On the irreversibility of fluctuations
Interrelationship of classical and quantum fluctuations is discussed. With
 energy dissipation being taken into account, fluctuations are shown to be
 irreversible even in the case of classical systems. An analogy to Kubo relation for classical fluctuations is derived. Fluctuations in b...
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| Опубліковано в: : | Condensed Matter Physics |
|---|---|
| Дата: | 2000 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
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Інститут фізики конденсованих систем НАН України
2000
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| Цитувати: | On the irreversibility of fluctuations / A.G. Sitenko // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 277-284. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860201016550490112 |
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| author | Sitenko, A.G. |
| author_facet | Sitenko, A.G. |
| citation_txt | On the irreversibility of fluctuations / A.G. Sitenko // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 277-284. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
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| description | Interrelationship of classical and quantum fluctuations is discussed. With
energy dissipation being taken into account, fluctuations are shown to be
irreversible even in the case of classical systems. An analogy to Kubo relation for classical fluctuations is derived. Fluctuations in both equilibrium and
nonequilibrium (though stationary) states of the system are considered.
Обговорюється взаємозв’язок між класичними та квантовими флюктуаціями. Вказано на незворотність (навіть у випадку класичних систем) флюктуацій за наявності дисипації енергії. Одержано аналог
співвідношення Кубо для класичних флюктуацій. Розглянуто флюктуації як у рівноважних, так і в нерівноважних (але стаціонарних) станах
системи.
|
| first_indexed | 2025-12-07T18:10:25Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 2(22), pp. 277–284
On the irreversibility of fluctuations
A.G.Sitenko
Bogolyubov Institute for Theoretical Physics
03143 Kyiv, Ukraine
Received March 6, 2000
Interrelationship of classical and quantum fluctuations is discussed. With
energy dissipation being taken into account, fluctuations are shown to be
irreversible even in the case of classical systems. An analogy to Kubo rela-
tion for classical fluctuations is derived. Fluctuations in both equilibrium and
nonequilibrium (though stationary) states of the system are considered.
Key words: classical and quantum fluctuations, irreversibility of
fluctuations, Kubo formula
PACS: 05.40, 42.50.L, 05.70.L, 52.25.G
1. Introduction
Basic theory of fluctuations of physical quantities in both classical and quantum
systems is discussed in the brilliant books [1,2]. In particular, these books give a
detailed treatment of electromagnetic fluctuations in equilibrium systems. The au-
thors of [1,2] introduce symmetrized binary correlation functions which are invariant
with respect to time reversal and employ this approach for a detailed consideration
of both classical and quantum fluctuations. However, they do not pay enough at-
tention to the question whether fluctuations are time-reversible or time-irreversible.
The pair correlation function for the fluctuations of classical quantities is assumed
to be symmetric with respect to time reversal (see equation (118.3) of [2]) and
hence fluctuations in classical systems are treated as time-reversible. This state-
ment contradicts the well known Kubo relation [3] which determines the relation of
the linear response of the system to external perturbations and the antisymmetrized
pair correlation function. Kubo relation for quantum fluctuations is derived in [2]
(equation (126.8) of [2]); it suggests an irreversible nature of quantum fluctuations.
Irreversibility of fluctuations even in the case of classical systems and applica-
bility of Kubo relation to the description of classical fluctuations are discussed in
this paper. Moreover, the consideration is extended to the case of nonequilibrium
stationary systems.
c© A.G.Sitenko 277
A.G.Sitenko
2. Correlation functions
We employ the notation introduced in [2] and consider time fluctuations of the
quantity x(t). We assume this quantity to be real and such that its mean value is
equal to zero in the absence of external perturbations. We introduce the relevant
Hermitian operator x̂(t) for this quantity and employ the Fourier transformation,
i.e.,
x̂(t) =
1
2π
∞
∫
−∞
dωe−iωtx̂ω, x̂ω =
∞
∫
−∞
dteiωtx̂(t). (1)
If the state of the system is stationary, then the correlation of fluctuations of this
quantity taken at different time instants, say 0 and t, depends only on the time
interval between these instants. We define the correlation functions
〈x(0)x(t)〉 ≡ 〈x2〉t, 〈x(t)x(0)〉 ≡ 〈x2〉−t, (2)
where angular brackets in the left-hand parts of the equations denote quantum and
statistical averaging. The relevant spectral distributions of fluctuations, 〈x2〉ω and
〈x2〉−ω, are determined by the Fourier transforms of the correlation functions (2).
