Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes
We give a brief review of applications of the method of the nonequilibrium statistical operator developed by D.N.Zubarev to some problems of nonequilibrium statistical mechanics.
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Інститут фізики конденсованих систем НАН України
1998
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| Цитувати: | Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes / V.G. Morozov, G. Röpke // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 673-686. — Бібліогр.: 47 назв. — англ. |
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irk-123456789-1198862017-06-11T03:03:36Z Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes Morozov, V.G. Röpke, G. We give a brief review of applications of the method of the nonequilibrium statistical operator developed by D.N.Zubarev to some problems of nonequilibrium statistical mechanics. Ми подаємо короткий огляд застосування методу нерівноважного статистичного оператора, розвинутого Д.М.Зубарєвим, до деяких проблем нерівноважної статистичної механіки. 1998 Article Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes / V.G. Morozov, G. Röpke // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 673-686. — Бібліогр.: 47 назв. — англ. 1607-324X DOI:10.5488/CMP.1.4.673 PACS: 05.20.Dd, 05.60.+w, 05.70.Ln, 52.25.Fi, 82.20.M http://dspace.nbuv.gov.ua/handle/123456789/119886 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We give a brief review of applications of the method of the nonequilibrium statistical operator developed by D.N.Zubarev to some problems of
nonequilibrium statistical mechanics. |
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Article |
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Morozov, V.G. Röpke, G. |
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Morozov, V.G. Röpke, G. Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes Condensed Matter Physics |
| author_facet |
Morozov, V.G. Röpke, G. |
| author_sort |
Morozov, V.G. |
| title |
Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes |
| title_short |
Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes |
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Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes |
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Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes |
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Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes |
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zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes |
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Інститут фізики конденсованих систем НАН України |
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1998 |
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Zubarev's method of a nonequilibrium statistical operator and some challenges in the theory of irreversible processes / V.G. Morozov, G. Röpke // Condensed Matter Physics. — 1998. — Т. 1, № 4(16). — С. 673-686. — Бібліогр.: 47 назв. — англ. |
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Condensed Matter Physics |
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| fulltext |
Condensed Matter Physics, 1998, Vol. 1, No. 4(16), p. 673–686
Zubarev’s method of a nonequilibrium
statistical operator and some
challenges in the theory of irreversible
processes
V.G.Morozov 1 , G.Röpke 2
1 MIREA, Physics Department,
Vernadsky Prospect, 78, 117454 Moscow, Russia
2 FB Physik, Universität Rostock, D-18051 Rostock, Germany
Received December 1, 1998
We give a brief review of applications of the method of the nonequilib-
rium statistical operator developed by D.N.Zubarev to some problems of
nonequilibrium statistical mechanics.
Key words: nonequilibrium statistical operator, irreversible processes,
kinetic theory, hydrodynamic fluctuations, transport coefficients
PACS: 05.20.Dd, 05.60.+w, 05.70.Ln, 52.25.Fi, 82.20.M
1. Introduction
Once Gibbs announced his method of statistical ensembles for equilibrium
many-particle systems, a serious effort was mounted to extend this method to
nonequilibrium situations. Nevertheless, it was a long time before such an ex-
tension was developed in a form general enough to study virtually any class of
nonequilibrium processes. One of the most compact and elegant approaches of
this kind is Zubarev’s method of a nonequilibrium statistical operator (NSOM).
This method is detailed in books [1,2] (see also Zubarev’s article [3] where the his-
tory of NSOM is presented). The Zubarev formulation of nonequilibrium statistical
mechanics turned out to be very convenient for concrete applications and at the
moment there exists extensive literature which shows NSOM in action. Here we
will not dwell on technical aspects and concrete applications of Zubarev’s method.
The reader may wish to refer to papers in this issue and in special literature. On
the other hand, since nonequilibrium statistical mechanics is on a stage of active
development, it makes sense to take a quick look at more general questions related
to NSOM, which may be considered as challenges and call for further investigation.
c© V.G.Morozov, G.Röpke 673
V.G.Morozov, G.Röpke
Some of these questions were discussed with D. N. Zubarev during our work on
books [2,4].
