Balance equations for a finite number of particles

Călin Vamoş
(original), paper

Abstract

In this article an abstract discrete system is considered, consisting of an arbitrary, finite numer of particles modelled as mathematical points to which analytic funcitons of time are attached. We prove that a space-time  average of these analytic functions can be defined, satisfying a relation of the same form with the balance equations in continuum mechanics.

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C. Vamos
Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy

A. Georgescu

N. Suciu
Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy

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C. Vamoş, A. Georgescu, N. Suciu (1996), Balance equations for a finite number of particles, Stud. Cerc. Mat., 48(1-2), 115-127.

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[3] C. Cercignani, Mathematical Metods in Kinetic Theory,  Plenum, New York, 1990.

[4] S. Chapman and T. G. Cowling,  The Mathematical Theory of Non-uniform Gases. University Press, Cambridge, 1953.

[5] E. G. D. Cohen,  Fifty years of kinetic theory. Physica A 194 (1993), 229-257.

[6] P. Glansdorff and  I. Prigogine,  Thermodynamic Theory of Structure, Stability and Fluctuations. Wiley, London, 1971.

[7] J. K. Irfinb and J. G. Kirkwood,  The statistical mechanical theory of transport processes, IV. The equations of hydrodynamics. J. Chem. Phys. 18 (1950) 817-40.

[8] A. Kolmogorov and S. Fomine,  Elements de la theorie des fonctions et de l’analyse fonctionelle. Mir. Moscou, 1974.

[9] J. Muller,  Thermodynamics.  Pitman, London, 1985.

[10] A. Sveshnikov and A. Tikhonov,  The Theory of Functions of a Complex Variable,  Mir. Moscou, 1978.

[11] C. Truesdell and R. A. Toupin,  The Classical Field Theories. In: Handbuch der Physik, III, Part. I, ed. S. Flugge, Springer, Berlin, 1960.

[12] C. Vamoș, A. Georgescu, N. Suciu and I. Turcu, Balance equations for physical systems with corpuscular structure.  In press at Physica A.

 

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1996b Suciu-Vamos-Georgescu - Balance equations for a finite number of particles

ACADEMIA ROMÂNĂ

STUDII SI CERCETARI MATEMATICE
(MATHEMATICAL REPORTS)

EXTRAS1-2

TOMUL 48
1996
ianuarie - aprilie

BALANCE EQUATIONS FOR A FINITE NUMBER OF PARTICLES

C. VAMOŞ, A. GEORGESCU and N. SUCIU

Abstract

In this article an abstract discrete system is considered, consisting of an arbitrary, finite number of particles modelled as mathematical points to which analytic functions of time are attached. We prove that a space-time average of these analytic functions can be defined, satisfying a relation of the same form with the balance equations in continuum mechanics.

1. INTRODUCTION

The balance equations are postulated relations for fundamental physical quantities (mass, momentum, energy, entropy etc.) valid for all continuous media [11]. We take over the differential expression of the balance equations from [9]. Let Ψ Ψ Psi\PsiΨ be a physical quantity additive with respect to space, associated to a continuous medium. That is, there exists a function ψ ψ psi\psiψ of space ( r → ) ( r → ) ( vec(r))(\vec{r})(r→) and time ( t ) ( t ) (t)(t)(t), called the volume density of Ψ Ψ Psi\PsiΨ, such that, for any volume V V VVV, the integral ∫ V ψ d r ¯ ∫ V   ψ d r ¯ int_(V)psid bar(r)\int_{V} \psi \mathrm{~d} \bar{r}∫Vψ dr¯ represents the amount of Ψ Ψ Psi\PsiΨ contained in V V VVV. The differential form of the balance equation at a regular point (i.e. without shocks or other discontinuities) is
(1) ∂ t ψ + ∑ α = 1 3 ∂ α ( Φ α + ψ v α ) − ( p + s ) = 0 (1) ∂ t ψ + ∑ α = 1 3   ∂ α Φ α + ψ v α − ( p + s ) = 0 {:(1)del_(t)psi+sum_(alpha=1)^(3)del_(alpha)(Phi_(alpha)+psiv_(alpha))-(p+s)=0:}\begin{equation*} \partial_{t} \psi+\sum_{\alpha=1}^{3} \partial_{\alpha}\left(\Phi_{\alpha}+\psi v_{\alpha}\right)-(p+s)=0 \tag{1} \end{equation*}(1)∂tψ+∑α=13∂α(Φα+ψvα)−(p+s)=0
where ∂ t ∂ t del_(t)\partial_{t}∂t is the temporal derivative, ∂ α ∂ α del_(alpha)\partial_{\alpha}∂α is the derivative with respect to the α α alpha\alphaα component of r ¯ , Φ α r ¯ , Φ α bar(r),Phi_(alpha)\bar{r}, \Phi_{\alpha}r¯,Φα is the α α alpha\alphaα component of the flux density of Ψ , ν α Ψ , ν α Psi,nu_(alpha)\Psi, \nu_{\alpha}Ψ,να is the α α alpha\alphaα component of the velocity, p p ppp is the production density of Ψ Ψ Psi\PsiΨ due to interior processes and s s sss is the supply density of Ψ Ψ Psi\PsiΨ controlled from the exterior of V V VVV. The quantities Φ ¯ , p Φ ¯ , p bar(Phi),p\bar{\Phi}, pΦ¯,p and s s sss 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 (e.g. [1] - [3]). 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 difficulties, the derivation of the balance equations and constitutive equations is possible only using certain hypotheses, approximations and simplifications [4]. So far, these results have been extended to hard sphere fluids, also a very idealized molecular model [5].
For a more complicated microscopic structure, the existence of the balance equations is implicity postulated and the problem of the statistical mechanics is reduced to the calculation of the constitutive equations.
In this article we show that the existence of mathematical relations of the form (1) can be proved in the more general framework of the kinematic description for the microscopic evolution of an arbitrary corpuscular physical system. The position and motion of the microscopic particles can be modelled as mathematical points, and the physical quantities characterizing the particles (mass, momentum, acceleration, kinetic momentum etc.) can be modelled as functions of time. Thus, the differential equations (the dynamic system) corresponding to the microscopic evolution of the corpuscular physical system are not known, but the existence of its solution is postulated.
In the following we shall consider an abstract mathematical model for the 'kinematic microscopic description discussed above. This mathematical discrete system consists of an arbitrary, finite number of mathematical points to which arbitrary functions of time are associated and it has an abstract nature because there is no physical specification for these time functions. Certainly, for a given corpuscular physical system, the functions of time can represent the mass of the particles, their momentum or any other physical quantity, but in general no specific physical quatity is assigned to the mathematical points. To be more concise, we shall use the name of "particles" or "material points" for these mathematical points together with the assigned abstract functions of time.
The particles can appear or disappear as a result of certain instantaneous processes. Every particle has a temporal interval of existence which can be different from the temporal interval over which we study the discrete system. We assume that a single existence interval corresponds to a given particle, i.e. its disappearance at one time precludes its reappearance. Even if a particle of the same type appears later on, it is considered as a new particle. For the abstract discrete system we do not impose any connection between the disappearance of some particles and the appearance of others. Certainly, in the case of processes like chemical reactions, such connections exist and mass, momentum and energy must be conserved.
The only dynamical requirement is that the evolution of the abstract discrete system (both the variation of the particles position and the associated functions) should be given by analitic functions of time. Under these circumstances we shall prove that a space-time average of the arbitrary functions of time has a.e. continuous partial derivatives. We emphasize that the averaging is an ordinary mathematical one, not a statistical average on an infinite ensemble of identical copies of the physical system. Although these averages preserve the discontinuities associated to the particles as discontinuity surfaces, they satisfy a relation of the form (1). To eliminate and posibility of confusion, we shall call the relations obtained for the abstract discrete system, the "discrete analogue" of the relations of continuum mechanics having a similar form.
We shall apply these results to a Hamiltonian system formed by a single type of particles which neither disappear, nor appear. Every particle is characterized by
a constant mass. For this case we shall write the discrete analogue of the balance equations for mass, momentum and energy. We shall briefly discuss the relation with the approach in nonequilibrium statistical mechanics.

