Abstract
The usual method in nonequilibrium statistical mechanics allows the derivation of the balance equations only for the collision invariants (mass, momentum, energy) and only by a complete knowledge of the microscopic structure. In this paper the existence of the balance equation for an arbitrary physical quantity is proved for any corpuscular system satisfying the local equilibrium assumption, if the microscopic components obey the classical mechanics principles and can be generated or destroyed as a result of some instantaneous processes. We discuss the fundamental equations of the continuum mechanics for mass and momentum.
Authors
Keywords
Nonequilibrium; balance equation; continuum mechanics
Cite this paper as
C. Vamoş, A. Georgescu, N. Suciu, I. Turcu (1996), Balance equations for physical systems with corpuscular structure, Physica A, 227, 81-92, doi: 10.1016/0378-4371(95)00373-8
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[7] E.G.D. Cohen, Physica A 194 (1993) 229.
[8] A. Sveshnikov and A. Tikhonov, The Theory of Functions of a Complex Variable (Mir, Moscow, 1978).
[9] C. Vamos, A. Georgescu and N. Suciu, St. Cerc. Mat. (1995), in press.
[10] A. Kolmogorov and S. Fomine, Elements de la Theorie des Fonctions et de Fanalyse Fonctionelle (Mir, Moscow, 1974).
[11] P. Glansdorff and 1. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley, London, 1971).
[12] A.I. Murdoch, Arch. Rational Mech. Anal. 88 (1985) 291.
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PHYSICA l!,,,,,
ELSEVIER Physica A 227 (1996) 81 92
Balance equations for physical systems with
corpuscular structure
C. Vamos a, A. Georgescu a, N. Suciu a, |. Turcu b'*
a Institute qfApplied Mathematics, Romanian Academy, PO Box 1-24, 70700 Bucharest 1, Romania
b Institute o['Isotopic and Molecular TechnoloKy', PO Box 700, 3400 Cluj-Napoca, Romania
Received 26 June 1995
Abstract
The usual method in nonequilibrium statistical mechanics allows the derivation of the
balance equations only for the collision invariants (mass. momentum, energy) and only by
a complete knowledge of the microscopic structure. In this paper the existence of the balance
equation for an arbitrary physical quantity is proved for any corpuscular system satisfying the
local equilibrium assumption, if the microscopic components obey the classical mechanics
principles and can be generated or destroyed as a result of some instantaneous processes. We
discuss the fundamental equations of the continuum mechanics for mass and momentum.
PACS: 03.40; 03.20; 05.20; 05,70
Kevwords: Nonequilibrium; Balance equation; Continuum mechanics
I. [nlroduction
The balance equations are postulated relations for fundamental physical quantities
(mass, momentum, energy, entropy, etc.) valid for all continuous media [1]. We take
over the differential expression of the balance equations from Ref. [2]. Let tp be an
additive physical quantity associated to a continuous medium. That is, there exists
a function 7~ of space (r) and time (t), called the volume density of qg such that, for any
volume V, the integral ~v tPdr represents the amount of ~P contained in V. The
differential form of the balance equation at a regular point (i.e. without shocks or
other discontinuities) is
3
8,~u + ~ ~%(4,~ + ~v~) - (p + s) -- 0,
(1)
*Corresponding author.
0378-4371..'96."$15.00 ~' 1996 Elsevier Science B.V. All rights reserved
SSDI 0378-4371(95)00373-8
82 c. Vamos et al. / Physica A 227 (1996) 81 92
where ~, is the temporal derivative, ~3,is the derivative with respect to the c~ component
of r, ~ is the ~ component of the flux density of 71, v~ is the ~ component of the
velocity, p is the production density of ~P due to interior processes and s is the supply
density of 7/controlled from the exterior of V. The quantities q~, p and s are expressed
by the constitutive equations characterizing the considered material.
The statistical method of derivation of the balance equations for a macroscopic
physical system from its microscopic structure was initiated by Boltzmann [3 5]. We
shall consider only the classical nonequilibrium statistical mechanics. This method
relies on the evolution equation of the probability density in the phase space for the
system consisting of all the microscopic components of the physical system (Liouville
equation). Even for the simple case of the ideal gas, because of mathematical difficul-
ties, the derivation of the balance equations and constitutive equations is possible only
using certain hypotheses, approximations and simplifications [-6]. So far, these results
have been extended to hard sphere fluids, also a very idealized molecular model [7].