We note that the average product of Fourier components of fluctuations x̂†
ω and x̂ω′
is related to the spectral distribution of fluctuations as given by
〈x̂†
ωx̂ω′〉 = 2πδ(ω − ω′)〈x2〉ω (3)
(the symbol † implies the Hermitian conjugation operation).
We calculate the mean value of the operators product x̂†
ωx̂ω′ for some quantum
state n with energy En and average the expression obtained over the statistical
distribution of quantum states f(En). Thus we obtain the spectral distributions of
fluctuations (2) in the form
〈
x2
〉
ω
= 2π
∑
m,n
f(En)|xnm|
2δ(ω − ωnm), (4)
〈
x2
〉
−ω
= 2π
∑
m,n
f(En)|xnm|
2δ(ω + ωnm), (5)
where xnm is the matrix element of the operator x̂ and ωnm = (En − Em)/h̄ is the
frequency of the transition between the states n and m.
The quantity 〈x2〉−ω determines the spectral distribution of fluctuations reversed
in time with respect to fluctuations described by the spectral distribution 〈x2〉ω. We
rename n ↔ m in (5) and employ the properties of the delta-function. Thus the
spectral distribution (5) takes the form
〈
x2
〉
−ω
= 2π
∑
m,n
f (En − h̄ω) |xnm|
2δ(ω − ωnm). (6)
If the system is in thermodynamic equilibrium, then the function of statistical dis-
tribution of states, f(En), should be taken in the Gibbs form, i.e.,
f(En) = e
F−En
T ,
278
On the irreversibility of fluctuations
where En is the energy of nth level, F is the free energy, and T is the temperature
of the system. Comparing (6) to (4) yields a relation for the spectral distributions
of unreversed and reversed fluctuations, i.e.,
〈
x2
〉
−ω
= e
h̄ω
T
〈
x2
〉
ω
. (7)
3. Energy dissipation and relationship with the correlation of
fluctuations
Correlation functions (4) and (6) can be related to the energy dissipated in the
system exposed to an external perturbation. If the perturbation is periodic with the
frequency ω and its energy is proportional to the quantity x, i.e.,
V̂ = −f(t)x̂, f(t) =
1
2
(
f0e
−iωt + f ∗
0 e
iωt
)
, (8)
where f0 is constant, then the mean energy absorbed by the system per unit time
is given by
Q =
ω
4h̄
|f0|
2
{
〈
x2
〉
−ω
−
〈
x2
〉
ω
}
. (9)
With the external perturbation (8) being present, the mean value 〈x(t)〉 is not
equal to zero; it is proportional to the perturbation f(t), i.e.,
〈x(t)〉 =
∞
∫
0
dτ α(τ)f(t− τ), (10)
where α(τ) is the linear response of the system. According to the causality principle,
integration in (10) extends over time previous to t. Having performed the Fourier
transformation, we find that
〈x〉ω = α(ω)fω, (11)
where
α(ω) =
∞
∫
0
dteiωtα(t).
The linear response α(ω) is an analytic function of the frequency ω.
The energy absorbed by the system is immediately expressed in terms of the
linear response, we have
Q =
ω
2
|f0|
2α′′(ω) (12)
where α′′(ω) is the imaginary part of the coefficient α(ω). Comparing this expression
to (9) yields the basic relation of the fluctuation theory, i.e.,
α′′(ω) =
1
2h̄
{
〈
x2
〉
−ω
−
〈
x2
〉
ω
}
, (13)
279
A.G.Sitenko
which relates the spectral distributions of time-reversed fluctuations with the dissi-
pative properties of the system.