2. THE DISCRETE ANALOGUE OF CONTINUOUS FIELDS

We study the evolution during the temporal interval I = [ 0 , T ] ⊂ R I = [ 0 , T ] ⊂ R I=[0,T]subRI=[0, T] \subset \mathbb{R}I=[0,T]⊂R of an abstract discrete system consisting of N N NNN particles (i.e. mathematical points with the assigned functions of time). We denote by I i = [ t i + , t i − ] ⊂ I I i = t i + , t i − ⊂ I I_(i)=[t_(i)^(+),t_(i)^(-)]sub II_{i}=\left[t_{i}^{+}, t_{i}^{-}\right] \subset IIi=[ti+,ti−]⊂I the existence inteval of the i i iii-th particle ( 1 ≤ i ≤ N 1 ≤ i ≤ N 1 <= i <= N1 \leq i \leq N1≤i≤N ). Obviously 0 ≤ t i + < T 0 ≤ t i + < T 0 <= t_(i)^(+) < T0 \leq t_{i}^{+}<T0≤ti+<T and 0 < t i − ≤ T 0 < t i − ≤ T 0 < t_(i)^(-) <= T0<t_{i}^{-} \leq T0<ti−≤T. If I i = I I i = I I_(i)=II_{i}=IIi=I, then the i i iii-th particle exists over the whole interval I I III. If I i ≠ I I i ≠ I I_(i)!=II_{i} \neq IIi≠I for one particle at least, then there are moments when the number of particles is smaller than the total number of particles N N NNN. Denoting by n ( t ) n ( t ) n(t)n(t)n(t) the number of the particles existing at the moment t ∈ I t ∈ I t in It \in It∈I, we have n ( t ) ≤ N n ( t ) ≤ N n(t) <= Nn(t) \leq Nn(t)≤N for each t ∈ I t ∈ I t in It \in It∈I. The equality holds only if I i = I I i = I I_(i)=II_{i}=IIi=I for all i ≤ N i ≤ N i <= Ni \leq Ni≤N, i.e. if no particles are generated or destroyed over the interval I I III.
Let φ i : I → R φ i : I → R varphi_(i):I rarrR\varphi_{i}: I \rightarrow \mathbb{R}φi:I→R be an arbitrary function of time characterizing the i i iii-th particle. If I i ≠ I I i ≠ I I_(i)!=II_{i} \neq IIi≠I, then φ i ( t ) = 0 φ i ( t ) = 0 varphi_(i)(t)=0\varphi_{i}(t)=0φi(t)=0 for all t ∈ I ∖ I i t ∈ I ∖ I i t in I\\I_(i)t \in I \backslash I_{i}t∈I∖Ii. We assume that the restriction φ i ∣ I i φ i ∣ I i varphi_(i)∣I_(i)\varphi_{i} \mid I_{i}φi∣Ii can be represented as a Taylor series, i.e. it is an analytic function. In the interval I i I i I_(i)I_{i}Ii the function φ i φ i varphi_(i)\varphi_{i}φi may take any real value, including zero (e.g. the velocity components of a motionless particle). Hence φ i φ i varphi_(i)\varphi_{i}φi is discontinuous at t i + t i + t_(i)^(+)t_{i}^{+}ti+and t i − t i − t_(i)^(-)t_{i}^{-}ti−if φ i ( t i + ) ≠ 0 φ i t i + ≠ 0 varphi_(i)(t_(i)^(+))!=0\varphi_{i}\left(t_{i}^{+}\right) \neq 0φi(ti+)≠0 and φ i ( t i − ) ≠ 0 φ i t i − ≠ 0 varphi_(i)(t_(i)^(-))!=0\varphi_{i}\left(t_{i}^{-}\right) \neq 0φi(ti−)≠0, respectively. Similarly, the derivatives of φ i φ i varphi_(i)\varphi_{i}φi at t i + t i + t_(i)^(+)t_{i}^{+}ti+and t i − t i − t_(i)^(-)t_{i}^{-}ti−may be continuous or discontinuous. The α α alpha\alphaα components of the radius vector r → i , x α i : I → R ( α = 1 , 2 , 3 ) r → i , x α i : I → R ( α = 1 , 2 , 3 ) vec(r)_(i),x_(alpha i):I rarrR(alpha=1,2,3)\vec{r}_{i}, x_{\alpha i}: I \rightarrow \mathbb{R}(\alpha=1,2,3)r→i,xαi:I→R(α=1,2,3), and of the velocity ξ ¯ i , ξ α i : I → R ( α = 1 , 2 , 3 ) ξ ¯ i , ξ α i : I → R ( α = 1 , 2 , 3 ) bar(xi)_(i),xi_(alpha i):I rarrR(alpha=1,2,3)\bar{\xi}_{i}, \xi_{\alpha i}: I \rightarrow \mathbb{R}(\alpha=1,2,3)ξ¯i,ξαi:I→R(α=1,2,3) may be treated as particular cases of functions φ i φ i varphi_(i)\varphi_{i}φi. The functions x α i x α i x_(alpha i)x_{\alpha i}xαi and ξ α i ξ α i xi_(alpha i)\xi_{\alpha i}ξαi supply a kinematic description of the motion of the discrete system.
Definition. For two arbitrary positive real parameters τ < T / 2 τ < T / 2 tau < T//2\tau<T / 2τ<T/2 and a a aaa, we define a function D φ : R 3 × ( τ , T − τ ) → R D φ : R 3 × ( τ , T − τ ) → R D_(varphi):R^(3)xx(tau,T-tau)rarrRD_{\varphi}: \mathbb{R}^{3} \times(\tau, T-\tau) \rightarrow \mathbb{R}Dφ:R3×(τ,T−τ)→R as
(2) D φ ( r → , t ) = 1 2 τ V ∑ i = 1 N ∫ t − τ t + τ φ i ( t ′ ) H + ( a 2 − ( r ¯ i ( t ′ ) − r ¯ ) 2 ) d t ′ (2) D φ ( r → , t ) = 1 2 τ V ∑ i = 1 N   ∫ t − τ t + τ   φ i t ′ H + a 2 − r ¯ i t ′ − r ¯ 2 d t ′ {:(2)D_(varphi)( vec(r)","t)=(1)/(2tau V)sum_(i=1)^(N)int_(t-tau)^(t+tau)varphi_(i)(t^('))H^(+)(a^(2)-( bar(r)_(i)(t^('))-( bar(r)))^(2))dt^('):}\begin{equation*} \mathrm{D}_{\varphi}(\vec{r}, t)=\frac{1}{2 \tau V} \sum_{i=1}^{N} \int_{t-\tau}^{t+\tau} \varphi_{i}\left(t^{\prime}\right) H^{+}\left(a^{2}-\left(\bar{r}_{i}\left(t^{\prime}\right)-\bar{r}\right)^{2}\right) \mathrm{d} t^{\prime} \tag{2} \end{equation*}(2)Dφ(r→,t)=12τV∑i=1N∫t−τt+τφi(t′)H+(a2−(r¯i(t′)−r¯)2)dt′
where V = 4 π a 3 / 3 V = 4 π a 3 / 3 V=4pia^(3)//3V=4 \pi a^{3} / 3V=4πa3/3 is the volume of the open sphere of center r ¯ r ¯ bar(r)\bar{r}r¯ and radius a a aaa denoted by S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a) and H + H + H^(+)H^{+}H+is the left continuous Heaviside function.
Since H + ( a 2 − ( r ¯ i ( t ′ ) − r ¯ ) 2 ) H + a 2 − r ¯ i t ′ − r ¯ 2 H^(+)(a^(2)-( bar(r)_(i)(t^('))-( bar(r)))^(2))H^{+}\left(a^{2}-\left(\bar{r}_{i}\left(t^{\prime}\right)-\bar{r}\right)^{2}\right)H+(a2−(r¯i(t′)−r¯)2) vanishes if the i i iii-th particle is located outside the sphere S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a) and φ i ( t ′ ) φ i t ′ varphi_(i)(t^('))\varphi_{i}\left(t^{\prime}\right)φi(t′) vanishes if t ′ ∈ I I i t ′ ∈ I I i t^(')in II_(i)t^{\prime} \in I I_{i}t′∈IIi, then a nonvanishing contribution to D φ D φ D_(varphi)D_{\varphi}Dφ is due only to particles which lie in S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a) over the interval ( t − τ , t + τ ) ( t − τ , t + τ ) (t-tau,t+tau)(t-\tau, t+\tau)(t−τ,t+τ). Therefore D φ ( r → , t ) D φ ( r → , t ) D_(varphi)( vec(r),t)\mathrm{D}_{\varphi}(\vec{r}, t)Dφ(r→,t) characterizes the mean distribution of φ φ varphi\varphiφ about the point of radius vector r → r → vec(r)\vec{r}r→ at the moment t t ttt, and it is a spatial average on the sphere S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a) and a temporal
one over the interval ( t − τ , t + τ ) ( t − τ , t + τ ) (t-tau,t+tau)(t-\tau, t+\tau)(t−τ,t+τ). Obviously, it also depends on the parameters τ τ tau\tauτ and a a aaa, but we are not interesed in this dependence.
The function H + ( a 2 − ( r → i ( t ′ ) − r → ) 2 ) H + a 2 − r → i t ′ − r → 2 H^(+)(a^(2)-( vec(r)_(i)(t^('))-( vec(r)))^(2))H^{+}\left(a^{2}-\left(\vec{r}_{i}\left(t^{\prime}\right)-\vec{r}\right)^{2}\right)H+(a2−(r→i(t′)−r→)2) in (2) takes only the values 0 and 1 . The jumps occur when the i i iii-th particle enters or leaves the open sphere S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a). These moments are among the solutions u i u i u_(i)u_{i}ui of the equation
(3) h i ( r ⇀ i , u i ) ≡ ( r ⇀ i ( u i ) − r ⇀ ) 2 − a 2 = 0 (3) h i r ⇀ i , u i ≡ r ⇀ i u i − r ⇀ 2 − a 2 = 0 {:(3)h_(i)(r^(⇀)_(i),u_(i))-=(r^(⇀)_(i)(u_(i))-(r^(⇀)))^(2)-a^(2)=0:}\begin{equation*} h_{i}\left(\stackrel{\rightharpoonup}{r}_{i}, u_{i}\right) \equiv\left(\stackrel{\rightharpoonup}{r}_{i}\left(u_{i}\right)-\stackrel{\rightharpoonup}{r}\right)^{2}-a^{2}=0 \tag{3} \end{equation*}(3)hi(r⇀i,ui)≡(r⇀i(ui)−r⇀)2−a2=0
where | h i ( r → , t ) | 1 / 2 h i ( r → , t ) 1 / 2 |h_(i)(( vec(r)),t)|^(1//2)\left|h_{i}(\vec{r}, t)\right|^{1 / 2}|hi(r→,t)|1/2 is the distance at the moment t t ttt between the i -th particle and the surface ∂ S ( r → , a ) ∂ S ( r → , a ) del S( vec(r),a)\partial S(\vec{r}, a)∂S(r→,a) of the sphere S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a). Since x α i x α i x_(alpha i)x_{\alpha i}xαi, and hence h i h i h_(i)h_{i}hi, are analytic functions with respect to u i u i u_(i)u_{i}ui, and I i I i I_(i)I_{i}Ii is a closed interval, then either equation (3) has a finite number of solutions or h i h i h_(i)h_{i}hi vanishes identically ([10], p 78 ). In the latter case the particle moves along the surface of S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a) and does not enter the sphere, hence H + ( a 2 − ( r ¯ i ( t ) − r ¯ ) 2 ) H + a 2 − r ¯ i ( t ) − r ¯ 2 H^(+)(a^(2)-( bar(r)_(i)(t)-( bar(r)))^(2))H^{+}\left(a^{2}-\left(\bar{r}_{i}(t)-\bar{r}\right)^{2}\right)H+(a2−(r¯i(t)−r¯)2) is identically zero and has no jumps. Since r ¯ i ( u i ) r ¯ i u i bar(r)_(i)(u_(i))\bar{r}_{i}\left(u_{i}\right)r¯i(ui) is a known function, then the isolated zeros of (3) are implicit functions u i ( r ¯ ) u i ( r ¯ ) u_(i)( bar(r))u_{i}(\bar{r})ui(r¯). The implicit function theorem can be applied only at interior points and it does not ensure the existence of u i ( r → ) u i ( r → ) u_(i)( vec(r))u_{i}(\vec{r})ui(r→) for u i = t i ± u i = t i ± u_(i)=t_(i)^(+-)u_{i}=t_{i}^{ \pm}ui=ti±, i.e. r → ∈ ∂ S ( r → i ( t i ± ) , a ) r → ∈ ∂ S r → i t i ± , a vec(r)in del S( vec(r)_(i)(t_(i)^(+-)),a)\vec{r} \in \partial S\left(\vec{r}_{i}\left(t_{i}^{ \pm}\right), a\right)r→∈∂S(r→i(ti±),a). This case will be discussed separately. For u i ∈ ( t 1 + , t 1 − ) u i ∈ t 1 + , t 1 − u_(i)in(t_(1)^(+),t_(1)^(-))u_{i} \in\left(t_{1}^{+}, t_{1}^{-}\right)ui∈(t1+,t1−), if
(4) ∂ h i ∂ u i = 2 ( r ¯ i ( u i ) − r ¯ ) ⋅ ξ ¯ i ( u i ) ≠ 0 (4) ∂ h i ∂ u i = 2 r ¯ i u i − r ¯ ⋅ ξ ¯ i u i ≠ 0 {:(4)(delh_(i))/(delu_(i))=2( bar(r)_(i)(u_(i))-( bar(r)))* bar(xi)_(i)(u_(i))!=0:}\begin{equation*} \frac{\partial h_{i}}{\partial u_{i}}=2\left(\bar{r}_{i}\left(u_{i}\right)-\bar{r}\right) \cdot \bar{\xi}_{i}\left(u_{i}\right) \neq 0 \tag{4} \end{equation*}(4)∂hi∂ui=2(r¯i(ui)−r¯)⋅ξ¯i(ui)≠0
then the function u i = r → u i = r → u_(i)= vec(r)u_{i}=\vec{r}ui=r→ exists in a neighborhood of r → r → vec(r)\vec{r}r→ and has the derivatives
(5) ∂ u i ∂ x α = − ∂ h i ∂ x α / ∂ h i ∂ u i = x α i ( u i ) − x α ( r → i ( u i ) − r → ) ⋅ ξ → i ( u i ) , α = 1 , 2 , 3 (5) ∂ u i ∂ x α = − ∂ h i ∂ x α / ∂ h i ∂ u i = x α i u i − x α r → i u i − r → ⋅ ξ → i u i , α = 1 , 2 , 3 {:(5)(delu_(i))/(delx_(alpha))=-(delh_(i))/(delx_(alpha))//(delh_(i))/(delu_(i))=(x_(alpha i)(u_(i))-x_(alpha))/(( vec(r)_(i)(u_(i))-( vec(r)))* vec(xi)_(i)(u_(i)))","alpha=1","2","3:}\begin{equation*} \frac{\partial u_{i}}{\partial x_{\alpha}}=-\frac{\partial h_{i}}{\partial x_{\alpha}} / \frac{\partial h_{i}}{\partial u_{i}}=\frac{x_{\alpha i}\left(u_{i}\right)-x_{\alpha}}{\left(\vec{r}_{i}\left(u_{i}\right)-\vec{r}\right) \cdot \vec{\xi}_{i}\left(u_{i}\right)}, \alpha=1,2,3 \tag{5} \end{equation*}(5)∂ui∂xα=−∂hi∂xα/∂hi∂ui=xαi(ui)−xα(r→i(ui)−r→)⋅ξ→i(ui),α=1,2,3
where x α x α x_(alpha)x_{\alpha}xα are the components of r → r → vec(r)\vec{r}r→. According to (4), the function u i ( r → ) u i ( r → ) u_(i)( vec(r))u_{i}(\vec{r})ui(r→) is not differentiable at the points of the discriminant surface of the family { ∂ S ( r → i ( t ) , a ) ; t ∈ I i } ∂ S r → i ( t ) , a ; t ∈ I i {del S( vec(r)_(i)(t),a);t inI_(i)}\left\{\partial S\left(\vec{r}_{i}(t), a\right) ; t \in I_{i}\right\}{∂S(r→i(t),a);t∈Ii}.