For a more complicated microscopic structure, the existence of the balance equations
is implicitly postulated and the problem of the statistical mechanics is reduced to the
calculation of the constitutive equations.
In this paper we show that the balance equations can be derived in three successive
stages. First we prove that for an arbitrary finite number of material points,
a spac~time average satisfying a mathematical relation of the form (1) can be defined.
The corpuscular structure of a physical system ensures the existence of mathematical
relations of the balance equations even if the space-time averages have discontinuous
first order partial derivatives. In the second stage we obtain the smoothness specific to
the continuous fields if the physical system verifies the local equilibrium assumption.
Thus, balance equation (1) follows only from two hypotheses: the corpuscular struc-
ture and the local equilibrium assumption. In the third stage, the information on the
microscopic structure is used to derive the constitutive equations.
The usual statistical approach does not allow the separation of these three stages,
all of them being simultaneously implied in the Liouville equation. The mathematical
form specific to balance equations is implicitly contained in the Liouville equation
which is a probability conservation law in phase space. Tile smoothness characteristic
to continuous fields is introduced by the probability density in phase space which is
defined as a continuous field. The Liouville equation can be written only if the
interaction forces of the microscopic components are known, i.e. the model of the
microscopic structure of the physical system has to be completely described.
A finite number of particles satisfying the classical mechanics principles is the
physical system with corpuscular structure considered in the following. The particles
are modelled as material points, i.e. all the physical quantities necessary to describe
the structure of the particles are assigned to mathematical points defining the position
of the particles. The particles can appear or disappear as a result of some instan-
taneous processes (e.g. chemical reactions). We assume that the evolution of any
physical quantity chacterizing the particles is given by an analytic function of time.
Under these circumstances we shall prove that a space-time average of an arbitrary
C. Vamos et al. / Physica A 227 (1996) 81 92 83
physical quantity has a.e. continuous partial derivatives. Although these averages
preserve the discontinuities associated to the particles as discontinuity surfaces, they
satisfy a relation of the form (1). In the following, such a relation will be called "the
discrete analogue" of a similar relation in continuum mechanics.
At the end, the balance equation for an arbitrary physical quantity is obtained using
the local equilibrium assumption. As an example, we prove the existence of the
balance equations for mass and momentum. These equations have the same form as in
continuum mechanics if the following hypotheses on the microscopic structure are
used: there is a single type of microscopic components and the interaction of the
particles has a spherical symmetry. The general problem of the derivation of the
constitutive equations from the microscopic structure is briefly discussed. It will be the
subject of future articles on physical systems with complex microscopic structure.
2. The discrete analogue of continuous fields
We study the evolution during the temporal interval I = [0, T ] c ~ of a discrete
system consisting on N particles obeying the principles of classical mechanics. We
denote by li = [t+,t:~] ~ I the existence interval of the ith particle (1 ~< i ~< N). Let
n(t) be the number of the particles existing at the moment t e I. The variations of n(t)
occur when some particles are generated or destroyed. It is obvious that n(t) <~ N for
each t ~ I, and n(t) = N for all t e I only if li = I for each i 4 N, i.e. if no particles are
generated or destroyed over the interval I.
Let ~p be an arbitrary physical quantity. The values of q~ characterizing the ith
particle are given by a function tpi: I ~ [~. If Ii ¢ I, then ~0/(t) = 0 for all t ~ l...li. We
assume that the restriction q)~ll/can be represented as a Taylor series, i.e., it is an
analytic function. In the interval I~ the quantity ~o/may take any real value, including
zero (e.g., the velocity components of a motionless particle). Hence (pi is discontinuous
at t + and t i if ~oi(t +) ~ 0 and tp~(tF) :/: 0, respectively. Similarly, the derivatives of
~0~ at t[ and tF may be continuous or discontinuous.
We assign to each particle a mathematical point indicating its position. The choice
of the particle mass center is always possible; other options may be also adopted
under the restriction that the mathematical point should be uniquely defined. To these
mathematical points we assign the microscopic quantities ~o~ describing the evolution
of the particles. The number and the type of these quantities depend on the properties
and the complexity of the particles. Thus the discrete system is modelled as a finite set
of material points. The e components of the radius vector ri, x~i: I --* R (:~ = 1,2,3t,
and of the velocity ~i, ~,i: I --* N (~ = 1,2, 3) may be treated as particular cases of
functions (p~. The functions x~ and ~i supply a kinematic description of the motion of
the discrete system. The assumption that each particle obeys the principles of classical
mechanics is necessary to ensure the uniqueness of the discrete system evolution and
thus, the existence of the functions ~pl. The moments when the solutions of Hamilton's
equations are not analytic may be considered moments when new particles are
84 C. Vamos et al. / Physica A 227 (1996) 81-92
generated. For example, an instantaneous perfectly elastic collision does not change
the type of the particles, but the velocity discontinuity may be associated with the
generation of new particles of the same type as the old ones with different velocities.
