Kinetic Equations of Granular Media

*Viktor Gerasimenko*

## **Abstract**

Approaches to the rigorous derivation of a priori kinetic equations, namely, the Enskog-type and Boltzmann-type kinetic equations, describing granular media from the dynamics of inelastically colliding particles are reviewed. We also consider the problem of potential possibilities inherent in describing the evolution of the states of a system of many hard spheres with inelastic collisions by means of a one-particle distribution function.

**Keywords:** granular media, inelastic collision, Boltzmann equation, Enskog equation

## **1. Introduction**

It is well known that the properties of granular media (sand, powders, cements, seeds, etc.) have been extensively studied, in the last decades, by means of experiments, computer simulations, and analytical methods, and a huge amount of physical literature on this topic has been published (for pointers to physical literature, see in [1–6]).

Granular media are systems of many particles that attract considerable interest not only because of their numerous applications but also as systems whose collective behavior differs from the statistical behavior of ordinary media, i.e., typical macroscopic properties of media, for example, gases. In particular, the most spectacular effects include with the phenomena of collapse or cooling effect at the kinetic scale or clustering at the hydrodynamical scale, spontaneous loss of homogeneity, modification of Fourier's law and non-Maxwellian equilibrium kinetic distributions [1–3].

In modern works [4–6], it is assumed that the microscopic dynamics of granular media is dissipative, and it is described by a system of many hard spheres with inelastic collisions. The purpose of this chapter is to review some advances in the mathematical understanding of kinetic equations of systems with inelastic collisions.

As is known [7], the collective behavior of many-particle systems can be effectively described by means of a one-particle distribution function governed by the kinetic equation derived from underlying dynamics in a suitable scaling limit. At present the considerable advance is observed in a problem of the rigorous derivation of the Boltzmann kinetic equation for a system of hard spheres in the Boltzmann–Grad scaling limit [7–10]. At the same time, many recent papers [5, 11] (and see references therein) consider the Boltzmann-type and the Enskog-type

kinetic equations for inelastically interacting hard spheres, modelling the behavior of granular gases, as the original evolution equations and the rigorous derivation of such kinetic equations remain still an open problem [12, 13].

and

2

Lint *j* 1, *j* 2 � �*bs* ≐ *σ*<sup>2</sup>

<sup>þ</sup> <sup>≐</sup> *<sup>η</sup>*<sup>∈</sup> <sup>3</sup>

where *<sup>ε</sup>* <sup>¼</sup> <sup>1</sup>�*<sup>e</sup>*

*<sup>α</sup>* <sup>¼</sup> <sup>⊕</sup><sup>∞</sup>

*<sup>n</sup>*¼<sup>0</sup>*α<sup>n</sup>* <sup>Ð</sup>

Let *L*<sup>1</sup>

*<sup>α</sup>* <sup>¼</sup> <sup>P</sup><sup>∞</sup>

*S* ∗

L∗ int *j* 1, *j* 2 � �*fs* <sup>≐</sup>*σ*<sup>2</sup>

**165**

*x*⋄ *j* 2

<sup>¼</sup> <sup>Y</sup>*<sup>n</sup> i*¼1 *S* ∗ <sup>1</sup> ð Þþ *t*, *i*

*<sup>n</sup>* ð Þ *t*, 1, … , *n*

∥*f* ∥*L*<sup>1</sup>

by *L*<sup>1</sup> <sup>0</sup> ⊂*L*<sup>1</sup>

*bs*ð ÞÞ *x*1, … , *xs δ qj*

*Kinetic Equations of Granular Media*

ð 2 þ

*DOI: http://dx.doi.org/10.5772/intechopen.90027*

1 � *qj* 2 <sup>þ</sup> *ση* � �,

jj j *η* ¼ 1, *η*, *pj*

<sup>2</sup> ∈ 0, <sup>1</sup> 2

*<sup>n</sup>*¼<sup>0</sup>*αnL*<sup>1</sup>

ð Þ¼ *<sup>b</sup>*, *<sup>f</sup>* <sup>X</sup><sup>∞</sup>

*n*¼0

ð*t*

0 *dτ* Y*n i*¼1 *S* ∗

ð 2 þ

1 � *qj* 2

where for *<sup>t</sup>*≥0 the operator <sup>L</sup><sup>∗</sup>

, … , *xn*Þ*δ qj*

determined by the expressions:

*dη η*, *pj*

symbol h i �, � means a scalar product, *δ* is the Dirac measure,

1 � *pj* 2 D E � � <sup>≥</sup><sup>0</sup>

> *p*∗ *j* 1 ¼ *pj* 1

> *p*∗ *j* 2 ¼ *pj* 2

*<sup>α</sup>* the everywhere dense set in *L*<sup>1</sup>

1 *n*! ð

differentiable functions with compact supports.

1 � *pj* 2 D E � � <sup>ð</sup>*bs <sup>x</sup>*1, … , *<sup>x</sup>*<sup>∗</sup>

respectively. In (2) and (3) the following notations are used: *x*<sup>∗</sup>

n o, and the post-collision momenta are

� ð Þ 1 � *ε η η*, *pj*

þ ð Þ 1 � *ε η η*, *pj*

functions *f <sup>n</sup>*ð Þ *x*1, … , *xn* defined on the phase space of *n* hard spheres that are symmetric with respect to the permutations of the arguments *x*1, … , *xn*, equal to zero on the set of forbidden configurations *<sup>n</sup>* and equipped with the norm:

On the space of integrable functions, the semigroup of operators *S* <sup>∗</sup>

tional is defined (the functional of mean values of observables):

3�<sup>3</sup> ð Þ*<sup>n</sup>*

adjoint to semigroup of operators (1) in the sense of the continuous linear func-

The adjoint semigroup of operators is defined by the Duhamel equation:

<sup>1</sup> ð Þ *<sup>t</sup>* � *<sup>τ</sup>*, *<sup>i</sup>* <sup>X</sup>*<sup>n</sup> j* <sup>1</sup> <*j* <sup>2</sup>¼1 L∗ int *j* 1, *j* 2 � �*S* <sup>∗</sup>

int *j* 1, *j* 2

1 � *pj* 2 D E � � � 1

<sup>þ</sup> *ση* � � � *<sup>f</sup> <sup>n</sup>*ð Þ *<sup>x</sup>*1, … , *xn <sup>δ</sup> qj*

collision momenta (solutions of equations (4)) are determined as follows:

*dη η*, *pj*

In (6) the notations similar to formula (3) are used, *x*<sup>⋄</sup>

� � and *<sup>e</sup>*∈ð � 0, 1 is a restitution coefficient [6].

