),
c [1 + O(e−2c)] , (c → ∞).
(41)
Both expansions (40) and (41) obviously agree with the expected behavior of ET
(13) and (14). Using definition (17) and expansion (40), we further have
ψ(c) =
{
(1/c)2 [1 − (1/3)c2 + O(c4)] , (c → 0),
4e−2c + O(e−4c), (c → ∞).
(42)
By means of equations (40)–(42) it is not difficult to determine the low- and high-c
asymptotes of all the ET derivatives (19)–(21); in particular, the first derivative (19)
shows the behavior, which completely satisfies the demands of classic theory at low
c and Nernst theorem at high c for the specific heat (26).
The exact p.f. ZQ(c) for the QHO (taking into account the zero-point energy)
was obtained by Einstein [29] in the form
ZQ(c) = e−c[1 − e−2c]−1 = [2 sh c]−1 = (1/2)[ψ(c)]1/2, (43)
having the asymptotes
ZQ(c) =
{
ZC(c)[1 − (1/6)c2 − O(c4)], (c → 0),
e−c + O(e−2c), (c → ∞).
(44)
Then according to definitions (2) and (3) the inner energy EQ(c) and heat capaci-
ty CQ
v (c) for QHO are given exactly by expressions (25) and (26) with f(c) and ϕ(c)
defined by equation (39); using equations (40)–(42) their asymptotes may be easily
compared with classic and semiclassic ones. Finally, according to equations (4), (25)
and definition (37) of σ0 the exact entropy of QHO is of the form
(1/kBα) SQ(c) = σ0 + [f(c) − 1] + (1/2) ln ψ(c), (45)
which well satisfies the standard form of the Nernst theorem (namely, SQ = 0 at
T = 0).
8. Bloch semiclassical approach
A remarkably interesting and useful example of the value of the ET concept was
actually given by Bloch [23], though Bloch himself did not explicitly employ this
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Towards the unified non-classical physics
concept. Later on his result was reproduced in somewhat different way in [18] and
[19], but, in our opinion, this remarkable result was not sufficiently commented from
the physical point of view.
Bloch combined quantum mechanical (Born) and statistical mechanical (Gibbs)
probabilistic descriptions on the general footing of von Neumann density operator
approach. Thus Bloch obtained the exact quantum d.f.s separately in coordinate q
and momentum p for the thermally equilibrium QHO.
Bloch’s calculation is rather non-trivial and in all variants [18,19,23] it includes
the exact procedure of quantum summation. But it appears that Bloch’s quantum
d.f.s PQ(q; T ) and PQ(p; T ) may be easily obtained from their classical counterparts
(32) and (33) by a simple use of the notion of Planck-Einstein ET. Indeed, making
the substitution T → T ∗(c) in the latter expressions we immediately obtain
PQ(q; T ) = PC[q; T ∗(c)], PQ(p; T ) = PC[p; T ∗(c)], (46)
where, naturally, αq,p(T ) → αq,p(T
∗) and the same is valid for σq,p(T ).
It is obvious, that in the high-T limit (T ∗ → T ) we return again to the classical
d.f.s (32) and (33) for CHO. But in the opposite low-T limit (T → 0, T ∗ → T0) we
obtain
αp(T0) = 1/~ω, αq(T0) = ω2αp(T0) = ω/~, (47)
and Bloch’s d.f.s (46) go over to the well known Gaussian quantum probability
densities for q and p in the ground state of QHO at T = 0.
It is worth noting that due to the inequality (16) one has from the definition
(35) that αq,p(T
∗) 6 αq,p(T ). Then for the dispersions σ2
q,p ≡ 1/2αq,p of d.f.’s (46) in
q and p one always has
σ2
q,p(T
∗) > σ2
q,p(T ), or [σQ
q,p]
2
> [σC
q,p]
2. (48)
This means that the quantum d.f.s (46) are always wider than their classical
counterparts (32) and (33), so the quantum fluctuations additionally to thermal ones
“smear out” the statistical distributions in q and p. It is especially important for the
limiting case T = 0, T ∗ = T0, when classical d.f.s degenerate into δ-functions with
zero dispersions, although quantum d.f.s remain Gaussian with finite dispersions:
σC
q,p(0) = 0, σQ
q (T0) = [~/2ω]1/2, σQ
p (T0) = [~ω/2]1/2. (49)
The difference between the classical (equal to zero) and pure quantum (finite and
proportional to ~) values of dispersion at T = 0 is, of course, completely due to the
ground-state (i.e., vacuum) quantum fluctuations.
