Hydrodynamic states of phonons in insulators
The Chapman-Enskog method is generalized for accounting the effect of kinetic modes on hydrodynamic evolution. Hydrodynamic states of phonon system of insulators have been studied in a small drift velocity approximation. For simplicity, the investigation was carried out for crystals of the cubic cla...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1203052017-06-12T03:05:32Z Hydrodynamic states of phonons in insulators Sokolovsky, S.A. The Chapman-Enskog method is generalized for accounting the effect of kinetic modes on hydrodynamic evolution. Hydrodynamic states of phonon system of insulators have been studied in a small drift velocity approximation. For simplicity, the investigation was carried out for crystals of the cubic class symmetry. It has been found that in phonon hydrodynamics, local equilibrium is violated even in the approximation linear in velocity. This is due to the absence of phonon momentum conservation law that leads to a drift velocity relaxation. Phonon hydrodynamic equations which take dissipative processes into account have been obtained. The results were compared with the standard theory based on the local equilibrium validity. Integral equations have been obtained for calculating the objects of the theory (including viscosity and heat conductivity). It has been shown that in low temperature limit, these equations are solvable by iterations. Steady states of the system have been considered and an expression for steady state heat conductivity has been obtained. It coincides with the famous result by Akhiezer in the leading low temperature approximation. It has been established that temperature distribution in the steady state of insulator satisfies a condition of heat source absence. Метод Чепмена-Енскога узагальнено для врахування впливу кiнетичних мод системи на гiдродинамiчну еволюцiю. У наближеннi малої дрейфової швидкостi вивчено гiдродинамiчнi стани фононної пiдсистеми дiелектрикa. Для спрощення, дослiдження проведено для кристалiв кубiчних класiв симетрiї. Встановлено, що у фононнiй гiдродинамiцi локальна рiвновага порушується навiть у лiнiйному наближеннi за швидкiстю. Це є наслiдком вiдсутностi закону збереження iмпульсу, що веде до релаксацiї дрейфової швидкостi. Отримано рiвняння фононної гiдродинамiки з урахуванням дисипативних процесiв. Результати порiвняно зi стандартною теорiєю, яка базується на наявностi локальної рiвноваги. Отримано iнтегральнi рiвняння для розрахунку об’єктiв теорiї (включаючи в’язкiсть та теплопровiднiсть). Показано, що у границi низьких температур цi рiвняння розв’язуються iтерацiями. Розглянуто стацiонарнi стани системи i отримано вираз для її стацiонарної теплопровiдностi. Показано, що вона у низькотемпературнiй границi спiвпадає з вiдомим результатом Ахiєзера. Встановлено, що розподiл температур у стацiонарному станi дiелектрика задовольняє умову вiдсутностi джерела тепла. 2012 Article Hydrodynamic states of phonons in insulators / S.A. Sokolovsky // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43007:1-10. — Бібліогр.: 13 назв. — англ. arXiv:1212.6140 PACS: 05.20.Dd, 51.10.+y, 63.20.-e, 63.20.Kr, 66.90.+r, 78.66.Nk. DOI:10.5488/CMP.15.43007 http://dspace.nbuv.gov.ua/handle/123456789/120305 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The Chapman-Enskog method is generalized for accounting the effect of kinetic modes on hydrodynamic evolution. Hydrodynamic states of phonon system of insulators have been studied in a small drift velocity approximation. For simplicity, the investigation was carried out for crystals of the cubic class symmetry. It has been found that in phonon hydrodynamics, local equilibrium is violated even in the approximation linear in velocity. This is due to the absence of phonon momentum conservation law that leads to a drift velocity relaxation. Phonon hydrodynamic equations which take dissipative processes into account have been obtained. The results were compared with the standard theory based on the local equilibrium validity. Integral equations have been obtained for calculating the objects of the theory (including viscosity and heat conductivity). It has been shown that in low temperature limit, these equations are solvable by iterations. Steady states of the system have been considered and an expression for steady state heat conductivity has been obtained. It coincides with the famous result by Akhiezer in the leading low temperature approximation. It has been established that temperature distribution in the steady state of insulator satisfies a condition of heat source absence. |
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Article |
| author |
Sokolovsky, S.A. |
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Sokolovsky, S.A. Hydrodynamic states of phonons in insulators Condensed Matter Physics |
| author_facet |
Sokolovsky, S.A. |
| author_sort |
Sokolovsky, S.A. |
| title |
Hydrodynamic states of phonons in insulators |
| title_short |
Hydrodynamic states of phonons in insulators |
| title_full |
Hydrodynamic states of phonons in insulators |
| title_fullStr |
Hydrodynamic states of phonons in insulators |
| title_full_unstemmed |
Hydrodynamic states of phonons in insulators |
| title_sort |
hydrodynamic states of phonons in insulators |
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Інститут фізики конденсованих систем НАН України |
| publishDate |
2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/120305 |
| citation_txt |
Hydrodynamic states of phonons in insulators / S.A. Sokolovsky // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43007:1-10. — Бібліогр.: 13 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT sokolovskysa hydrodynamicstatesofphononsininsulators |
| first_indexed |
2025-07-08T17:37:43Z |
| last_indexed |
2025-07-08T17:37:43Z |
| _version_ |
1837101235578077184 |
| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 4, 43007: 1–10
DOI: 10.5488/CMP.15.43007
http://www.icmp.lviv.ua/journal
Hydrodynamic states of phonons in insulators
S.A. Sokolovsky
Prydniprovs’ka State Academy of Civil Engineering and Architecture
Received July 3, 2012, in final form October 1, 2012
The Chapman-Enskog method is generalized for accounting the effect of kinetic modes on hydrodynamic evolu-
tion. Hydrodynamic states of phonon system of insulators have been studied in a small drift velocity approxima-
tion. For simplicity, the investigation was carried out for crystals of the cubic class symmetry. It has been found
that in phonon hydrodynamics, local equilibrium is violated even in the approximation linear in velocity. This
is due to the absence of phonon momentum conservation law that leads to a drift velocity relaxation. Phonon
hydrodynamic equations which take dissipative processes into account have been obtained. The results were
compared with the standard theory based on the local equilibrium validity. Integral equations have been ob-
tained for calculating the objects of the theory (including viscosity and heat conductivity). It has been shown that
in low temperature limit, these equations are solvable by iterations. Steady states of the system have been con-
sidered and an expression for steady state heat conductivity has been obtained. It coincides with the famous
result by Akhiezer in the leading low temperature approximation. It has been established that temperature
distribution in the steady state of insulator satisfies a condition of heat source absence.