1 � *pj* 2 D E � � ,

1 � *pj* 2 D E � � ,

*<sup>n</sup>* be the space of sequences *<sup>f</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>0</sup>, *<sup>f</sup>* <sup>1</sup>, … , *<sup>f</sup> <sup>n</sup>*, … � � of integrable

*dx*<sup>1</sup> … *dxn*∣*f <sup>n</sup>*ð Þ *x*1, … , *xn* ∣, where *α*>1 is a real number. We denote

*dx*<sup>1</sup> … *dxn bn*ð Þ *x*1, … , *xn f <sup>n</sup>*ð Þ *x*1, … , *xn :*

� � is determined by the formula

ð Þ <sup>1</sup> � <sup>2</sup>*<sup>ε</sup>* <sup>2</sup> *<sup>f</sup> <sup>n</sup>*ð*x*1, … , *<sup>x</sup>*<sup>⋄</sup>

1 � *qj* 2 � *ση* � ��

*<sup>j</sup>* � *qj*

*<sup>α</sup>* of finite sequences of continuously

*j* 1 , … , *x*<sup>∗</sup> *j* 2 , … , *xs*

� ��

(3)

(4)

*<sup>j</sup>* � *qj*

, *p*<sup>∗</sup> *j* � �, the

*<sup>n</sup>* ð Þ*t* , *t*≥0,

*<sup>n</sup>* ð Þ *<sup>τ</sup>*, 1, … , *<sup>n</sup>* , (5)

*j* 1 , … ,

, *p*<sup>⋄</sup> *j*

*:*

� �, and the pre-

(6)

Hereinafter, an approach will be formulated, which makes it possible to rigorously justify the kinetic equations previously introduced a priori for the description of granular media, namely, the Enskog-type and Boltzmann-type kinetic equations. In addition, we will consider the problem of potential possibilities inherent in describing the evolution of the states of a system of many hard spheres with inelastic collisions by means of a one-particle distribution function.

## **2. Dynamics of hard spheres with inelastic collisions**

As mentioned above, the microscopic dynamics of granular media is described by a system of many hard spheres with inelastic collisions. We consider a system of a non-fixed, i.e., arbitrary, but finite average number of identical particles of a unit mass with the diameter *σ* >0, interacting as hard spheres with inelastic collisions. Every particle is characterized by the phase coordinates: *qi* , *pi* � � � *xi* <sup>∈</sup> <sup>3</sup> � 3 , *i*≥ 1*:*

Let *C<sup>γ</sup>* be the space of sequences *b* ¼ ð Þ *b*0, *b*1, … , *bn*, … of bounded continuous functions *bn* ∈*Cn* defined on the phase space of *n* hard spheres that are symmetric with respect to the permutations of the arguments *x*1, … , *xn*, equal to zero on the set of forbidden configurations *<sup>n</sup>* ≐f *q*1, … , *qn* � �<sup>∈</sup> <sup>3</sup>*<sup>n</sup>*k*qi* � *qj* ∣<*σ* for at least one pair ð Þ *<sup>i</sup>*, *<sup>j</sup>* : *<sup>i</sup>* 6¼ *<sup>j</sup>*∈ð Þg 1, … , *<sup>n</sup>* and equipped with the norm: <sup>∥</sup>*b*∥*<sup>C</sup><sup>γ</sup>* <sup>¼</sup> max *<sup>n</sup>*≥<sup>0</sup> *<sup>γ</sup><sup>n</sup> <sup>n</sup>*! ∥*bn*∥*Cn* ¼ max *<sup>n</sup>*≥<sup>0</sup> *<sup>γ</sup><sup>n</sup> <sup>n</sup>*! sup*<sup>x</sup>*1, … ,*xn* ∣*bn*ð Þ *x*1, … , *xn* ∣. We denote the set of continuously differentiable functions with compact supports by *Cn*,0 ⊂*Cn*.

We introduce the semigroup of operators *Sn*ð Þ*t* , *t*≥0, that describes dynamics of *n* hard spheres. It is defined by means of the phase trajectories of a hard sphere system with inelastic collisions almost everywhere on the phase space <sup>3</sup>*<sup>n</sup>* � <sup>3</sup>*<sup>n</sup>*n*<sup>n</sup>* � �, namely, outside the set <sup>0</sup> *<sup>n</sup>* of the zero Lebesgue measure, as follows [14]:

$$(\mathcal{S}\_n(t)b\_n)(\mathbf{x}\_1, \dots, \mathbf{x}\_n) \equiv \mathcal{S}\_n(t, \mathbf{1}, \dots, n) b\_n(\mathbf{x}\_1, \dots, \mathbf{x}\_n) \doteq$$

$$\begin{cases} b\_n(X\_1(t), \dots, X\_n(t)), & \text{if } (\mathbf{x}\_1, \dots, \mathbf{x}\_n) \in \left(\mathbb{R}^{3n} \backslash \mathbb{W}\_n\right) \times \mathbb{R}^{3n}, \\ \mathbf{0}, & \text{if } \left(q\_1, \dots, q\_n\right) \in \mathbb{W}\_n, \end{cases} \tag{1}$$

where the function *Xi*ðÞ� *t Xi*ð Þ *t*, *x*1, … , *xn* is a phase trajectory of *i*th particle constructed in [7] and the set <sup>0</sup> *<sup>n</sup>* consists from phase space points-specified initial data *x*1, … , *xn* that generate multiple collisions during the evolution.

On the space *Cn* one-parameter mapping (1) is a bounded ∗ -weak continuous semigroup of operators, and ∥*Sn*ð Þ*t* ∥*Cn* <1.

The infinitesimal generator L*<sup>n</sup>* of the semigroup of operators (1) is defined in the sense of a ∗ -weak convergence of the space C*n*, and it has the structure L*<sup>n</sup>* ¼ P*<sup>n</sup> <sup>j</sup>*¼<sup>1</sup>Lð Þþ*<sup>j</sup>* <sup>P</sup>*<sup>n</sup> j* <sup>1</sup> <*j* <sup>2</sup>¼<sup>1</sup>Lint *<sup>j</sup>* 1, *j* 2 � �, , and the operators Lð Þ*<sup>j</sup>* and <sup>L</sup>int *<sup>j</sup>* 1, *j* 2 � � are defined by formulas:

$$\mathcal{L}(j) \doteq \left\langle p\_j, \frac{\partial}{\partial q\_j} \right\rangle,\tag{2}$$

and

kinetic equations for inelastically interacting hard spheres, modelling the behavior of granular gases, as the original evolution equations and the rigorous derivation

Hereinafter, an approach will be formulated, which makes it possible to rigorously justify the kinetic equations previously introduced a priori for the description of granular media, namely, the Enskog-type and Boltzmann-type kinetic equations. In addition, we will consider the problem of potential possibilities inherent in describing the evolution of the states of a system of many hard spheres with

As mentioned above, the microscopic dynamics of granular media is described by a system of many hard spheres with inelastic collisions. We consider a system of a non-fixed, i.e., arbitrary, but finite average number of identical particles of a unit mass with the diameter *σ* >0, interacting as hard spheres with inelastic collisions.