Some more relations may be also of use from the viewpoint of interrelations
between the dynamic and thermodynamic descriptions. Forming the product of the
dispersions σq and σp, one obtains for the C-regime
(σC
q σC
p )2 = (1/2~
C
T)2, ~
C
T ≡ ~/c, (50)
while for the Q-regime
(σC
q σC
p )2 = (1/2~
Q
T)2, ~
Q
T ≡ ~ϕ(c). (51)
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Yu.G.Rudoy, A.D.Sukhanov
In these expressions we have introduced the corresponding effective, or “thermal”
Planck constants ~T (strictly alike the introduction of “thermal” de Broglie wave
length λT) for both regimes considered. Note that ~
C
T is rather formal quantity and
may even not contain ~ itself – this is just the case for HO, because due to definitions
(50) and (8)
~
C
T = ~/c = ~(T/T0) = T (2/kω). (52)
On the other hand, taking into account the asymptotic properties (40) of ϕ(c) one
obtains
~
Q
T =
{
~
C
T[1 + (1/3)c2 + O(c3)], (c → 0),
~[1 + O(e−2c)], (c → ∞),
(53)
so that ~
Q
T ≈ ~
C
T at high T , but ~
Q
0 = ~ > 0 in contrast to ~
C
0 = 0 at T = 0.
Note in conclusion that on the grounds of the result (51) thermally equilibrium
states of HO described by the Bloch d.f.s (46) at T = 0 go over into coherent states
with the minimal property (σQ
q σQ
p )2 = (~/2)2 (see papers [8] in this connection).
The remarkable simplicity in obtaining Bloch’s results for quantum d.f.s from
classical ones via the concept of ET makes it natural to raise the following question:
will the calculation of p.f.17 using the same “recipe” of the substitution of ET in
place of T ,
ZC(T ) → ZC[T ∗(c)], (54)
also lead to the exact p.f. ZQ(T ) (or, at least, differ from it not very strong)?
Unfortunately, the answer is rather disappointing: the p.f. in rhs of equation (54),
being of course different from ZC(T ) given by equation (36), does not coincide with
ZQ(T ) given by equation (43). That is why p.f. in rhs of equation (54) should be
considered only as an intermediate, or semiclassical one:
ZC[T ∗(c)] ≡ ZSC(c), ZSC(c) = ZC(c)f(c) = (1/2)ϕ(c); (55)
nevertheless, we consider it as worth studying.
In particular, due to equation (40) ZSC(c) possesses the following asymptotes
(where we use the notation ZSC
min(∞) = 1/2):
ZSC(c) =
{
ZC(c)[1 + (1/3)c2 + O(c4)], (c → 0),
ZSC
min(∞)[1 + O(e−2c)], (c → ∞).
(56)
On the other hand, it is easy to see that the exact p.f. ZQ(c) for QHO from
equation (43) may be written as
ZQ(c) = ZSC(c)Ψ(c), (57)
where
Ψ(c) =
{
1 − [ZSC
min(∞)/ZSC(c)]2
}1/2
. (58)
17In order to avoid any confusion note that Bloch [23] did not calculate the p.f. of QHO at all.
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Towards the unified non-classical physics
Evidently, the factor (58) can by no means be “guessed” or anyhow deduced from
equation (36) by means of ET (see also the footnote 10). The reason of such a
discrepancy between equations (56) and (57) lies quite deep and concerns the very
essence of the quantum description.
First of all let us note that the simple substitution meant in (54) implicitly
suggests the fulfillment of the following rather strong (and in fact inadmissible in
Q-regime) Ansatz:
The joint semiclassical d.f. in q and p should be of the same multiplicative form
as it is in equation (29) for C-regime with the only change T → T ∗(c). In other
words, one should accept that
P SC(q, p; T ) = PQ(q; T )PQ(p; T ) = PC[q; T ∗(c)]PC[p; T ∗(c)], (59)
because only in this case the transition from (30) to (54) would be justified.