We denote the moments when the i i iii-th particle enters (leaves) the sphere S ( r ¯ , a ) S ( r ¯ , a ) S( bar(r),a)S(\bar{r}, a)S(r¯,a) by t i + < u i 1 ′ < u i 2 ′ < … < u i n ′ ′ < t i − ( t i + < u i 1 ′ ′ < u i 2 ′ ′ < … < u i n ′ ′ ′ ′ < t i − ) t i + < u i 1 ′ < u i 2 ′ < … < u i n ′ ′ < t i − t i + < u i 1 ′ ′ < u i 2 ′ ′ < … < u i n ′ ′ ′ ′ < t i − t_(i)^(+) < u_(i1)^(') < u_(i2)^(') < dots < u_(in^('))^(') < t_(i)^(-)(t_(i)^(+) < u_(i1)^('') < u_(i2)^('') < dots < u_(in^(''))^('') < t_(i)^(-))t_{i}^{+}<u_{i 1}^{\prime}<u_{i 2}^{\prime}<\ldots<u_{i n^{\prime}}^{\prime}<t_{i}^{-}\left(t_{i}^{+}<u_{i 1}^{\prime \prime}<u_{i 2}^{\prime \prime}<\ldots<u_{i n^{\prime \prime}}^{\prime \prime}<t_{i}^{-}\right)ti+<ui1′<ui2′<…<uin′′<ti−(ti+<ui1′′<ui2′′<…<uin′′′′<ti−). Since the sphere S ( r ¯ , a ) S ( r ¯ , a ) S( bar(r),a)S(\bar{r}, a)S(r¯,a) is open, H + ( a 2 − ( r → i ( t ) − r ¯ ) 2 ) H + a 2 − r → i ( t ) − r ¯ 2 H^(+)(a^(2)-( vec(r)_(i)(t)-( bar(r)))^(2))H^{+}\left(a^{2}-\left(\vec{r}_{i}(t)-\bar{r}\right)^{2}\right)H+(a2−(r→i(t)−r¯)2) as a function of t t ttt is left (right) continuous when the particle enters (leaves). Hence for t ∈ I i t ∈ I i t inI_(i)t \in I_{i}t∈Ii, we have
(6) H + ( a 2 − ( r → i ( t ) − r ¯ ) 2 ) = H + ( a 2 − ( r ¯ i ( t i + ) − r ¯ ) 2 ) + + ∑ k ′ = 1 n ′ H + ( t − u i k ′ ′ ) − ∑ k ′ ′ = 1 n ′ ′ H − ( t − u i k ′ ′ ′ ′ ) (6) H + a 2 − r → i ( t ) − r ¯ 2 = H + a 2 − r ¯ i t i + − r ¯ 2 + + ∑ k ′ = 1 n ′   H + t − u i k ′ ′ − ∑ k ′ ′ = 1 n ′ ′   H − t − u i k ′ ′ ′ ′ {:[(6)H^(+)(a^(2)-( vec(r)_(i)(t)-( bar(r)))^(2))=H^(+)(a^(2)-( bar(r)_(i)(t_(i)^(+))-( bar(r)))^(2))+],[+sum_(k^(')=1)^(n^('))H^(+)(t-u_(ik^('))^('))-sum_(k^('')=1)^(n^(''))H^(-)(t-u_(ik^(''))^(''))]:}\begin{gather*} H^{+}\left(a^{2}-\left(\vec{r}_{i}(t)-\bar{r}\right)^{2}\right)=H^{+}\left(a^{2}-\left(\bar{r}_{i}\left(t_{i}^{+}\right)-\bar{r}\right)^{2}\right)+ \tag{6}\\ +\sum_{k^{\prime}=1}^{n^{\prime}} H^{+}\left(t-u_{i k^{\prime}}^{\prime}\right)-\sum_{k^{\prime \prime}=1}^{n^{\prime \prime}} H^{-}\left(t-u_{i k^{\prime \prime}}^{\prime \prime}\right) \end{gather*}(6)H+(a2−(r→i(t)−r¯)2)=H+(a2−(r¯i(ti+)−r¯)2)++∑k′=1n′H+(t−uik′′)−∑k′′=1n′′H−(t−uik′′′′)
where H − H − H^(-)H^{-}H−is the right continuous Heaviside jump function. The first term in the right-hand side of (6) vanishes if the i i iii-th particle is generated in the exterior of S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a) and equals 1 otherwise. If u i = t i + u i = t i + u_(i)=t_(i)^(+)u_{i}=t_{i}^{+}ui=ti+, then the particle can only enter the sphere after its generation, hence u i 1 ′ = t i + u i 1 ′ = t i + u_(i1)^(')=t_(i)^(+)u_{i 1}^{\prime}=t_{i}^{+}ui1′=ti+and relation (6) holds. If u i = t i − u i = t i − u_(i)=t_(i)^(-)u_{i}=t_{i}^{-}ui=ti−, then the particle could only leave the sphere before its disappearance, hence u i n ′ ′ ′ ′ = t i − u i n ′ ′ ′ ′ = t i − u_(in^(''))^('')=t_(i)^(-)u_{i n^{\prime \prime}}^{\prime \prime}=t_{i}^{-}uin′′′′=ti− and relation (6) holds again. So (6) contains all the possible situations if we take t i + ≤ u i 1 ′ t i + ≤ u i 1 ′ t_(i)^(+) <= u_(i1)^(')t_{i}^{+} \leq u_{i 1}^{\prime}ti+≤ui1′ and u i n ′ ′ ′ ′ ≤ t i − u i n ′ ′ ′ ′ ≤ t i − u_(in^(''))^('') <= t_(i)^(-)u_{i n^{\prime \prime}}^{\prime \prime} \leq t_{i}^{-}uin′′′′≤ti−. The following notation will be used:
(7) U i ′ = { u i 1 ′ , u i 2 ′ , … , u i n ′ ′ } and U i ′ ′ = { u i 1 ′ ′ , u i 2 ′ ′ , … , u i n ′ ′ ′ ′ } . (7) U i ′ = u i 1 ′ , u i 2 ′ , … , u i n ′ ′  and  U i ′ ′ = u i 1 ′ ′ , u i 2 ′ ′ , … , u i n ′ ′ ′ ′ . {:(7)U_(i)^(')={u_(i1)^('),u_(i2)^('),dots,u_(in^('))^(')}" and "U_(i)^('')={u_(i1)^(''),u_(i2)^(''),dots,u_(in^(''))^('')}.:}\begin{equation*} U_{i}^{\prime}=\left\{u_{i 1}^{\prime}, u_{i 2}^{\prime}, \ldots, u_{i n^{\prime}}^{\prime}\right\} \text { and } U_{i}^{\prime \prime}=\left\{u_{i 1}^{\prime \prime}, u_{i 2}^{\prime \prime}, \ldots, u_{i n^{\prime \prime}}^{\prime \prime}\right\} . \tag{7} \end{equation*}(7)Ui′={ui1′,ui2′,…,uin′′} and Ui′′={ui1′′,ui2′′,…,uin′′′′}.
Proposition 1. The function D φ D φ D_(varphi)D_{\varphi}Dφ defined by (2) has partial derivatives continuous a.e. in R 3 × ( τ , T − τ ) R 3 × ( τ , T − τ ) R^(3)xx(tau,T-tau)\mathbb{R}^{3} \times(\tau, T-\tau)R3×(τ,T−τ).
Proof. To study the differentiability of D φ D φ D_(varphi)D_{\varphi}Dφ, we consider the function g i : R 3 × ( τ , T − τ ) → R g i : R 3 × ( τ , T − τ ) → R g_(i):R^(3)xx(tau,T-tau)rarrRg_{i}: \mathbb{R}^{3} \times(\tau, T-\tau) \rightarrow \mathbb{R}gi:R3×(τ,T−τ)→R
(8) g i ( r → , t ) = ∫ t − τ t + τ φ i ( t ′ ) H + ( a 2 − ( r → i ( t ′ ) − r ¯ ) 2 ) d t . (8) g i ( r → , t ) = ∫ t − τ t + τ   φ i t ′ H + a 2 − r → i t ′ − r ¯ 2 d t . {:(8)g_(i)( vec(r)","t)=int_(t-tau)^(t+tau)varphi_(i)(t^('))H^(+)(a^(2)-( vec(r)_(i)(t^('))-( bar(r)))^(2))dt.:}\begin{equation*} g_{i}(\vec{r}, t)=\int_{t-\tau}^{t+\tau} \varphi_{i}\left(t^{\prime}\right) H^{+}\left(a^{2}-\left(\vec{r}_{i}\left(t^{\prime}\right)-\bar{r}\right)^{2}\right) \mathrm{d} t . \tag{8} \end{equation*}(8)gi(r→,t)=∫t−τt+τφi(t′)H+(a2−(r→i(t′)−r¯)2)dt.
For a fixed r → r → vec(r)\vec{r}r→, the integrand
(9) G i ( r ⇀ , t ) = φ ( t ) H + ( a 2 − ( r ⇀ i ( t ) − r ⇀ ) 2 ) (9) G i ( r ⇀ , t ) = φ ( t ) H + a 2 − r ⇀ i ( t ) − r ⇀ 2 {:(9)G_(i)(r^(⇀)","t)=varphi(t)H^(+)(a^(2)-(r^(⇀)_(i)(t)-(r^(⇀)))^(2)):}\begin{equation*} G_{i}(\stackrel{\rightharpoonup}{r}, t)=\varphi(t) H^{+}\left(a^{2}-\left(\stackrel{\rightharpoonup}{r}_{i}(t)-\stackrel{\rightharpoonup}{r}\right)^{2}\right) \tag{9} \end{equation*}(9)Gi(r⇀,t)=φ(t)H+(a2−(r⇀i(t)−r⇀)2)
is a continuous function, except a finite number of jump discontinuities { t i + , t i − } ∪ U i ′ ∪ U i ′ ′ t i + , t i − ∪ U i ′ ∪ U i ′ ′ {t_(i)^(+),t_(i)^(-)}uuU_(i)^(')uuU_(i)^('')\left\{t_{i}^{+}, t_{i}^{-}\right\} \cup U_{i}^{\prime} \cup U_{i}^{\prime \prime}{ti+,ti−}∪Ui′∪Ui′′. Hence G i G i G_(i)G_{i}Gi is Riemann integrable and g i g i g_(i)g_{i}gi has partial derivative with respect to t t ttt a.e. in ( τ , T − τ ) ( τ , T − τ ) (tau,T-tau)(\tau, T-\tau)(τ,T−τ), equal to
(10) ∂ t g i ( r ⇀ , t ) = G i ( r ⇀ , t + τ ) − G i ( r ⇀ , t − τ ) . (10) ∂ t g i ( r ⇀ , t ) = G i ( r ⇀ , t + τ ) − G i ( r ⇀ , t − τ ) . {:(10)del_(t)g_(i)(r^(⇀)","t)=G_(i)(r^(⇀)","t+tau)-G_(i)(r^(⇀)","t-tau).:}\begin{equation*} \partial_{t} g_{i}(\stackrel{\rightharpoonup}{r}, t)=G_{i}(\stackrel{\rightharpoonup}{r}, t+\tau)-G_{i}(\stackrel{\rightharpoonup}{r}, t-\tau) . \tag{10} \end{equation*}(10)∂tgi(r⇀,t)=Gi(r⇀,t+τ)−Gi(r⇀,t−τ).
The discontinuities of ∂ t g i ∂ t g i del_(t)g_(i)\partial_{t} g_{i}∂tgi with respect to ( r → , t r → , t vec(r),t\vec{r}, tr→,t ) are related to those of G i G i G_(i)G_{i}Gi. From (9) it follows that G i G i G_(i)G_{i}Gi is discontinuous when φ i φ i varphi_(i)\varphi_{i}φi is discontinuous and H + H + H^(+)H^{+}H+ nonvanishing, or conversely, when H + H + H^(+)H^{+}H+is discontinuous and φ i φ i varphi_(i)\varphi_{i}φi nonvanishing. In the first case the i -th particle appears or disappears in S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a), i.e. t = t i ± t = t i ± t=t_(i)^(+-)t=t_{i}^{ \pm}t=ti±and r → ∈ S ( r → i ( t i ± ) , a ) r → ∈ S r → i t i ± , a vec(r)in S( vec(r)_(i)(t_(i)^(+-)),a)\vec{r} \in S\left(\vec{r}_{i}\left(t_{i}^{ \pm}\right), a\right)r→∈S(r→i(ti±),a), and in the second case the i i iii-th particle lies in the surface of S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a), i.e. t ∈ I i t ∈ I i t inI_(i)t \in I_{i}t∈Ii and r → ∈ ∂ S ( r → i ( t ) , a ) r → ∈ ∂ S r → i ( t ) , a vec(r)in del S( vec(r)_(i)(t),a)\vec{r} \in \partial S\left(\vec{r}_{i}(t), a\right)r→∈∂S(r→i(t),a).
Hence the derivative (10) is not continuous over
(11) Ω i ′ = { ( r → , t ) ∣ t ∈ { t i ± − τ , t i ± + τ } ∩ ( τ , T − τ ) and r → ∈ S ( r → i ( t i ± ) , a ) } ∪ ∪ { ( r → , t ) ∣ t ˙ ∈ [ t i + ± τ , t i − ± τ ] ∩ ( τ , T − τ ) and r → ∈ ∂ S ( r → i ( t ∓ τ ) , a ) } . (11) Ω i ′ = ( r → , t ) ∣ t ∈ t i ± − τ , t i ± + τ ∩ ( τ , T − τ )  and  r → ∈ S r → i t i ± , a ∪ ∪ ( r → , t ) ∣ t ˙ ∈ t i + ± τ , t i − ± τ ∩ ( τ , T − τ )  and  r → ∈ ∂ S r → i ( t ∓ τ ) , a . {:[(11)Omega_(i)^(')={(( vec(r)),t)∣t in{t_(i)^(+-)-tau,t_(i)^(+-)+tau}nn(tau,T-tau)" and "( vec(r))in S( vec(r)_(i)(t_(i)^(+-)),a)}uu],[ uu{(( vec(r)),t)∣(t^(˙))in[t_(i)^(+)+-tau,t_(i)^(-)+-tau]nn(tau,T-tau)" and "( vec(r))in del S( vec(r)_(i)(t∓tau),a)}.]:}\begin{align*} & \Omega_{i}^{\prime}=\left\{(\vec{r}, t) \mid t \in\left\{t_{i}^{ \pm}-\tau, t_{i}^{ \pm}+\tau\right\} \cap(\tau, T-\tau) \text { and } \vec{r} \in S\left(\vec{r}_{i}\left(t_{i}^{ \pm}\right), a\right)\right\} \cup \tag{11}\\ & \cup\left\{(\vec{r}, t) \mid \dot{t} \in\left[t_{i}^{+} \pm \tau, t_{i}^{-} \pm \tau\right] \cap(\tau, T-\tau) \text { and } \vec{r} \in \partial S\left(\vec{r}_{i}(t \mp \tau), a\right)\right\} . \end{align*}(11)Ωi′={(r→,t)∣t∈{ti±−τ,ti±+τ}∩(τ,T−τ) and r→∈S(r→i(ti±),a)}∪∪{(r→,t)∣t˙∈[ti+±τ,ti−±τ]∩(τ,T−τ) and r→∈∂S(r→i(t∓τ),a)}.
The set Ω i ′ Ω i ′ Omega_(i)^(')\Omega_{i}^{\prime}Ωi′ has null Lebesgue measure in R 3 × ( τ , T − τ ) R 3 × ( τ , T − τ ) R^(3)xx(tau,T-tau)\mathbb{R}^{3} \times(\tau, T-\tau)R3×(τ,T−τ), hence ∂ t g i ∂ t g i del_(t)g_(i)\partial_{t} g_{i}∂tgi is a.e. continuous.
In the appendix we show that the derivative of g i g i g_(i)g_{i}gi with respect to x α x α x_(alpha)x_{\alpha}xα exists and is continuous a.e. in R 3 × ( τ , T − τ ) R 3 × ( τ , T − τ ) R^(3)xx(tau,T-tau)\mathbb{R}^{3} \times(\tau, T-\tau)R3×(τ,T−τ). Using definition (2) and relations (10), (11), (A3) and (A4), it follows the a.e. continuity of the partial derivatives of D φ D φ D_(varphi)D_{\varphi}Dφ, given by
(12) ∂ t D φ ( r ⇀ , t ) = 1 2 τ V ∑ i = 1 N [ G i ( r ⇀ , t + τ ) − G i ( r ⇀ , t − τ ) ] (12) ∂ t D φ ( r ⇀ , t ) = 1 2 τ V ∑ i = 1 N   G i ( r ⇀ , t + τ ) − G i ( r ⇀ , t − τ ) {:(12)del_(t)D_(varphi)(r^(⇀)","t)=(1)/(2tau V)sum_(i=1)^(N)[G_(i)((r^(⇀)),t+tau)-G_(i)((r^(⇀)),t-tau)]:}\begin{equation*} \partial_{t} D_{\varphi}(\stackrel{\rightharpoonup}{r}, t)=\frac{1}{2 \tau V} \sum_{i=1}^{N}\left[G_{i}(\stackrel{\rightharpoonup}{r}, t+\tau)-G_{i}(\stackrel{\rightharpoonup}{r}, t-\tau)\right] \tag{12} \end{equation*}(12)∂tDφ(r⇀,t)=12τV∑i=1N[Gi(r⇀,t+τ)−Gi(r⇀,t−τ)]
for ( r ⇀ , t ) ∈ R 3 × ( τ , T − τ ) ∖ ⋃ i = 1 N Ω i ′ ( r ⇀ , t ) ∈ R 3 × ( τ , T − τ ) ∖ ⋃ i = 1 N   Ω i ′ (r^(⇀),t)inR^(3)xx(tau,T-tau)\\uuu_(i=1)^(N)Omega_(i)^(')(\stackrel{\rightharpoonup}{r}, t) \in \mathbb{R}^{3} \times(\tau, T-\tau) \backslash \bigcup_{i=1}^{N} \Omega_{i}^{\prime}(r⇀,t)∈R3×(τ,T−τ)∖⋃i=1NΩi′, and
(13) ∂ α D φ ( r → , t ) = 1 2 τ V ∑ i = 1 N ∑ u ∈ U i φ i ( u ) x α i ( u ) − x α | ( r → i ( u ) − r → ) ⋅ ξ → i ( u ) | (13) ∂ α D φ ( r → , t ) = 1 2 τ V ∑ i = 1 N   ∑ u ∈ U i   φ i ( u ) x α i ( u ) − x α r → i ( u ) − r → ⋅ ξ → i ( u ) {:(13)del_(alpha)D_(varphi)( vec(r)","t)=(1)/(2tau V)sum_(i=1)^(N)sum_(u inU_(i))varphi_(i)(u)(x_(alpha i)(u)-x_(alpha))/(|( vec(r)_(i)(u)-( vec(r)))* vec(xi)_(i)(u)|):}\begin{equation*} \partial_{\alpha} D_{\varphi}(\vec{r}, t)=\frac{1}{2 \tau V} \sum_{i=1}^{N} \sum_{u \in U_{i}} \varphi_{i}(u) \frac{x_{\alpha i}(u)-x_{\alpha}}{\left|\left(\vec{r}_{i}(u)-\vec{r}\right) \cdot \vec{\xi}_{i}(u)\right|} \tag{13} \end{equation*}(13)∂αDφ(r→,t)=12τV∑i=1N∑u∈Uiφi(u)xαi(u)−xα|(r→i(u)−r→)⋅ξ→i(u)|
for ( r → , t ) ∈ R 3 × ( τ , T − τ ) ∖ ⋃ i = 1 N Ω i ′ ′ ( r → , t ) ∈ R 3 × ( τ , T − τ ) ∖ ⋃ i = 1 N   Ω i ′ ′ ( vec(r),t)inR^(3)xx(tau,T-tau)\\uuu_(i=1)^(N)Omega_(i)^('')(\vec{r}, t) \in \mathbb{R}^{3} \times(\tau, T-\tau) \backslash \bigcup_{i=1}^{N} \Omega_{i}^{\prime \prime}(r→,t)∈R3×(τ,T−τ)∖⋃i=1NΩi′′.