2. Nonequilibrium statistical ensembles and irreversible ther-
modynamics
It is a matter of common knowledge that statistical mechanics must provide
microscopic foundations for the thermodynamic description of many-particle sys-
tems. For equilibrium, the situation seems to be wholly satisfactory since Gibbsian
statistical ensembles give rigorous formulations of thermodynamic quantities and
relations. Following this line of reasoning, it is hoped that an extension of Gibbs’
method to the nonequilibrium case would justify the basic postulates of irreversible
thermodynamics. On this way, however, one meets a number of fundamental prob-
lems. Some of them are yet to be solved. Here we will briefly discuss this point in
the context of NSOM.
For orientation, we first recall the basic ideas of NSOM which are required in
further considerations. In Zubarev’s approach, the starting point for the construc-
tion of the nonequilibrium statistical operator1 ̺(t) is the Liouville equation with
an infinitesimally small source term that breaks the time reversal symmetry of the
equation [1,2]:
∂̺(t)
∂t
+ iL̺(t) = −ε {̺(t) − ̺rel(t)} , ε → +0, (2.1)
where L is the Liouville operator determined by the quantum or the classical Pois-
son bracket with the Hamiltonian Ĥ: iL̺(t) = [̺(t), Ĥ]/i~ (the quantum case) and
iL̺(t) = {̺(t), Ĥ} (the classical case). The source term on the right-hand side of
equation (2.1) selects a retarded solution of the Liouville equation, which coin-
cides with an auxiliary statistical operator ̺rel(t) in a distant past. This relevant
statistical operator (another frequently used term is a quasi-equilibrium statistical
operator) is intimately related to the thermodynamic description of the system.
Here we would like to emphasize the trivial but sometimes overlooked fact that
the Liouville equation, being a differential equation, does not uniquely determine
the statistical operator, so that one has to introduce some initial or boundary
conditions describing the physical situation.
A standard procedure for constructing ̺rel(t) is based on the extremum of the
information entropy for the given average values
Pm(t) = 〈P̂m〉
t ≡ Tr
{
P̂m ̺(t)
}
(2.2)
of some relevant dynamical variables {P̂m} describing the macroscopic state of the
system [5]. In many concrete problems one deals with local variables P̂i(r) which
1Since a major part of our considerations will be concerned with classical and quantum systems,
we use the term “statistical operator” for classical systems as well, although in this case ̺(t) is a
function in phase space.
674
Zubarev’s method of NSO
depend on space. In such cases, the composite index m = {i, r} contains spatial
coordinates of the point, and the summation over m includes integration over r.
The extremum condition for the information entropy leads to the following
explicit expression for the relevant statistical operator:
̺rel(t) = exp
{
−Φ(t)−
∑
m
Fm(t) P̂m
}
, (2.3)
where Fm(t) are Lagrange multipliers determined by the self-consistency conditions
Pm(t) = Tr
{
P̂m ̺rel(t)
}
(2.4)
for the given values Pm(t). The quantity
Φ(t) = lnTr exp
{
−
∑
m
Fm(t) P̂m
}
(2.5)
is called the Massieu-Planck function (functional).
Starting from equation (2.1) with the relevant statistical operator given by
equation (2.3), one can derive a formally closed set of the evolution equations for
the observables Pm(t). The corresponding mathematics is presented, for example,
in book [2].
Formulation of the boundary condition for ̺(t) in terms of the relevant statis-
tical operator is just the point where a thermodynamic description is introduced.
It can easily be shown that the Massieu-Planck function (2.5) can be regarded
as a nonequilibrium thermodynamic potential in the variables {Fm(t)}. Indeed,
varying the both sides of equation (2.5) with respect to Fm(t) and then taking
the self-consistency conditions (2.4) into account, we obtain the thermodynamic
relations
Pm(t) = −
δΦ(t)
δFm(t)
. (2.6)
We can also go to the description in the variables {Pm(t)} by means of a nonequi-
librium generalization of the Legendre transformation. The potential conjugated
to Φ(t) is the nonequilibrium thermodynamic entropy S(t) which is defined as the
information entropy in the relevant statistical ensemble, i.e.