It should be noted that in C-regime the property (29) is quite acceptable due to
two reasons: the first (general) is the very possibility of simultaneous definition (i.e.,
measuring) of q and p, which is necessary in order to define the joint d.f.; the second
(specific) is the separable structure of the Hamilton function (5), which causes the
statistical independence of q and p fully expressed by equations (27)–(29).
However, in Q-regime these both reasons are violated and so the Ansatz (59) may
be viewed at best as only giving some convenient “test” joint d.f. (and therefore also
the p.f.) if – as is generally the case – one does not have the exact p.f. at one’s
disposal. In the considered here case of HO the exact p.f. in the form (43) fortunate-
ly exists, so the equations (55) and (59) are employed mainly for methodological
purposes.
The physical reason of non-applicability of the Ansatz (59) as an exact one
in Q-regime is called forth by the presence of quantum uncertainties σQ
q and σQ
p ,
which according to equation (53) are of “saturated” Heisenberg form σQ
q σQ
p = ~/2
at T = 0. Due to this the quantities q and p are exactly defined only in their
own representations, but they cannot be defined (and/or measured) simultaneously
in any one of them. So the d.f.s (46) may well exist separately, but one should
be very careful and thoroughly control the accuracy of results when using them
simultaneously18 in the simple multiplicative form (60).
Summing up, the coordinates q and momenta p in Q-regime are somehow inher-
ently correlated and so the Ansatz (60) is evidently not appropriate in the whole
interval of c (besides, maybe, only the “classical” high-T , or low-c region). From
equations (54)–(56) it is clear that an adequate (but of course quite formal) con-
sideration of the quantum correlation between q and p may be achieved, if one
introduces the factor (58) into equation (59) and defines the full quantum d.f.19:
PQ(q, p; T ) = P SC(q, p; T )[Ψ(c)]−1. (60)
18Note that the described in section 3 rather crude “cellular” method of considering the quantum
indeterminacies is now sufficient and leads only to the correct form of classical ZC (see (35)).
19Certainly, the representation (57) is only a formal trick because it demands to know already
the exact quantum result (41), and in this case the whole idea of inventing some SC-approach of
the type (52) becomes superfluous.
527
Yu.G.Rudoy, A.D.Sukhanov
Though not quite justified, we may adopt the p.f. (56) as a tool of desired inter-
mediate or semiclassical description, because the results for its derivatives according
to equations (2)–(4) may have (and does have) more sense than ZSC(c) itself. It is
important that having an exact expression ZQ(c) we may easily check the accuracy
of the “semiclassical” results on each step.
Due to asymptotic expansion (40) of ϕ(c) one has according to (56) in the high-T
(low-c) region ZSC(c) À ZSC
min(∞), so the factor (58) is very near to unity, and thus
(up to corrections of the order O(c2))
ZC(c) > ZSC(c) > ZQ(c) (c → 0). (61)
But in low-T (high-c) region ZSC(c) → ZSC
min(∞) = 1/2 and the factor (58) plays
a decisive role forcing ZQ(c) (in contrast to ZSC(c)) to go to zero exponentially fast.
Paradoxically enough, but in the low-T region ZC has the same limiting value (equal
to zero) as ZQ has, but with wrong asymptotes, whereas ZSC, on the contrary, has
a wrong (finite=1/2) limiting value, but correct asymptotic behavior.
The thermodynamic functions (2)–(4) in our “quasi-Bloch’s” SC-approach are
of the form
ESC(c) = E0 [ψ(c)/ϕ(c)] , CSC
V (c) = CQ
V (c)[ϕ2(c) + 1],
(1/kB)SSC(c) = σ0 + [cψ(c)/ϕ(c) − 1] + ln ϕ(c); (62)
their asymptotes may be easily obtained by means of expansions (40)–(42) and it
may be seen that quantities (62) behave much better – i.e., resemble more (though
not fully) the exact quantities (25), (26) and (45) than do the classical ones (23)
and (37) in the most important high-c region (in particular, CSC
V → 0 and SSC →
σ0 − 1 = − ln 2 as T → 0).