Key words: phonons of insulator, Umklapp processes, relaxation degrees of freedom, local equilibrium, the
Chapman-Enskog method, phonon hydrodynamics, small drift velocity, low temperatures, steady states
PACS: 05.20.Dd, 51.10.+y, 63.20.-e, 63.20.Kr, 66.90.+r, 78.66.Nk.
1. Introduction
A lot of papers are devoted to investigations of the phonon system of insulators. We mean hydrody-
namic states described by densities of additive conserved values or their functions (for a system with
broken symmetry, order parameters are added). The description of hydrodynamic processes in insula-
tors is still more complicated due to the presence of Umklapp processes in which the momentum is not
conserved. A pioneering research into the role of such processes in the theory of thermal conductivity of
insulators belongs to Peierls [1]. Significant contribution to the study of the problem has been made by
Akhiezer [2] who calculated the thermal conductivity of dielectric crystals in steady states at low tempe-
ratures.
In phonon hydrodynamics, local temperature T (x, t) and drift velocity un(x, t) introduced with the
help of the local equilibrium distribution, are used instead of the densities of energy ε(x, t) and mo-
mentum πn (x, t) (see, for example, books [3, 4] and reviews [5–7] containing the standard theory). The
mentioned works substantiated that the velocity decays with time (slowly at low temperatures) but this
relaxation process is not considered to be a manifestation of the kinetic mode of a system. During the
recent years, great attention has been paid to the investigation of the effect of kinetic modes on the sys-
tem dynamics. Especially the role of the Lviv school on statistical physics (Mryglod, Bryk, Tokarchuk,
Omelyan, et al.) should be mentioned. See in this regard, for example, the paper [8] devoted to the effect
of relaxation on hydrodynamic processes in a two-component system, and the review [9] on the general
theory. Similar studies of dielectrics are unknown for the author and it was the motive of the investiga-
tion. In the investigation we observed that the local equilibrium distribution at un (x, t), 0 does not give
the leading approximation for the nonequilibrium distribution function. Similar result concerning the
violation of the local equilibrium was obtained in our paper [10] on the polaron kinetics due to the pres-
ence of velocity and temperature relaxation in the electron-phonon system. Such a result has not been
obtained in [8] since the dynamics of the system was studied in terms of component energies.
© S.A. Sokolovsky, 2012 43007-1
http://dx.doi.org/10.5488/CMP.15.43007
http://www.icmp.lviv.ua/journal
S.A. Sokolovsky
Researchers usually put the Chapman-Enskog method in the basis of hydrodynamic state considera-
tion. In the general formulation of this method, the particle distribution function fαp (x, t) (α is the num-
ber of species of particles, pn is momentum) is considered to be a functional fαp (x,ξ(t)) of parameters
ξµ(x, t) describing the hydrodynamic state (see, for example, [11]). Such state is observed after the time
τ0 that depends on the initial state of the system and the problem to be solved
fαp (x, t)−−−−−−→
t≫τ0
fαp (x,ξ(t)). (1)
Relation (1) is referred to as the functional hypothesis and is a natural assumption allowing one to obtain
a closed set of equations for parameters ξµ(x, t) and to express all the observed values through these
parameters. In the known paper [2], Akhiezer actually made the same assumption concerning the struc-
ture of the nonequilibrium phonon distribution function in his consideration of stationary states of the
phonon system of insulators.