We assume that the instantaneous discontinuous processes are of finite number over
the interval I = [0, T ].
We intend to characterize the mean distribution of ~o about the point of radius
vector r at the moment t. Therefore we average the values q~, corresponding to the
particles lying in the open sphere of center r and radius a denoted S(r, a), over the
interval (t - ~, t + 27). Here 27 < T and a are arbitrary positive real parameters. We
define the mean distribution of ~o as a function D,: ~ 3 X (27, T - 27) -* ~ with
t+~
/L(r, t) = ~-~ ~o,(t')/-/+ (a 2 - (r,(t') - ri2)dt ', (2)
i=l
where V = 4~a3/3 is the volume of S(r,a) and H + is the left continuous Heaviside
function. Since H + (a 2 - (r,(t') - r) 2) vanishes if the ith particle is located outside the
sphere S(r, a) and %(t') vanishes if t' e 1\ I,, then a nonvanishing contribution to D~ is
due only to particles which lie in S(r, a) over the interval (t - 27,t + z).
The function H + (a 2 - (ri(t) - r) 2) in (2) takes only the values 0 and 1. The jumps
occur when the ith particle enters or leaves the open sphere S(r, a). These moments are
among the solutions u, of the equation
hi(r, ui) - (ri(ui) - r) 2 - a 2 = O, (3)
where Ih,(r,t)l 1/2 is the distance at the moment t between the ith particle and the
surface OS(r, a) of the sphere S(r, a). Since x~,, and hence h,, are analytic functions with
respect to u,, and li is closed interval, then either Eq. (3) has a finite number of
solutions or h, vanishes identically [8].
In the latter case the particle moves along the surface of S(r, a) and does not enter
the sphere, hence H+(a 2 -(r,(t)- r) 2) is identically zero and has no jumps. Since
ri(u,) is a known function, then the isolated zeros of(3) are implicit functions u,(r). The
implicit function theorem can be applied only at interior points and it does not ensure
the existence of u,(r) for u, = t,+ , i.e. re OS(r,(t,+-),a). This case will be discussed
separately. For u, ~ (t/+ , t{), if
Ohi/Oui = 2(ri(ui) - r). ~i(ui) # O, i4)
then the function u,(r) exists in a neighborhood of r and has the derivatives
ou, Oh, ~Oh, _ x,, u,l - x,
Ox, - Ox~/ Ou, (ri(u,) - r).~i(ui) ' ~ = 1,2,3, (5)
where x, are the components of r. According to (4), the function ui(r) is not different,-
able at the points of the discriminant surface of the family {OS(r,(t),a); t ~ Ii}.
C. Vamos et al. / Physica A 227 (1996) 81 92 85
We denote the moments when the ith particle enters (leaves) the sphere S(r,a) by
t t t tp t/
t + < Uil < u~2 < "" < u~,, < tf (t + < u~ < ui2 < "'" < u~,,, < t { ) . Since the sphere
S(r, a) is open, H + (a 2 - (ri(t) - r) 2) as a function of t is left (right) continuous when the
particle enters (leaves). Hence for t e I~, we have
H + (a 2 - (rift) --r):) = H + (a 2 - (ri(t +) - r) 2)
n' n"
+ ~ H+(t - ui~,)- ~', H (t--u',~,,,), (6~
k,=l k"=l
where H is the right continuous Heaviside jump function. The first term in the
right-hand side of (6) vanishes if the ith particle is generated in the exterior of S(r, ai
and equals 1 otherwise. If u~ = t + , then the particle can only enter the sphere after its
generation, hence u'~ = t + and relation (6) holds. If ui = t/-, then the particle could
only leave the sphere before its disappearance, hence u'[,,, = t~ and relation (6) holds
again. So (6) contains all the possible situations if we take t + ~< ui~ and u'{,,, ~< t~. The
following notation will be used:
u, {,,,u,:, .. ,u,.,}, u, = {u,l,u,,_ .... "
t = t ! . ! tt tt t¢ , ~in" ] "
(7)
To study the differentiability of D~, we consider the function gi: E 3 x (r, T r) ~ R
t+r
gi(r,t) = f qgi(t')H+ (a 2 -(ri(t')- r)2)dt '.