3. THE DISCRETE ANALOGUE OF BALANCE EQUATION

PROPOSITION 2. When the derivatives (12) and (13) exist, the function D φ D φ D_(varphi)D_{\varphi}Dφ satisfies the following relation
(14) ∂ t D φ + ∑ α = 1 3 ∂ α D φ ξ α = D φ ˙ + ( ∂ t D φ ) g (14) ∂ t D φ + ∑ α = 1 3   ∂ α D φ ξ α = D φ ˙ + ∂ t D φ g {:(14)del_(t)D_(varphi)+sum_(alpha=1)^(3)del_(alpha)D_(varphixi_(alpha))=D_(varphi^(˙))+(del_(t)D_(varphi))_(g):}\begin{equation*} \partial_{t} D_{\varphi}+\sum_{\alpha=1}^{3} \partial_{\alpha} D_{\varphi \xi_{\alpha}}=D_{\dot{\varphi}}+\left(\partial_{t} D_{\varphi}\right)_{g} \tag{14} \end{equation*}(14)∂tDφ+∑α=13∂αDφξα=Dφ˙+(∂tDφ)g
where ( ∂ t D φ ) g ∂ t D φ g (del_(t)D_(varphi))_(g)\left(\partial_{t} D_{\varphi}\right)_{g}(∂tDφ)g is determined by the particles generation.
Proof. We use a theorem stating that every function with bounded variation may be uniquely split into a sum of two functions: one continuous and a jump function ([8], p331). We apply this theorem to G i G i G_(i)G_{i}Gi given by (9) considered as a function of t t ttt. But except a finite number of jump discontinuities, G i G i G_(i)G_{i}Gi is analytic on I I III and then its continuous part G i ′ G i ′ G_(i)^(')G_{i}^{\prime}Gi′ is also absolutely continuous. Hence we may write G i = G i ′ + G i ′ ′ G i = G i ′ + G i ′ ′ G_(i)=G_(i)^(')+G_(i)^('')G_{i}=G_{i}^{\prime}+G_{i}^{\prime \prime}Gi=Gi′+Gi′′, where G i ′ ′ G i ′ ′ G_(i)^('')G_{i}^{\prime \prime}Gi′′ is the jump function. Replacing this relation in (12), it follows that ∂ t D φ ∂ t D φ del_(t)D_(varphi)\partial_{t} D_{\varphi}∂tDφ can also be written as a two term sum
(15) ∂ t D φ = ( ∂ t D φ ) ′ + ( ∂ t D φ ) ′ ′ (15) ∂ t D φ = ∂ t D φ ′ + ∂ t D φ ′ ′ {:(15)del_(t)D_(varphi)=(del_(t)D_(varphi))^(')+(del_(t)D_(varphi))^(''):}\begin{equation*} \partial_{t} D_{\varphi}=\left(\partial_{t} D_{\varphi}\right)^{\prime}+\left(\partial_{t} D_{\varphi}\right)^{\prime \prime} \tag{15} \end{equation*}(15)∂tDφ=(∂tDφ)′+(∂tDφ)′′
According to Lebesgue theorem, the absolutely continuous part of G i G i G_(i)G_{i}Gi is equal to
(16) G i ′ ( r → , t + τ ) − G i ′ ( r → , t − τ ) = ∫ t − τ t + τ φ ˙ i ( t ′ ) H + ( a 2 − ( r → i ( t ′ ) − r → ) 2 ) d t ′ (16) G i ′ ( r → , t + τ ) − G i ′ ( r → , t − τ ) = ∫ t − τ t + τ   φ ˙ i t ′ H + a 2 − r → i t ′ − r → 2 d t ′ {:(16)G_(i)^(')( vec(r)","t+tau)-G_(i)^(')( vec(r)","t-tau)=int_(t-tau)^(t+tau)varphi^(˙)_(i)(t^('))H^(+)(a^(2)-( vec(r)_(i)(t^('))-( vec(r)))^(2))dt^('):}\begin{equation*} G_{i}^{\prime}(\vec{r}, t+\tau)-G_{i}^{\prime}(\vec{r}, t-\tau)=\int_{t-\tau}^{t+\tau} \dot{\varphi}_{i}\left(t^{\prime}\right) H^{+}\left(a^{2}-\left(\vec{r}_{i}\left(t^{\prime}\right)-\vec{r}\right)^{2}\right) \mathrm{d} t^{\prime} \tag{16} \end{equation*}(16)Gi′(r→,t+τ)−Gi′(r→,t−τ)=∫t−τt+τφ˙i(t′)H+(a2−(r→i(t′)−r→)2)dt′
Dividing (16) by 2 τ V 2 τ V 2tau V2 \tau V2τV, summing up with respect to i i iii, taking into account (9), (12) and (2) we obtain
(17) ( ∂ t D φ ) ′ = D φ ˙ . (17) ∂ t D φ ′ = D φ ˙ . {:(17)(del_(t)D_(varphi))^(')=D_(varphi^(˙)).:}\begin{equation*} \left(\partial_{t} D_{\varphi}\right)^{\prime}=D_{\dot{\varphi}} . \tag{17} \end{equation*}(17)(∂tDφ)′=Dφ˙.
From (12), the discontinuous part of ∂ t D φ ∂ t D φ del_(t)D_(varphi)\partial_{t} D_{\varphi}∂tDφ can be written as
(18) ( ∂ t D φ ) ′ ′ ( r ⇀ , t ) = 1 2 τ V ∑ i = 1 N [ G i ′ ′ ( r ⇀ , t + τ ) − G i ′ ′ ( r ⇀ , t − τ ) ] (18) ∂ t D φ ′ ′ ( r ⇀ , t ) = 1 2 τ V ∑ i = 1 N   G i ′ ′ ( r ⇀ , t + τ ) − G i ′ ′ ( r ⇀ , t − τ ) {:(18)(del_(t)D_(varphi))^('')(r^(⇀)","t)=(1)/(2tau V)sum_(i=1)^(N)[G_(i)^('')((r^(⇀)),t+tau)-G_(i)^('')((r^(⇀)),t-tau)]:}\begin{equation*} \left(\partial_{t} D_{\varphi}\right)^{\prime \prime}(\stackrel{\rightharpoonup}{r}, t)=\frac{1}{2 \tau V} \sum_{i=1}^{N}\left[G_{i}^{\prime \prime}(\stackrel{\rightharpoonup}{r}, t+\tau)-G_{i}^{\prime \prime}(\stackrel{\rightharpoonup}{r}, t-\tau)\right] \tag{18} \end{equation*}(18)(∂tDφ)′′(r⇀,t)=12τV∑i=1N[Gi′′(r⇀,t+τ)−Gi′′(r⇀,t−τ)]
It contains the discontinuous variations of G i G i G_(i)G_{i}Gi during the temporal interval [ t − τ , t + τ ] [ t − τ , t + τ ] [t-tau,t+tau][t-\tau, t+\tau][t−τ,t+τ]. As proved in the preceding section, ∂ t D φ ∂ t D φ del_(t)D_(varphi)\partial_{t} D_{\varphi}∂tDφ exists if G i G i G_(i)G_{i}Gi is not discontinuous at t + τ t + τ t+taut+\taut+τ and t − τ t − τ t-taut-\taut−τ (see expression (11)), therefore we consider only the jumps occuring at the interior points of [ t − τ , t + τ ] [ t − τ , t + τ ] [t-tau,t+tau][t-\tau, t+\tau][t−τ,t+τ], i.e. in ( t − τ , t + τ t − τ , t + τ t-tau,t+taut-\tau, t+\taut−τ,t+τ ). From ( 9 ) it follows that such a variation can take place if the particle is generated inside the sphere S ( r ¯ , a ) S ( r ¯ , a ) S( bar(r),a)S(\bar{r}, a)S(r¯,a) during the temporal interval ( t − τ , t + τ t − τ , t + τ t-tau,t+taut-\tau, t+\taut−τ,t+τ ). Hence the jump of G i G i G_(i)G_{i}Gi is equal to
Δ + G i = φ i ( t i + ) H + ( a 2 − ( r ¯ i ( t i + ) − r ¯ ) 2 ) ( H + ( t + τ − t i + ) − H − ( t − τ − t i + ) ) . Δ + G i = φ i t i + H + a 2 − r ¯ i t i + − r ¯ 2 H + t + τ − t i + − H − t − τ − t i + . Delta^(+)G_(i)=varphi_(i)(t_(i)^(+))H^(+)(a^(2)-( bar(r)_(i)(t_(i)^(+))-( bar(r)))^(2))(H^(+)(t+tau-t_(i)^(+))-H^(-)(t-tau-t_(i)^(+))).\Delta^{+} G_{i}=\varphi_{i}\left(t_{i}^{+}\right) H^{+}\left(a^{2}-\left(\bar{r}_{i}\left(t_{i}^{+}\right)-\bar{r}\right)^{2}\right)\left(H^{+}\left(t+\tau-t_{i}^{+}\right)-H^{-}\left(t-\tau-t_{i}^{+}\right)\right) .Δ+Gi=φi(ti+)H+(a2−(r¯i(ti+)−r¯)2)(H+(t+τ−ti+)−H−(t−τ−ti+)).
Similarly, the discontinuous variation of G i G i G_(i)G_{i}Gi related to the destruction of a particle is
Δ − G i = − φ i ( t i − ) H + ( a 2 − ( r ¯ i ( t i − ) − r ¯ ) 2 ) ( H + ( t + τ − t i − ) − H − ( t − τ − t i − ) ) . Δ − G i = − φ i t i − H + a 2 − r ¯ i t i − − r ¯ 2 H + t + τ − t i − − H − t − τ − t i − . Delta^(-)G_(i)=-varphi_(i)(t_(i)^(-))H^(+)(a^(2)-( bar(r)_(i)(t_(i)^(-))-( bar(r)))^(2))(H^(+)(t+tau-t_(i)^(-))-H^(-)(t-tau-t_(i)^(-))).\Delta^{-} G_{i}=-\varphi_{i}\left(t_{i}^{-}\right) H^{+}\left(a^{2}-\left(\bar{r}_{i}\left(t_{i}^{-}\right)-\bar{r}\right)^{2}\right)\left(H^{+}\left(t+\tau-t_{i}^{-}\right)-H^{-}\left(t-\tau-t_{i}^{-}\right)\right) .Δ−Gi=−φi(ti−)H+(a2−(r¯i(ti−)−r¯)2)(H+(t+τ−ti−)−H−(t−τ−ti−)).
The function G i G i G_(i)G_{i}Gi also has discontinuous variations when the particle enters or leaves the sphere S ( r ¯ , a ) S ( r ¯ , a ) S( bar(r),a)S(\bar{r}, a)S(r¯,a)
(19) G i ′ ′ ( r ⇀ , t + τ ) − G i ′ ′ ( r ⇀ , t − τ ) = Δ + G i + Δ − G i + ∑ u ∈ W i ′ φ i ( u ) − ∑ u ∈ W i ′ ′ φ i ( u ) (19) G i ′ ′ ( r ⇀ , t + τ ) − G i ′ ′ ( r ⇀ , t − τ ) = Δ + G i + Δ − G i + ∑ u ∈ W i ′   φ i ( u ) − ∑ u ∈ W i ′ ′   φ i ( u ) {:(19)G_(i)^('')(r^(⇀)","t+tau)-G_(i)^('')(r^(⇀)","t-tau)=Delta^(+)G_(i)+Delta^(-)G_(i)+sum_(u inW_(i)^('))varphi_(i)(u)-sum_(u inW_(i)^(''))varphi_(i)(u):}\begin{equation*} G_{i}^{\prime \prime}(\stackrel{\rightharpoonup}{r}, t+\tau)-G_{i}^{\prime \prime}(\stackrel{\rightharpoonup}{r}, t-\tau)=\Delta^{+} G_{i}+\Delta^{-} G_{i}+\sum_{u \in W_{i}^{\prime}} \varphi_{i}(u)-\sum_{u \in W_{i}^{\prime \prime}} \varphi_{i}(u) \tag{19} \end{equation*}(19)Gi′′(r⇀,t+τ)−Gi′′(r⇀,t−τ)=Δ+Gi+Δ−Gi+∑u∈Wi′φi(u)−∑u∈Wi′′φi(u)