S(t) = −Tr {̺rel(t) ln ̺rel(t)} = Φ(t) +
∑
m
Fm(t)Pm(t), (2.7)
where again self-consistency conditions have been used. From equations (2.6) and
(2.7) it follows at once that
Fm(t) =
δS(t)
δPm(t)
. (2.8)
Thus, the parameters Fm(t) are thermodynamically conjugate to the observables
Pm. The corresponding Gibbs-Helmholtz relations are given by
S(t) = Φ(t)−
∑
m
Fm(t)
δΦ(t)
δFm(t)
, Φ(t) = S(t)−
∑
m
Pm(t)
δS(t)
δPm(t)
. (2.9)
675
V.G.Morozov, G.Röpke
Note that the conditions (2.4) may be interpreted as nonequilibrium equations of
state.
Differentiating entropy (2.7) with respect to time and then using equation (2.8),
we find that
dS(t)
dt
=
∑
m
δS(t)
δPm(t)
∂Pm(t)
∂t
=
∑
m
Fm(t)
∂Pm(t)
∂t
. (2.10)
This equation is well known from irreversible thermodynamics [6]. It gives the en-
tropy production in terms of the thermodynamic forces Fm(t) and the macroscopic
fluxes ∂Pm(t)/∂t.
The crucial point in the above formalism is the choice of the set of relevant
variables {P̂m} which is adequate for the description of a nonequilibrium system.
One frequently used guide-line for choosing relevant variables is the so-called “hi-
erarchy” of widely spaced relaxation times in many-particle systems. By this is
meant that nonequilibrium processes typically proceed on different stages, each
characterized by a proper time scale. Different time scales can be determined by
the characteristic time intervals ∆t at which the change of the system state occurs.
The larger is the characteristic time interval, the smaller set of slowly varying ob-
servables can be used. This idea of a reduced description of nonequilibrium systems
was first formulated and developed by Bogoliubov [7] in the context of the classical
kinetic theory and now is, in fact, the heart of all the methods in nonequilibrium
statistical mechanics.
In Zubarev’s and many other approaches, the level of the reduced description
is determined by the set of dynamical variables {P̂m} in the relevant statistical
operator. For example, the usual hydrodynamic description implies that the vari-
ables P̂m ≡ P̂j(r) correspond to locally conserved quantities, such as the densities
of energy, mass, and momentum. It should be emphasized, however, that, in prin-
ciple, nothing prevents us from using variables which are not “slow” variables on
a chosen time scale. An example is the Grad method in the classical kinetic the-
ory [8], where the fluxes of hydrodynamic quantities are treated as independent
variables in constructing normal solutions of the Boltzmann equation.
Note that, depending on the choice of relevant variables, we have in fact differ-
ent thermodynamic descriptions of the same nonequilibrium system. However, this
“versatility” of the formalism brings up some questions. First, what is the physical
meaning of thermodynamic parameters Fm for different sets of relevant variables ?
Second, what is the relation between the evolution equations for different sets of
relevant variables and how these equations are related to the observable quanti-
ties ? Third, is there a general rule for the choice of P̂m if one uses slow variables
as well as fast variables ?
At present we know enough to solve the above questions only for linear irre-
versible processes when the state of a system is close to total equilibrium. It can
be shown (for a detailed discussion see, e.g., [4]) that in the limit ε → +0 all sets
of relevant variables are equivalent in the sense that they give the same results
for observable quantities (transport coefficients and susceptibilities), if one deals
with exact expressions. On the other hand, using the same approximation for dif-
676
Zubarev’s method of NSO
ferent sets of relevant variables, say the Born approximation in time correlation
functions, different results for transport coefficients and susceptibilities can be ob-
tained. Roughly speaking, the more dynamical variables {P̂m} are included into
the relevant statistical operator, the closer we get to the exact results for transport
coefficients and susceptibilities. Thus it appears reasonable to consider different
sets {P̂m} as trial sets of variables within some variational principle. This inter-
pretation can be justified rigorously in the special case of stationary processes [4].