A general study of hydrodynamic states of the phonon system of dielectric crystal is conducted in the
standard theory [3–7] using the Chapman-Enskog method. Herewith, the coincidence of the distribution
function fαp (x,ξ(t)) in the zero approximation in the gradients of hydrodynamic variables f(0)
αp (x,ξ(t))
and the Planck distribution with the drift velocity un (x, t) and temperature T (x, t)
nαp (ξ(x, t)), nαp (ξ) ≡
[
e(εαp−pn un)/T
−1
]−1
(2)
is substantiated [ξµ(x, t): ξ0(x, t) ≡ T (x, t), ξn(x, t) ≡ un (x, t)]. The relation f(0)
αp (x,ξ) = nαp (ξ(x)) means
that the local equilibrium is valid for the system. At low temperatures, this result is grounded with great
accuracy by the fact that the number of the Umklapp collisions is small compared to the number of the
normal ones. In the present paper, the zero approximation in the gradients f(0)
αp (x,ξ) for the distribution
function fαp (x,ξ) is calculated not only for low temperatures. This is done with the appropriate gener-
alization of the Chapman-Enskog method and, in the case of small drift velocity un , it is shown that
f(0)
αp (x,ξ), nαp (ξ(x)) but f(0)
αp (x,ξ)|u=0 =
o
nαp (ξ(x)) (
o
nαp (ξ) ≡ [eεαp /T −1]−1 is the local Planck distribution
without velocity). In all the mentioned works [3–7], the drift velocity of phonons un is also considered to
be a small value but our consideration yields nonlinear hydrodynamics for a small drift velocity [it con-
tains nonlinear terms in un(x, t)]. The established violation of the local equilibrium has principal physical
meaning although at low temperatures it is small.
In the present paper, insulators of a cubic symmetry classes are considered. Even in this case, some
crystal quantities have a complicated structure (for example, tensors of the fourth rank). Using a small
drift velocity approximation, let us avoid a complicated symmetry research. Among other things, terms
in estimations have a complicated tensor structure in all our series expansions in the velocity.
In some papers, devoted to phonon hydrodynamics issues, crystalline solids are considered in an
isotropic model [6, 7, 12]. This leads to some simplification of the equations obtained in the present paper
that will be analyzed elsewhere. A version of a τ-approximation for phonon collision integral proposed
in [12] allows one to simplify the equations of the theory too. Such an approximation is not used in the
present paper. Moreover, a τ-approximation approach can be improved by the method of orthogonal
polynomials developed in [7].
The outline of the paper is as follows. In section 2, the basic equations of the theory are presented. In
section 3, the basic equations of the theory are solved in a double perturbation theory in small gradients
of hydrodynamic variables and small velocity. As a result, expressions for the fluxes of energy and mo-
mentum of the phonons with allowance for dissipative processes are obtained and equations of phonon
hydrodynamics are derived. Section 4 discusses the relation of the results with the standard phonon hy-
drodynamics based on the validity of the local equilibrium. In section 5, an iteration procedure is built for
solving integral equations of the theory in the approximation of low temperatures. In the final section 6,
the steady states of the system are considered.
43007-2
Hydrodynamics of phonons
2. Basic equations of the theory
A kinetic equation for phonons is put
∂fαp (x, t)
∂t
=−
∂εαp
∂pn
∂fαp (x, t)
∂xn
+ Iαp (f(x, t)) (3)
in the basis of the consideration. The main contribution to the collision integral corresponds to three-
phonon processes
Iαp (f) =
∑
α1α2α3
∫
B
d3p1d3p2d3p3|Φ(12,3)|2δ(ε1 +ε2 −ε3)
∑
n
δ(p1 +p2 −p3 −ħbn )
×
[
δαα3δ(p −p3)−δαα1δ(p −p1)−δαα2δ(p −p2)
][
f1f2(1+ f3)− (1+ f1)(1+ f2)f3
]
(4)
[abbreviations of the type εαi pi
≡ εi , fαi pi
≡ fi , Φ(α1p1;α2p2,α3p3) ≡Φ(12,3) are used; α, pn are polar-
ization and quasi-momentum of a phonon; the integrals are taken over the basic cell of a reciprocal lattice
B ; for simplicity, the term “momentum” is used instead of “quasi-momentum”]. The collision integral is
divided into the contribution of normal processes and Umklapp ones Iαp (f) = I N
αp (f)+ IU
αp (f) in the usual
manner [in (4) summand with n = 0 gives I N
αp (f)]. The hydrodynamic equations are a consequence of the
phonon energy conservation law and the law of system momentum change
∂ε(x, t)
∂t
=−
∂ql (x, t)
∂xl
,
∂πn(x, t)
∂t
=−
∂tnl (x, t)
∂xl
+Rn (x, t). (5)
Densities of energy and momentum ε(x, t), πn(x, t) of the phonon system, corresponding flux densities
ql (x, t), tnl (x, t), and that of a frictional force Rn(x, t) are given by formulas
ε(x, t) ≡
∑
α
∫
B
dτpεαp fαp (x, t), πn(x, t) ≡
∑
α
∫
B
dτp pnfαp (x, t),
ql (x, t) ≡
∑
α
∫
B
dτpεαp
∂εαp
∂pl
fαp (x, t), tnl (x, t) ≡
∑
α
∫
B
dτp pn
∂εαp
∂pl
fαp (x, t),
Rn (x, t) ≡
∑
α
∫
B
dτp pn IU
αp (f(x, t)),
[
dτp ≡
1
(2πħ)3
d3p
]
. (6)
Temperature T (x, t) and phonon drift velocity un (x, t) [variables ξµ(x, t)] are used in this paper in-
stead of densities ε(x, t), πn(x, t) as independent variables in hydrodynamics. In the standard theory
[3–7], the densities are expressed through ξµ(x, t) with the definitions
ε(ξ) =
∑
α
∫
B
dτpεαp nαp (ξ), πn(ξ) =
∑
α
∫
B
dτp pnnαp (ξ) (7)
where nαp (ξ) is the Planck distribution (2). Below, we use the details connected with these relations only
while comparing the presented theory with the standard theory according to the above remarks of ours
on the violation of the local equilibrium for un , 0.
