Kinetic Derivation of The Hydrodynamic Equations For Capillary Fluids
Kinetic Derivation of The Hydrodynamic Equations For Capillary Fluids
Kinetic Derivation of The Hydrodynamic Equations For Capillary Fluids
Based on the generalized kinetic equation for the one-particle distribution function with a small source, the
transition from the kinetic to the hydrodynamic description of many-particle systems is performed. The basic
feature of this interesting technique to obtain the hydrodynamic limit is that the latter has been partially
incorporated into the kinetic equation itself. The hydrodynamic equations for capillary fluids are derived from
the characteristic function for the local moments of the distribution function. Fick’s law appears as a conse-
quence of the transformation law for the hydrodynamic quantities under time inversion.
冋 册
considerations, it appears interesting to explore the possibil-
V ␦共%F兲 ity of a derivation from kinetic theory of the general hydro-
+ 共V · 兲V = − , 共2兲 dynamic picture thus discussed. The analysis performed in
t ␦% this paper can be outlined as follows: starting from the equa-
where F共% , ␣兲 is a function of the density % and of ␣ tion for the one-particle distribution function, we consider a
= 共1 / 2兲兩 ⵜ 兩2 and V is the current velocity [1]. This formula- stochastic contribution from an additional collision term
tion of the van der Waals theory was originally due to (small source). To estimate this term, we use the Kramers-
Korteweg [2], who proposed a continuum mechanical model Moyal expansion truncated at second order (diffusion pro-
in which the Cauchy stress tensor apart from the standard cesses). The final step constituting the hydrodynamic mar-
Cauchy-Poisson term contains an additional term defined as ginalization is performed in the third section. Taking into
account a special class of diffusion processes, the so-called
T = 共− p + ␣ⵜ2%共x;t兲 + 兩 %共x;t兲兩2兲1 + ␦ %共x;t兲 Nelson diffusion [11,8], this step recovers as expected the
丢 %共x;t兲 + ␥共 丢 兲%共x;t兲, 共3兲 standard time-reversal invariance of hydrodynamic equa-
tions.
where 1 is the unit tensor. As already mentioned by Dunn
and Serrin [3], the modern terminology concerning the
Korteweg model refers to elastic materials of grade n, where II. GENERAL FRAMEWORK
the particular case of n = 3 has been well studied in recent
years [4]. The starting point of our analysis is the equation for the
Equations (1) and (2) have been linked recently [5,6] to a microscopic phase space density NM 共x , p ; t兲
nonlinear Schrödinger equation and its hydrodynamics coun-
terpart, i.e., nonlinear Madelung fluid [7,8]. This link be-
N M 1
tween capillarity and the Schrödinger equation can shed + · 共pNM 兲 + ជ p · 关F M 共x,p;t兲NM 兴 = 0, 共4兲
more light onto the so-called quantumlike approach to many- t m
particle systems such as beams in particle accelerators and
beam-plasma systems. The standard procedure in this direc- for a system consisting of N particles, which occupies vol-
tion is to approximate the physical systems characterized by ume V in the configuration space. Here x and p are the co-
an overall interaction with a suitable mean field theory. To ordinates and the canonically conjugate momenta, m is the
particle mass, and F M 共x , p ; t兲 is the microscopic force, which
apart from the external force includes a part specifying the
*Electronic address: demartino@sa.infn.it type of interaction between particles. Suppose that at some
†
Electronic address: rosfal@sa.infn.it initial time t0 the microscopic phase space density is known
‡
Electronic address: giuliana.lauro@unina2.it to be N M0共x , p ; t0兲. Then, the formal solution of Eq. (4) for
§
Electronic address: tzenov@sa.infn.it arbitrary time t can be written as
1 共G兲 T
exp关− 共z − 具z典x,t 兲
冑兩det Ĉ兩
where for simplicity the explicit dependence on the momen- G共z;t兩x兲 =
tum variables p has been suppressed. To take into account 3/2
the initial preparation of the system, one has to displace the 共G兲
⫻Ĉ−1共x;t兲共z − 具z典x,t 兲兴. 共14兲
initial time t0 at −⬁ and perform an average over the past
history of the system. Then Eq. (4) becomes [9] The quantities
N M 1 1 共G兲 共G兲
+ · 共pNM 兲 + ជ p · 关F M 共x,p;t兲N M 兴 = 共Ñ M − NM 兲. 具zk典x,t , 具zkzl典x,t , 共15兲
t m
共7兲 are the first and the second moment of z at the instant of time
t + , provided that z measured at the instant t equals x [i.e.,
Since the collision time is supposed to be much smaller than
z共t兲 = x]. In addition, Ĉ共x ; t兲 is the covariance matrix defined
the time , the standard collision integral which appears in
as
kinetic theory can be dropped and the kinetic equation for
the one-particle distribution function f共x , p ; t兲 can be written 共G兲
Ckl共x,t兲 = 2关具zkzl典x,t 共G兲
− 具zk典x,t 共G兲
具zl典x,t 兴. 共16兲
as
The generalized kinetic equation (8) has a form analogous to
f p 1
+ · f + F共x,p;t兲 · ជ p f = 共f̃ − f兲. 共8兲 the Bhatnagar-Gross-Krook (BGK) equation, widely used in
t m the kinetic theory of gases [10]. There is, however, an im-
The right-hand-side of Eq. (8) is regarded as a “collision portant conceptual difference between the two equations. In
integral” and using the Kramers-Moyal expansion, it can be the BGK equation the function f̃ should be replaced by the
expressed as equilibrium distribution function f 0 describing the global
⬁ l
equilibrium and the characteristic time should be replaced
1 共− 1兲l
兺 兺
by the corresponding relaxation time. The smoothed distri-
共f̃ − f兲 =
l=1 n1,n2,. . .,nk=0 n 1 ! n 2 ! ¯ n k! bution function in Eq. (8) characterizes a local quasiequilib-
n1+n2+¯+nk=l
rium state within the smallest unit cell of continuous me-
dium, while is the corresponding time scale.
