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Bulletin of PFUR Series Mathematics. Information Sciences. Physics. No 3 (1). 2010. Pp. 94-97
UDC 536.1
Two Approaches to Quantum Generalization of
Thermodynamics
O. N. Golubjeva, A. D. Sukhanov, Zh. V. Umanskaja
Center for Science Education Peoples' Friendship University of Russia 6, Miklukho-Maklaya str., Moscow, 117198, Russia
For the describing of a system under the combined quantum and thermal influences, it is offered two approaches each of them is a thermofield analogue of Clausius'classical and Einstein's statistical thermodynamics accordingly. We call them as Thermofield Thermodynamics and Thermofield Statistical Thermodynamics. We start from the thermofield vacuum and make a consent between basic notions of the quantum mechanics and thermodynamics. We suppose to consider the thermofield vacuum as an effective thermobath and ground stationary state as an thermal equilibrium state. On this ground we introduce such notions as effective temperature and effective entropy and get possibility to describe systems under combined quantum and thermal influences.
Key words and phrases: effective temperature, effective entropy, stochastic influence, combined quantum and thermal influences, thermofield vacuum.
For the describing of a system under the combined quantum and thermal influences, it is offered, as the first step, to start from the thermofield vacuum as the main idea of Umezawa's thermofield dynamics (TfD) [1]. To formulate an entire theory of fluctuations it is necessary to make a consent between two ways of description: the quantum and thermodynamics languages. So we suppose to consider the thermofield vacuum as an effective thermobath (ETB) [2] and ground stationary state as an thermal equlibrium state [3].
Taking the first way, we use the Bogolyubov's (u, ^-transformation to find a wave function of quantum oscillator in the thermal correlated-coherent state (TCCS) in the form [4]
exp {iy (q,T)}.
Its amplitude and phase depend on frequency w and temperature T. We note the phase plays an essential role even at the high temperatures when ip(q) = mwq2/2h. As a result Sroedinger's uncertainties relation "coordinate- momentum" takes the form of equality at any temperature
(Ap)2(Aq)2 = 1 (|{Ap, Aif}|)2 + ^ = ^ coth2 hu
4XK iU/ 4 4 2kBT'
that corresponds to presence of a thermal noise in the pure state. Here we can formally enter the "effective quantum of action"
, Hu
hef = h coth > h.
2kb ±
Taking the second way we constructed a model of the system environment in the form of the ETB filled with the thermal radiation which is the most natural object with really infinite number of freedom degrees. It contains the infinite set of weakly bounded quantum oscillators, each of them corresponds to a harmonic mode of radiation at frequency uj and average energy (e), determined by Planck's formula
. . Ku hu
(e) = — coth
2 2kB T
Received 29th December, 2009.
We would like to thank Yu.P. Rybakov who is our such attentive and helpful colleague.
at any temperature. So this model is fundamentally differed from the standard model of thermobath used in quantum statistical mechanics (QSM) consisting from weakly bounded classical oscillators with
<£> = kB T.
Thus, it is accepted that an effective temperature of an ETB mode (Bloch's temperature [5])
- (e>
Uf = -¡—. kB
Its limiting expressions are Kelvin's temperature T at kBT ^ Kh/2, which is the same for all modes and a specific quantum temperature Tq = ftLj/2kB (Wigner's temperature [6]) at kBT ^ Kuj/2.
Essentially, that the initial imbedding of any system into the ETB allows consistently constructing two sections of the new theory each of them does not use such a notion as a particles number and, in a certain sense, is a thermofield analogue of Clausius'classical and Einstein's statistical thermodynamics accordingly. We call them as Thermofield Thermodynamics (TfT) and Thermofield Statistical Thermodynamics (TfSD).
In the TfT section (as it is accepted in thermodynamics) non concrete representations about a system concerning its mass and structure are required and the condition of thermal equilibrium of system with the ETB mode is kept its validity in the standard form of the Zero Law _
Te f = Te f
. The First and the Second Laws of equilibrium thermodynamics thus formally do not change, but we suggest generalizing the concept of transferred heat
&Qe f = Te fA5e f = (Ce f)v ATe f.
Here, a more general characteristic, such as energy capacity (Cef)v, is entered. Its
limiting values has the expressions Cy and ^Pw where pw is the spectral density of
radiation. Finally, the Third Law and efficiency of Carnot cycle accordingly have the forms
lim = = 0; V = 1 - .
t^ 0 (Te f)i
In the TfST section both types of uncontrollable influences on a system are considered correspondingly to an simultaneously account of quantum and thermal fluctuations of system macroparameters, including its effective temperature as well. The thermal equilibrium is defined now by the generalized Zero Law
Tef = Tef ± ATef.
According to Gibbs-Einstein and Blokhintsev, in contrary to Boltzmann's assembly of particles, we enter the ensemble [7], containing an infinite set of complete system copies under certain external conditions. This choice allows us to apply the obtained relations to a single microparticle as well. TfST is based on Gibbs'-Einstein's canonical initial distribution in the space of macroparameters at the module of distribution
associating by the effective temperature Tef. It takes the form [8]
p(£) = eXp{^f} n(£).
Expanding the structural function fi(e)into a series up to the second order, it is possible to find fluctuations of any system macroparameters, including effective temperature as well. In the most simple case at fi(e) = 1, from the given distribution,
96
Golubjeva O. N., Sukhanov A. D., Umanskaja Zh. V.
the expression for effective entropy follows in the form
(e> - Fet
Sef = -ks J dep(e) ln/5(e)
Tef
Here effective free energy
Fef = -kBTef ln coth ; ^
2 kBT
and /5(e), in contrary to p(e), is a dimensionless density of probability. Formulae for fluctuations of energy and temperature of a system are similar to Einstein's formulae, but they additionally show that fluctuations of frequency Aw = (w) and energy Ae = (e ) always take place including the ground state.