Investigating fundamental issues of phonon kinetics, we restrict our consideration to the crystals of
cubic symmetry classes O, Oh , Th for simplicity. In the present paper, the phonon drift velocity un is
assumed to be small compared to the second sound velocity (let λ be a corresponding small parameter).
In this case, the energy and momentum densities of the phonon gas can be expanded in powers of un as
it follows
ε(ξ) ≡ ε(T,u) =
o
ε(T )+a(T )u2
+O
(
λ4
)
, πn(ξ) ≡πn(T,u) =σ(T )un +O
(
λ3
)
(8)
where functions
o
ε(T ), σ(T ), a(T ) are considered to be known. In the standard theory [3–7], they are
calculated from (7). The terms given in (8) by estimations have a complicated tensor structure even for
43007-3
S.A. Sokolovsky
the mentioned symmetry classes. Their complete investigation can be made using the ideas developed,
for example, in [13].
Hydrodynamic equations for the variables ξµ(x, t) have the structure
∂ξµ(x, t)
∂t
= Mµ
(
x, f(ξ(t))
)
,
[
∂sξµ(x, t)
∂xn1 . . .∂xns
∼ g s , g ≪ 1
]
(9)
and, as usual, the gradients of these variables are small [in (9) g = l/L, where l is a mean free path of
particles, L is a characteristic size of inhomogeneities in the system]. Functions Mµ(x, f) can be expressed
through the distribution function fαp (x,ξ) with the help of relations (5), (6), and (8). According to (1) and
(9), the functional fαp (x,ξ) satisfies the integro-differential equation
∑
µ
∫
d3x′
δfαp (x,ξ)
δξµ(x′)
Mµ(x′, f(ξ)) =−
∂εαp
∂pn
∂fαp (x,ξ)
∂xn
+ Iαp (f(x,ξ)) (10)
and additional conditions
∑
α
∫
B
dτpεαp fαp (x,ξ) = ε(ξ(x)),
∑
α
∫
B
dτp pnfαp (x,ξ) =πn(ξ(x)) (11)
that define the phonon gas temperature and drift velocity together with (8).
3. Phonon hydrodynamics
Equations (10), (11) are solved by us in the double perturbation theory in gradients (the small param-
eter g ) and in the drift velocity (the small parameter λ)
fαp (x,ξ) = f(0)
αp + f(1)
αp +O
(
g 2
)
, f(s)
αp = f(s,0)
αp + f(s,1)
αp +O
(
g sλ2
)
; f(0,0)
αp =
o
nαp . (12)
The Planck distribution without velocity
o
nαp is the main contribution to the phonon distribution func-
tion. Our calculation is based on the estimations
T ∼λ0g 0, un ∼λ1g 0,
∂T
∂xn
∼ λ0g 1,
∂un
∂xl
∼ g 1λ1. (13)
The simplest and the most important contributions to the theory have the following structure
f(0,1)
αp =
o
nαp
(
1+
o
nαp
)
Aαn(p)un , f(0,2)
αp =
o
nαp
(
1+
o
nαp
)
Bαnl (p)unul ,
f
(1,0)
αp =
o
nαp
(
1+
o
nαp
)
Cαn(p)
∂T
∂xn
, f
(1,1)
αp =
o
nαp
(
1+
o
nαp
)
[
Dαnl (p)
∂un
∂xl
+Eαnl (p)
∂T
∂xn
ul
]
. (14)
These formulas contain functions Aαn(p), Cαn(p), Dαnl (p) that satisfy equations
ν
σ
Aαn(p)=
∑
α′
∫
B
d3p ′Kαα′
(
p, p ′
)
Aα′n(p ′), 〈pn Aαn(p)〉 = 3σ; (15)
Aαn(p)
1
σ
(
νT +
∂p
∂T
)
−
1
T 2
εαp
∂εαp
∂pn
=
∑
α′
∫
B
d3p ′Kαα′
(
p, p ′
)
Cα′n
(
p ′
)
, 〈pnCαn(p)〉 = 0; (16)
h
cT 2
εαpδnl +
ν
σ
Dαnl (p)−
∂εαp
∂pl
Aαn(p)=
∑
α′
∫
B
d3p ′Kαα′
(
p, p ′
)
Dα′nl
(
p ′
)
,
〈εαp Dαnn(p)〉 = 0. (17)
43007-4
Hydrodynamics of phonons
Here, heat capacity c, pressure p of the phonon gas and a special average 〈gαp〉
c =
∂
o
ε
∂T
=
〈ε2
αp〉
T 2
, p =
1
3
∫
dτp pn
∂εαp
∂pn
o
nαp ; 〈gαp〉 ≡
∑
α
∫
B
dτp
o
nαp
(
1+
o
nαp
)
gαp (18)
are introduced for an arbitrary function gαp . Equations for Bαnl (p) and Eαnl (p) are not written due to
their complexity. The kernel Kαα′(p, p ′) of the integral equations (15)–(17) is defined by relations
Iαp
(
o
n +δf
)
=
∑
α′
∫
d3p ′Mαα′(p, p ′)δfα′p′ +O
(
δf2
)
,
o
nαp
(
1+
o
nαp
)
Kαα′
(
p, p ′
)
=−Mαα′
(
p, p ′
) o
nα′p′
(
1+
o
nα′p′
)
. (19)