t-r
For a fixed r, the integrand
Gdr, t) = q)i(t)H + (a 2 - (ri(t) - r) 2)
is a continuous function, except a
(8)
{ t{, tF l w U'; w U'i'. Hence Gi is Riemann integrable and gi has partial derivative with
respect to t a.e. in (r, T - r), equal to
~,,q~(r,t) = Gdr, t + z) - G~(r,t - r). (10)
The discontinuities of 3tg~ with respect to (r, t) are related to those of G~. From (9) it
follows that G; is discontinuous when ~oi is discontinuous and H + nonvanishing, or
conversely, when H + is discontinuous and ~o~ nonvanishing. In the first case the ith
particle appears or disappears in S(r,a), i.e. t = t~ + and r ~ S(ri(t~ +-),a), and in the
second case the ith particle lies in the surface of S(r,a), i.e. t ~ li and r ~ ?S(rAt),aL
Hence the derivative (10) is not continuous over
f2'i = {(r,t)lt ~ {ti + - r, ti + + z}cNz, T - ~) and r ~ S(ri(ti+),a)}
u{(r,t)lt~[t~- ++_z, t y ++_r]c~(r,T-~)andre~S(rdtT-r),a)}. (11)
The set f2'i has null Lebesgue measure in ~3 × (r, T - z), hence Otgi is a.e. continuous.
(9)
finite number of jump discontinuities
86 c Vamos et al. /Physica A 227 (1996) 81 92
Although (6) holds only for t • li, it may be substituted in (8) because q)~ vanishes for
t • I \ I~. The obtained expression allows the study of the differentiability with respect
to r of the function g,. There exist three possible situations, namely: 23 ~< t£ - t +,
< tF - t + < 23 and t£ - ti + ~< 3. For each of them the discussion, although elemen-
tary, is rather long and involved [9]. Here we give only the result. The set where the
function gi is not differentiable with respect to r is
QI' = {(r,t)l t e (t [ - 3, t[- + 3] c~(r, T - 3) and r• aS(ri(ti~),a) }
u {(r, t)[ t • [ti - 3, ti + r) a(3, T - r) and r • c?S(ri(t[), a) }
u{(r,t)lt •(t[ + r, tF _+ 3)n(z,T- 3) and re OS(ri(t T- 3),a)}
w{(r,t)[ t e(r,r - 3), exists t' e(t - r,t + z)c~Ii such that
r • OS(ri(t'), a) and (ri(t') - r). ~i(t') = 0}, (12)
and it is of null Lebesgue measure in 1~ 3 x (r, T -- r). The derivative of g~ with respect
to x~, denoted by ~?~gi, is
c~,g,(,',t) = y~ ,p,(u) x,,(u) - x,
.~v, I(ri(u) -- r)-¢i(u)] ' (13)
where Ui = (UiwUi')c~(t - z,t + z).
Using definition (2) and relations (10)-(13), it follows the a.e. continuity of the
partial derivatives of Do, given by
a, Do(r, t) = f~ [Gi(r, t + T) - Gi(r, t - 3)],
i=1
(14)
for (r,t) • ~3 x (r, T - 3)\ U/N=1 Q'i, and
~Do(", t) = ~-W Y, q)i(u)
i= 1 u~Ui
x~i(u) - x~
I(*,(u) -r)'¢i(u)l'
(15)
for (r, t)e N3x (v, T- 3)\U/~= 1 (21 '. The a.e. continuity of the partial derivatives en-
sures the continuity of D o with respect to (r,t)• Nax (r, T- 3) and then D o is
a continuous field.
3. The discrete analogue of the balance equation
In the following we shall derive the relation satisfied by D o. We use a theorem
stating that every function with bounded variation may be uniquely split into a sum of
C. Vamos et al. / Physica A 227 (1996) 81-92 87
two functions: one continuous and a jump function [10]. We apply this theorem to
G~ given by (9) considered as a function of t. But except a finite number of jump
discontinuities, Gi is analytic on I and then its continuous part G'~ is also absolutely
continuous. Hence we may write Gi = G'i + GI', where GI' is the jump function.