where W i ′ = U i ′ ∩ ( t − τ , t + τ ) W i ′ = U i ′ ∩ ( t − τ , t + τ ) W_(i)^(')=U_(i)^(')nn(t-tau,t+tau)W_{i}^{\prime}=U_{i}^{\prime} \cap(t-\tau, t+\tau)Wi′=Ui′∩(t−τ,t+τ) and W i ′ ′ = U i ′ ′ ∩ ( t − τ , t + τ ) W i ′ ′ = U i ′ ′ ∩ ( t − τ , t + τ ) W_(i)^('')=U_(i)^('')nn(t-tau,t+tau)W_{i}^{\prime \prime}=U_{i}^{\prime \prime} \cap(t-\tau, t+\tau)Wi′′=Ui′′∩(t−τ,t+τ). The sign of φ i ( u ) φ i ( u ) varphi_(i)(u)\varphi_{i}(u)φi(u) is positive (negative) if the particle enters (leaves) the sphere S ( r ¯ , a ) S ( r ¯ , a ) S( bar(r),a)S(\bar{r}, a)S(r¯,a), and it is given by the sign of the expression − ( r ¯ i ( u ) − r ¯ ) ⋅ ξ ¯ i ( u ) − r ¯ i ( u ) − r ¯ ⋅ ξ ¯ i ( u ) -( bar(r)_(i)(u)-( bar(r)))* bar(xi)_(i)(u)-\left(\bar{r}_{i}(u)-\bar{r}\right) \cdot \bar{\xi}_{i}(u)−(r¯i(u)−r¯)⋅ξ¯i(u) which is proportional to the interior normal component of ξ → i ξ → i vec(xi)_(i)\vec{\xi}_{i}ξ→i to the surface of S ( r ¯ , a ) S ( r ¯ , a ) S( bar(r),a)S(\bar{r}, a)S(r¯,a) at the moment u u uuu. Hence we may use a single sum in (19) if we denote U i = W i ′ ∪ W i ′ ′ U i = W i ′ ∪ W i ′ ′ U_(i)=W_(i)^(')uuW_(i)^('')U_{i}=W_{i}^{\prime} \cup W_{i}^{\prime \prime}Ui=Wi′∪Wi′′. Replacing (19) in (18) we obtain
(20) ( ∂ t D φ ) ′ ′ = ( ∂ t D φ ) g − 1 2 τ V ∑ i = 1 N ∑ u ∈ U i φ i ( u ) ( r ¯ i ( u ) − r ¯ ) ⋅ ξ ¯ i ( u ) | ( r ¯ i ( u ) − r ¯ ) ⋅ ξ ¯ i ( u ) | (20) ∂ t D φ ′ ′ = ∂ t D φ g − 1 2 τ V ∑ i = 1 N   ∑ u ∈ U i   φ i ( u ) r ¯ i ( u ) − r ¯ ⋅ ξ ¯ i ( u ) r ¯ i ( u ) − r ¯ ⋅ ξ ¯ i ( u ) {:(20)(del_(t)D_(varphi))^('')=(del_(t)D_(varphi))_(g)-(1)/(2tau V)sum_(i=1)^(N)sum_(u inU_(i))varphi_(i)(u)(( bar(r)_(i)(u)-( bar(r)))* bar(xi)_(i)(u))/(|( bar(r)_(i)(u)-( bar(r)))* bar(xi)_(i)(u)|):}\begin{equation*} \left(\partial_{t} D_{\varphi}\right)^{\prime \prime}=\left(\partial_{t} D_{\varphi}\right)_{g}-\frac{1}{2 \tau V} \sum_{i=1}^{N} \sum_{u \in U_{i}} \varphi_{i}(u) \frac{\left(\bar{r}_{i}(u)-\bar{r}\right) \cdot \bar{\xi}_{i}(u)}{\left|\left(\bar{r}_{i}(u)-\bar{r}\right) \cdot \bar{\xi}_{i}(u)\right|} \tag{20} \end{equation*}(20)(∂tDφ)′′=(∂tDφ)g−12τV∑i=1N∑u∈Uiφi(u)(r¯i(u)−r¯)⋅ξ¯i(u)|(r¯i(u)−r¯)⋅ξ¯i(u)|
where
(21) ( ∂ t D φ ) g = 1 2 τ V ∑ i = 1 N ( Δ + G i + Δ − G i ) . (21) ∂ t D φ g = 1 2 τ V ∑ i = 1 N   Δ + G i + Δ − G i . {:(21)(del_(t)D_(varphi))_(g)=(1)/(2tau V)sum_(i=1)^(N)(Delta^(+)G_(i)+Delta^(-)G_(i)).:}\begin{equation*} \left(\partial_{t} D_{\varphi}\right)_{g}=\frac{1}{2 \tau V} \sum_{i=1}^{N}\left(\Delta^{+} G_{i}+\Delta^{-} G_{i}\right) . \tag{21} \end{equation*}(21)(∂tDφ)g=12τV∑i=1N(Δ+Gi+Δ−Gi).
Since φ i ( t ) φ i ( t ) varphi_(i)(t)\varphi_{i}(t)φi(t) and ξ α i ( t ) ξ α i ( t ) xi_(alpha i)(t)\xi_{\alpha i}(t)ξαi(t) are analytic functions with respect to time, then their product is also analytic and relation (20) can be written as
(22) ( ∂ t D φ ) ′ ′ = − ∑ α = 1 3 ∂ α D φ α g + ( ∂ t D φ ) g . (22) ∂ t D φ ′ ′ = − ∑ α = 1 3   ∂ α D φ α g + ∂ t D φ g . {:(22)(del_(t)D_(varphi))^('')=-sum_(alpha=1)^(3)del_(alpha)D_(varphi_(alpha)^(g))+(del_(t)D_(varphi))_(g).:}\begin{equation*} \left(\partial_{t} D_{\varphi}\right)^{\prime \prime}=-\sum_{\alpha=1}^{3} \partial_{\alpha} D_{\varphi_{\alpha}^{g}}+\left(\partial_{t} D_{\varphi}\right)_{g} . \tag{22} \end{equation*}(22)(∂tDφ)′′=−∑α=13∂αDφαg+(∂tDφ)g.
The physical quantity φ i ξ → φ i ξ → varphi_(i) vec(xi)\varphi_{i} \vec{\xi}φiξ→ represents the transport of φ φ varphi\varphiφ by the i i iii-th particle, and the space-time average D φ → ξ → D φ → ξ → D_( vec(varphi) vec(xi))D_{\vec{\varphi} \vec{\xi}}Dφ→ξ→ represents the mean flux of φ φ varphi\varphiφ.
Relation (14) follows from (15), (17) and (22).
In contrast to balance equation (1), relation (14) does not contain a quantity equivalent to the velocity v → v → vec(v)\vec{v}v→. The velocity is not a volume density, but an average quantity. To define a discrete analogue, we must divide D φ D φ D_(varphi)D_{\varphi}Dφ by the number of the particles contributing to D φ D φ D_(varphi)D_{\varphi}Dφ. Let D 1 D 1 D_(1)D_{1}D1 be the density D φ D φ D_(varphi)D_{\varphi}Dφ corresponding to φ i ( t ) = 1 φ i ( t ) = 1 varphi_(i)(t)=1\varphi_{i}(t)=1φi(t)=1 for all t ∈ I i t ∈ I i t inI_(i)t \in I_{i}t∈Ii and i ≤ N i ≤ N i <= Ni \leq Ni≤N. Since D 1 D 1 D_(1)D_{1}D1 characterizes the average number of particles per unit volume, the discrete average of φ φ varphi\varphiφ is defined as
(23) φ ¯ ( r ⇀ , t ) = D φ ( r ⇀ , t ) / D 1 ( r ⇀ , t ) (23) φ ¯ ( r ⇀ , t ) = D φ ( r ⇀ , t ) / D 1 ( r ⇀ , t ) {:(23) bar(varphi)(r^(⇀)","t)=D_(varphi)(r^(⇀)","t)//D_(1)(r^(⇀)","t):}\begin{equation*} \bar{\varphi}(\stackrel{\rightharpoonup}{r}, t)=D_{\varphi}(\stackrel{\rightharpoonup}{r}, t) / D_{1}(\stackrel{\rightharpoonup}{r}, t) \tag{23} \end{equation*}(23)φ¯(r⇀,t)=Dφ(r⇀,t)/D1(r⇀,t)
if D 1 ( r → , t ) ≠ 0 D 1 ( r → , t ) ≠ 0 D_(1)( vec(r),t)!=0D_{1}(\vec{r}, t) \neq 0D1(r→,t)≠0 and it vanishes if D 1 ( r → , t ) = 0 D 1 ( r → , t ) = 0 D_(1)( vec(r),t)=0D_{1}(\vec{r}, t)=0D1(r→,t)=0. It is easy to show that φ ¯ ― = φ ¯ , φ 1 + φ 2 ― = φ 1 ― + φ 2 ― φ ¯ ¯ = φ ¯ , φ 1 + φ 2 ¯ = φ 1 ¯ + φ 2 ¯ bar(bar(varphi))= bar(varphi), bar(varphi_(1)+varphi_(2))= bar(varphi_(1))+ bar(varphi_(2))\overline{\bar{\varphi}}=\bar{\varphi}, \overline{\varphi_{1}+\varphi_{2}}=\overline{\varphi_{1}}+\overline{\varphi_{2}}φ¯―=φ¯,φ1+φ2―=φ1―+φ2― and λ φ ― = λ φ ¯ λ φ ¯ = λ φ ¯ bar(lambda varphi)=lambda bar(varphi)\overline{\lambda \varphi}=\lambda \bar{\varphi}λφ―=λφ¯, where λ λ lambda\lambdaλ is a real function of r ¯ r ¯ bar(r)\bar{r}r¯ and t t ttt.
The mean motion of the particle is given by the discrete average of the velocity ξ ¯ ― ξ ¯ ¯ bar(bar(xi))\overline{\bar{\xi}}ξ¯― with the components ξ ¯ α ξ ¯ α bar(xi)_(alpha)\bar{\xi}_{\alpha}ξ¯α. To introduce ξ ¯ α ξ ¯ α bar(xi)_(alpha)\bar{\xi}_{\alpha}ξ¯α in (14), we write
D φ ξ α = D φ [ ξ ¯ α + ( ξ α − ξ ¯ α ) ] = ξ ¯ α D φ + ( Φ ¯ φ ′ ) α D φ ξ α = D φ ξ ¯ α + ξ α − ξ ¯ α = ξ ¯ α D φ + Φ ¯ φ ′ α D_(varphixi_(alpha))=D_(varphi[ bar(xi)_(alpha)+(xi_(alpha)- bar(xi)_(alpha))])= bar(xi)_(alpha)D_(varphi)+( bar(Phi)_(varphi)^('))_(alpha)D_{\varphi \xi_{\alpha}}=D_{\varphi\left[\bar{\xi}_{\alpha}+\left(\xi_{\alpha}-\bar{\xi}_{\alpha}\right)\right]}=\bar{\xi}_{\alpha} D_{\varphi}+\left(\bar{\Phi}_{\varphi}^{\prime}\right)_{\alpha}Dφξα=Dφ[ξ¯α+(ξα−ξ¯α)]=ξ¯αDφ+(Φ¯φ′)α
where Φ ¯ φ ′ Φ ¯ φ ′ bar(Phi)_(varphi)^(')\bar{\Phi}_{\varphi}^{\prime}Φ¯φ′ is the discrete analogue of the kinetic part of the flux density
Φ ¯ φ ′ = ∑ α = 1 3 D ˙ φ ( ξ α − ξ ¯ α ) e ¯ α Φ ¯ φ ′ = ∑ α = 1 3   D ˙ φ ξ α − ξ ¯ α e ¯ α bar(Phi)_(varphi)^(')=sum_(alpha=1)^(3)D^(˙)_(varphi(xi_(alpha)- bar(xi)_(alpha))) bar(e)_(alpha)\bar{\Phi}_{\varphi}^{\prime}=\sum_{\alpha=1}^{3} \dot{D}_{\varphi\left(\xi_{\alpha}-\bar{\xi}_{\alpha}\right)} \bar{e}_{\alpha}Φ¯φ′=∑α=13D˙φ(ξα−ξ¯α)e¯α
e → α e → α vec(e)_(alpha)\vec{e}_{\alpha}e→α being the unit vectors in ordinary three-dimensional space. Then (14) becomes
(24) ∂ t D φ + ∇ ⋅ ( D φ ξ ¯ ― ) + ∇ ⋅ Φ ¯ φ ′ = D φ ˙ + ( ∂ t D φ ) g (24) ∂ t D φ + ∇ ⋅ D φ ξ ¯ ¯ + ∇ ⋅ Φ ¯ φ ′ = D φ ˙ + ∂ t D φ g {:(24)del_(t)D_(varphi)+grad*(D_(varphi) bar(bar(xi)))+grad* bar(Phi)_(varphi)^(')=D_(varphi^(˙))+(del_(t)D_(varphi))_(g):}\begin{equation*} \partial_{t} D_{\varphi}+\nabla \cdot\left(D_{\varphi} \overline{\bar{\xi}}\right)+\nabla \cdot \bar{\Phi}_{\varphi}^{\prime}=D_{\dot{\varphi}}+\left(\partial_{t} D_{\varphi}\right)_{g} \tag{24} \end{equation*}(24)∂tDφ+∇⋅(Dφξ¯―)+∇⋅Φ¯φ′=Dφ˙+(∂tDφ)g
This is the discrete analogue of the balance equation (1).