Equations (15)–(17) contain the values a, σ defined by (8) and depending on T . The meaning of vari-
ables ν, νT , h (they are also functions of the temperature) in these equations becomes clear from the
following expressions for the fluxes of momentum, energy, and frictional force density
tn l = pδnl +µnl ,ms um us −ηnl ,ms
∂um
∂xs
−αnl ,ms us
∂T
∂xm
+O
(
g 0λ3, g 1λ2
)
,
ql = hul −κ
∂T
∂xl
+O
(
g 0λ3, g 1λ2
)
, Rl =−νul −νT
∂T
∂xl
+R(2,1)
l
+O
(
g 0λ3, g 1λ2, g 2λ2
)
. (20)
These expressions are obtained in the perturbation theory from the definitions (6) taking into account the
contributions (14) to the phonon distribution function fαp (x,ξ). The term R(2,1)
l
is included in (20) because
it gives contribution of the same order as a viscous momentum flux t
(1,1)
nl
in hydrodynamic equation for
the velocity [see (24)]. Contributions f
(2,0)
αp , f
(2,1)
αp to the function fαp (x,ξ) are required in order to calculate
R(2,1)
l
(they will be analyzed in another paper). In (20) ν, νT are damping rates, κ is a thermal conductivity,
ηnl ,ms is a viscosity tensor, h is a coefficient of the drift energy transfer, µnl ,ms is a coefficient of the drift
momentum transfer, αnl ,ms is a coefficient of convective momentum transfer. Their values are given by
formulas
h =
1
3
〈
εαp
∂εαp
∂pn
Aαn(p)
〉
, κ=−
1
3
〈
εαp
∂εαp
∂pn
Cαn(p)
〉
,
ηnl ,ms = −
〈
pn
∂εαp
∂pl
Dαms (p)
〉
, µnl ,ms =
〈
pn
∂εαp
∂pl
Bαms (p)
〉
,
αnl ,ms = −
〈
pn
∂εαp
∂pl
Eαms(p)
〉
; ν=
1
3
{
pn , Aαn(p)
}U
, νT =
1
3
{
pn ,Cαn(p)
}U
. (21)
Here, damping rates are written in terms of a bilinear form defined by the relation
{
gαp ,hαp
}N ,U
= −
1
(2πħ)3
∑
αα′
∫
B
d3pd3p ′gαp M
N ,U
αα′
(
p, p ′
) o
nα′p′
(
1+
o
nα′p′
)
hα′p′ ,
{
gαp ,hαp
}
=
{
gαp ,hαp
}N
+
{
gαp ,hαp
}U
(22)
where M N
αα′(p, p ′), MU
αα′(p, p ′) are contributions to the kernel Mαα′(p, p ′) associated with the normal and
Umklapp processes, correspondingly. Forms of this type are very useful in the kinetic theory and have
the properties
{
gαp ,hαp
}N ,U
=
{
hαp , gαp
}N ,U
,
{
gαp , gαp
}N ,U
Ê 0 (23)
that can be proved through the usual way for arbitrary functions gαp , hαp (see, for example, [3]).
Expressions (21) for ν and νT follow from the integral equations (15), (16) taking into account the
additional condition for Aαn(p) from (15) and expression (18) for pressure p. Formula (21) for h follows
from equations (17) taking into account the definition of the heat capacity c (18) and the symmetry of the
43007-5
S.A. Sokolovsky
bilinear form {gαp ,hαp } (23). Therefore, equations (15)–(17) can be solved without taking into account the
expressions (21) for ν, νT and h.
The equations of phonon hydrodynamics (9) can be written in the form
c
∂T
∂t
=
{
−h
∂un
∂xn
}
−
∂h
∂T
∂T
∂xn
un +
2av
σ
un
[
un +
1
ν
(
νT +
∂p
∂T
)
∂T
∂xn
]
−
∂q (1,0)
n
∂xn
+O
(
g 0λ3, g 1λ2, g 2λ2
)
,
σ
∂un
∂t
=
{
−νun −
∂p
∂T
∂T
∂xn
}
−νT
∂T
∂xn
−
∂t (1,1)
nl
∂xl
+R(2,1)
n −
1
c
∂σ
∂T
un
∂q (1,0)
l
∂xl
+O
(
g 0λ3, g 1λ2, g 2λ3
)
(24)
with accounting relations (5), (6), (12), (14). Only the terms given in curly brackets were obtained, for
example, in book [3] that discusses the standard theory [in the standard theory, h and ν should be taken
from (28)]. Other terms in (24) are obtained in the present paper (the corresponding contributions of
the standard theory were not found in [3]). Dissipative fluxes q
(1,0)
n , t
(1,1)
nl
are given in (20). The obtained
equations of the phonon hydrodynamics (24) fully take into account all the dissipative processes.