Some advantage of TfST over QSM ground on the different choice of thermobath models. Studying the entropy in the frame of TfST we can take the quantum oscillator in the TCCS as a macrosystem in the thermal equilibrium. Its effective entropy Sef has the form
Sef = kB l 1 + ln coth
2 kBTj
in which the contribution of the energy fluctuations in the ground state is included. But according to QSM, we have the essentially other expression
S = k* { ^ (exp kW - l)-' - "> (1 - exp (-kW ))}•
We note the entropies Sef and S are different. Particulary, at T ^ 0 : Sef ^ kB = 0 while S ^ 0 but at T ^ ftw/2kB we have Sef & S• The condition of applicability of thermodynamics ATef = Tef at T ^ 0 in TfST, in contrary to QSM, are satisfied.
The legitimacy of our results can be also proved with the help of the TfD calculations. We define the effective entropy on the ground of the wave function ^(q,T) obtained by us
Sef = -ks^ J dqp(q) ln/5((?) + y dpp(p) ln/5(p)j ,
where p(q) and p(p) are dimentionless densities of probability. This expression might be rearranged through the "effective quantum of action"
Sef = kB {1 + ln(ftefA)} = kB {1 + lnW},
where W — is a number of microstates in the given macrostate.
As a result, Fenyes' idea [9] about an opportunity of the Nature description simultaneously at micro- and macrolevels by the means of two alternative ways proves to be true. We have demonstrated the language of TfST allows using the generalized diffusion equation for density pn(q,Tef), at the effective coefficient of diffusion
Def = ~^Tef, mw
while the language of TfD allows using the Shroedinger equation for wave function ^(q, T). Moreover the both ways of the description give us the equivalent results, and they are applicable even up to the ground state of oscillator equal in rights (at T ^ 0).
We believe that using TfST principles and the main TfT ideas together can be propagated on other systems (relic radiation in the early Universe, quark-gluon plasma, quantum computers, nanostructures etc.). Then we can obtain their adequate description at any temperatures and avoid some difficulties which appear in the frame of QSM
for any system at
(e > = kBTQ > kBT.
We hope that efficiency of the suggested theory at the description of nearly perfect fluids [10] can be confirmed in the near future both at the analysis of quark - gluon plasma and the further studying of He-4 superfluidity and Bose-Einstein condensates.
References
1. Umezava H. Advanced Field Theory: Micro-, Macro-, and Thermal Physics. — New York: AIP, 1993. — 240 p.
2. Sukhanov A. D. Towards a Quantum Generalization of Equilibrium Statistical Thermodynamics: Effective Macroparameters // TMP. — 2008. — Vol. 154(1). — Pp. 153-164.
3. Sukhanov A. D., Golubjeva O. N. Towards a Quantum Generalization of Equilibrium Statistical Thermodynamics: (H, fc)-Dynamics // TMP. — 2009. — Vol. 160(2). — Pp. 1177-1189.
4. Sukhanov A. D. Quantum Oscillator in the Thermostat as a Model in the Thermodynamics of Open Quantum System. — ArXiv: cond-mat/0509113v1 [cond-mat.stat-mech] 5 Sept 2005.
5. Bloch F. // Z.Phys. A. — 1932. — Vol. 74. — P. 295.
6. Wigner E. // Phys. Rev. — 1932. — Vol. 40. — P. 749.
7. Sukhanov A. D. Statistical Thermodynamics Ideas of Einstein in Modern Physical Picture of World // Phys. Part. Nucl. — 2005. — Vol. 36(6). — P. 667.
8. Sukhanov A. D., Rudoy Y. G. About One Non-noticed Idea of Gibbs // Phys. Usp. — 2006. — Vol. 49. — Pp. 531-535.
9. Fenyes I. Theoretical Probability Proof and Interpretation of Quantum Mechanics // Z.Phys. — 1952. — Vol. 132. — Pp. 81-112.
10. Schaefer T., Teaney D. Nearly Perfect Fluidity: from Cold Atoms to Hot Quark Gluon Plasmas. — ArXiv: 0904.3107 v2 [hep-ph]. — 2009.
УДК 536.1
Два подхода к квантовому обобщению термодинамики
О. Н. Голубева, А. Д. Суханов, Ж. В. Уманская
Центра естественнонаучного образования Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Россия, 117198
Предлагаются два подхода к описанию системы, находящей под одновременным квантовым и тепловым стохастическим воздействием, каждый из которых является термополевым аналогом классической термодинамики Клаузиуса и статистической термодинамики Эйнштейна соответственно. Мы называем их термополевой термодинамикой и термополевой статистической термодинамикой. Исходя из с термополевого вакуума устанавливается соответствие между основными понятиями квантовой механики и термодинамики. Предлагается рассматривать термополевой вакуум как эффективный термостат и основное состояние как состояние теплового равновесия. На этом основании вводятся такие понятия как эффективная температурная и эффективная энтропия и открывается возможность описания системы, находящейся под одновременным квантовым и тепловым воздействием.
Ключевые слова: эффективная температура, эффективная энтропия, комбинированное квантовое и тепловое воздействие, стохастическое воздействие, термополевой вакуум.