Replacing this relation in (14), it follows that g,D,p can also be written as a two-term
sum
~tD,p = (O,D~o)' + (O, Dq,)". (16)
According to the Lebesgue theorem, the absolutely continuous part of G; is equal to
the integral of the derivative of G~
t+r
f ~Gi a'i(r, t + r) - ai(r, t - ~) = ~- (r, t') dt'
t-r
t+~
t'*
(gi(t')H+ (a z -- (ri(t') - r)Z)dt '. (17)
Id
t
Dividing (17) by 2rV, summing up with respect to i, taking into account (9), (14) and
(2) we obtain the part of ~?tD~ resulting from G'i,
(OtD,p)' = D+. (18)
From (14), the discontinuous part of ~tD,p can be written as
(~,D~)"(r, t) = ~ [61'(r, t + r) - Gi'(r, t - ~)] 119)
i=l
It contains the discontinuous variations of G~ during the temporal interval
[t -- r, t + r]. As proved in the preceding section, gtD~ exists if Gi is not discontinuous
at t + r and t - r (see expression (11)), therefore we consider only the jumps occurring
at the interior points of [t - r, t + r], i.e. in (t - r, t + r). From (9) it follows that such
a variation can take place if the particle is generated inside the sphere S(r, a) during the
temporal interval (t - r, t + r). Hence the jump of G~ is equal to
A+ Gi = (pi(t+)H+ (a 2 - (ri(t +) - r)2)(H+ (t + z - t +) - H- (t - r - t? )).
Similarly, the discontinuous variation of G~ related to the distruction of a particle is
A-G~ = - q)i(tT)H+(a 2 - (r~(ti ) -r)Z)(H+(t + ~ - tT ) - H-(t- r - t~)).
The function G~ also has discontinuous variations when the particle enters or leaves
the sphere S(r, a),
GT(r,t + z)- G;(r,t- z) = A+G, + A-Gi + ~ q)i(u)- ~ ~oi(u), (201
u~W~ uEW"
88 C. Vamos et al. / Physica A 227 (1996) 81 92
where W'i = U~n(t - z, t + z) and W'i' = Ui' c~(t - z, t + z). The sign of qgi(u) is posi-
tive (negative) if the particle enters (leaves) the sphere S(r, a), and it is given by the sign
of the expression - (r~(u) - r). ¢i(u) which is proportional to the interior normal
component of ~ to the surface of S(r, a) at the moment u. Hence we may use a single
sum in (20) if we denote, as in (13), U~ = W'~wW'f. Replacing (20) in (19) we obtain
1 N (ri(u) --r).~i(u)
(a'Dr)" =(a'Dr)g -- 2z~ ~ ~ ¢pi(u) (21)
i=1 .~, I(ri(u) -r)'¢i(u)l'
where
1 N
(a'Dr)° - 2rV y' (A+Gi + A-Gi). (22)
i=1
Relation (15) for the physical quantity ~i reads
1 N .¥~i(u) - x~
i= u~U i I(ri(u) - r). ¢i(u)i
Comparing this relation with (21) we obtain
3
(a,Dr)" = - ~ a~Dr¢ ~ + (#tDr)g. (23)
From (16), (18) and (23) follows the relation
3
~,D r + ~ 0~Dr¢ ~ = D~ + (0,Dr)g. (24)
a=l
This relation holds only when the derivatives (14) and (15) exist.
In contrast to balance equation (1), relation (24) does not contain a quantity
equivalent to the velocity v. The velocity is not a volume density, but an average
quantity. To define a discrete analogue, we must divide D r by the number of the
particles contributing to D r. Let D1 be the density D r corresponding to ¢pi(t) = 1 for
all t e Ii and i ~< N. Since D1 characterizes the average number of particles per unit
volume, the discrete average of q~ is defined as
q3(r, t) = Dr(r, t)/Dl(r, t) (25)
if D~(r,t)¢:O, and it vanishes if Dl(r,t)=O. It is easy to show that q3=q3,
(/91 -[- (/)2 = (~1 -[- (,02 and 2q~ = ,~0, where 2 is a real function of r and t.