Let *C<sup>γ</sup>* be the space of sequences *b* ¼ ð Þ *b*0, *b*1, … , *bn*, … of bounded continuous functions *bn* ∈*Cn* defined on the phase space of *n* hard spheres that are symmetric with respect to the permutations of the arguments *x*1, … , *xn*, equal to zero on the set

We introduce the semigroup of operators *Sn*ð Þ*t* , *t*≥0, that describes dynamics of *n* hard spheres. It is defined by means of the phase trajectories of a hard sphere system with inelastic collisions almost everywhere on the phase space <sup>3</sup>*<sup>n</sup>* �

ð Þ *<sup>i</sup>*, *<sup>j</sup>* : *<sup>i</sup>* 6¼ *<sup>j</sup>*∈ð Þg 1, … , *<sup>n</sup>* and equipped with the norm: <sup>∥</sup>*b*∥*<sup>C</sup><sup>γ</sup>* <sup>¼</sup> max *<sup>n</sup>*≥<sup>0</sup> *<sup>γ</sup><sup>n</sup>*

ð Þ *Sn*ð Þ*t bn* ð Þ� *x*1, … , *xn Sn*ð Þ *t*, 1, … , *n bn*ð Þ *x*1, … , *xn* ≐ *bn*ð Þ *<sup>X</sup>*1ð Þ*<sup>t</sup>* , … , *Xn*ð Þ*<sup>t</sup>* , ifð Þ *<sup>x</sup>*1, … , *xn* <sup>∈</sup> <sup>3</sup>*<sup>n</sup>*n*<sup>n</sup>*

where the function *Xi*ðÞ� *t Xi*ð Þ *t*, *x*1, … , *xn* is a phase trajectory of *i*th particle

On the space *Cn* one-parameter mapping (1) is a bounded ∗ -weak continuous

The infinitesimal generator L*<sup>n</sup>* of the semigroup of operators (1) is defined in the

� �, , and the operators Lð Þ*<sup>j</sup>* and <sup>L</sup>int *<sup>j</sup>*

, *∂ ∂qj* \* +

sense of a ∗ -weak convergence of the space C*n*, and it has the structure L*<sup>n</sup>* ¼

Lð Þ*j* ≐ *pj*

0, if *q*1, … , *qn*

data *x*1, … , *xn* that generate multiple collisions during the evolution.

� �<sup>∈</sup> <sup>3</sup>*<sup>n</sup>*k*qi* � *qj*

� �∈*n*,

∣*bn*ð Þ *x*1, … , *xn* ∣. We denote the set of continuously differentia-

*<sup>n</sup>* of the zero Lebesgue measure, as fol-

*<sup>n</sup>* consists from phase space points-specified initial

� � � <sup>3</sup>*<sup>n</sup>*,

1, *j* 2 � � are defined

, (2)

, *pi*

� � � *xi* <sup>∈</sup> <sup>3</sup> �

∣<*σ* for at least one pair

*<sup>n</sup>*! ∥*bn*∥*Cn* ¼

(1)

of such kinetic equations remain still an open problem [12, 13].

inelastic collisions by means of a one-particle distribution function.

**2. Dynamics of hard spheres with inelastic collisions**

Every particle is characterized by the phase coordinates: *qi*

of forbidden configurations *<sup>n</sup>* ≐f *q*1, … , *qn*

ble functions with compact supports by *Cn*,0 ⊂*Cn*.

*<sup>n</sup>*! sup*<sup>x</sup>*1, … ,*xn*

*Progress in Fine Particle Plasmas*

8 < :

constructed in [7] and the set <sup>0</sup>

*j* <sup>1</sup> <*j*

semigroup of operators, and ∥*Sn*ð Þ*t* ∥*Cn* <1.

<sup>2</sup>¼<sup>1</sup>Lint *<sup>j</sup>*

1, *j* 2

� �, namely, outside the set <sup>0</sup>

3 , *i*≥ 1*:*

max *<sup>n</sup>*≥<sup>0</sup> *<sup>γ</sup><sup>n</sup>*

<sup>3</sup>*<sup>n</sup>*n*<sup>n</sup>*

lows [14]:

P*<sup>n</sup>*

**164**

*<sup>j</sup>*¼<sup>1</sup>Lð Þþ*<sup>j</sup>* <sup>P</sup>*<sup>n</sup>*

by formulas:

$$\begin{split} \mathcal{L}\_{\text{int}}(j\_1, j\_2) b\_{\boldsymbol{s}} & \doteq \sigma^2 \int\_{S^2\_+} d\eta \Big/ \eta, \left( p\_{j\_1} - p\_{j\_2} \right) \Big/ (b\_{\boldsymbol{s}} \{ \mathbf{x}\_1, \dots, \mathbf{x}\_{j\_1}^\*, \dots, \mathbf{x}\_{j\_2}^\*, \dots, \mathbf{x}\_{\boldsymbol{s}} \} - \\ b\_{\boldsymbol{s}} (\mathbf{x}\_1, \dots, \mathbf{x}\_{\boldsymbol{s}})) \delta \Big( q\_{j\_1} - q\_{j\_2} + \sigma \eta \Big), \end{split} \tag{3}$$

respectively. In (2) and (3) the following notations are used: *x*<sup>∗</sup> *<sup>j</sup>* � *qj* , *p*<sup>∗</sup> *j* � �, the symbol h i �, � means a scalar product, *δ* is the Dirac measure, 2 <sup>þ</sup> <sup>≐</sup> *<sup>η</sup>*<sup>∈</sup> <sup>3</sup> jj j *η* ¼ 1, *η*, *pj* 1 � *pj* 2 D E � � <sup>≥</sup><sup>0</sup> n o, and the post-collision momenta are determined by the expressions:

$$\begin{aligned} p\_{j\_1}^\* &= p\_{j\_1} - (\mathbf{1} - \boldsymbol{\varepsilon}) \eta \langle \eta, \left( p\_{j\_1} - p\_{j\_2} \right) \rangle, \\ p\_{j\_2}^\* &= p\_{j\_2} + (\mathbf{1} - \boldsymbol{\varepsilon}) \eta \langle \eta, \left( p\_{j\_1} - p\_{j\_2} \right) \rangle, \end{aligned} \tag{4}$$

where *<sup>ε</sup>* <sup>¼</sup> <sup>1</sup>�*<sup>e</sup>* <sup>2</sup> ∈ 0, <sup>1</sup> 2 � � and *<sup>e</sup>*∈ð � 0, 1 is a restitution coefficient [6].

Let *L*<sup>1</sup> *<sup>α</sup>* <sup>¼</sup> <sup>⊕</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*αnL*<sup>1</sup> *<sup>n</sup>* be the space of sequences *<sup>f</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>0</sup>, *<sup>f</sup>* <sup>1</sup>, … , *<sup>f</sup> <sup>n</sup>*, … � � of integrable functions *f <sup>n</sup>*ð Þ *x*1, … , *xn* defined on the phase space of *n* hard spheres that are symmetric with respect to the permutations of the arguments *x*1, … , *xn*, equal to zero on the set of forbidden configurations *<sup>n</sup>* and equipped with the norm: ∥*f* ∥*L*<sup>1</sup> *<sup>α</sup>* <sup>¼</sup> <sup>P</sup><sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*α<sup>n</sup>* <sup>Ð</sup> *dx*<sup>1</sup> … *dxn*∣*f <sup>n</sup>*ð Þ *x*1, … , *xn* ∣, where *α*>1 is a real number. We denote by *L*<sup>1</sup> <sup>0</sup> ⊂*L*<sup>1</sup> *<sup>α</sup>* the everywhere dense set in *L*<sup>1</sup> *<sup>α</sup>* of finite sequences of continuously differentiable functions with compact supports.