l
⫻ 关Dn共l兲n ¯n 共x;t兲f兴, 共9兲
xn11 ¯ xnk k 1 2 k
III. HYDRODYNAMIC APPROXIMATION
where Rather than following the standard approach in deriving
冕
1 the hydrodynamic picture, we introduce the characteristic
Dn共l兲n 共x;t兲 = d3z⌬zn11⌬zn22 ¯ ⌬znk kG共z;t兩x兲 function
1 2¯nk
1
共G兲
= 具⌬zn11⌬zn22 ¯ ⌬znk k典x,t , 共10兲 G共x,w;t兲 = 冕 d3pf共x,p;t兲e−iw·p , 共17兲
with ⌬z = z − x. As a first very interesting step, we consider instead. It is straightforward to verify that G satisfies the
the diffusion approximation following equation:
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sented according to
lG
具pn11 pn22 ¯ pnk k典 = il , 共19兲 % 1
wn11wn22 ¯ wnk k − · 关%共− V共−兲 + A兲兴 = − ⵜkⵜl共Bkl%兲. 共29兲
w=0
t 2
where n1 + n2 + ¯ + nk = l. The well-known hydrodynamic
quantities, such as the mass density %, the mean velocity V共+兲 Summing up and subtracting Eqs. (26) and (27), we obtain
the continuity equation
of a fluid element, and the hydrodynamic stress tensor ⌸kl
can be defined as %
+ · 共%V兲 = 0, 共30兲
%共x;t兲 = mnG共x,0;t兲 = mn 冕 3
d pf共x,p;t兲, 共20兲
and the Fick’s law
t
冏 冏 冕
Here
n 2G n
⌸kl共x;t兲 = − = 3
d ppk pl f共x,p;t兲, 1 1
m w k w l w=0 m V = 共V共+兲 + V共−兲兲, U = 共V共+兲 − V共−兲兲, 共32兲
共22兲 2 2
Here, n = limN,V→⬁共N / V兲 implies the thermodynamic limit. are the current and the osmotic velocity, respectively. It is
Defining also the deviation from the mean velocity as worthwhile to mention that since the mean velocity of a fluid
element is a generic function of time t, it can be split into
mc共+兲 = p − mV共+兲 , 共23兲 odd and even parts. Note that from Eq. (32) it follows that
V共+兲 = V + U, where V is the odd part, while U is the even
and using the evident relation
part. Equation (27) for the balance of momentum can be
冕 d3pc共+兲共x,p;t兲f共x,p;t兲 = 0,
written alternatively as
V共+兲k Fk 1
we can represent the stress tensor ⌸mn according to the rela- + V共−兲lⵜlV共+兲k = + AlⵜlV共+兲k − ⵜlPkl
t m %
tion
Bln
⌸mn = %V共+兲mV共+兲n + Pmn . 共24兲 + ⵜlⵜnV共+兲k . 共33兲
2
Here
After performing a time inversion in Eq. (33), we obtain
% Vk Fk 1
共%V共+兲k兲 + ⵜl共%V共+兲kV共+兲l兲 = Fk − · 共A%V共+兲k兲 − ⵜlPkl + V lⵜ lV k = + AlⵜlUk − ⵜlP̄lk + UlⵜlUk
t m t m %
1 Bln
+ ⵜlⵜn共Bln%V共+兲k兲. 共27兲 + ⵜ lⵜ nU k , 共35兲
2 2
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the approach is that Fick’s law emerges naturally from the
V 2 ⵜ2冑% transformation properties of the hydrodynamic quantities un-
+ 共V · 兲V = − ␣ ln % − , 共39兲
t 2 冑% der time inversion. The osmotic velocity is uniquely speci-
fied by the first two infinitesimal moments of the smoothing
where ␣ = 3kBT / m. Thus the hydrodynamic equations de- function and in a sense is a measure of the irreversibility.
scribing a free capillary fluid have been recovered. The main result of the analysis performed in this paper,
In the case, where an external force is applied, the the hydrodynamic equations for free capillary fluids, has
Korteweg stress tensor contains an additional term propor- been derived from kinetic theory. If an external force is
tional to the drift coefficient A. On the other hand, from the present, the Korteweg stress tensor has to be modified ac-
principle of detailed balance, it follows that the drift coeffi- cordingly. An additional term proportional to the drift coef-
cient is proportional to the external force. The physical im- ficient emerges implying a coupling between the external
plication of the latter is that the additional term in the Ko- field and the mean field of purely hydrodynamical origin.
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[3] J. E. Dunn and J. B. Serrin, Arch. Ration. Mech. Anal. 88, 95 Princeton, NJ, 1985).
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