We introduce the symmetrized correlation function
〈
x2
〉s
ω
=
1
2
{
〈
x2
〉
ω
+
〈
x2
〉
−ω
}
(14)
and make use of relation (7). Thus, directly from (13), we obtain the fluctuation-
dissipation relation for an equilibrium system [4], i.e.,
〈
x2
〉s
ω
= h̄ cth
h̄ω
2T
α′′(ω). (15)
In a similar manner we find that spectral distributions of fluctuations (9) and (6)
for the equilibrium state of the system are given by
〈
x2
〉
ω
=
〈
x2
〉s
ω
− h̄α′′(ω),
〈
x2
〉
−ω
=
〈
x2
〉s
ω
+ h̄α′′(ω). (16)
4. Kubo relation and irreversibility of fluctuations
Since the linear response α(ω) is an analytic function without singularities in the
upper halfplane of complex frequency ω [2], we have
1
π
∞
∫
−∞
dω′ α′′(ω′)
ω′ − ω − i0
= α(ω)− α(∞).
Within the context of the relation
1
π
∞
∫
−∞
dω′ 〈x
2〉−ω′ − 〈x2〉ω′
ω′ − ω − i0
= 2i
∞
∫
0
dteiωt
{
〈x2〉−t − 〈x2〉t
}
we obtain from (13) the Kubo formula [3], i.e.,
α(ω)− α(∞) =
i
h̄
∞
∫
0
dteiωt {〈x(t)x(0)〉 − 〈x(0)x(t)〉} . (17)
By definition, 〈x2〉ω and 〈x2〉−ω are spectral distributions of fluctuations reversed
in time. According to (13), their difference is determined by the linear response
imaginary part α′′(ω) which characterizes the energy dissipation in the system. The
observation that spectral distributions 〈x2〉ω and 〈x2〉−ω are not equal implies that
fluctuations in a dissipative system are irreversible.
5. Irreversibility of classical fluctuations and classical analogy
to Kubo formula
The above conclusion concerning the time-irreversibility of fluctuations has been
formally derived in terms of quantum mechanics. Meanwhile, fluctuations are irre-
versible in classical systems as well. Indeed, though for h̄ → 0 (i.e., for a classical
280
On the irreversibility of fluctuations
system) spectral distributions of time-reversed fluctuations,
〈
x2
〉cl
ω
= lim
h̄→0
〈
x2
〉
ω
, and
〈
x2
〉cl
−ω
= lim
h̄→0
〈
x2
〉
−ω
,
are equal within the context of (4) and (6), their difference nevertheless determines
the imaginary part of the linear response for the classical system. In the limiting
case h̄ → 0, equation (6) yields
〈
x2
〉
−ω
≃
h̄≪1
〈
x2
〉
ω
− h̄ω
∂
∂E
〈
x2
〉
ω
, (18)
where ∂〈x2〉ω/∂E is the correlation function averaged over the derivative of the
distribution function,
∂
∂E
〈
x2
〉
ω
≡ 2π
∑
m,n
∂f(En)
∂En
|xmn|
2 δ(ω − ωnm). (19)
Thus, in the case of classical fluctuations we have
1
2
lim
h̄→0
〈x2〉−ω − 〈x2〉ω
h̄
= −
ω
2
∂
∂E
〈
x2
〉
ω
,
and hence the condition of fluctuation irreversibility reduces to
α′′(ω) = −
ω
2
∂
∂E
〈
x2
〉
ω
. (20)
Since the linear response is analytical, we obtain from (20)
α(ω)− α(∞) = −
1
2π
∞
∫
−∞
dω′ω
′ ∂
∂E
〈x2〉ω′
ω′ − ω − i0
, (21)
which implies that the linear response is determined by the time-irreversible part of
the correlation function. If the system is in the state of thermodynamic equilibrium,
then we have
∂
∂E
〈
x2
〉
ω
= −
1
T
〈
x2
〉s
ω
.