The physical meaning of the variational principle is very simple: as long as the
parameter ε in equation (2.1) is kept finite, the exact values of transport coeffi-
cients and susceptibilities correspond to the maximum of the entropy production in
the system. This resembles the situation in the kinetic theory of rarefied gases [9]
where the exact solution of the Chapman-Enskog equations gives transport coef-
ficients which correspond to the maximum of the entropy production (the Kohler
variational principle). For non-stationary linear irreversible processes, the relation
between NSOM and variational principles calls for further investigation.
It goes without saying that in far-from-equilibrium situations the problem of
the choice of proper relevant variables becomes much more complicated. At the
phenomenological level, different sets of state variables are introduced in the so-
called extended irreversible thermodynamics [10,11] which now is established as a
formal closed theory. Recently a serious effort was made to justify the extended
irreversible thermodynamics on the basis of general principles of statistical me-
chanics and, in particular, on the basis of NSOM (for a detailed discussion and
references see, e.g., [12,13]).
A rather general scheme for the selection of relevant variables can be formu-
lated [12,14] in the case when the total Hamiltonian admits a decomposition into
two parts, Ĥ = Ĥ0+ Ĥ ′, where the secular part Ĥ0 includes kinetic energies of the
particles and, may be, some strong interactions associated with very short relax-
ation times on the chosen time scale. The other part, Ĥ ′, is considered as a small
perturbation describing slow relaxation processes in the system. Then it is assumed
that appropriate variables P̂m satisfy a closure condition (in the quantum case)
1
i~
[P̂m, Ĥ
0] =
∑
n
iΩmn P̂n, (2.11)
where Ωmn are some c-numbers. This condition plays an important role in the
kinetic theory (see, e.g., [1,2,15,16]) where Ĥ0 describes noninteracting particles
(or quasiparticles), while Ĥ ′ is an interaction term in the Hamiltonian. In the
context of the kinetic theory, the relevant dynamical variables, P̂m = a†lal′ , are
bilinear in the creation and annihilation operators corresponding to some one-
particle quantum states |l〉.
If the variables P̂m and (or) the secular part Ĥ0 of the Hamiltonian are not
bilinear in the creation and annihilation operators, condition (2.11) produces, in
general, an infinite chain of variables coupled through their commutator with Ĥ0.
Therefore, in concrete calculations one is led to apply a truncation procedure [14]
which resembles the variational methods of the kinetic theory but is difficult to
677
V.G.Morozov, G.Röpke
formulate explicitly. Note also that in many cases of interest the decomposition of
the Hamiltonian into the secular part and the interaction term is not unique since
this decomposition can depend on the order of the approximation (see, e.g., [17]).
In summary one may conclude that now the choice of relevant variables for
highly nonequilibrium systems is based essentially on heuristic arguments sup-
ported by explicit calculations for some concrete models. The most natural way
to a more general scheme is to look for a variational principle applicable to a wide
range of far-from-equilibrium situations. We believe that the works along this line
are of great importance for the development of NSOM.
3. Correlations in the kinetic theory
The kinetic theory is a traditional field of application of the nonequilibrium
statistical mechanics. In a broad sense, any problem where the evolution of the
single-particle distribution f1(t) (the classical distribution function or the quantum
density matrix) is of interest may be assigned to the kinetic theory. Over many
years it has been agreed that the main objective of the kinetic theory is to derive a
closed kinetic equation for f1(t). In the context of NSOM this means that the dy-
namical variable f̂1 determined by the relation f1(t) = 〈f̂1〉
t is the only variable P̂m
in the relevant statistical operator (2.3). Physically, the corresponding boundary
condition for the Liouville equation is a complete uncoupling of all many-particle
distribution functions to one-particle functions in the distant past. This is nothing
but Bogoliubov’s boundary condition in the kinetic theory of dilute gases [7]. It
is clear that for moderately dense gases and liquids such a boundary condition
is not adequate since the relevant statistical operator describing an ideal gas dif-
fers greatly from the true nonequilibrium statistical operator which must involve
many-particle correlations. These correlations are associated, for instance, with
the bound states of particles, conservation laws, etc. The shortcoming of Bogoli-
ubov’s boundary condition for correlated systems is manifested in strong memory
effects in the kinetic equation. Several modifications of Bogoliubov’s boundary
conditions have been proposed to include relevant correlations, thus preserving
the Markovian character of the kinetic equation. Unfortunately, for the most part
these modifications are based on heuristic considerations adapted to special cases.