On the space of integrable functions, the semigroup of operators *S* <sup>∗</sup> *<sup>n</sup>* ð Þ*t* , *t*≥0, adjoint to semigroup of operators (1) in the sense of the continuous linear functional is defined (the functional of mean values of observables):

$$\pi(b, f) = \sum\_{n=0}^{\infty} \frac{1}{n!} \int\_{\left(\mathbb{R}^3 \times \mathbb{R}^3\right)^{\pi}} d\boldsymbol{\infty}\_1 \dots d\boldsymbol{\infty}\_n b\_n(\boldsymbol{\infty}\_1, \dots, \boldsymbol{\infty}\_n) f\_n(\boldsymbol{\infty}\_1, \dots, \boldsymbol{\infty}\_n).$$

The adjoint semigroup of operators is defined by the Duhamel equation:

$$\begin{aligned} &S\_{\pi}^{\*\prime}(t,1,\ldots,n) \\ &=\prod\_{i=1}^{n}\mathcal{S}\_{1}^{\*}(t,i) + \int\_{0}^{t} d\tau \prod\_{i=1}^{n} \mathcal{S}\_{1}^{\*}(t-\tau,i) \sum\_{j\_{1}$$

where for *<sup>t</sup>*≥0 the operator <sup>L</sup><sup>∗</sup> int *j* 1, *j* 2 � � is determined by the formula

$$\begin{split} \mathcal{L}\_{\text{int}}^{\*}(\boldsymbol{j}\_{1}, \boldsymbol{j}\_{2}) \boldsymbol{f}\_{s} & \doteq \sigma^{2} \Big\|\_{\mathbb{S}\_{+}^{2}} d\eta \Big\langle \eta, \left( \boldsymbol{p}\_{\boldsymbol{j}\_{1}} - \boldsymbol{p}\_{\boldsymbol{j}\_{2}} \right) \rangle \Big( \frac{\mathbf{1}}{(1 - 2\varepsilon)^{2}} \boldsymbol{f}\_{n}(\mathbf{x}\_{1}, \ldots, \mathbf{x}\_{\boldsymbol{j}\_{1}}^{\boldsymbol{\bullet}}, \ldots, \mathbf{1}) \\\\ \boldsymbol{\mathfrak{x}}\_{\boldsymbol{j}\_{2}}^{\boldsymbol{\bullet}}, \ldots, \boldsymbol{\mathfrak{x}}\_{\boldsymbol{n}}) \delta \Big( \boldsymbol{q}\_{\boldsymbol{j}\_{1}} - \boldsymbol{q}\_{\boldsymbol{j}\_{2}} + \sigma \eta \Big) \ - \boldsymbol{f}\_{n}(\mathbf{x}\_{1}, \ldots, \mathbf{x}\_{\boldsymbol{n}}) \delta \Big( \boldsymbol{q}\_{\boldsymbol{j}\_{1}} - \boldsymbol{q}\_{\boldsymbol{j}\_{2}} - \sigma \eta \Big) \Big). \end{split} \tag{6}$$

In (6) the notations similar to formula (3) are used, *x*<sup>⋄</sup> *<sup>j</sup>* � *qj* , *p*<sup>⋄</sup> *j* � �, and the precollision momenta (solutions of equations (4)) are determined as follows:

$$\begin{aligned} p\_{j\_1}^\circ &= p\_{j\_1} - \frac{\mathbf{1} - \boldsymbol{\varepsilon}}{\mathbf{1} - \mathbf{2}\boldsymbol{\varepsilon}} \eta \Big/ \eta , \left( p\_{j\_1} - p\_{j\_2} \right) \Big), \\\ p\_{j\_2}^\circ &= p\_{j\_2} + \frac{\mathbf{1} - \boldsymbol{\varepsilon}}{\mathbf{1} - \mathbf{2}\boldsymbol{\varepsilon}} \eta \Big/ \eta , \left( p\_{j\_1} - p\_{j\_2} \right) \Big/ \boldsymbol{\varepsilon}. \end{aligned} \tag{7}$$

<sup>A</sup>1þ*n*ð Þ *<sup>t</sup>*, f g *<sup>Y</sup>*n*<sup>X</sup>* ,*<sup>X</sup>* <sup>≐</sup> <sup>X</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.90027*

*Kinetic Equations of Granular Media*

<sup>1</sup> � 1, *j*

*B*ð Þ<sup>1</sup>

statement is true. If *<sup>B</sup>*ð Þ¼ <sup>0</sup> *<sup>B</sup>*0, *<sup>B</sup>*<sup>0</sup>

*<sup>α</sup>* <sup>¼</sup> ⊕∞ *<sup>s</sup>*¼<sup>0</sup>*α<sup>s</sup> L*1

> 1 *s*! ð

� � <sup>¼</sup> *<sup>B</sup>*ð Þ<sup>1</sup> ð Þ <sup>0</sup> , *F t*ð Þ � � <sup>¼</sup>

X∞ *n*¼0

1 *n*! ð

3�<sup>3</sup> ð Þ*<sup>n</sup>*

for mean values of marginal observables, the following equality is true:

3�<sup>3</sup> ð Þ*<sup>s</sup>*

<sup>1</sup> ð Þ 0, *x*<sup>1</sup> , 0, …

*s*¼0

generalized solution.

<sup>1</sup> , … , *F*<sup>0</sup> *<sup>s</sup>* , … � �∈*L*<sup>1</sup>

ð Þ¼ *B t*ð Þ, *<sup>F</sup>*ð Þ <sup>0</sup> <sup>X</sup><sup>∞</sup>

observables *<sup>B</sup>*ð Þ<sup>1</sup> ð Þ¼ <sup>0</sup> 0, *<sup>B</sup>*ð Þ<sup>1</sup>

the series expansion [10]

*F*1ð Þ¼ *t*, *x*<sup>1</sup>

and the generating operator A<sup>∗</sup>

*<sup>B</sup>*ð Þ<sup>1</sup> ð Þ*<sup>t</sup>* , *<sup>F</sup>*ð Þ <sup>0</sup>

defined by the formula: *θ*ð Þ¼ f g *Y*n*Z Y*n*Z*.

and *Y* � ð Þ 1, … , *s* , *Z* � *j*

*Y*n*Z* ¼ 1, … , *j*

the form

1, *F*<sup>0</sup>

**167**

P:ð Þ¼ f g *Y*n*X* ,*X* ∪*iXi*

<sup>1</sup> þ 1, … , *j*

<sup>1</sup>, … , *j n*

*<sup>n</sup>* � 1, *j*

observables of certain structure, namely, the marginal observable *<sup>B</sup>*ð Þ<sup>1</sup> <sup>¼</sup> ð Þ 0, *b*1ð Þ *x*<sup>1</sup> , 0, … corresponds to the additive-type observable, and the marginal observable *<sup>B</sup>*ð Þ*<sup>k</sup>* <sup>¼</sup> <sup>ð</sup>0, … , 0, *bk*ð*x*1, … , *xk*Þ, 0, … <sup>Þ</sup> corresponds to the *<sup>k</sup>*-ary-type observable. If as initial data (9) we consider the marginal observable of additive type, then the decomposition structure of solution (10) is simplified and takes

*<sup>s</sup>* ð Þ¼ *t*, *x*1, … , *xs* A*s*ð Þ *t*, 1, … , *s*

ð Þ �<sup>1</sup> <sup>∣</sup>P∣�<sup>1</sup>

of the partition P such that ∣P∣ ¼ 1; the mapping *θ*ð Þ� is a declusterization operator

We note that one component sequences of marginal observables correspond to

On the space *C<sup>γ</sup>* for abstract initial-value problem (8) and (9), the following

nitely differentiable functions with compact supports, then the sequence of functions (10) is a classical solution, and for arbitrary initial data *B*ð Þ 0 ∈*C<sup>γ</sup>* , it is a