The integral equation (15) is an equation for eigenfunction Aαn(p) and eigenvalue ν/σ. The solution of
this spectral problem describes the kinetic mode announced above. According to equation (24) the velocity
really decays if the ratio ν/σ is positive. Due to (23), the positivity of the coefficient ν/σ follows from the
relation
ν
σ
〈
Aαn(p)2
〉
=
{
Aαn(p), Aαn(p)
}
(25)
that can be derived from equation (15) and definitions (18) and (22). This result is obtained in the frame-
work of the considered theory which implies that the phonon drift velocity is small and has no obvious
restrictions for the temperature.
The considered hydrodynamic states occur when t ≫ τ0 where time τ0, introduced by the functional
hypothesis (1), should be much smaller than the drift velocity decay time τu [τu =σ/ν due to (24)].
4. Comparison with the standard phonon hydrodynamics
In the standard theory [3–7], the hydrodynamic distribution function in the zero order approximation
in gradients f(0)
αp (x,ξ) coincides with the local Planck distribution with velocity nαp (ξ(x)). The functions
Aαn(p), Bαnl (p) according to (2), (12), and (14) are given by expressions
Ao
αn(p)=
1
T
pn , Bo
αnl (p)=
1
2T 2
(
1+2
o
nαp
)
pn pl . (26)
These functions are not solutions of the integral equation (15) and the corresponding equation for Bαnl (p),
i.e. Ao
αn(p), Aαn(p), Bo
αnl
(p),Bαnl (p). The first inequality means that even in the linear approximation
in drift velocity, f(0)
αp (x,ξ) , nαp (ξ(x)) and, therefore, the local equilibrium is violated in the phonon hy-
drodynamics. The result Ao
αn(p) , Aαn(p), Bo
αnl
(p) , Bαnl (p) follows from the next section where it is
shown that expressions (26) give only the leading contribution to Aαn(p), Bαnl (p) in the low temperature
approximation.
In the standard theory, temperature and drift velocity are defined by formulas (7) giving the following
expressions for values σ and a in (8)
σo
=
1
3T
〈
p2
〉
, ao
=
1
6T 2
〈
εαp (1+2
o
nαp )p2
〉
(27)
(hereinafter we denote the value A of the standard theory by Ao).
Formulas (21) can be also obtained in the standard approach but not all corresponding contributions
to phonon hydrodynamic equations are discussed in [3–7]. Therefore, (21) and (26) lead to expressions
for the damping rates ν, νT and drift transfer coefficients h, µnl ,ms
νo
=
1
3T
{
pn , pn
}U
, νo
T =
1
3T
{
pn ,C o
νn
}U
, ho
=
1
3T
〈
εαp
∂εαp
∂pn
pn
〉
= T
∂p
∂T
=
o
ε+p,
43007-6
Hydrodynamics of phonons
µo
nl ,ms =
1
2T 2
〈
pn
∂εαp
∂pl
pm ps
(
1+2
o
nαp
)
〉
. (28)
Equations (16)–(17) for the functions Cαn(p) and Dαnl (p) are simplified by the first formula in (26).
Therefore, according to equation (16) and expression (21) the phonon thermal conductivity is given in
the standard theory by formula
κo
=
T 2
3
{
C o
αn(p),C o
αn(p)
}
. (29)
5. Solution of the integral equations of the theory at low temperatures
At low temperatures T ≪ TD, the following estimates of the kernels of integral equations (15)–(17) are
valid
K N
αα′
(
p, p ′
)
∼µ0, K U
αα′
(
p, p ′
)
∼µ1,
(
µ≡ e−TD/T
)
(30)
(see, for example [2, 3]) where TD is Debye temperature which is equal to the maximal energy of a
phonon.
The solutions of equations (15)–(17) for functions Aαn(p),Cαn(p), and Dαnl (p) and the corresponding
equation for Bαnl (p), damping rates ν,νT are found in the form
Aαn = A[0]
αn + A[1]
αn +O
(
µ2
)
, ν= ν[1]
+ν[2]
+O
(
µ3
)
; Bαnl = B [0]
αnl
+B [1]
αnl
+O
(
µ2
)
;
Cαn = C [0]
αn +C [1]
αn +O
(
µ2
)
, νT = ν[1]
T
+ν[2]
T
+O
(
µ3
)
; Dαnl = D[0]
αnl
+D[1]
αnl
+O
(
µ2
)
(31)
(A[s] is a contribution of the order µs to the quantity A).
It is easily to understand that the mentioned main equations of the developed theory (15)–(17) are sol-
vable by a simple iteration procedure. At each step, we obtain an integral equation of the type
gα(p)=
∑
α′
∫
B
d3p ′K N
αα′
(
p, p ′
)
hα′
(
p ′
)
(32)
for a function hα(p) with additional conditions that eliminate arbitrariness in hα(p) of the form cn pn +
c εαp [gα(p) is a known function]. Values ν,νT , and h in this equations and kinetic coefficients of the
system are calculated using the formulas (21). The mentioned arbitrariness is related to conservation
of energy and momentum in normal phonon processes. Further analysis of equations of the type (32)
requires additional information on the spectrum εαp of the phonons and amplitude of their interaction
Φ(α1p1;α2p2,α3p3). It should be also stressed that necessary solutions of equation (32) are vectors and
tensors of second rank depending on momentum. Even for crystals of the considered cubic symmetry
classes O, Oh , Th , they have a rather complicated structure.