9. Wigner semiclassical approach
Wigner [22] originally introduced (with the assistance of Szilard) the so-called
quasiprobability, or Wigner function W (q, p) in phase space thus assuming from the
very beginning the possibility of simultaneously determining p and q; after suitable
integration this function gives in principle the exact d.f.s PQ(q; T ) and PQ(p; T ).
But by means of W (q, p) Wigner elaborated only the expansions20 (formal) of
the statistical d.f.s and p.f. in powers of ~. In contrast to Bloch, Wigner was not
restricted by the case of HO, but considered the general case (5) with an arbitrary
potential U(q), which should be only an analytic function of q.
Wigner’s approach proves to be in fact not only a “small-h” one, but (which is
perhaps more important) rather a high-T (or low-c) expansion. This is not surprising,
20In search of maximal physical transparence we present here just the original version of Wigner’s
approach [22] in the convenient form given to it by Landau and Lifshitz [18] (for more refined form
see [24,25]. Unfortunately, Wigner’s and Bloch’s results were not compared in [18] with each other
nor with the exact Planck-Einstein result even in the simplest case of quantum harmonic oscillator
(QHO).
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Towards the unified non-classical physics
since according to definition (8) c ∼ E0, where always E0 → 0 at ~ → 0 (e.g.,
E0 = O(~) for HO, E0 = O(~2) for free particle in a box etc.). By this consideration
it becomes clear that the opposite region of low-T (or high-c) cannot in principle
be described by such an expansion (see, however, extensive general discussion about
more general Wigner-like expansions in [18,19]).
Nevertheless, the original Wigner’s approach is rather simple and highly inter-
esting and leads to the results cited below. The d.f.s PW(q; T ) and PW(p; T ) in
coordinates and momenta are of the following form:
PW(q; T ) ∼ exp{−(1/kBT )UW(q; T )}, (63)
PW(p; T ) ∼ exp
{
−(p2/2kBT )
[
1 − [~2/12(kBT )3
]
〈(∂U(q)/∂q)2〉C]
}
, (64)
(for simplicity we present them only for one particle in the external potential U(q)
and only up to terms O(~2) omitting O(h4)).
Here classical Gibbs averaging with account of equation (6) means as usual
〈· · ·〉C = [ZC(T )]−1
∫
dqdp/(2π~)s exp[−H(q, p)/kBT ]; (65)
finally, the Wigner’s effective potential UW(q; T ) has the form
UW(q; T ) = U(q) + [~2/12(kBT )][∂2U(q)/∂q2 − (1/2kBT )(∂U(q)/∂q)2]. (66)
Note that in general both d.f.s (63) and (64) depend upon the potential deriva-
tives, which is quite alien to the classical statistical mechanics. But in quantum
statistical mechanics it reflects the growth of kinetic energy (described by the oper-
ator (∼ h2[∂2/∂q2]) in q-representation) in those space regions, where the potential
energy U(q) changes sharply enough21.
With the d.f.s (63) and (64) the joint semiclassical d.f. in Wigner’s approach –
quite similar to Bloch’s one (59) – may be written as their product:
P SC
W (q, p; T ) = PW(q; T )PW(p; T ). (67)
Then the Wigner’s p.f. has the form
ZSC
W (T ) = ZC(T )[1 + ζ(T ) + O(~4)], (68)
where in obtaining the correction term
ζ(T ) = (1/kBT ) 〈UW(q; T ) − U(q)〉C = −(~2/24)(1/kBT )
〈
∂2U(q)/∂q2
〉
C (69)
21Note, that it is possible to write the exact quantum Bloch coordinate d.f. (46) in terms of
effective potential in the form like (59) with the substitution UW(q, T ) → UB(q, T ) = U(q)[f(c)]−1
(where f(c) is defined by equation (38)); clearly, no gradients of U(q) enter explicitly UB(q, T ).
But it appears more appropriate to rewrite, on the contrary, equation (59) in terms of Wigner’s
ET (defined below in equation (74)) as it also takes place in Wigner’s momenta d.f. (64) written
in the form (69).
529
Yu.G.Rudoy, A.D.Sukhanov
we used the following integration by parts from equation (65) relation
〈
(∂U(q)/∂q)2
〉
C = kBT
〈
∂2U(q)/∂q2
〉
C. (70)
Thus the sign of the correction term (69) depends on the form of U(q).