We remark that expansion (10) can be also represented in the form of the weak

The mean value of the marginal observable *B t*ðÞ¼ ð Þ *B*0, *B*1ð Þ*t* , … , *Bs*ð Þ*t* , … ∈C*<sup>γ</sup>* in

*dx*<sup>1</sup> … *dxs Bs*ð Þ *<sup>t</sup>*, *<sup>x</sup>*1, … , *xs <sup>F</sup>*<sup>0</sup>

ð

where the one-particle marginal distribution function *F*1ð Þ *t*, *x*<sup>1</sup> is determined by

*dx*<sup>2</sup> … *dxn*þ<sup>1</sup> <sup>A</sup><sup>∗</sup>

of adjoint semigroups of hard spheres with inelastic collisions. In the general case

3�<sup>3</sup>

<sup>1</sup> , … , *B*<sup>0</sup> *<sup>s</sup>* , … � �∈*C*<sup>0</sup>

formulation of the perturbation (iteration) series as a result of the applying of analogs of the Duhamel equation to cumulants of semigroups of operators (11).

initial state specified by a sequence of marginal distribution functions *F*ð Þ¼ 0

In particular, functional (12) of mean values of the additive-type marginal

� � takes the form:

*<sup>n</sup>* <sup>þ</sup> 1, … , *<sup>s</sup>* � �, i.e., this set is a connected subset

ð Þ jPj�1 !

X*s j*¼1

*b*<sup>1</sup> *xj*

*<sup>s</sup>* is determined by the following functional:

*dx*<sup>1</sup> *B*ð Þ<sup>1</sup>

<sup>1</sup>þ*<sup>n</sup>*ð Þ*<sup>t</sup> <sup>F</sup>*<sup>0</sup>

<sup>1</sup>þ*<sup>n</sup>*ð Þ*t* of this series is the 1ð Þ þ *n th*-order cumulant

� �, *s* ≥1*:*

*<sup>γ</sup>* ⊂*C<sup>γ</sup>* is finite sequence of infi-

*<sup>s</sup>* ð Þ *x*1, … , *xs :* (12)

<sup>1</sup> ð Þ 0, *x*<sup>1</sup> *F*1ð Þ *t*, *x*<sup>1</sup> ,

<sup>1</sup>þ*<sup>n</sup>*ð Þ *<sup>x</sup>*1, … , *xn*þ<sup>1</sup> ,

Y *Xi* ⊂P

� �⊂*Y*, f g *<sup>Y</sup>*n*<sup>Z</sup>* is the set consisting of one element

*S*∣*θ*ð Þ *Xi* <sup>∣</sup>ð Þ *t*, *θ*ð Þ *Xi* , (11)

Hence an infinitesimal generator of the adjoint semigroup of operators *S* <sup>∗</sup> *<sup>n</sup>* ð Þ*t* is defined on *L*<sup>1</sup> 0,*<sup>n</sup>* as the operator, <sup>L</sup><sup>∗</sup> *<sup>n</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> <sup>j</sup>*¼1L<sup>∗</sup> ð Þþ*<sup>j</sup>* <sup>P</sup>*<sup>n</sup> j* <sup>1</sup> <*j* 2¼1L<sup>∗</sup> int *j* 1, *j* 2 � �, where the operator adjoint to free motion operator (2) <sup>L</sup><sup>∗</sup> ð Þ*<sup>j</sup>* <sup>≐</sup> � *pj* , *∂ ∂qj* D E was introduced.

On the space *L*<sup>1</sup> *<sup>n</sup>* the one-parameter mapping defined by Eq. (5) is a bounded strong continuous semigroup of operators.

## **3. The dual hierarchy of evolution equations for observables**

It is well known [7] that many-particle systems are described by means of states and observables. The functional for mean value of observables determines a duality of states and observables, and, as a consequence, there exist two equivalent approaches to describing the evolution of systems of many particles. Traditionally, the evolution is described in terms of the evolution of states by means of the BBGKY hierarchy for marginal distribution functions. An equivalent approach to describing evolution is based on marginal observables governed by the dual BBGKY hierarchy. In the same framework, the evolution of particles with the dissipative interaction, namely, hard spheres with inelastic collisions, is described [14].

Within the framework of observables, the evolution of a system of hard spheres is described by the sequences *B t*ðÞ¼ *B*0, *B*1ð Þ *t*, *x*<sup>1</sup> , … , *Bs*ð Þ *t*, *x*1, … , *xs* ð Þ , … ∈*C<sup>γ</sup>* of the marginal observables *Bs*ð Þ *t*, *x*1, … , *xs* defined on the phase space of *s*≥1 hard spheres that are symmetric with respect to the permutations of the arguments *x*1, … , *xn*, equal to zero on the set *s*, and for *t*≥0 they are governed by the Cauchy problem of the weak formulation of the dual BBGKY hierarchy [14]:

$$\begin{split} \frac{\partial}{\partial t} B\_{i}(t, \mathbf{x}\_{1}, \dots, \mathbf{x}\_{i}) &= \left( \sum\_{j=1}^{s} \mathcal{L}(j) B\_{i}(t) + \sum\_{j\_{1} < j\_{2} = 1}^{s} \mathcal{L}\_{\text{int}}(j\_{1}, j\_{2}) B\_{i}(t) \right) (\mathbf{x}\_{1}, \dots, \mathbf{x}\_{i}) \\ &+ \sum\_{j\_{1} \neq j\_{2} = 1}^{s} \left( \mathcal{L}\_{\text{int}}(j\_{1}, j\_{2}) B\_{t-1}(t) \right) (\mathbf{x}\_{1}, \dots, \mathbf{x}\_{j\_{1}-1}, \mathbf{x}\_{j\_{1}+1}, \dots, \mathbf{x}\_{i}), \end{split} \tag{8}$$
 
$$B\_{i}(t, \mathbf{x}\_{1}, \dots, \mathbf{x}\_{i})|\_{t=0} = B\_{i}^{0}(\mathbf{x}\_{1}, \dots, \mathbf{x}\_{i}), \quad s \ge 1, \tag{9}$$

where on the set *Cs*,0 ⊂*Cs* the free motion operator Lð Þ*j* and the operator of inelastic collisions Lint *j* 1, *j* 2 � � are defined by formulas (2) and (3), respectively. We refer to recurrence evolution equation (8) as the dual BBGKY hierarchy for hard spheres with inelastic collisions.