4. THE HAMILTONIAN SYSTEMS

In this section we consider a Hamiltonian system consisting of a single type of particles. The abstract particles considered till now become real particles with mass m m mmm, satisfying the principles of classical mechanics. Obviously, the mass m m mmm is constant in time and the same for all the particles. Since we have a single type of particles which are not generated or destroyed, the generating term (21) vanishes.
The relation (24) for mass is obtained if φ i ( t ) = m φ i ( t ) = m varphi_(i)(t)=m\varphi_{\mathrm{i}}(t)=mφi(t)=m for all t ∈ I i t ∈ I i t inI_(i)t \in I_{i}t∈Ii and i ≤ N i ≤ N i <= Ni \leq Ni≤N. Then D φ = D m = m D 1 D φ = D m = m D 1 D_(varphi)=D_(m)=mD_(1)D_{\varphi}=D_{m}=m D_{1}Dφ=Dm=mD1 is the discrete analogue of the mass density. Moreover, φ ¯ = m φ ¯ = m bar(varphi)=m\bar{\varphi}=mφ¯=m, Φ ¯ m ′ = 0 , φ ˙ i = 0 Φ ¯ m ′ = 0 , φ ˙ i = 0 bar(Phi)_(m)^(')=0,varphi^(˙)_(i)=0\bar{\Phi}_{m}^{\prime}=0, \dot{\varphi}_{i}=0Φ¯m′=0,φ˙i=0 and (24) becomes the discrete analogue of the continuity equation
(25) ∂ t D m + ∇ ⋅ ( D m ξ ¯ ― ) = 0 (25) ∂ t D m + ∇ ⋅ D m ξ ¯ ¯ = 0 {:(25)del_(t)D_(m)+grad*(D_(m) bar(bar(xi)))=0:}\begin{equation*} \partial_{t} D_{m}+\nabla \cdot\left(D_{m} \overline{\bar{\xi}}\right)=0 \tag{25} \end{equation*}(25)∂tDm+∇⋅(Dmξ¯―)=0
For the α α alpha\alphaα component of momentum we have φ i = p α i = m ξ α i φ i = p α i = m ξ α i varphi_(i)=p_(alpha i)=mxi_(alpha i)\varphi_{i}=p_{\alpha i}=m \xi_{\alpha i}φi=pαi=mξαi and D p α = D m ξ α = D m ξ ¯ α D p α = D m ξ α = D m ξ ¯ α D_(p_(alpha))=D_(mxi_(alpha))=D_(m) bar(xi)_(alpha)D_{p_{\alpha}}=D_{m \xi_{\alpha}}=D_{m} \bar{\xi}_{\alpha}Dpα=Dmξα=Dmξ¯α. The discrete analogue of the kinetic part of the flux density
takes the form of a symmetric tensor
(26) σ α β ′ = − ( Φ ¯ p α ′ ) β = − m 2 τ V ∑ i = 1 N ∫ t − τ t + τ ( ξ α i ( t ′ ) − ξ ¯ α ) ( ξ β i ( t ′ ) − ξ ¯ β ) ⋅ H + ( a 2 − ( r ¯ i ( t ′ ) − r ¯ ) 2 ) d t ′ (26) σ α β ′ = − Φ ¯ p α ′ β = − m 2 τ V ∑ i = 1 N   ∫ t − τ t + τ   ξ α i t ′ − ξ ¯ α ξ β i t ′ − ξ ¯ β ⋅ H + a 2 − r ¯ i t ′ − r ¯ 2 d t ′ {:[(26)sigma_(alpha beta)^(')=-( bar(Phi)_(p_(alpha))^('))_(beta)=-(m)/(2tau V)sum_(i=1)^(N)int_(t-tau)^(t+tau)(xi_(alpha i)(t^('))- bar(xi)_(alpha))(xi_(beta i)(t^('))- bar(xi)_(beta))],[*H^(+)(a^(2)-( bar(r)_(i)(t^('))-( bar(r)))^(2))dt^(')]:}\begin{gather*} \sigma_{\alpha \beta}^{\prime}=-\left(\bar{\Phi}_{p_{\alpha}}^{\prime}\right)_{\beta}=-\frac{m}{2 \tau V} \sum_{i=1}^{N} \int_{t-\tau}^{t+\tau}\left(\xi_{\alpha i}\left(t^{\prime}\right)-\bar{\xi}_{\alpha}\right)\left(\xi_{\beta i}\left(t^{\prime}\right)-\bar{\xi}_{\beta}\right) \tag{26}\\ \cdot H^{+}\left(a^{2}-\left(\bar{r}_{i}\left(t^{\prime}\right)-\bar{r}\right)^{2}\right) \mathrm{d} t^{\prime} \end{gather*}(26)σαβ′=−(Φ¯pα′)β=−m2τV∑i=1N∫t−τt+τ(ξαi(t′)−ξ¯α)(ξβi(t′)−ξ¯β)⋅H+(a2−(r¯i(t′)−r¯)2)dt′
The derivative φ ˙ i φ ˙ i varphi^(˙)_(i)\dot{\varphi}_{i}φ˙i is the a component of the force f → i f → i vec(f)_(i)\vec{f}_{i}f→i acting on the i i iii-th particle and relation (24) becomes
(27) ∂ t ( D m ξ ¯ α ) + ∑ β = 1 3 ∂ β ( D m ξ ¯ α ξ ¯ β ) − ∑ β = 1 3 ∂ β σ α β ′ = D f α (27) ∂ t D m ξ ¯ α + ∑ β = 1 3   ∂ β D m ξ ¯ α ξ ¯ β − ∑ β = 1 3   ∂ β σ α β ′ = D f α {:(27)del_(t)(D_(m) bar(xi)_(alpha))+sum_(beta=1)^(3)del_(beta)(D_(m) bar(xi)_(alpha) bar(xi)_(beta))-sum_(beta=1)^(3)del_(beta)sigma_(alpha beta)^(')=D_(f_(alpha)):}\begin{equation*} \partial_{t}\left(D_{m} \bar{\xi}_{\alpha}\right)+\sum_{\beta=1}^{3} \partial_{\beta}\left(D_{m} \bar{\xi}_{\alpha} \bar{\xi}_{\beta}\right)-\sum_{\beta=1}^{3} \partial_{\beta} \sigma_{\alpha \beta}^{\prime}=D_{f_{\alpha}} \tag{27} \end{equation*}(27)∂t(Dmξ¯α)+∑β=13∂β(Dmξ¯αξ¯β)−∑β=13∂βσαβ′=Dfα
Making additional hypotheses on the interaction between particles, one can prove that equation (27) is the discrete analogue of the momentum equation in continuum mechanics [7].
Choosing as physical quantity the kinetic energy of the particles φ i = E i = 1 2 m ξ i 2 φ i = E i = 1 2 m ξ i 2 varphi_(i)=E_(i)=(1)/(2)mxi_(i)^(2)\varphi_{i}=E_{i}=\frac{1}{2} m \xi_{i}^{2}φi=Ei=12mξi2, we obtain
D E = 1 2 m D [ ξ → + ( ξ → − ξ → ) ] 2 = 1 2 m D ξ → 2 + 1 2 m D ξ → ― ( ξ ¯ − ξ ¯ ) + 1 2 m D ( ξ → − ξ ¯ ) 2 D E = 1 2 m D [ ξ → + ( ξ → − ξ → ) ] 2 = 1 2 m D ξ → 2 + 1 2 m D ξ → ¯ ( ξ ¯ − ξ ¯ ) + 1 2 m D ( ξ → − ξ ¯ ) 2 D_(E)=(1)/(2)mD_([ vec(xi)+( vec(xi)- vec(xi))]^(2))=(1)/(2)mD_( vec(xi)^(2))+(1)/(2)mD_( bar(vec(xi))( bar(xi)- bar(xi)))+(1)/(2)mD_(( vec(xi)- bar(xi))^(2))D_{E}=\frac{1}{2} m D_{[\vec{\xi}+(\vec{\xi}-\vec{\xi})]^{2}}=\frac{1}{2} m D_{\vec{\xi}^{2}}+\frac{1}{2} m D_{\overline{\vec{\xi}}(\bar{\xi}-\bar{\xi})}+\frac{1}{2} m D_{(\vec{\xi}-\bar{\xi})^{2}}DE=12mD[ξ→+(ξ→−ξ→)]2=12mDξ→2+12mDξ→―(ξ¯−ξ¯)+12mD(ξ→−ξ¯)2
where we used the linearity of D φ D φ D_(varphi)D_{\varphi}Dφ with respect to φ φ varphi\varphiφ, i.e. D φ 1 + φ 2 = D φ 1 + D φ 2 D φ 1 + φ 2 = D φ 1 + D φ 2 D_(varphi_(1)+varphi_(2))=D_(varphi_(1))+D_(varphi_(2))D_{\varphi_{1}+\varphi_{2}}=D_{\varphi_{1}}+D_{\varphi_{2}}Dφ1+φ2=Dφ1+Dφ2. The second term of the expression vanishes D ξ → , ( ξ → − ξ → ) = ξ ¯ ― ⋅ D ξ → − ξ → = ξ ¯ ― ⋅ ( D ξ → − D ξ → ) = 0 D ξ → , ( ξ → − ξ → ) = ξ ¯ ¯ ⋅ D ξ → − ξ → = ξ ¯ ¯ ⋅ D ξ → − D ξ → = 0 D_( vec(xi),( vec(xi)- vec(xi)))= bar(bar(xi))*D_( vec(xi)- vec(xi))= bar(bar(xi))*(D_( vec(xi))-D_( vec(xi)))=0D_{\vec{\xi},(\vec{\xi}-\vec{\xi})}=\overline{\bar{\xi}} \cdot D_{\vec{\xi}-\vec{\xi}}=\overline{\bar{\xi}} \cdot\left(D_{\vec{\xi}}-D_{\vec{\xi}}\right)=0Dξ→,(ξ→−ξ→)=ξ¯―⋅Dξ→−ξ→=ξ¯―⋅(Dξ→−Dξ→)=0. The last term is the discrete analogue of the kinetic energy density of the thermal motion, since 1 2 m ( ξ ¯ − ξ ¯ ― ) 2 1 2 m ( ξ ¯ − ξ ¯ ¯ ) 2 (1)/(2)m( bar(xi)- bar(bar(xi)))^(2)\frac{1}{2} m(\bar{\xi}-\overline{\bar{\xi}})^{2}12m(ξ¯−ξ¯―)2 represents the kinetic energy of the relative motion of the i i iii-th particle with respect to the mean motion of the particles in the sphere S ( r ¯ , a ) S ( r ¯ , a ) S( bar(r),a)S(\bar{r}, a)S(r¯,a), over ( t − τ , t + τ ) ( t − τ , t + τ ) (t-tau,t+tau)(t-\tau, t+\tau)(t−τ,t+τ). We denote this term by
ε = 1 2 m D ( ξ → − ξ → ) 2 = 1 2 ∑ α = 1 3 σ α α ′ ε = 1 2 m D ( ξ → − ξ → ) 2 = 1 2 ∑ α = 1 3   σ α α ′ epsi=(1)/(2)mD_(( vec(xi)- vec(xi))^(2))=(1)/(2)sum_(alpha=1)^(3)sigma_(alpha alpha)^(')\varepsilon=\frac{1}{2} m D_{(\vec{\xi}-\vec{\xi})^{2}}=\frac{1}{2} \sum_{\alpha=1}^{3} \sigma_{\alpha \alpha}^{\prime}ε=12mD(ξ→−ξ→)2=12∑α=13σαα′
the last equality following directly from (26). Similarly
( Φ ¯ E ) α = 1 2 m D ξ 2 ( ξ α − ξ ¯ α ) = 1 2 ∑ β = 1 3 ξ β σ α β ′ + ( Φ ¯ ε ) α Φ ¯ E α = 1 2 m D ξ 2 ξ α − ξ ¯ α = 1 2 ∑ β = 1 3   ξ β σ α β ′ + Φ ¯ ε α ( bar(Phi)_(E))_(alpha)=(1)/(2)mD_(xi^(2)(xi_(alpha)- bar(xi)_(alpha)))=(1)/(2)sum_(beta=1)^(3)xi_(beta)sigma_(alpha beta)^(')+( bar(Phi)_(epsi))_(alpha)\left(\bar{\Phi}_{E}\right)_{\alpha}=\frac{1}{2} m D_{\xi^{2}\left(\xi_{\alpha}-\bar{\xi}_{\alpha}\right)}=\frac{1}{2} \sum_{\beta=1}^{3} \xi_{\beta} \sigma_{\alpha \beta}^{\prime}+\left(\bar{\Phi}_{\varepsilon}\right)_{\alpha}(Φ¯E)α=12mDξ2(ξα−ξ¯α)=12∑β=13ξβσαβ′+(Φ¯ε)α
where the flux of the discrete analogue of the kinetic energy of the thermal motion is
Φ ¯ ε = 1 2 m D ( ξ → − ξ ¯ ) 2 ( ξ → − ξ ¯ ) Φ ¯ ε = 1 2 m D ( ξ → − ξ ¯ ) 2 ( ξ → − ξ ¯ ) bar(Phi)_(epsi)=(1)/(2)mD_(( vec(xi)- bar(xi))^(2)( vec(xi)- bar(xi)))\bar{\Phi}_{\varepsilon}=\frac{1}{2} m D_{(\vec{\xi}-\bar{\xi})^{2}(\vec{\xi}-\bar{\xi})}Φ¯ε=12mD(ξ→−ξ¯)2(ξ→−ξ¯)
The time derivative of the kinetic energy can not be written in a simple form, so the balance equation (24) for kinetic energy becomes
(28) ∂ t ( 1 2 D m ξ ¯ ― 2 + ε ) + ∇ ⋅ [ ( 1 2 D m ξ ¯ ― 2 + ε ) ξ ¯ ― ] + ∇ ⋅ Φ ¯ ε + ∑ α , β = 1 3 ∂ α ( ξ ¯ β σ α β ′ ) = D E ˙ . (28) ∂ t 1 2 D m ξ ¯ ¯ 2 + ε + ∇ ⋅ 1 2 D m ξ ¯ ¯ 2 + ε ξ ¯ ¯ + ∇ ⋅ Φ ¯ ε + ∑ α , β = 1 3   ∂ α ξ ¯ β σ α β ′ = D E ˙ . {:(28)del_(t)((1)/(2)D_(m) bar(bar(xi))^(2)+epsi)+grad*[((1)/(2)D_(m) bar(bar(xi))^(2)+epsi) bar(bar(xi))]+grad* bar(Phi)_(epsi)+sum_(alpha,beta=1)^(3)del_(alpha)( bar(xi)_(beta)sigma_(alpha beta)^('))=D_(E^(˙)).:}\begin{equation*} \partial_{t}\left(\frac{1}{2} D_{m} \overline{\bar{\xi}}^{2}+\varepsilon\right)+\nabla \cdot\left[\left(\frac{1}{2} D_{m} \overline{\bar{\xi}}^{2}+\varepsilon\right) \overline{\bar{\xi}}\right]+\nabla \cdot \bar{\Phi}_{\varepsilon}+\sum_{\alpha, \beta=1}^{3} \partial_{\alpha}\left(\bar{\xi}_{\beta} \sigma_{\alpha \beta}^{\prime}\right)=D_{\dot{E}} . \tag{28} \end{equation*}(28)∂t(12Dmξ¯―2+ε)+∇⋅[(12Dmξ¯―2+ε)ξ¯―]+∇⋅Φ¯ε+∑α,β=13∂α(ξ¯βσαβ′)=DE˙.
In addition to relations (25), (27) and (28), discrete analogues of balance equations for any physical quantity are possible.

5. CONCLUSION

Relation (24) is not a balance equation, but an identity of the same form with a balance equation. It has been derived under general conditions, for an arbitrary, finite number of mathematical points to which analytic functions of time were attached. Due to this very general approach, the results can be applied to a large number of corpuscular physical systems. For example, the discrete analogue of the balance equation (24) is valid for an arbitrary physical quantity, and for an arbitrary number of particles (even very small). Also, since the dynamical equations for the microscopic evolution have not been used explicity, relation (24) holds for any microscopic interaction forces satisfying the analycity condition. In this article we have considered the case of the Hamiltonian systems consisting of a single type of particles which can not be generated or destroyed.
To transform the function D φ D φ D_(varphi)D_{\varphi}Dφ defined by (2) into a continuous field, and relation (24) into a balance equation, a statistical average on an ensemble formed by a very large number of identical copies of the considered corpuscular system is needed. Although, if the number of particles contributing to the value of D φ D φ D_(varphi)D_{\varphi}Dφ is large enough, then D φ D φ D_(varphi)D_{\varphi}Dφ approximates closely the continuous field corresponding to the physical quantity φ φ varphi\varphiφ. That is, if the physical system satisfies the local equilibrium principle [6], then the parameters a a aaa and τ τ tau\tauτ can be chosen so that the particles lying in the sphere S ( r → , a ) S ( r → , a ) S( vec(r),a)S(\vec{r}, a)S(r→,a) over the interval ( t − τ , t + τ ) ( t − τ , t + τ ) (t-tau,t+tau)(t-\tau, t+\tau)(t−τ,t+τ) should form a near-equilibrium thermodynamical system [12]. Obviously, in this case the total number of particles N N NNN can no longer be arbitrary, but it must be large enough to ensure the validity of the thermodynamical limit.
The balance equation in continuum mechanics can also be obtained as the limit for a → 0 a → 0 a rarr0a \rightarrow 0a→0 and τ → 0 τ → 0 tau rarr0\tau \rightarrow 0τ→0 of the statistical average of relation (24). Both methods to obtain the balance equation (1) from (24) will be discuss in another article.