Let us now turn to the special case of HO, where
U(q) = (1/2)ω2q2, ∂U(q)/∂q = ω2q, ∂2U(q)/∂q2 = ω2 = const. (71)
In this case the effective potential (66) (omitting the term independent of q, which
effects only the beginning of the energy scale and so the normalization constant in
p.f.’s) takes the form:
UW(q; T ) = U(q)[1 − h2ω2/12(kBT )2] = U(q)[1 − (1/3)c2]. (72)
Then the d.f. (63) may be written in an equivalent form:
PW(q; T ) = PC[q; T ∗
W(c)], (73)
where instead of Wigner’s effective potential (72) one may now introduce Wigner’s
effective temperature
T ∗
W(c) ≡ T [1 + (1/3)c2]. (74)
Further, let us note that
〈
(∂U(q)/∂q)2
〉
C
= ω4〈q2〉C , 〈U(q)〉C = (1/2) ω2〈q2〉C = (1/2)kBT,
where in the latter equality we have used the classical theorem (22) of energy equipar-
tition (for HO, the contribution of potential and kinetic energies are equal).
So we have
〈q2〉C = kBT/ω2,
〈
(∂U(q)/∂q)2
〉
C
= kBT ω2, (75)
and finally the d.f. (64) obtains the form quite analogous to equation (73):
PW(p; T ) = PC[p; T ∗
W(c)] (76)
with the same Wigner’s ET (74).
Using definition (16) and upper line of expansion (41) one can immediately see
that the Wigner’s (approximate) ET is nothing else but high-T expansion of the
Planck-Einstein (exact) ET (39) (also coinciding with ET (46) used by Bloch).
It is further obvious that on the grounds of the last expression in (75) the cor-
rection term (69) for the p.f. may be written as
ζ(T ) = −(1/6)c2, (c = ~ω/2kBT ). (77)
Thus Wigner’s p.f. (68) for QHO coincides with the first high-T (low-c) correction
term of the exact ZQ (see the upper line of expansion (44)), but not Bloch’s semi-
classical ZSC
B . Indeed, with equation (77) the expression (68) differs from the upper
line (high-T ) of expansion (56) in sign and numerical coefficient 2.
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Towards the unified non-classical physics
Let us remark in conclusion that Wigner’s approach based originally on “small h”
perturbation theory is, in general, of great value in quasiclassical dynamic problems
(see, e.g., [18,19]), but has much more restricted efficiency in the statistical mechani-
cal, or thermodynamic ones. The origin of this seems to lie in an essentially different
nature of dimensionless parameters, which one may consider as small in these two
cases.
More specific, in dynamic case the role of such a parameter is played by the ratio
of actions ~/S, where S is the characteristic value of the action in the considered
problem. With ~ being the minimal value, or “quantum” of action this parameter
is always of the order of unity or less; for the objects in almost classical regime, S
is large, so ~/S ¿ 1, which formally may be used as the limit of ~ → 0.
However, in statistical mechanical problems, the action ceases to be a relevant
physical quantity, so ~ is no more a single fundamental constant. As follows from the
idea of effective temperature (see especially sections 4 and 5), in these problems the
true physical dimensionless parameter is the ratio of energies c = E0/Eth (introduced
in equation (8)), where E0 is of the order ~ and characterizes the object itself.
But the quantity Eth = kBT depends upon the object’s macroscopic surroundi-
ngs, or thermostat, through its temperature T and plays the role of some average
thermal energy per one degree of thermostat’s freedom; note, that by means of Eth
another fundamental (Boltzmann) constant k comes into play.
Contrary to the dynamic action “S” the thermal energy Eth possesses no natural
non-zero limitations from below, whereas E0 sometimes may be very small (but non-
zero). The point is that E0, though being proportional to ~, is itself not a universal
constant, thus allowing the values Eth 6 E0, or T 6 T0, the classical regime for
the quantum object is possible when Eth À E0, c ¿ 1, and so this regime may be
reproduced formally by means of the limit ~ > 0.