The solution *B t*ðÞ¼ *B*0, *B*1ð Þ *t*, *x*<sup>1</sup> , … , *Bs*ð Þ *t*, *x*1, … , *xs* ð Þ , … of the Cauchy problem (8),(9) is determined by the expansions [10]:

$$B\_{i}(t, \mathbf{x}\_{1}, \ldots, \mathbf{x}\_{i}) = \sum\_{n=0}^{t-1} \frac{1}{n!} \sum\_{j\_{1} \neq \ldots \neq j\_{n} = 1}^{t} \mathfrak{A}\_{1+n}(t, \{Y\backslash Z\}, Z) B\_{s-n}^{0} \tag{10}$$

$$(\mathfrak{x}\_{1}, \ldots, \mathfrak{x}\_{j\_{1}-1}, \mathfrak{x}\_{j\_{1}+1}, \ldots, \mathfrak{x}\_{j\_{n}-1}, \mathfrak{x}\_{j\_{n}+1}, \ldots, \mathfrak{x}\_{s}),$$

where the 1ð Þ þ *n th*-order cumulant of semigroups of operators (1) of hard spheres with inelastic collisions is defined by the formula

*Kinetic Equations of Granular Media DOI: http://dx.doi.org/10.5772/intechopen.90027*

*p*⋄ *j* 1 ¼ *pj* 1

*p*⋄ *j* 2 ¼ *pj* 2 þ

0,*<sup>n</sup>* as the operator, <sup>L</sup><sup>∗</sup>

strong continuous semigroup of operators.

operator adjoint to free motion operator (2) <sup>L</sup><sup>∗</sup> ð Þ*<sup>j</sup>* <sup>≐</sup> � *pj*

defined on *L*<sup>1</sup>

*∂ ∂t*

**166**

*Bs*ð Þ¼ *t*, *x*1, … , *xs*

inelastic collisions Lint *j*

spheres with inelastic collisions.

*Bs*ð Þ¼ *t*, *x*1, … , *xs*

On the space *L*<sup>1</sup>

*Progress in Fine Particle Plasmas*

� <sup>1</sup> � *<sup>ε</sup>*

1 � *ε*

*<sup>n</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

**3. The dual hierarchy of evolution equations for observables**

namely, hard spheres with inelastic collisions, is described [14].

of the weak formulation of the dual BBGKY hierarchy [14]:

Lð Þ*<sup>j</sup> Bs*ðÞþ*<sup>t</sup>* <sup>X</sup>*<sup>s</sup>*

Lint *j* 1, *j* 2 � �*Bs*�<sup>1</sup>ð Þ*<sup>t</sup>* � � *<sup>x</sup>*1, … , *xj*

*Bs*ð Þj *<sup>t</sup>*, *<sup>x</sup>*1, … , *xs <sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> *<sup>B</sup>*<sup>0</sup>

*j* <sup>1</sup> <*j* <sup>2</sup>¼1

where on the set *Cs*,0 ⊂*Cs* the free motion operator Lð Þ*j* and the operator of

refer to recurrence evolution equation (8) as the dual BBGKY hierarchy for hard

The solution *B t*ðÞ¼ *B*0, *B*1ð Þ *t*, *x*<sup>1</sup> , … , *Bs*ð Þ *t*, *x*1, … , *xs* ð Þ , … of the Cauchy problem

<sup>1</sup><sup>þ</sup>1, … , *xj*

� �,

Lint *j* 1, *j* 2 � �*Bs*ð Þ*<sup>t</sup>*

� � are defined by formulas (2) and (3), respectively. We

<sup>A</sup><sup>1</sup>þ*<sup>n</sup>*ð Þ *<sup>t</sup>*, f g *<sup>Y</sup>*n*<sup>Z</sup>* , *<sup>Z</sup> <sup>B</sup>*<sup>0</sup>

*<sup>n</sup>*�1, *xj*

1

<sup>1</sup>�1, *xj*

*<sup>s</sup>* ð Þ *x*1, … , *xs* , *s* ≥1, (9)

*s*�*n*

(10)

*<sup>n</sup>*<sup>þ</sup>1, … , *xs*

Að Þ *x*1, … , *xs*

<sup>1</sup><sup>þ</sup>1, … , *xs* � �, (8)

X*s j*¼1

<sup>þ</sup> <sup>X</sup>*<sup>s</sup> j* <sup>1</sup>6¼*j* <sup>2</sup>¼1

> 1, *j* 2

> > X*s*�1 *n*¼0

spheres with inelastic collisions is defined by the formula

1 *n*!

*x*1, … , *xj*

*j* <sup>1</sup>6¼ … 6¼*j <sup>n</sup>*¼1

X*s*

<sup>1</sup>�1, *xj*

where the 1ð Þ þ *n th*-order cumulant of semigroups of operators (1) of hard

(8),(9) is determined by the expansions [10]:

0 @

of states and observables, and, as a consequence, there exist two equivalent approaches to describing the evolution of systems of many particles. Traditionally, the evolution is described in terms of the evolution of states by means of the BBGKY hierarchy for marginal distribution functions. An equivalent approach to describing evolution is based on marginal observables governed by the dual BBGKY hierarchy. In the same framework, the evolution of particles with the dissipative interaction,

<sup>1</sup> � <sup>2</sup>*<sup>ε</sup> η η*, *pj*

<sup>1</sup> � <sup>2</sup>*<sup>ε</sup> η η*, *pj*

It is well known [7] that many-particle systems are described by means of states and observables. The functional for mean value of observables determines a duality

Within the framework of observables, the evolution of a system of hard spheres is described by the sequences *B t*ðÞ¼ *B*0, *B*1ð Þ *t*, *x*<sup>1</sup> , … , *Bs*ð Þ *t*, *x*1, … , *xs* ð Þ , … ∈*C<sup>γ</sup>* of the marginal observables *Bs*ð Þ *t*, *x*1, … , *xs* defined on the phase space of *s*≥1 hard spheres that are symmetric with respect to the permutations of the arguments *x*1, … , *xn*, equal to zero on the set *s*, and for *t*≥0 they are governed by the Cauchy problem

Hence an infinitesimal generator of the adjoint semigroup of operators *S* <sup>∗</sup>

1 � *pj* 2

1 � *pj* 2

*<sup>j</sup>*¼1L<sup>∗</sup> ð Þþ*<sup>j</sup>* <sup>P</sup>*<sup>n</sup>*

*<sup>n</sup>* the one-parameter mapping defined by Eq. (5) is a bounded

D E � �

D E � �

,

(7)

*<sup>n</sup>* ð Þ*t* is

was introduced.

*:*

, *∂ ∂qj* D E

*j* <sup>1</sup> <*j* 2¼1L<sup>∗</sup> int *j* 1, *j* 2 � �, where the

$$\mathfrak{A}\_{1+\mathfrak{n}}(t,\{Y|X\},X) \doteq \sum\_{\mathbb{P}: (\{Y|X\},X) = \cup\_i X\_i} (-\mathbf{1})^{|\mathbb{P}|-1} (|\mathcal{P}|-\mathbf{1})! \prod\_{X\_i \subset \mathcal{P}} \mathbb{S}\_{|\theta(X\_i)|}(t,\theta(X\_i)), \tag{11}$$

and *Y* � ð Þ 1, … , *s* , *Z* � *j* <sup>1</sup>, … , *j n* � �⊂*Y*, f g *<sup>Y</sup>*n*<sup>Z</sup>* is the set consisting of one element *Y*n*Z* ¼ 1, … , *j* <sup>1</sup> � 1, *j* <sup>1</sup> þ 1, … , *j <sup>n</sup>* � 1, *j <sup>n</sup>* <sup>þ</sup> 1, … , *<sup>s</sup>* � �, i.e., this set is a connected subset of the partition P such that ∣P∣ ¼ 1; the mapping *θ*ð Þ� is a declusterization operator defined by the formula: *θ*ð Þ¼ f g *Y*n*Z Y*n*Z*.