It should be noted, however, that in changing Bogoliubov’s boundary condition it
would be reasonable to proceed from a physical principle applicable to as many
real systems as possible. Here we shall discuss the arguments put forward in [18].
The existence of long-lived correlations in a system is connected with the fact
that there are some collective dynamical variables Ĉj which vary slowly in time.
The locally conserved hydrodynamic variables may serve as an example. It is signif-
icant that Ĉj cannot all be expressed exactly in terms of a one-particle distribution
(say, the energy density). Thus, in addition to f̂1, one has from the beginning to
include Ĉj in the set of relevant variables. This simple reasoning is conceptually
in line with the Bogoliubov principle of the reduced description of nonequilibrium
systems [7]. The only new aspect is that we consider a sequence of decreasing time
678
Zubarev’s method of NSO
scales rather than a sequence of increasing ones, as it was done by Bogoliubov.
To clarify the meaning of this statement, we assume that the system of interest
possesses a hierarchy of relaxation times τi, (i = 0, 1, . . .), and τi ≪ τj+1. The
shortest relaxation time, τ0, defines the so-called dynamic stage of evolution when
no reduced description can be used (for gases τ0 can be estimated as the collision
time). Let M(i) = {P̂
(i)
m } (i > 1) be a set of relevant dynamical variables which
are slowly varying variables on the time scale with the characteristic time interval
∆ti such that τi ≪ ∆ti ≪ τi+1. Then the Bogoliubov formulation of the princi-
ple of a reduced description is that M(i+1) is a subset of M(i), and that M(1) is
sufficient for the most detailed description of the system. Stated more precisely,
it is assumed that the equations of motion for 〈P̂
(1)
m 〉t can be taken as basic evo-
lution equations for all the time scales. This is the case in the kinetic theory of
dilute gases, where 〈P̂
(1)
m 〉t may be interpreted as a total set of moments of the
single-particle distribution function f1(t), and the Boltzmann equation plays the
role of the basic evolution equation. Then, if the characteristic time interval ∆t
is taken to be much larger than the mean free time of a particle, one can use
only a few moments of f1(t) (the hydrodynamic variables) to describe the system.
Unfortunately, even for moderate gases we do not know a closed kinetic equation
which is applicable to all the time scales, so that we have to start directly from
the N-particle Liouville equation and the above scheme of a reduced description
becomes ineffective. Let us, however, consider the characteristic time scales in the
reverse order. The largest time scale is determined by the characteristic interval
∆t > τeq, where τeq is the time necessary for the relaxation of the system to total
equilibrium. On this scale, only the average values of the integrals of motion, 〈Ĉj〉,
are needed to describe the system. The hydrodynamic time scale is associated with
the characteristic interval such that τloc ≪ ∆t ≪ τeq, where τloc is the time for the
establishment of local equilibrium in a volume that is small but still contains a
large number of particles. Now the state of the system can be described by the av-
eraged densities of the integrals of motion (or by their spatial Fourier components
〈Ĉj,k〉
t). Note that a new set of the relevant dynamical variables {Ĉj,k} includes
all the variables Ĉj = Ĉj,k=0 of the previous set. This rule is quite general since
any variable which is slowly varying on some scale of time is also slowly varying
on shorter scales.
Generally speaking, the set of slowly varying variables on the hydrodynamic
scale is not exhausted with Ĉj,k , because an arbitrary high order product, like
Ĉj1,k1
Ĉj2,k2
· · · Ĉjs,ks
, is also a slowly varying variable. Therefore, to obtain the
Markovian equations of evolution, hydrodynamic fluctuations should be included
into the set of relevant variables. In doing so, one can construct the thermody-
namics of fluctuations [19] and derive the hydrodynamic Fokker-Planck equation
for the distribution functional [20]. If there exist other slow relaxation processes
in the system (for instance, the formation of long-lived bound states or the relax-
ation of internal degrees of freedom of the particles), the corresponding dynamical
variables should be incorporated along with the hydrodynamic variables. As one
goes to shorter time scales, some new dynamical variables can also be taken to
679
V.G.Morozov, G.Röpke
construct the relevant statistical operator determining the boundary condition for
the true nonequilibrium statistical operator of the system.