Note also that at low temperatures, the long-wavelength acoustic phonons give the main contribu-
tion to thermodynamic and kinetic properties of a crystal (see, for example, [6]). This allows one to find
phonon spectrum εap and phonon interaction amplitudeΦ(α1p1;α2p2,α3p3) based on the elasticity the-
ory. This leads to some simplification of the above obtained equations that will be analyzed elsewhere
(the standard theory in this approach has been constructed in [6, 7]).
Let us present some results of our calculations in the main low temperature approximation:
A[0]
αn(p)=
3σ
〈
p2
〉 pn , ν[1]
=
σ
〈
p2
〉
{
pn , pn
}U
, h[0]
=
3σT 2
〈
p2
〉
∂p
∂T
;
ν[1]
T
=
1
3T
{
pn ,C [0]
αn(p)
}U
, κ[0]
=
T 2
3
{
C [0]
αn(p),C [0]
αn(p)
}N
,
η[0]
nl ,ms
=
〈
p2
〉
3σ
{
D[0]
αnl
(p),D[0]
αms(p)
}N
. (33)
Among other things, these formulas give positively defined expressions for the heat conductivity and
viscosity of the system. Some results of the main approximation coincide with the corresponding results
of the standard theory
A[0]
αn(p)= Ao
αn(p), B [0]
αnl
(p) = Bo
αnl (p);
43007-7
S.A. Sokolovsky
ν[1]
= νo , h[0]
= ho , µ[0]
nl ,ms
= µo
nl ,ms , (at σ=σo , a = ao). (34)
Functions C o
αn , Do
αn l
satisfy the integral equations (16), (17) with quantities Ao
αn ,σo ,ho ,νo ,νo
T taken from
(26)–(28) instead of Aαn ,σ,h,ν,νT . These equations can be solved at low temperatures in the perturba-
tion theory in µ that gives the following results
C [0]
αn(p) =C o[0]
αn (p), D[0]
αnl
(p)= Do[0]
αn l
(p);
κ[1]
T
=κo[1]
T
, κ[0]
=κo[0], η[0]
nl ,ms
= ηo[0]
nl ,ms
, (at σ=σo , a = ao) (35)
(notation like Ao[s] gives a contribution of the order µs to a quantity of the standard theory Ao). So, results
of the developed theory and the standard theory coincide with one another in the main low temperature
approximation (at σ=σo , a = ao , i.e., at the standard definition of the drift velocity and temperature).
The first two formulas (34) show that at low temperatures, the difference between the phonon dis-
tribution function in the zeroth approximation in gradients f(0)
αp (x,ξ) and the local Planck distribution
nαp (ξ(x)) is estimated by f(0)
αp (x,ξ) = nαp (ξ(x))+O(λ1µ1,λ2µ1,λ3). Therefore, at low temperatures T ≪ TD
and small drift velocity, the violation of the local equilibrium in the phonon hydrodynamics is exponentially
small. This result confirms the applicability of the standard theory at low temperatures.
6. The steady states of insulators
Investigation of steady states of the system is of great interest because their properties are easier to
be analyzed experimentally. The hydrodynamic equation for the drift velocity (24) should be solved in
the steady state with respect to the drift velocity un in the form of a series in temperature gradients that
gives
un =−
1
ν
(
νT +
∂p
∂T
)
∂T
∂xn
+O
(
g 3
)
. (36)
The accuracy of this result is limited by the accuracy estimation in (24). It should be noted that an equation
of the type (36) appears in the standard theory of steady processes in dielectrics too [see, for example, [3]
where (36) is obtained for T ≪ T0 without a term with νT that is exponentially small according to (31)].
Taking into account the result (36), the expression (20) for phonon energy flux yields
qn =−κ̃
∂T
∂xn
+O
(
g 3
)
, κ̃≡ κ+
h
ν
(
νT +
∂p
∂T
)
(37)
where we have introduced thermal conductivity in a steady state κ̃. The formula shows that in the ab-
sence of the Umklapp processes, conductivity κ̃ =∞ [according to (21), in this case ν = 0, νT = 0]. This
well-known result is not surprising since an isolated system does not have nonequilibrium steady states.
Only the presence of Umklapp processes that lead to a nonconservation of the phonon momentum of a di-
electric crystal makes steady states possible.
There is an interesting issue concerning the conditions of stationarity of the phonon temperature of
such a system. Substitution of the expression for the drift velocity in a steady state (36) into equation (24)
for temperature, yields a condition
∂
∂xn
(
κ̃
∂T
∂xn
)
= 0 (38)
(in the second order of the perturbation theory in gradients). The meaning of this result as the condition
of heat source absence in a steady state is clear. At the same time, the issue regarding the temperature
distribution in isolated insulators is not discussed in the literature, although the nature of steady states
in such systems is quite unusual.