We note that one component sequences of marginal observables correspond to observables of certain structure, namely, the marginal observable *<sup>B</sup>*ð Þ<sup>1</sup> <sup>¼</sup> ð Þ 0, *b*1ð Þ *x*<sup>1</sup> , 0, … corresponds to the additive-type observable, and the marginal observable *<sup>B</sup>*ð Þ*<sup>k</sup>* <sup>¼</sup> <sup>ð</sup>0, … , 0, *bk*ð*x*1, … , *xk*Þ, 0, … <sup>Þ</sup> corresponds to the *<sup>k</sup>*-ary-type observable. If as initial data (9) we consider the marginal observable of additive type, then the decomposition structure of solution (10) is simplified and takes the form

$$B\_s^{(1)}(t, \mathbf{x}\_1, \dots, \mathbf{x}\_s) = \mathfrak{A}\_s(t, \mathbf{1}, \dots, s) \sum\_{j=1}^s b\_1(\mathbf{x}\_j), \quad s \ge \mathbf{1}.$$

On the space *C<sup>γ</sup>* for abstract initial-value problem (8) and (9), the following statement is true. If *<sup>B</sup>*ð Þ¼ <sup>0</sup> *<sup>B</sup>*0, *<sup>B</sup>*<sup>0</sup> <sup>1</sup> , … , *B*<sup>0</sup> *<sup>s</sup>* , … � �∈*C*<sup>0</sup> *<sup>γ</sup>* ⊂*C<sup>γ</sup>* is finite sequence of infinitely differentiable functions with compact supports, then the sequence of functions (10) is a classical solution, and for arbitrary initial data *B*ð Þ 0 ∈*C<sup>γ</sup>* , it is a generalized solution.

We remark that expansion (10) can be also represented in the form of the weak formulation of the perturbation (iteration) series as a result of the applying of analogs of the Duhamel equation to cumulants of semigroups of operators (11).

The mean value of the marginal observable *B t*ðÞ¼ ð Þ *B*0, *B*1ð Þ*t* , … , *Bs*ð Þ*t* , … ∈C*<sup>γ</sup>* in initial state specified by a sequence of marginal distribution functions *F*ð Þ¼ 0 1, *F*<sup>0</sup> <sup>1</sup> , … , *F*<sup>0</sup> *<sup>s</sup>* , … � �∈*L*<sup>1</sup> *<sup>α</sup>* <sup>¼</sup> ⊕∞ *<sup>s</sup>*¼<sup>0</sup>*α<sup>s</sup> L*1 *<sup>s</sup>* is determined by the following functional:

$$\mathbb{P}\left(B(t), F(0)\right) = \sum\_{\iota=0}^{\infty} \frac{1}{\mathfrak{s}!} \int\_{\left(\mathbb{R}^{\triangleright} \times \mathbb{R}^{\triangleright}\right)'} d\mathfrak{x}\_1 \dots d\mathfrak{x}\_\iota B\_\iota(t, \mathfrak{x}\_1, \dots, \mathfrak{x}\_\iota) F\_\iota^0(\mathfrak{x}\_1, \dots, \mathfrak{x}\_\iota). \tag{12}$$

In particular, functional (12) of mean values of the additive-type marginal observables *<sup>B</sup>*ð Þ<sup>1</sup> ð Þ¼ <sup>0</sup> 0, *<sup>B</sup>*ð Þ<sup>1</sup> <sup>1</sup> ð Þ 0, *x*<sup>1</sup> , 0, … � � takes the form:

$$\left(B^{(1)}(t), F(\mathbf{0})\right) = \left(B^{(1)}(\mathbf{0}), F(t)\right) = \int\_{\mathbb{R}^{\beta}\times\mathbb{R}^{\beta}} d\boldsymbol{\pi}\_1 B\_1^{(1)}(\mathbf{0}, \boldsymbol{\pi}\_1) F\_1(t, \boldsymbol{\pi}\_1),$$

where the one-particle marginal distribution function *F*1ð Þ *t*, *x*<sup>1</sup> is determined by the series expansion [10]

$$F\_1(t, \mathbf{x}\_1) = \sum\_{n=0}^{\infty} \frac{1}{n!} \int\_{\left(\mathbb{R}^3 \times \mathbb{R}^3\right)^n} d\mathbf{x}\_2 \dots d\mathbf{x}\_{n+1} \mathfrak{A}\_{1+n}^\*(t) F\_{1+n}^0(\mathbf{x}\_1, \dots, \mathbf{x}\_{n+1}),$$

and the generating operator A<sup>∗</sup> <sup>1</sup>þ*<sup>n</sup>*ð Þ*t* of this series is the 1ð Þ þ *n th*-order cumulant of adjoint semigroups of hard spheres with inelastic collisions. In the general case for mean values of marginal observables, the following equality is true:

$$(B(t), F(\mathbf{0})) = (B(\mathbf{0}), F(t)),$$

where the sequence *F t*ðÞ¼ ð Þ 1, *F*1ð Þ*t* , … , *Fs*ð Þ*t* , … is a solution of the Cauchy problem of the BBGKY hierarchy of hard spheres with inelastic collisions [14]. The last equality signifies the equivalence of two pictures of the description of the evolution of hard spheres by means of the BBGKY hierarchy [7] and the dual BBGKY hierarchy (8).

Hereinafter we consider initial states of hard spheres specified by a one-particle marginal distribution function, namely,

$$F\_s^{(\varepsilon)}(\mathbf{x}\_1, \dots, \mathbf{x}\_s) = \prod\_{i=1}^s F\_1^0(\mathbf{x}\_i) \mathcal{X}\_{\mathbb{R}^k \backslash \mathbb{W}\_\varepsilon}, \quad s \ge 1,\tag{13}$$

P

¼0

*DOI: http://dx.doi.org/10.5772/intechopen.90027*

*n*�*m*1� … �*mj*þ*s*þ1�*i*

*s*þ*n*�*m*1� … �*mj*þ2�*i <sup>j</sup>*

<sup>A</sup>b1þ*n*ð Þ *<sup>t</sup>*, f g *<sup>Y</sup>* ,*X*n*<sup>Y</sup>* <sup>≐</sup> <sup>A</sup><sup>∗</sup>

of hard spheres with inelastic collisions.

X∞ *n*¼0

� � ¼ � *<sup>p</sup>*1, *<sup>∂</sup>*

forbidden configurations.

<sup>þ</sup>*σ*<sup>2</sup> ð

1 *n*! ð

where the generating operator A<sup>∗</sup>

3�<sup>3</sup> ð Þ*<sup>n</sup>*

*∂q*1 � �*F*<sup>1</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*<sup>1</sup>

> 3�<sup>2</sup> þ

(14) in the case of *<sup>s</sup>* <sup>¼</sup> 2, and the expressions *<sup>p</sup>*<sup>⋄</sup>

*<sup>s</sup>*þ*n*�*m*Q<sup>1</sup>� … �*mj ij*¼1

> *j* �*k j*

order scattering cumulant by the operator Ab1þ*n*ð Þ*t* :

<sup>1</sup> � *mj*, *<sup>k</sup> <sup>j</sup>*

ated by dynamics of a hard sphere system with inelastic collisions.

*k j*

*n*�*m*1� … �*mj*þ*s*þ2�*i*

*j*

1

� �!