However, in the opposite case, when Eth ¿ E0 (so that T is low enough) the
parameter c À 1 and thus no kind of expansion in ~ is in principle allowed and appli-
cable. Let us emphasize once more that the failure of Wigner’s original semiclassical
approach in the low-T region of quantum regime is due to the presence of the outer
(relative to the described object) parameter T , which may be made arbitrary small
(though staying non-negative).
In this region the exact (Q) results for thermodynamic functions differ from
the classical (C) ones by the correction terms of the order O(e−c), which are non-
analytical in c at c → ∞ (low-T limit). Clearly, in this limit the Planck constant ~
can in no way (even formally) be considered as a “small” one.
In conclusion, it is instructive to note the following. As is well known [22,24,25],
the Wigner’s d.f.s for the dynamic quantum systems (e.g., for QHO) possess another
hard disadvantage: they are negative for some excited states of QHO, whereas for
the ground-state the Wigner’s d.f.s are strongly positive and coincide with Bloch
ones.
However, for the thermodynamic quantum object (i.e, QHO in thermal contact
with thermostat) the Wigner’s d.f.s (63) and (64) are always positive. This can be
considered as if the thermostat induces the additional incoherence and so makes
531
Yu.G.Rudoy, A.D.Sukhanov
Wigner’s (as well as Bloch’s, see equation (48)) d.f.s more smooth and wide spread
at the same time.
10. Conclusions
In this paper we investigated the applicability of the concept of universal effective
temperature (ET) anticipated by Kirchhoff and Wien and first deduced and applied
by Planck and Einstein to the thermal radiation. The latter authors studied the
exact (in the sense of quantum statistical mechanics) description of the set of har-
monic oscillator being in thermal equilibrium with the thermostat at temperature
T . Much earlier this problem was in fact formulated by Maxwell and Boltzmann on
the grounds of classical statistical mechanics description; their results proved to be
approximate and valid only in the high-T region.
The aim of our paper was to establish some intermediate, or semiclassical de-
scription following the seminal ideas of Bloch and Wigner. Namely, we considered
the thermodynamic consequences of substituting the Planck-Einstein ET instead of
T into Maxwell-Boltzmann formulas, which is equivalent to Bloch-Wigner approach.
We have shown that semiclassical thermodynamic quantities, i.e., inner energy,
heat capacity and entropy obtained in this way, agree qualitatively with their exact
quantum counterparts much better than the classical ones. So one may consider the
concept of effective temperature as the effective tool for approximate “quantization”
of thermodynamic functions (i.e., partially considering the quantum fluctuations
additionally to thermal ones) obtained on the grounds of Gibbs classical statistical
mechanics.
Note, that the more general question of failure of the Gibbs approach itself at very
low T due to very large temperature fluctuations was raised recently [16]; evidently,
this may demand to go over from Gibbs to Einstein approach (or, in other terms,
from SC- to NC- theory on the T-level, see section 2).
Nevertheless, we hope that the method suggested by us will appear useful not
only for the well-known example of harmonic oscillator (which fortunately possesses
the exact solution and permits to test the accuracy of any approximation), but also
for other less simple objects.
In particular it may well be applied to compound systems, consisting of some
dynamically non-interacting subsystems of quantum origin, each of them having its
own ground-state energy. As a good example of this kind may serve the microscopic
particle with inner degrees of freedom (besides the translational ones) – e.g., the
diatomic molecule with rotations, vibrational and electronic degrees of freedom.
In general we assume that the method of ET is applicable to arbitrary quantum
systems though it is delicate enough and in every case deserves a careful analysis.
The applicability of various approximations should be verified and their relative
errors accurately evaluated which is a far not trivial task in the absence of the exact
solution. But, as we hope, an important specific case of the harmonic oscillator
studied here will give us some valuable hints due to universal applicability of Planck-
Einstein ET at least, it is valid for the large class of objects, possessing no upper
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Towards the unified non-classical physics
bound for their energy and so not permitting the negative values of T (such as objects
ore, e.g., the spin systems). Indeed, it appears even in the quantum generalizations
of the Langevin equation and the fluctuation-dissipation theorem (see, e.g., [3,18]).