APPENDIX

Here we study the differentiability with respect to r → r → vec(r)\vec{r}r→ of the function g i g i g_(i)g_{i}gi defined by (8). Although (6) holds only for t ∈ I i t ∈ I i t inI_(i)t \in I_{i}t∈Ii, it may be substituted in (8) because φ i φ i varphi_(i)\varphi_{i}φi
vanishes for t ∈ I ∖ I i t ∈ I ∖ I i t in I\\I_(i)t \in I \backslash I_{i}t∈I∖Ii, and we obtain *
(A1) g i ( r → ′ , t ) = H + ( a 2 − ( r → i ( t i + ) − r → ) 2 ) t + τ ∫ t − τ φ i ( t ′ ) d t ′ + ∑ u ∈ U i ′ ∫ t − τ t + τ φ i ( t ′ ) H + ( t ′ − u ) d t ′ − − ∑ u ∈ U i ′ ′ ∫ t − τ t + τ φ i ( t ′ ) H − ( t ′ − u ) d t ′ (A1) g i r → ′ , t = H + a 2 − r → i t i + − r → 2 t + τ ∫ t − τ   φ i t ′ d t ′ + ∑ u ∈ U i ′   ∫ t − τ t + τ   φ i t ′ H + t ′ − u d t ′ − − ∑ u ∈ U i ′ ′   ∫ t − τ t + τ   φ i t ′ H − t ′ − u d t ′ {:[(A1)g_(i)( vec(r)^('),t)=H^(+)(a^(2)-( vec(r)_(i)(t_(i)^(+))-( vec(r)))^(2))^(t+tau)int_(t-tau)varphi_(i)(t^('))dt^(')+sum_(u inU_(i)^('))int_(t-tau)^(t+tau)varphi_(i)(t^('))H^(+)(t^(')-u)dt^(')-],[-sum_(u inU_(i)^(''))int_(t-tau)^(t+tau)varphi_(i)(t^('))H^(-)(t^(')-u)dt^(')]:}\begin{gather*} g_{i}\left(\vec{r}^{\prime}, t\right)=H^{+}\left(a^{2}-\left(\vec{r}_{i}\left(t_{i}^{+}\right)-\vec{r}\right)^{2}\right)^{t+\tau} \int_{t-\tau} \varphi_{i}\left(t^{\prime}\right) \mathrm{d} t^{\prime}+\sum_{u \in U_{i}^{\prime}} \int_{t-\tau}^{t+\tau} \varphi_{i}\left(t^{\prime}\right) H^{+}\left(t^{\prime}-u\right) \mathrm{d} t^{\prime}- \tag{A1}\\ -\sum_{u \in U_{i}^{\prime \prime}} \int_{t-\tau}^{t+\tau} \varphi_{i}\left(t^{\prime}\right) H^{-}\left(t^{\prime}-u\right) \mathrm{d} t^{\prime} \end{gather*}(A1)gi(r→′,t)=H+(a2−(r→i(ti+)−r→)2)t+τ∫t−τφi(t′)dt′+∑u∈Ui′∫t−τt+τφi(t′)H+(t′−u)dt′−−∑u∈Ui′′∫t−τt+τφi(t′)H−(t′−u)dt′
where U i ′ U i ′ U_(i)^(')U_{i}^{\prime}Ui′ and U i ′ ′ U i ′ ′ U_(i)^('')U_{i}^{\prime \prime}Ui′′ are defined in (7). First, we consider 2 τ < t i − − t i + 2 τ < t i − − t i + 2tau < t_(i)^(-)-t_(i)^(+)2 \tau<t_{i}^{-}-t_{i}^{+}2τ<ti−−ti+. The following cases are possible:
(a) t ≤ t i + − τ t ≤ t i + − τ t <= t_(i)^(+)-taut \leq t_{i}^{+}-\taut≤ti+−τ. Then ( t − τ , t + τ ) ∩ I i = ∅ ( t − τ , t + τ ) ∩ I i = ∅ (t-tau,t+tau)nnI_(i)=O/(t-\tau, t+\tau) \cap I_{i}=\varnothing(t−τ,t+τ)∩Ii=∅ and φ i φ i varphi_(i)\varphi_{i}φi vanishes in the integration intervals in (A1), such that g i ( r → , t ) = 0 g i ( r → , t ) = 0 g_(i)( vec(r),t)=0g_{i}(\vec{r}, t)=0gi(r→,t)=0 for all r → ∈ R 3 r → ∈ R 3 vec(r)inR^(3)\vec{r} \in \mathbb{R}^{3}r→∈R3.
(b) t ∈ ( t i + − τ , t i + + τ ] t ∈ t i + − τ , t i + + τ t in(t_(i)^(+)-tau,t_(i)^(+)+tau]t \in\left(t_{i}^{+}-\tau, t_{i}^{+}+\tau\right]t∈(ti+−τ,ti++τ]. Then ( t − τ , t + τ ) ∩ I i = [ t i + , t + τ ) ( t − τ , t + τ ) ∩ I i = t i + , t + τ (t-tau,t+tau)nnI_(i)=[t_(i)^(+),t+tau)(t-\tau, t+\tau) \cap I_{i}=\left[t_{i}^{+}, t+\tau\right)(t−τ,t+τ)∩Ii=[ti+,t+τ) and the integral in (A1) have the same limits. The first term in (A1) depends on r → r → vec(r)\vec{r}r→ through the function H + ( a 2 − ( r ¯ i ( t i + ) − r ¯ ) 2 ) H + a 2 − r ¯ i t i + − r ¯ 2 H^(+)(a^(2)-( bar(r)_(i)(t_(i)^(+))-( bar(r)))^(2))H^{+}\left(a^{2}-\left(\bar{r}_{i}\left(t_{i}^{+}\right)-\bar{r}\right)^{2}\right)H+(a2−(r¯i(ti+)−r¯)2) which can take only the values 0 and 1 . Hence, when this function is continuous with respect to r ¯ r ¯ bar(r)\bar{r}r¯, its derivative exists and equals zero. Then the first term in (A1) is not differentiable if H + ( a 2 − ( r ¯ i ( t i + ) − r ¯ ) 2 ) H + a 2 − r ¯ i t i + − r ¯ 2 H^(+)(a^(2)-( bar(r)_(i)(t_(i)^(+))-( bar(r)))^(2))H^{+}\left(a^{2}-\left(\bar{r}_{i}\left(t_{i}^{+}\right)-\bar{r}\right)^{2}\right)H+(a2−(r¯i(ti+)−r¯)2) is discontinuous, i.e. r → ∈ ∂ S ( r → i ( t i + ) , a ) r → ∈ ∂ S r → i t i + , a vec(r)in del S( vec(r)_(i)(t_(i)^(+)),a)\vec{r} \in \partial S\left(\vec{r}_{i}\left(t_{i}^{+}\right), a\right)r→∈∂S(r→i(ti+),a). The others terms in (A1) depend on r → r → vec(r)\vec{r}r→ through the moments u u uuu defined by (3). These terms are not differentiable either if u u uuu is not differentiable (i.e. relation (4) is not satisfied) of if the moments u u uuu coincide with the integration limits (i.e. r → ∈ ∂ S ( r → i ( t i + ) , a ) r → ∈ ∂ S r → i t i + , a vec(r)in del S( vec(r)_(i)(t_(i)^(+)),a)\vec{r} \in \partial S\left(\vec{r}_{i}\left(t_{i}^{+}\right), a\right)r→∈∂S(r→i(ti+),a) or r → ∈ ∂ S ( r → i ( t + τ ) , a ) r → ∈ ∂ S r → i ( t + τ ) , a vec(r)in del S( vec(r)_(i)(t+tau),a)\vec{r} \in \partial S\left(\vec{r}_{i}(t+\tau), a\right)r→∈∂S(r→i(t+τ),a) ). In this case the integration intervals have discontinuous variations with respect to r → r → vec(r)\vec{r}r→.
(c) t ∈ ( t i + + τ , t i − − τ ) t ∈ t i + + τ , t i − − τ t in(t_(i)^(+)+tau,t_(i)^(-)-tau)t \in\left(t_{i}^{+}+\tau, t_{i}^{-}-\tau\right)t∈(ti++τ,ti−−τ). Then ( t − τ , t + τ ) ∩ I i = ( t − τ , t + τ ) ( t − τ , t + τ ) ∩ I i = ( t − τ , t + τ ) (t-tau,t+tau)nnI_(i)=(t-tau,t+tau)(t-\tau, t+\tau) \cap I_{i}=(t-\tau, t+\tau)(t−τ,t+τ)∩Ii=(t−τ,t+τ) and the integrals with u ≤ t − τ u ≤ t − τ u <= t-tauu \leq t-\tauu≤t−τ are equal to the integral of the first term. Using the expression
H + ( a 2 − ( r ¯ i ( t i + ) − r ¯ ) 2 ) + ∑ u ∈ W 1 ′ H + ( t − u ) − ∑ u ∈ W 1 ′ ′ H − 1 ( t − u ) = H + ( a 2 − ( r ¯ i ( t − τ ) − r ¯ ) 2 ) H + a 2 − r ¯ i t i + − r ¯ 2 + ∑ u ∈ W 1 ′   H + ( t − u ) − ∑ u ∈ W 1 ′ ′   H − 1 ( t − u ) = H + a 2 − r ¯ i ( t − τ ) − r ¯ 2 H^(+)(a^(2)-( bar(r)_(i)(t_(i)^(+))-( bar(r)))^(2))+sum_(u inW_(1)^('))H^(+)(t-u)-sum_(u inW_(1)^(''))H^(-1)(t-u)=H^(+)(a^(2)-( bar(r)_(i)(t-tau)-( bar(r)))^(2))H^{+}\left(a^{2}-\left(\bar{r}_{i}\left(t_{i}^{+}\right)-\bar{r}\right)^{2}\right)+\sum_{u \in W_{1}^{\prime}} H^{+}(t-u)-\sum_{u \in W_{1}^{\prime \prime}} H^{-1}(t-u)=H^{+}\left(a^{2}-\left(\bar{r}_{i}(t-\tau)-\bar{r}\right)^{2}\right)H+(a2−(r¯i(ti+)−r¯)2)+∑u∈W1′H+(t−u)−∑u∈W1′′H−1(t−u)=H+(a2−(r¯i(t−τ)−r¯)2)
where W 1 ′ = U i ′ ∩ [ t i + , t − τ ] W 1 ′ = U i ′ ∩ t i + , t − τ W_(1)^(')=U_(i)^(')nn[t_(i)^(+),t-tau]W_{1}^{\prime}=U_{i}^{\prime} \cap\left[t_{i}^{+}, t-\tau\right]W1′=Ui′∩[ti+,t−τ] and W 1 ′ ′ = U i ′ ′ ∩ ( t i + , t − τ ] W 1 ′ ′ = U i ′ ′ ∩ t i + , t − τ W_(1)^('')=U_(i)^('')nn(t_(i)^(+),t-tau]W_{1}^{\prime \prime}=U_{i}^{\prime \prime} \cap\left(t_{i}^{+}, t-\tau\right]W1′′=Ui′′∩(ti+,t−τ], relation (A1) becomes
(A2) g i ( r → , t ) = H + ( a 2 − ( r ¯ i ( t − τ ) − r ¯ ) 2 ) ∫ t − τ t + τ φ i ( t ′ ) d t ′ + ∑ u ∈ V 2 ′ t − τ t + τ φ i ( t ′ ) H + ( t ′ − u ) d t ′ − − ∑ u ∈ V 2 ′ ′ ∫ t − τ t + τ φ i ( t ′ ) H − ( t ′ − u ) d t ′ (A2) g i ( r → , t ) = H + a 2 − r ¯ i ( t − τ ) − r ¯ 2 ∫ t − τ t + τ   φ i t ′ d t ′ + ∑ u ∈ V 2 ′ t − τ t + τ   φ i t ′ H + t ′ − u d t ′ − − ∑ u ∈ V 2 ′ ′   ∫ t − τ t + τ   φ i t ′ H − t ′ − u d t ′ {:[(A2)g_(i)( vec(r)","t)=H^(+)(a^(2)-( bar(r)_(i)(t-tau)-( bar(r)))^(2))int_(t-tau)^(t+tau)varphi_(i)(t^('))dt^(')+sum_(u inV_(2)^(')t-tau)^(t+tau)varphi_(i)(t^('))H^(+)(t^(')-u)dt^(')-],[-sum_(u inV_(2)^(''))int_(t-tau)^(t+tau)varphi_(i)(t^('))H^(-)(t^(')-u)dt^(')]:}\begin{gather*} g_{i}(\vec{r}, t)=H^{+}\left(a^{2}-\left(\bar{r}_{i}(t-\tau)-\bar{r}\right)^{2}\right) \int_{t-\tau}^{t+\tau} \varphi_{i}\left(t^{\prime}\right) \mathrm{d} t^{\prime}+\sum_{u \in V_{2}^{\prime} t-\tau}^{t+\tau} \varphi_{i}\left(t^{\prime}\right) H^{+}\left(t^{\prime}-u\right) \mathrm{d} t^{\prime}- \tag{A2}\\ -\sum_{u \in V_{2}^{\prime \prime}} \int_{t-\tau}^{t+\tau} \varphi_{i}\left(t^{\prime}\right) H^{-}\left(t^{\prime}-u\right) \mathrm{d} t^{\prime} \end{gather*}(A2)gi(r→,t)=H+(a2−(r¯i(t−τ)−r¯)2)∫t−τt+τφi(t′)dt′+∑u∈V2′t−τt+τφi(t′)H+(t′−u)dt′−−∑u∈V2′′∫t−τt+τφi(t′)H−(t′−u)dt′
where W 2 ′ = U i ′ ∩ ( t − τ , t i − ) W 2 ′ = U i ′ ∩ t − τ , t i − W_(2)^(')=U_(i)^(')nn(t-tau,t_(i)^(-))W_{2}^{\prime}=U_{i}^{\prime} \cap\left(t-\tau, t_{i}^{-}\right)W2′=Ui′∩(t−τ,ti−)and W 2 ′ ′ = U i ′ ′ ∩ ( t − τ , t i − ] W 2 ′ ′ = U i ′ ′ ∩ t − τ , t i − W_(2)^('')=U_(i)^('')nn(t-tau,t_(i)^(-)]W_{2}^{\prime \prime}=U_{i}^{\prime \prime} \cap\left(t-\tau, t_{i}^{-}\right]W2′′=Ui′′∩(t−τ,ti−]. As for (b), the first term is not differentiable if r → ∈ ∂ S ( r → i ( t − τ ) , a ) r → ∈ ∂ S r → i ( t − τ ) , a vec(r)in del S( vec(r)_(i)(t-tau),a)\vec{r} \in \partial S\left(\vec{r}_{i}(t-\tau), a\right)r→∈∂S(r→i(t−τ),a) and the other terms if r → ∈ ∂ S ( r → i ( t + τ ) , a ) r → ∈ ∂ S r → i ( t + τ ) , a vec(r)in del S( vec(r)_(i)(t+tau),a)\vec{r} \in \partial S\left(\vec{r}_{i}(t+\tau), a\right)r→∈∂S(r→i(t+τ),a), or if relation (4) is not satisfied.