The above procedure has no restriction to its generality and does not require
that a basic kinetic equation like the Boltzmann equation be known, as it is neces-
sary when one follows the original Bogoliubov scheme of the reduced description.
The remaining problem, of course, is the choice of a proper set of relevant variables
on small time scales.
Turning back to the kinetic theory, we note that the characteristic relaxation
time for the single-particle distribution is distinctly smaller than the hydrodynamic
relaxation time τloc. Thus, strictly speaking, kinetic processes must be considered
together with the evolution of the locally conserved quantities and other “slow vari-
ables”. This is not clearly seen in the case of dilute gases since all the quantities
of interest can be expressed approximately in terms of the single-particle distribu-
tion. For dense and strongly correlated systems, however, the interplay between
the kinetic and long-lived collective modes is of the utmost significance. Linear
irreversible processes in classical fluids [21–23] and plasmas [24] are the examples.
The inclusion of long-lived correlations in the kinetic theory fits naturally into
the scheme based on NSOM. Having constructed the relevant statistical operator
with the kinetic and collective dynamical variables, one can use equation (2.1)
to obtain the corresponding boundary conditions for the classical or quantum
hierarchy. For a more detailed discussion of this point see our paper in this issue [25]
and [2,4].
4. Nonequilibrium statistical mechanics of relativistic systems
Recent experimental studies of the ultra-relativistic heavy-ion collisions have
attracted considerable attention to the theory of nonequilibrium processes in rel-
ativistic quantum systems. It is notable in this connection that one of the first
Zubarev’s papers on the method of the nonequilibrium statistical operator was
devoted to the relativistic transport theory [26] (see also [1]). In this short paper
Zubarev derived Green-Kubo formulae for transport coefficients and paved the
way for nonequilibrium statistical mechanics of quantum fields. Later, a covariant
formulation of the method was developed [27,28] and transport coefficients were
calculated for some models [29].
The existing version of the relativistic NSOM works well in the case when
disturbances are slowly varying on space and time scales associated with the mi-
croscopic correlation length and correlation time (for low-density systems these
are the mean free path and the mean collision time). The Markovian evolution
equations derived within the statistical approach coincide with the equations of
the phenomenological relativistic hydrodynamics [30]. If one attempts to study
the processes involving steep gradients and rapid variations in time, the effects
of nonlocality and memory should be taken into account. Note that in the rel-
ativistic context these effects have a fundamental significance. The point is that
the customary (Markovian) equations of relativistic hydrodynamics are parabolic
680
Zubarev’s method of NSO
differential equations and, therefore, predict infinite speed of propagation of ther-
mal disturbances in contradiction with the principle of causality. Formally, the
generalized transport equations derived within NSOM [26,27] or other equivalent
approaches [31] are nonlocal in space and time, but the structure of the transport
coefficients is very complicated to be used in concrete problems. So, the task is to
construct a suitable perturbation scheme for evaluating non-Markovian corrections
to the hydrodynamic equations.
Another possible way to improve the customary relativistic hydrodynamics on
shorter time scales is to use an extended set of variables describing the system.
An analogous phenomenological version of irreversible relativistic thermodynam-
ics was proposed by Israel [32]. It is based on the assumption that the entropy
density contains extra terms which depend on the heat flux and viscous stresses.
Such a theory is in line with the molecular hydrodynamics developed for classi-
cal fluids [21] and predicts a wave-like propagation of thermal modes, which does
not violate causality. Israel’s extended relativistic thermodynamics was then con-
firmed by a generalized Grad method applied to the relativistic Boltzmann equa-
tion [33,34]. Note, however, that the relativistic Boltzmann equation is inadequate
for many of the field models, so that it might be good to use a more fundamen-
tal evolution equation. In the context of NSOM, one again faces the problem of
the choice of a proper set of relevant variables on space and time scales, where
relaxation processes are considered on the same footing as hydrodynamic ones.