Steady state thermal conductivity of the phonon system of an insulator κ̃ can be calculated at low
temperatures based on the results of the previous section that in the main approximation gives
κ̃= κ̃[−1]
+O
(
µ0
)
, κ̃[−1]
=
3T 2
{
pn , pn
}U
(
∂p
∂T
)2
. (39)
43007-8
Hydrodynamics of phonons
This result coincides with the result by Akhiezer [2] and does not depend on the definition of the drift veloc-
ity (8). The thermal conductivity κ̃[−1] is exponentially large which eliminates the problem of calculating
the corrections to it at T ≪ TD.
7. Conclusions
In this paper, the Chapman-Enskog method is generalized to take into account relaxation processes
(kinetic modes) in the hydrodynamic theory, i.e., the processes that can be present in a spatially homoge-
neous state of a system. On this basis:
• Nonlinear hydrodynamics of the phonon system has been built in the approximation of small
phonon drift velocity for insulators with cubic symmetry of the lattice .
• It is proved that a small phonon drift velocity decays.
• In the perturbation theory in the drift velocity it is found, that in the phonon hydrodynamics the
distribution function of phonons in the zeroth approximation in the gradients is different from the
Planck distribution with the velocity. At the same time, at zero drift velocity, these distributions
coincide. So, it is established that the local equilibrium in the phonon hydrodynamics is violated.
• It is shown, that at low temperatures the integral equations of the theory are solvable iteratively.
• It is shown, that at low temperatures and at small drift velocity, the local equilibrium in the hydro-
dynamic phonon system takes place with exponential accuracy.
• The obtained results can be applied to the analysis of hydrodynamic processes in the system of
phonons of the insulator at intermediate and high temperatures.
• The relation between usual hydrodynamic thermal conductivity and thermal conductivity of phonons
in the steady state has been established. It is shown that the Akhiezer expression gives the main
low temperature contribution to the thermal conductivity of an insulator in its steady states.
• The thermal conductivity of an insulator is impossible to be calculated without taking into account
the Umklapp processes because nonequilibrium steady states in a closed system do not exist with-
out them.
Acknowledgements
This work was supported in part by the State Foundation for Basic Research of Ukraine (project No.
25.2/102).
43007-9
S.A. Sokolovsky
References
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3. Lifshitz E.M., Pitaevskii L.P., Physical Kinetics. Pergamon Press, Oxford, 1981.
4. Gurevich V.L., Transport in Phonon Systems. North-Holland, Amsterdam, 1988.
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6. Akhiezer A., Alexin V., Khodusov V., Low Temp. Phys., 1994, 20, No. 12, 939; doi:10.1063/1.592848 .
7. Akhiezer A., Alexin V., Khodusov V., Low Temp. Phys.,1995, 21, No. 1, 1; doi:10.1063/1.593042.
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11. Akhiezer A.I., Peletminsky S.V., Methods of Statistical Physics. Pergamon Press, Oxford, 1981.
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13. Schouten J.A., Tensor Analysis for Physicists. Dover Publications, New York, 2011.
Гiдродинамiчнi стани фононiв дiелектрикiв
С.О. Соколовський
Приднiпровська державна академiя будiвництва та архiтектури
Метод Чепмена-Енскога узагальнено для врахування впливу кiнетичних мод системи на гiдродинамiчну
еволюцiю. У наближеннi малої дрейфової швидкостi вивчено гiдродинамiчнi стани фононної пiдсисте-
ми дiелектрикa. Для спрощення, дослiдження проведено для кристалiв кубiчних класiв симетрiї. Вста-
новлено, що у фононнiй гiдродинамiцi локальна рiвновага порушується навiть у лiнiйному наближеннi
за швидкiстю. Це є наслiдком вiдсутностi закону збереження iмпульсу, що веде до релаксацiї дрейфової
швидкостi. Отримано рiвняння фононної гiдродинамiки з урахуванням дисипативних процесiв. Резуль-
тати порiвняно зi стандартною теорiєю, яка базується на наявностi локальної рiвноваги. Отримано iнте-
гральнi рiвняння для розрахунку об’єктiв теорiї (включаючи в’язкiсть та теплопровiднiсть). Показано, що у
границi низьких температур цi рiвняння розв’язуються iтерацiями. Розглянуто стацiонарнi стани системи
i отримано вираз для її стацiонарної теплопровiдностi. Показано, що вона у низькотемпературнiй границi
спiвпадає з вiдомим результатом Ахiєзера. Встановлено, що розподiл температур у стацiонарному станi
дiелектрика задовольняє умову вiдсутностi джерела тепла.
Ключовi слова: фонони дiелектрика, процеси перекидання, релаксацiйнi ступенi вiльностi, локальна
рiвновага, метод Чепмена-Енскога, фононна гiдродинамiка, мала дрейфова швидкiсть, низькi
температури, стацiонарнi стани
43007-10
http://dx.doi.org/10.1002/andp.19293950803
http://dx.doi.org/10.1070/PU1968v011n02ABEH003815
http://dx.doi.org/10.1063/1.592848
http://dx.doi.org/10.1063/1.593042
http://dx.doi.org/10.1007/s11232-011-0093-z
http://dx.doi.org/10.1016/0003-4916(92)90089-5
Introduction
Basic equations of the theory
Phonon hydrodynamics
Comparison with the standard phonon hydrodynamics
Solution of the integral equations of the theory at low temperatures
The steady states of insulators
Conclusions
|