We substitute the latter expression in (21) and carry out integration over the fre-
quency ω′. Thus we obtain a classical analogy to Kubo formula for an equilibrium
system. It is given by
α(ω)− α(∞) = −
1
2T
∞
∫
0
dteiωt
∂
∂t
{〈x(t)x(0)〉 + 〈x(0)x(t)〉} . (22)
281
A.G.Sitenko
6. Inversion of the fluctuation-dissipation relation
It is clear that in order to calculate the linear response from (21), taking into
account the interaction in the system in the linear approximation, it is sufficient
to substitute the correlation function in the right-hand part disregarding the self-
consistent interaction. Thus, in the linear approximation we have
α(ω)− α(∞) = −
1
2π
∞
∫
−∞
dω′ω
′ ∂
∂E
〈x2〉
0
ω′
ω′ − ω − i0
, (23)
where ∂
∂E
〈x2〉
0
ω is the correlation function of the system without interaction. For an
equilibrium system, in particular, we have
α(ω)− α(∞) =
1
2π
1
T
∞
∫
−∞
dω′ ω′ 〈x2〉
0
ω′
ω′ − ω − i0
. (24)
Relations (23) and (24) can be treated as an inversion of the fluctuation-dissipation
relation [5,6].
7. Electromagnetic fluctuations
Now we extend the consideration to the case of fluctuations of spatially dis-
tributed quantities. For example we consider electromagnetic fluctuations in a me-
dium, in particular, fluctuations of the current density j(r, t). We assume the system
to be spatially homogeneous and stationary and put the average current to be equal
to zero, 〈j(r, t)〉 = 0.
We introduce the space-time Fourier transformation for the current fluctuations,
jkω =
∞
∫
−∞
dteiωtjk(t), jk(t) =
∫
dre−ikrj(r, t), (25)
and calculate the average product of fluctuation components
〈
j†i (k, ω)jj(k
′, ω′)
〉
≡ (2π)4δ(ω − ω′)δ(k− k′) 〈jijj〉kω , (26)
Thus we obtain an expression for the spectral distribution of fluctuations given by
〈jijj〉kω = 2π
∑
m,n
f(En)ji(k)
∗
mnjj(k)mnδ(ω − ωnm), (27)
where ji(k)mn is the matrix element of the operator jk(t) for the transition from n
to m. We note that
j†(k)nm ≡ j(k)∗mn = j(−k)nm. (28)
The spectral distribution of fluctuations reversed in time with respect to the
fluctuations described by the distribution (27) is given by
〈jijj〉−k−ω
= 2π
∑
m,n
f(En − h̄ω)ji(k)mnjj(k)
∗
mnδ(ω − ωnm). (29)
282
On the irreversibility of fluctuations
It is not difficult to verify that
〈jijj〉
∗
−k−ω
= 〈jjji〉−k−ω
. (30)
For the equilibrium state we have
〈jjji〉−k−ω
= e
h̄ω
T 〈jijj〉kω . (31)
We take the perturbation energy to be described by the expression
V̂ = −
∫
drA(r, t)j(r, t),
where A(r, t) is a potential with harmonic time dependence, i.e.,
A(r, t) =
1
2
∑
k
{
Akωe
ikr−iωt +A∗
kωe
−ikr+iωt
}
. (32)
The external perturbation (32) gives rise to a current j(r, t) with nonzero mean value
proportional to the potential A(r, t). The linear relation for these quantities may be
written as
ji(k, ω) = αij(ω,k)Aj(k, ω), (33)
where αij(ω,k) are macroscopic coefficients which determine the dissipative proper-
ties of the system.
The mean energy absorbed by the system, on the one hand is expressed in terms
of the correlation functions (27) and (29), and on the other hand it is determined by
the dissipative linear response. We equate the relevant expressions and thus obtain
the basic relation of the theory of electromagnetic fluctuations, i.e.,
1
h̄
{
〈jijj〉
∗
−k−ω
− 〈jijj〉kω
}
= i
{
α∗
ij(ω,k)− αji(ω,k)
}
. (34)
For an isotropic medium, the coefficients αij(ω,k) are symmetric with respect to
the indices i and j,
αij(ω,k) = αji(ω,k).
In this case the difference of spectral distributions of time-reversed electromagnetic
fluctuations is directly determined by the imaginary part of the dissipative coefficient
αij(ω,k). We have
1
2h̄
{
〈jijj〉
∗
−k−ω
− 〈jijj〉kω
}
= α′′
ij(ω,k). (35)
Making use of (30) and (31), we obtain from (34) or (35) the fluctuation-dissipation
relation for the symmetrized correlation function for the case of equilibrium system
[6].