(d) t ∈ [ t i − τ , t i − + τ ) t ∈ t i − τ , t i − + τ t in[t_(i)^(-)tau,t_(i)^(-)+tau)t \in\left[t_{i}^{-} \tau, t_{i}^{-}+\tau\right)t∈[ti−τ,ti−+τ). Then ( t − τ , t + τ ) ∩ I i = ( t − τ , t i − ] ( t − τ , t + τ ) ∩ I i = t − τ , t i − (t-tau,t+tau)nnI_(i)=(t-tau,t_(i)^(-)](t-\tau, t+\tau) \cap I_{i}=\left(t-\tau, t_{i}^{-}\right](t−τ,t+τ)∩Ii=(t−τ,ti−]and the expression for g i ( r → , t ) g i ( r → , t ) g_(i)( vec(r),t)g_{i}(\vec{r}, t)gi(r→,t) is identic with (A2), except the upper integration limit is t i − t i − t_(i)^(-)t_{i}^{-}ti−. So g i ( r → , t ) g i ( r → , t ) g_(i)( vec(r),t)g_{i}(\vec{r}, t)gi(r→,t) is not differentiable if r → ∈ ∂ S ( r → i ( t − τ ) , a ) r → ∈ ∂ S r → i ( t − τ ) , a vec(r)in del S( vec(r)_(i)(t-tau),a)\vec{r} \in \partial S\left(\vec{r}_{i}(t-\tau), a\right)r→∈∂S(r→i(t−τ),a) or r → ∈ ∂ S ( r → i ( t i − ) , a ˙ ) r → ∈ ∂ S r → i t i − , a ˙ vec(r)in del S( vec(r)_(i)(t_(i)^(-)),(a^(˙)))\vec{r} \in \partial S\left(\vec{r}_{i}\left(t_{i}^{-}\right), \dot{a}\right)r→∈∂S(r→i(ti−),a˙) or (4) is not satisfied.
(e) t ≥ t i − + τ t ≥ t i − + τ t >= t_(i)^(-)+taut \geq t_{i}^{-}+\taut≥ti−+τ. Then ( t − τ , t + τ ) ∩ I i = ∅ ( t − τ , t + τ ) ∩ I i = ∅ (t-tau,t+tau)nnI_(i)=O/(t-\tau, t+\tau) \cap I_{i}=\varnothing(t−τ,t+τ)∩Ii=∅ and g i ( r → , t ) = 0 g i ( r → , t ) = 0 g_(i)( vec(r),t)=0g_{i}(\vec{r}, t)=0gi(r→,t)=0.
If τ < t i − − t i + < 2 τ τ < t i − − t i + < 2 τ tau < t_(i)^(-)-t_(i)^(+) < 2tau\tau<t_{i}^{-}-t_{i}^{+}<2 \tauτ<ti−−ti+<2τ, then the possible cases for(A1) are t ≤ t i + − τ , t ∈ ( t i + − τ , t i − − τ ) t ≤ t i + − τ , t ∈ t i + − τ , t i − − τ t <= t_(i)^(+)-tau,t in(t_(i)^(+)-tau,t_(i)^(-)-tau)t \leq t_{i}^{+}-\tau, t \in\left(t_{i}^{+}-\tau, t_{i}^{-}-\tau\right)t≤ti+−τ,t∈(ti+−τ,ti−−τ), t ∈ [ t i − − τ , t i + + τ ] , t ∈ ( t i + + τ , t i − + τ ) t ∈ t i − − τ , t i + + τ , t ∈ t i + + τ , t i − + τ t in[t_(i)^(-)-tau,t_(i)^(+)+tau],t in(t_(i)^(+)+tau,t_(i)^(-)+tau)t \in\left[t_{i}^{-}-\tau, t_{i}^{+}+\tau\right], t \in\left(t_{i}^{+}+\tau, t_{i}^{-}+\tau\right)t∈[ti−−τ,ti++τ],t∈(ti++τ,ti−+τ) and t ≥ t i − + τ t ≥ t i − + τ t >= t_(i)^(-)+taut \geq t_{i}^{-}+\taut≥ti−+τ and the discussion is similar. Finally, if t i − − t i + ≤ τ t i − − t i + ≤ τ t_(i)^(-)-t_(i)^(+) <= taut_{i}^{-}-t_{i}^{+} \leq \tauti−−ti+≤τ, then other five intervals for t t ttt exist. Taking into account that t ∈ ( τ , T − τ ) t ∈ ( τ , T − τ ) t in(tau,T-tau)t \in(\tau, T-\tau)t∈(τ,T−τ), the set where the function g i g i g_(i)g_{i}gi is not differentiable with respect to r → r → vec(r)\vec{r}r→ is
(A3) Ω i ′ ′ = { ( r ¯ , t ) ∣ t ∈ ( t i + − τ , t i + + τ ] ∩ ( τ , T − τ ) and r → ∈ ∂ S ( r ¯ i ( t i + ) , a ) } ∪ ∪ { ( r → , t ) ∣ t ∈ [ t i − − τ , t i − + τ ) ∩ ( τ , T − τ ) and r → ∈ ∂ S ( r ¯ i ( t i − ) , a ) } ∪ ∪ { ( r → , t ) ∣ t ∈ ( t i + ± τ , t i − ± τ ) ∩ ( τ , T − τ ) and r → ∈ ∂ S ( r ¯ i ( t ∓ τ ) , a ) } ∪ ∪ { ( r → , t ) ∣ t ∈ ( τ , T − τ ) , exists t ′ ∈ ( t − τ , t + τ ) ∩ I i such that r → ∈ ∂ S ( r ¯ i ( t ′ ) , a ) and ( r → i ( t ′ ) − r → ) ⋅ ξ → i ( t ′ ) = 0 } (A3) Ω i ′ ′ = r ¯ , t ∣ t ∈ t i + − τ , t i + + τ ∩ ( τ , T − τ )  and  r → ∈ ∂ S r ¯ i t i + , a ∪ ∪ ( r → , t ) ∣ t ∈ t i − − τ , t i − + τ ∩ ( τ , T − τ )  and  r → ∈ ∂ S r ¯ i t i − , a ∪ ∪ r → , t ∣ t ∈ t i + ± τ , t i − ± τ ∩ ( τ , T − τ )  and  r → ∈ ∂ S r ¯ i ( t ∓ τ ) , a ∪ ∪ ( r → , t ) ∣ t ∈ ( τ , T − τ ) ,  exists  t ′ ∈ ( t − τ , t + τ ) ∩ I i  such that  r → ∈ ∂ S r ¯ i t ′ , a  and  r → i t ′ − r → ⋅ ξ → i t ′ = 0 {:[(A3)Omega_(i)^('')={( bar(r)_(,)t)∣t in(t_(i)^(+)-tau,t_(i)^(+)+tau]nn(tau,T-tau)" and "( vec(r))in del S( bar(r)_(i)(t_(i)^(+)),a)}uu],[ uu{(( vec(r)),t)∣t in[t_(i)^(-)-tau,t_(i)^(-)+tau)nn(tau,T-tau)" and "( vec(r))in del S( bar(r)_(i)(t_(i)^(-)),a)}uu],[ uu{( vec(r)_(,)t)∣t in(t_(i)^(+)+-tau,t_(i)^(-)+-tau)nn(tau,T-tau)" and "( vec(r))in del S( bar(r)_(i)(t∓tau),a)}uu],[ uu{(( vec(r)),t)∣t in(tau,T-tau)," exists "t^(')in(t-tau,t+tau)nnI_(i):}" such that "],[{:( vec(r))in del S( bar(r)_(i)(t^(')),a)" and "( vec(r)_(i)(t^('))-( vec(r)))* vec(xi)_(i)(t^('))=0}]:}\begin{align*} \Omega_{i}^{\prime \prime} & =\left\{\left(\bar{r}_{,} t\right) \mid t \in\left(t_{i}^{+}-\tau, t_{i}^{+}+\tau\right] \cap(\tau, T-\tau) \text { and } \vec{r} \in \partial S\left(\bar{r}_{i}\left(t_{i}^{+}\right), a\right)\right\} \cup \tag{A3}\\ & \cup\left\{(\vec{r}, t) \mid t \in\left[t_{i}^{-}-\tau, t_{i}^{-}+\tau\right) \cap(\tau, T-\tau) \text { and } \vec{r} \in \partial S\left(\bar{r}_{i}\left(t_{i}^{-}\right), a\right)\right\} \cup \\ & \cup\left\{\left(\vec{r}_{,} t\right) \mid t \in\left(t_{i}^{+} \pm \tau, t_{i}^{-} \pm \tau\right) \cap(\tau, T-\tau) \text { and } \vec{r} \in \partial S\left(\bar{r}_{i}(t \mp \tau), a\right)\right\} \cup \\ & \cup\left\{(\vec{r}, t) \mid t \in(\tau, T-\tau), \text { exists } t^{\prime} \in(t-\tau, t+\tau) \cap I_{i}\right. \text { such that } \\ & \left.\vec{r} \in \partial S\left(\bar{r}_{i}\left(t^{\prime}\right), a\right) \text { and }\left(\vec{r}_{i}\left(t^{\prime}\right)-\vec{r}\right) \cdot \vec{\xi}_{i}\left(t^{\prime}\right)=0\right\} \end{align*}(A3)Ωi′′={(r¯,t)∣t∈(ti+−τ,ti++τ]∩(τ,T−τ) and r→∈∂S(r¯i(ti+),a)}∪∪{(r→,t)∣t∈[ti−−τ,ti−+τ)∩(τ,T−τ) and r→∈∂S(r¯i(ti−),a)}∪∪{(r→,t)∣t∈(ti+±τ,ti−±τ)∩(τ,T−τ) and r→∈∂S(r¯i(t∓τ),a)}∪∪{(r→,t)∣t∈(τ,T−τ), exists t′∈(t−τ,t+τ)∩Ii such that r→∈∂S(r¯i(t′),a) and (r→i(t′)−r→)⋅ξ→i(t′)=0}
This is a set of null Lebesgue measure in R 3 × ( τ , T − τ ) R 3 × ( τ , T − τ ) R^(3)xx(tau,T-tau)\mathbb{R}^{3} \times(\tau, T-\tau)R3×(τ,T−τ).
Only the terms in (A1) which contain u u uuu in the integration interval have a nonvanishing contribution to the derivative of g i g i g_(i)g_{i}gi with respect to x α x α x_(alpha)x_{\alpha}xα, denoted by ∂ α g i ∗ ∂ α g i ∗ del_(alpha)g_(i^(**))\partial_{\alpha} g_{i^{*}}∂αgi∗. Using (5) and taking into account that the sign of the terms in (A1) coincides with the sign of the expression − ( r ¯ i ( u ) − r ¯ ) ⋅ ξ ¯ i ( u ) − r ¯ i ( u ) − r ¯ ⋅ ξ ¯ i ( u ) -( bar(r)_(i)(u)-( bar(r)))* bar(xi)_(i)(u)-\left(\bar{r}_{i}(u)-\bar{r}\right) \cdot \bar{\xi}_{i}(u)−(r¯i(u)−r¯)⋅ξ¯i(u), we obtain
(A4) ∂ α g i ( r → , t ) = ∑ u ∈ U i φ i ( u ) x α i ( u ) − x α | ( r → i ( u ) − r → ) ⋅ ξ → i ( u ) | (A4) ∂ α g i ( r → , t ) = ∑ u ∈ U i   φ i ( u ) x α i ( u ) − x α r → i ( u ) − r → ⋅ ξ → i ( u ) {:(A4)del_(alpha)g_(i)( vec(r)","t)=sum_(u inU_(i))varphi_(i)(u)(x_(alpha i)(u)-x_(alpha))/(|( vec(r)_(i)(u)-( vec(r)))* vec(xi)_(i)(u)|):}\begin{equation*} \partial_{\alpha} g_{i}(\vec{r}, t)=\sum_{u \in U_{i}} \varphi_{i}(u) \frac{x_{\alpha i}(u)-x_{\alpha}}{\left|\left(\vec{r}_{i}(u)-\vec{r}\right) \cdot \vec{\xi}_{i}(u)\right|} \tag{A4} \end{equation*}(A4)∂αgi(r→,t)=∑u∈Uiφi(u)xαi(u)−xα|(r→i(u)−r→)⋅ξ→i(u)|
where U i = ( U i ′ ∪ U i ′ ′ ) ∩ ( t − τ , t + τ ) U i = U i ′ ∪ U i ′ ′ ∩ ( t − τ , t + τ ) U_(i)=(U_(i)^(')uuU_(i)^(''))nn(t-tau,t+tau)U_{i}=\left(U_{i}^{\prime} \cup U_{i}^{\prime \prime}\right) \cap(t-\tau, t+\tau)Ui=(Ui′∪Ui′′)∩(t−τ,t+τ). It is obvious that ∂ α g i ∂ α g i del_(alpha)g_(i)\partial_{\alpha} g_{i}∂αgi is continuous over R 3 × ( τ , T − τ ) ∖ Ω i ′ ′ R 3 × ( τ , T − τ ) ∖ Ω i ′ ′ R^(3)xx(tau,T-tau)\\Omega_(i)^('')\mathbb{R}^{3} \times(\tau, T-\tau) \backslash \Omega_{i}^{\prime \prime}R3×(τ,T−τ)∖Ωi′′.
Acknowledgement. We are grateful to Dr. I. Turcu for several fruitful discussions.

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Received March 1, 1993
Revised form March 13, 1995
Romanian Academy
Institute of Applied Mathematics P. O. Box 1-24, 70700 Bucharest I
Romania
1996

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