From the practical point of view it is very significant at what stage ET should
be inserted into classical thermodynamic functions in order to obtain their correct
asymptotic behavior in both limits of high (T/T0 → ∞) as well as low (T/T0 → 0)
thermostat temperatures. For example, if the substitution T → T ∗ is performed
in the partition function (as is made in the paper), in the high-T limit all three
approaches – classical, semiclassical and quantum – give approximately the same
results, differing only by small correction terms of the order (T/T0)
2.
On the other hand, at low-T limit quantum as well as – which is especially
important – semiclassical approaches give results, that are exponentially small in
(T0/T ) and thus radically different from the classical one, for which T0 → 0. In this
way the important physical requirement – namely, the fulfillment of Nernst theorem
– is properly guaranteed. One should note that the same substitution T → T ∗ in
the classical expressions for inner energy leads (probably accidentally) to the exact
result, whereas such a procedure for the specific heat capacity is not possible at all
because CC
V does not contain T .
In conclusion let us comment on the choice between the Wigner and Bloch forms
of the semiclassical (in the sense of considering not thermal but only quantum fluc-
tuations). As was shown in [24], the modified Wigner’s approach is eventually equiv-
alent to the Bloch’s one; this is not surprising, because the presentation in [24] is
based on the phase space interpretation of Bloch equation for von Neumann density
operator.
In our opinion, the Bloch’s approach is something more practical and conveni-
ent. But its difficulty consists in the necessity of knowing not only the form of the
object’s potential energy, but also of the full Hamiltonian eigenvalues needed for the
calculation of Gibbs factors entering the quantum partition function.
In any case, the original Wigner’s approach (which is available in valid textbooks)
is applicable only in the high-T limit, and this fact is not always sufficiently stressed.
On the other hand, the use of Wigner’s approach for “warm” quantum objects is
more justified because the effect of the thermostat eliminates one of the greatest
difficulties in this approach. Namely, the thermal fluctuations induced by thermostat
lead to additional incoherence and thus “smooth out” and “rectify” Wigner functions
restoring their positivity.
In the subsequent papers we shall try to apply the semiclassical approach in the
form of effective temperature to some more examples of “warm” quantum objects.
Moreover, we intend to study the generalized case when the thermostat itself is
partially quantized by means of its own effective temperature.
11. Acknowledgement
The authors devote this paper in honour of Prof. Dr. Igor R. Yukhnovsky on
the occasion of his 80th birthday. We evaluate very much the fruitful efforts, which
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Yu.G.Rudoy, A.D.Sukhanov
Prof. Yukhnovsky has spent to continue and elaborate the ideas of N.N.Bogoliubov’s
school in the field of modern statistical mechanics. The present authors are also
indebted to the late Prof. Dr. Yu.L.Klimontovich with whom we have had interesting
discussions on the theory of open quantum systems.
References
1. Rudoy Yu.G., Sukhanov A.D., Uspekhi Fiz. Nauk, 2000, 170, 1265 (in Russian).
Engl. transl.: Phys. Uspekhi, 2000, 43(12), 1169.
2. Giulini D., Joos E., Zeh H.D., Decoherence and the Appearance of Classical World in
Quantum Theory. Springer, Heidelberg, 1996.
3. Klimontovich Yu.L., The Statistical Theory of Open Quantum Systems, vol. 1. Yanus,
Moscow, 1995 (in Russian). Engl. transl.: Kluwer Academic Publishers, Dordrecht,
1995.
4. Kadomtsev B.B., Dynamics and Information. UFN Publishing House, Moscow, 1997
(in Russian).
5. Sukhanov A.D., Golubeva O.N., Physical Education in High School, 2001, 7, 16 (in
Russian).
6. Einstein A., Ann. Phys. Leipzig, 1903, 11, 170.
7. Lavenda B., Int. J. Theor. Phys., 1987, 26, 1069; Schlogl F., J. Phys. Chem. Sol.,
1988, 49, 679; Uffink J., van Lith J., Found. Phys., 1988, 28, 323.
8. Sukhanov A.D., Theor. and Math. Phys., 2000, 125, 1489; Physics of Particles and
Nuclei, 2001, 32, 619.
9. Szilard L., Z. Phys., 1925, 32, 753.
10. Mandelbrot B., Ann. Math. Stat., 1962, 33, 1021; J. Math. Phys., 1964, 5, 164.
11. Tisza L., Quay P.M., Ann. Phys., 1963, 25, 48.
12. Petrov N., Brankov Y., The Modern Problems of Thermodynamics. Mir, Moscow,
1986 (in Russian).