The mean motion of the particles is given by the discrete average of the velocity
with the components ~. To introduce ~-~ in (24), we write
Dr~ ~ = DrtL+(¢ ~ ¢~)1 = ~-~D r + (~b~,)~,
where q~ is the discrete analogue of the kinetic part of the flux density
3
~'r = ~ D~o~¢ _ ~,)e~,
C. Vamos et al. / Physica A 227 (1996) 81 92 89
e~ being the unit vectors in ordinary three-dimensional space. Then (22) becomes
~,D~ + V.(D,p~) + V.O'~ = D+ + (~,D~) o. (26)
This is the discrete analogue of the balance equation (1).
4. The continuous modelling
The discrete analogues of continuous fields (2) and of balance equations (26) have
an intermediate status between the microscopic discrete description of a physical
system and its macroscopic description by means of continuous fields, having charac-
teristics in common with both.
As in the microscopic discrete description, we imposed no limits on the number of
particles, whereas it is known that the continuous modelling is possible only for
a large number of particles. Then, the complete knowledge of the function D,p given by
(2) {including the points where it is not differentiable) is equivalent to the knowledge of
the evolution of the microscopic state for all the N particles. On the contrary, the
macroscopic continuous fields are characterized by smearing out the discontinuities of
microscopic nature.
The main similarities between our results and the macroscopic continuous descrip-
tion are the following. Using only microscopic physical quantities, we have construc-
ted a physical quantity satisfying relation (26), formally identical with the macroscopic
balance equation (1). Moreover, relation (26) has the same generality as its macro-
scopic correspondent, being valid for any physical quantity and for any physical
system with corpuscular structure.
In order to obtain the balance equation (1), we have to eliminate from (26) the
discontinuities specific to the microscopic scale. In statistical physics these discon-
tinuities are eliminated from the very beginning, since the probability density in phase
space has the smoothness properties of a continuous field. The usual statistical
average may be applied to relation (26) and the balance equation (1) is obtained for
T --* 0 and a --* 0. In this article we do not present this approach, but a shorter and
more intuitive one, although less rigorous.
The linear thermodynamics of irreversible processes can be applied to far-from-
equilibrium states if the local equilibrium assumption is satisfied [11]. Then at any
time and for any point there exists a microscopic part of the macroscopic body acting
as a near-equilibrium thermodynamic system. In our case we choose the parameters
a and r such that over the interval (t - r, t + T) the particles in the sphere S(r, a) should
form the near-equilibrium thermodynamic system. It is obvious that the space time
average (2) and the discrete average (25) become the corresponding continuous fields
and the relation (26) becomes a balance equation. The local equilibrium assumption
eliminates the microscopic discontinuities due to the large number of particles
contributing to the value of D,p. This condition was also used by Murdoch [12] to
90 C. Vamos et al. / Physica A 227 (1996) 81 92
smooth the microscopic discontinuities by a space-time averaging. The main differ-
ence is that in our approach the smoothing is made after a relation of the form (1) was
obtained. Similarly to the usual statistical method, Murdoch obtains at the same time
both the mathematical form of the balance equation and the smoothness character-
istic to macroscopic continuous fields.
On the other hand, the parameters a and z must be smaller than the spatial and
temporal characteristic magnitude of the considered physical process. Otherwise the
modelled process is distorted. For example, in the case of a plane wave, the
wavelength and the period must be much larger than a and z, respectively.
Thus the existence of the balance equation for any physical quantity has been
proved, but its explicit form depends on the microscopic structure of the physical
system. As an example, in the following we derive the fundamental equations of
continuum mechanics, the balance equation for mass and momentum, in the simple
case of identical particles. Otherwise the equations would become much more in-
volved because of additional diffusive terms. Since there is only one type of particles,
processes like chemical reactions are excluded. Thus the appearance or disappearance
of the particles may be related only to the discontinuous variations of certain physical
quantities as velocity, momentum, etc. This situation occurs, for example, if the
collisions of the particles can be modelled as instantaneous processes of discontinuous
variations of the momentum or if the particles interact instantaneously with exterior
fields (e.g., photons in the Compton effect).
First, for each microscopic component we must uniquely define the mathematical
point determining its position and the physical quantities necessary to describe its
state. The relation (26) can be written for each of these physical quantities. Whatever
the structure of the microscopic components, their mass and position are always
uniquely defined, and hence their momentum too. Thus the balance equations for
mass and momentum exist for any corpuscular physical system.
The relation (26) for mass is obtained if ~oi(t) = m for all t ~ I i and i ~< N, where m is
the mass of the particles. Then D~ = D,, = mD1 is the discrete analogue of the mass
density. Since there is only one type of particles, the appearance or disappearance of
the particles does not imply mass variations, hence (~tD,,)o = 0. Moreover, q5 = m,
q~, = 0, ~bi = 0 and (26) becomes
OtD,, + V°(Om~) = 0. (27)
This is the discrete analogue of the continuity equation.