We see that, setting aside questions specific to the Lorentz symmetry, etc.,
further development of the nonequilibrium relativistic statistical mechanics is clo-
sely related to the solution of the problems discussed in section 2. What is needed
is a sufficiently justified procedure for constructing relevant statistical operators
on short space and time scales.
5. Connection between NSOM and some other methods
As a final topic of our brief review, we compare NSOM with some other methods
based on “first principles” of statistical mechanics.
At the level of a linear response of equilibrium systems to mechanical perturba-
tions, the first general theory was developed in the pioneering work by Kubo [35].
In Kubo’s theory, it is assumed that in the distant past the system is in equilibrium
with a heat bath and then its evolution is described by the Liouville equation with
the Hamiltonian involving external mechanical perturbations. Within the frame-
work of NSOM, Kubo’s response formulae are obtained by setting ̺rel(t) = ̺eq,
where ̺eq is an equilibrium statistical operator [4]. In a certain sense Kubo’s scheme
may be considered as a particular case of NSOM when the set of relevant vari-
ables contains only the integrals of motion. It should be emphasized that Kubo’s
theory describes the response of an isolated system to external mechanical pertur-
bations. In general, this response need not be equal to the response of a system
being in contact with a heat bath during the evolution. That is why Kubo’s theory
meets with some difficulties in the case of static perturbations [36]. The formalism
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V.G.Morozov, G.Röpke
based on NSOM seems to be more satisfactory, because the use of the relevant
statistical operator ̺rel(t) to formulate the boundary condition for the Liouville
equation can be treated as violating a complete isolation of the system. Although
the static (thermodynamic) susceptibilities can be obtained from Kubo’s formulae
if the limits ω → 0 and ε → +0 are taken in the correct order [4], these formulae
are not always convenient in concrete problems since they require a partial sum-
mation of the perturbation series. On the other hand, with a proper set of relevant
variables, NSOM gives a reasonable approximation for susceptibilities and kinetic
coefficients using rather crude approximations for correlation functions appearing
in the response equations.
A linear response to thermal (internal) perturbations is conveniently studied in
the memory function formalism (it is also called the Mori-Zwanzig formalism [37,
38]) based on the projection method. The basic equations of the memory function
formalism follow from the response equations derived by NSOM, when external
mechanical perturbations are absent [4,39]. The advantage of NSOM is that it can
also be used to study a linear response to mechanical perturbations and the cross
effects.
The projection formalism was generalized to far-from-equilibrium situations
by Zwanzig [40], Robertson [41], and Kawasaki and Gunton [42]. In the context
of NSOM, Zwanzig’s approach corresponds to the following representation of the
relevant statistical operator: ̺rel(t) = P̺(t), where P is a time-independent projec-
tor. In the original Zwanzig formulation, P selects the diagonal part of ̺(t) with
respect to eigenstates of the main (secular) part H0 of the Hamiltonian. Note,
however, that Zwanzig’s projector can also be constructed for the cases where the
reduced description of nonequilibrium states is not limited to using the diagonal
part of the statistical operator.
Robertson’s method [41] is based on the fact that the time derivative of the
relevant statistical operator (2.3) can be represented in the form ∂̺rel(t)/∂t =
PR(t) ∂̺(t)/∂t, where PR(t) is some time-dependent projector. This property al-
lows one to derive a formally closed evolution equation for ̺rel(t) (the gener-
alized master equation), which, in turn, leads to a closed set of equations for
the observables 〈P̂m〉
t = 〈P̂m〉
t
rel. Robertson’s scheme was improved by Kawasaki
and Gunton [42], who introduced a new projector Prel(t) which has the property
Prel(t) ̺(t) = ̺rel(t). This property is analogous to the property of Zwanzig’s pro-
jector P.
It can be shown (see, e.g., [2]) that all the above approaches are, in fact, different
formulations of NSOM, except that sometimes one uses the initial condition for the
nonequilibrium statistical operator instead of the boundary condition in a distant
past.