For classical systems (in the limiting case h̄ → 0), the relation of fluctuation
irreversibility (34) reduces to the form
−ω
∂
∂E
〈jijj〉kω = i
{
α∗
ij(ω,k)− αji(ω,k)
}
, (36)
where ∂
∂E
〈jijj〉kω is the correlation function (27) averaged over the derivative of
the distribution function. The relevant Kubo relation or the inverse fluctuation-
dissipation relation follow immediately from (34) or (36).
283
A.G.Sitenko
References
1. Landau L.D., Lifshitz E.M. Electrodynamics of Continuous Media. Moscow, GITTL,
1957 (in Russian).
2. Landau L.D., Lifshitz E.M. Statistical Physics. Pergamon Press, Oxford, 1969.
3. Kubo R. Statistical-mechanical theory of irreversible processes. // Journ. Phys. Society
of Japan, 1957, vol. 12, p. 570–586.
4. Callen H., Welton T. Irreversibility and generalized noise. // Phys. Rev., 1951, vol. 83,
p. 34–46.
5. Sitenko A.G. Fluctuation-dissipative ratio for non-equilibrium systems. // Ukr. Journ.
Phys., 1966, vol. 11, p. 1161–1166.
6. Sitenko A.G. Electromagnetic Fluctuations in Plasma. Academic Press, New York,
1967.
Про незворотність флюктуацій у часі
О.Г.Ситенко
Інститут теоретичної фізики ім. М.М. Боголюбова НАН України,
03143 Київ, вул. Метрологічна, 14б
Отримано 6 березня 2000 р.
Обговорюється взаємозв’язок між класичними та квантовими флюк-
туаціями. Вказано на незворотність (навіть у випадку класичних си-
стем) флюктуацій за наявності дисипації енергії. Одержано аналог
співвідношення Кубо для класичних флюктуацій. Розглянуто флюкту-
ації як у рівноважних, так і в нерівноважних (але стаціонарних) станах
системи.
Ключові слова: класичні і квантові флюктуації, незворотність
флюктуацій, формула Кубо
PACS: 05.40, 42.50.L, 05.70.L, 52.25.G
284
|
| id | nasplib_isofts_kiev_ua-123456789-121024 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T18:10:25Z |
| publishDate | 2000 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Sitenko, A.G. 2017-06-13T13:10:50Z 2017-06-13T13:10:50Z 2000 On the irreversibility of fluctuations / A.G. Sitenko // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 277-284. — Бібліогр.: 6 назв. — англ. 1607-324X DOI:10.5488/CMP.3.2.277 PACS: 05.40, 42.50.L, 05.70.L, 52.25.G https://nasplib.isofts.kiev.ua/handle/123456789/121024 Interrelationship of classical and quantum fluctuations is discussed. With
 energy dissipation being taken into account, fluctuations are shown to be
 irreversible even in the case of classical systems. An analogy to Kubo relation for classical fluctuations is derived. Fluctuations in both equilibrium and
 nonequilibrium (though stationary) states of the system are considered. Обговорюється взаємозв’язок між класичними та квантовими флюктуаціями. Вказано на незворотність (навіть у випадку класичних систем) флюктуацій за наявності дисипації енергії. Одержано аналог
 співвідношення Кубо для класичних флюктуацій. Розглянуто флюктуації як у рівноважних, так і в нерівноважних (але стаціонарних) станах
 системи. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the irreversibility of fluctuations Про незворотність флюктуацій у часі Article published earlier |
| spellingShingle | On the irreversibility of fluctuations Sitenko, A.G. |
| title | On the irreversibility of fluctuations |
| title_alt | Про незворотність флюктуацій у часі |
| title_full | On the irreversibility of fluctuations |
| title_fullStr | On the irreversibility of fluctuations |
| title_full_unstemmed | On the irreversibility of fluctuations |
| title_short | On the irreversibility of fluctuations |
| title_sort | on the irreversibility of fluctuations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/121024 |
| work_keys_str_mv | AT sitenkoag ontheirreversibilityoffluctuations AT sitenkoag pronezvorotnístʹflûktuacíiučasí |