13. Gibbs J.W., Elementary Principles in Statistical Mechanics. Scribner, NY, 1902.
14. The Conceptual Foundations of Quantum Field Theory, ed. T.Y.Cao. Cambridge Uni-
versity Press, Cambridge, 1999.
15. Penrose R., The Emperor’s New Mind. Cambridge University Press, Cambridge, 1989.
16. Wu J., Widom A., Phys. Rev. E, 1998, 57, 5178; Nieuwanheuizen Th.M., Al-
lahverdyan A.E., Phys. Rev. E, 2002, 66, 036102.
17. Einstein A., Ann. Phys. Leipzig, 1910, 33, 1275.
18. Landau L.D., Lifshitz E.M., Statistical Physics. GTTI, Moscow, 1939 (in Russian).
English transl.: Addison-Wesley, Reading, Mass., 1958, Ch. III.
19. Leontovich M.A., Statistical Physics. GTTI, Moscow, 1944, 2nd ed.: Nauka, Moscow,
1983 (in Russian), Ch. III.
20. Zubarev D.N., Nonequilibrium Statistical Thermodynamics. Nauka, Moscow, 1971
(in Russian). German transl.: D.N.Zubarev, Statistische Thermodynamic der Nicht-
gleichgewicht. Akademie Verlag, Berlin, 1976.
21. Huang K., Statistical Mechanics. John Wiley, New York- London, 1963, Ch. 10.
22. Wigner E.P., Phys. Rev., 1932, 40, 749.
23. Bloch F., Z. Phys., 1932, 74, 295.
24. Hillery M., O’Connell R.F., Scully M.O., Wigner E.P., Phys. Rep., 1984, 106, 121.
534
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25. O’Connell R.F., Found. Phys., 1983, 13, 83; Lee H.W., Scully M.O., Found. Phys.,
1983, 13, 61.
26. Sukhanov A.D., On the global interrelation between quantum dynamics and thermo-
dynamics. – In: Proc. of the XI Int. Conf. “Problems of Quantum Field Theory”, 1998.
Dubna, JINR, 232, 1999.
27. Planck M., Verhandl. Deutsch. Phys. Ges., 1900, 2, 202, 237; Ann. Phys., 1901, 4,
553.
28. Planck M., Verhandl. Deutsch. Phys. Ges., 1911, 13, 138.
29. Einstein A., Verhandl. Deutsch. Phys. Ges., 1914, 16, 820.
30. Wien W., Sitzungsber. Deutsch. Akad. Wiss. Berlin, (vom 9 Februar 1893, S. 55);
Ann. Phys., 1901, 4, 553.
31. Kirchhoff G., Ann. Phys., 1860, 19, 275.
До єдиної некласичної фізики: врахування
квантових флуктуацій в рівноважній термодинаміці
через ефективну температуру
Ю.Г.Рудой, А.Д.Суханов
Університет дружби народів ,
117198, вул. Міклухи-Маклая, 6,
Москва, Росія
Отримано 13 травня 2005 р.
Концепція ефективної температури (ЕТ) T ∗(T0, T ) використовуєть-
ся для того, щоб наближено “квантувати” термодинамічні функції
динамічного об’єкту, який є в термічній рівновазі з термостатом при
постійній температурі T (T0 = E0/kB, де E0 – енергія основного
стану, kB – постійна Больцмана, – є характеристичною “квантовою”
температурою самої системи). Базуючись на цьому, здійснено екс-
тенсивне порівняння дослідження “стандартної моделі” статистичної
механіки – одновимірного гармонійного осцилятора (ГО). Розгляну-
то три добре відомі підходи та добре вивчено їх термодинамічні
наслідки, а саме: точний квантовий або некласичний підхід Планка-
Ейнштейна, проміжний або напівкласичний підхід Блоха-Вігнера і,
нарешті, чисто класичний або підхід Максвелла-Больцмана.
Ключові слова: рівноважна термодинаміка, ефективна
температура, квантові флуктуації
PACS: 05.70.-a, 05.30.-d, 05.40.-a
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