The boundary conditions for the Liouville equation can be formulated as the
so-called ergodic conditions which are a far-reaching generalization of Bogoliubov’s
boundary condition of the weakening of initial correlations [2,43]. They reflect the
fact that during the evolution of a macroscopic system its initial statistical oper-
ator (say, ̺rel(t)) must tend to the statistical operator ̺(t) which is an integral of
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Zubarev’s method of NSO
the Liouville equation. In [15,16] the ergodic conditions were used as the basis for
constructing nonequilibrium statistical operators. Zubarev and Kalashnikov [43]
(see also chapter 2 in [2]) showed that ergodic conditions lead to the same ex-
pression for the nonequilibrium statistical operator as the Liouville equation with
the broken time-reversal symmetry, equation (2.1). In other words, the method of
ergodic conditions is equivalent to NSOM.
Finally, we would like to mention a further scheme for constructing the non-
equilibrium statistical operator. It is based on the principle of the extremum of
the information entropy, including memory effects. The supplementary conditions
have the form
Pm(t
′) = Tr
{
P̂m̺(t
′)
}
= Tr
{
P̂m(t
′ − t)̺(t)
}
(5.12)
for −∞ < t′ 6 t. These conditions include information about the evolution of the
system, whereas self-consistency conditions (2.4) include information only about
the state of the system at the given time t. The conditional extremum of the
information entropy with memory effects leads to the nonequilibrium statistical
operator in the so-called “exponential form”, which coincides with the statistical
operator obtained as a retarded solution of the Liouville equation for the oper-
ator σ(t) = − ln ̺(t) [1,44]. The above derivation is not without appeal since it
shows a connection with irreversible thermodynamics, including memory effects.
Recently an extremum-entropy approach to NSOM was used to construct a family
of nonequilibrium statistical operators which differ in the representation of the
Lagrange parameters [14,45]. Each of the representations corresponds to a special
choice of the infinitesimal source term in the Liouville equation and, therefore, the
question arises as to whether all these representations are equivalent in the sense
that they give the same evolution equations for the observables. This seems to be
the case because the source term is designed only to select the retarded solution
of the Liouville equation and after all this the term tends to zero. Nevertheless, it
is not inconceivable that there are exceptions to this rule. As far as we know, the
role of the form of the source term has never been investigated.
To summarize, one can say that, physically, the central ideas of the existing
general approaches to nonequilibrium statistical mechanics are closely related to
one another and fit naturally into the scheme of NSOM. It should be emphasized,
however, that some of the fundamental problems concerning all these approaches
are still open. Little is known, for instance, about the behaviour of nonequilibrium
averages and transport coefficients in the thermodynamic limit. Another important
problem is to construct nonequilibrium ensembles representing the state of an open
system in a strong contact with other nonequilibrium systems, accompanied by
fluxes of matter, energy, and entropy. In the case of weakly coupled systems we
have a well-developed theory based on NSOM or other similar methods, but many
of the most intriguing self-organization phenomena in open systems are beyond
this weak-coupling limit.
We have not discussed the connection between NSOM and the nonequilibrium
Green function formalism (see, e.g., [46,47]) which is often considered an alterna-
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V.G.Morozov, G.Röpke
tive to the methods based on the Liouville equation for the statistical operator.
This point is discussed in our paper [25]. Suffice it to note here that, in fact,
these approaches complement each other and their combination appears to have a
considerable promise.
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Метод нерівноважного статистичного оператора
Д.М.Зубарєва та деякі проблемні питання теорії
необоротніх процесів
В.Г.Морозов 1 , Ґ.Репке 2
1 Московський інститут радіоелектроніки та автоматики,
просп. Вернадського, 78, 117454 Москва, Росія
2 Університет м. Ростока, фізичний факультет,
D-18051 Росток, Німеччина
Отримано 1 грудня 1998 р.
Ми подаємо короткий огляд застосування методу нерівноважного
статистичного оператора, розвинутого Д.М.Зубарєвим, до деяких
проблем нерівноважної статистичної механіки.
Ключові слова: нерівноважний статистичний оператор, необоротні
процеси, кінетична теорія, гідродинамічні флюктуації, коефіцієнти
переносу
PACS: 05.20.Dd, 05.60.+w, 05.70.Ln, 52.25.Fi, 82.20.M
686
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