For the ~ component of momentum we have ~oi =p,i= m~,~ and Dp, =
Dine, = Dm(~. The discrete analogue of the kinetic part of the flux density takes the
form of a symmetric tensor
/+r
T~p (~'p~)~ 2rV (~i(t ) -- - ' --
i=1
t-'c
x H + (a 2 -- (ri(t') -- r)2)dt ' . (28)
C. Vamos et al. / Physica A 227 (1996) 81 92 91
The derivative ~bi is the ~ component of the force Z acting on the ith particle and
relation (26) becomes
3 3
~,(Dmf,) + ~ O,(Dm~-~-p) - ~. ~,T',p = Dy, + (,?,D,,) o . (29)
This relation takes a form identical with the balance equation for momentum in
continuum mechanics only if additional hypotheses on the microscopic structure are
made.
As discussed above, the parameters a and z are chosen so that the local equilibrium
assumption should hold and space-time average (2) should not distort the modelled
process. Then the space-time averages D1 and D m become the concentration c(r, t) and
the mass density p(r,t), respectively, which are continuous fields and the discrete
average of the velocity ~- becomes the continuous field of velocity v(r, t). Then from
relation (27) we obtain the continuity equation
~,p + V.(pv) = 0. (30)
The symmetric tensor (28) becomes the kinetic contribution to stress tensor a'~, [13].
The force term in (29) is the sum of the contributions due to the external potential
U(r, t) and interaction forces between the particles. Since the values of a and ~ were
chosen so that the variation of U in the sphere S(r,a) over the interval (t - r, t + r)
should be negligible, the first contribution equals - c VU. Then relation (29) becomes
3 3
~t(pv~) + ~ ~.p(pv,vp) -- ~ (~pa',~ -- cF~ = -- c<~U + P~, (31)
B=I /~=1
where F(r, t) is the continuous field representing the mean force acting at the moment t
on a particle placed at r due to the interaction with the other particles, and P(r, t) is the
momentum production at the time t and the point r.
The number of the continuous fields in the balance equations (30) and (31) can be
decreased if further hypotheses on the microscopic structure are introduced. For
example, if the interaction potential is a continuous function, then the momentum
variations of the particles are continuous and P = 0. If, in addition, the interparticle
forces have spherical symmetry, then the term cF in (31) is the divergence of a symmet-
ric tensor [13] and (31) becomes the usual momentum balance equation of continuum
mechanics.
5. Conclusion
The existence of balance equations has been proved for corpuscular physical
systems satisfying the local equilibrium assumption. We have obtained this result
under circumstances similar to the most general ones in continuum mechanics: the
microscopic components can be generated or destroyed as a result of some instan-
taneous processes (chemical reactions, collisions, etc.) and the balance equation is
derived for an arbitrary physical quantity.
92 c. Vamos et al. /Physica A 227 (1996) 81 92
As in kinetic theory, we have assumed that the microscopic components obey the
classical mechanics principles, but in kinetic theory the balance equations are derived
under more restrictive conditions [3 5]. The derivation of the Boltzmann equation
from the Liouville equation requires a series of assumptions: the molecular chaos
hypothesis, biparticle collisions, central interaction forces, etc. Besides, the balance
equations can be obtained only for the collision invariants (mass, momentum, energy).
We avoided these limitations since not all the information on the microscopic
structure was introduced from the very beginning. In the first stage we have showed
that for a finite number of material points the space time average (2) satisfies
a mathematical relation of the same form as the balance equations. In the second stage
we have used the local equilibrium assumption which ensures the necessary smooth-
ness for the continuous fields. Only in the last stage the particular microscopic
structure of the physical system has been taken into account. In this article the
microscopic structure has been specified only as far as to obtain the fundamental
momentum balance equation of the continuum mechanics, i.e. we have considered
only one type of particles with spherical symmetry.
The method exposed above may present a supplementary interest in the case of
a complex microscopic structure. The more complex the microscopic structure, the
more numerous are the simplifications and approximations necessary to obtain the
constitutive equations. The discrete analogue of the balance equation (26) is a frame
for the verification of the compatibility and the completeness of these approximations.
This problem will be the subject of one of our future papers.
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