*j*

*<sup>t</sup>*, *ij*, *<sup>s</sup>* <sup>þ</sup> *<sup>n</sup>* � *<sup>m</sup>*<sup>1</sup> � … � *mj* <sup>þ</sup> <sup>1</sup><sup>þ</sup> �

*<sup>n</sup>*�*m*1� … �*mj*þ*s*þ<sup>1</sup> � 0, and we denote the 1ð Þ þ *n th*-

*j*

Y*s*þ*n i*¼1 A∗ <sup>1</sup> ð Þ *<sup>t</sup>*, *<sup>i</sup>* �<sup>1</sup> ,

> Y*n*þ1 *i*¼1 *F*0

<sup>1</sup>þ*<sup>n</sup>*ð Þ *<sup>t</sup>*, 1, … , *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> is the 1ð Þ <sup>þ</sup> *<sup>n</sup> th*-

<sup>2</sup>j*F*1ð Þ*<sup>t</sup>* � �

<sup>1</sup> , (17)

<sup>2</sup> are the pre-collision

<sup>1</sup> ð Þ *xi* , (15)

<sup>1</sup> , *<sup>q</sup>*<sup>1</sup> � *ση*, *<sup>p</sup>*<sup>⋄</sup>

*n*�*m*1� … �*mj*þ*s*þ2�*ij*

*s*þ*n*�*m*1� … �*m <sup>j</sup>*þ1�*ij*

�

Þ,

*<sup>n</sup>*�*m*1� … �*mj*þ*s*þ1�*ij* � *<sup>k</sup>*

, … , *s* þ *n* � *m*<sup>1</sup> � … � *mj* þ *k*

<sup>1</sup>þ*n*ð ÞX *<sup>t</sup>*, f g *<sup>Y</sup>* ,*X*n*<sup>Y</sup>* 3ð Þ *<sup>s</sup>*þ*<sup>n</sup>* <sup>n</sup>*s*þ*<sup>n</sup>*

We emphasize that in fact functionals (14) characterize the correlations gener-

The second element of the sequence *F t*ð Þ j*F*1ð Þ*t* , i.e., the one-particle marginal

<sup>1</sup>þ*<sup>n</sup>*ðÞ� *<sup>t</sup>* <sup>A</sup><sup>∗</sup>

For *t* ≥0 the one-particle marginal distribution function (15) is a solution of the following Cauchy problem of the non-Markovian Enskog kinetic equation [14, 15]:

� � � � 1

*<sup>F</sup>*1ð Þj *<sup>t</sup> <sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> *<sup>F</sup>*<sup>0</sup>

momenta of hard spheres with inelastic collisions (7), i.e., solutions of Eq. (4). We note that the structure of collision integral of the non-Markovian Enskog equation for granular gases (16) is such that the first term of its expansion is the collision integral of the Boltzmann–Enskog kinetic equation and the next terms describe all possible correlations which are created by hard sphere dynamics with inelastic collisions and by the propagation of initial correlations connected with the

where the collision integral is determined by the marginal functional of the state

We remark also that based on the non-Markovian Enskog equation (16), we can

For the abstract Cauchy problem of the non-Markovian Enskog kinetic equation (16), (17) in the space of integrable functions , the following statement is true [14]. A global in time solution of the Cauchy problem of the non-Markovian Enskog

formulate the Markovian Enskog kinetic equation with inelastic collisions [14].

�*F*<sup>2</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*1, *<sup>q</sup>*<sup>1</sup> <sup>þ</sup> *ση*, *<sup>p</sup>*2j*F*1ð Þ*<sup>t</sup>* � �Þ, (16)

<sup>1</sup> and *p*<sup>⋄</sup>

order cumulant of adjoint semigroups of hard spheres with inelastic collisions.

� �

*dp*2*dη η*, *p*<sup>1</sup> � *p*<sup>2</sup>

distribution function *F*1ð Þ*t* , is determined by the following series expansion:

*dx*<sup>2</sup> … *dxn*þ<sup>1</sup> <sup>A</sup><sup>∗</sup>

<sup>1</sup>þ*<sup>n</sup>*ð Þ*t* is the 1ð Þ þ *n th*-order cumulant of adjoint semigroups

<sup>1</sup>þ*<sup>n</sup>*ð ÞX*<sup>t</sup>* 3 1ð Þ <sup>þ</sup>*<sup>n</sup>* <sup>n</sup>1þ*<sup>n</sup>*

ð Þ <sup>1</sup> � <sup>2</sup>*<sup>ε</sup>* <sup>2</sup> *<sup>F</sup>*<sup>2</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*<sup>⋄</sup>

*n*�*m*1� … �*m j*þ*s*

<sup>A</sup>b1þ*<sup>k</sup> j*

*k j*

where it means that *k <sup>j</sup>*

and the operator A<sup>∗</sup>

*F*1ð Þ¼ *t*, *x*<sup>1</sup>

*∂ ∂t*

**169**

*F*<sup>1</sup> *t*, *q*1, *p*<sup>1</sup>

*n*�*m*1� … �*mj*þ*s*�1

*Kinetic Equations of Granular Media*

*k j*

*k j*

where X 3*<sup>s</sup>* <sup>n</sup>*<sup>s</sup>* � X*<sup>s</sup> <sup>q</sup>*1, … , *qs* � � is a characteristic function of allowed configurations <sup>3</sup>*<sup>s</sup>* <sup>n</sup>*<sup>s</sup>* of *<sup>s</sup>* hard spheres and *<sup>F</sup>*<sup>0</sup> <sup>1</sup> <sup>∈</sup>*L*<sup>1</sup> <sup>3</sup> � <sup>3</sup> � �. Initial data (13) is intrinsic for the kinetic description of many-particle systems because in this case all possible states are described by means of a one-particle marginal distribution function.

## **4. The non-Markovian Enskog kinetic equation**

In the case of initial states (13), the dual picture of the evolution to the picture of the evolution by means of observables of a system of hard spheres with inelastic collisions governed by the dual BBGKY hierarchy (8) for marginal observables is the evolution of states described by means of the non-Markovian Enskog kinetic equation and a sequence of explicitly defined functionals of a solution of such kinetic equation.

Indeed, in view of the fact that the initial state is completely specified by a oneparticle marginal distribution function on allowed configurations (13), for mean value functional (12), the following representation holds [14, 15]:

$$\left(B(t), F^{(c)}\right) = \left(B(\mathbf{0}), F(t|F\_1(t))\right),$$

where *<sup>F</sup>*ð Þ*<sup>c</sup>* <sup>¼</sup> 1, *<sup>F</sup>*ð Þ*<sup>c</sup>* <sup>1</sup> , … , *F*ð Þ*<sup>c</sup> <sup>s</sup>* , … � � is the sequence of initial marginal distribution functions (13) and the sequence *F t*ð Þ¼ j*F*1ð Þ*t* ð Þ 1, *F*1ð Þ*t* , *F*2ð Þ *t*j*F*1ð Þ*t* , … , *Fs*ð Þ *t*j*F*1ð Þ*t* is a sequence of the marginal functionals of the state *Fs t*, *x*1, … , *xs* ð Þ j*F*1ð Þ*t* represented by the series expansions over the products with respect to the one-particle marginal distribution function *F*1ð Þ*t* :

$$F\_{\mathbf{r}}(\mathbf{t}, \mathbf{x}\_1, \dots, \mathbf{x}\_r | F\_1(\mathbf{t})) \doteq \sum\_{n=0}^{\infty} \frac{1}{n!} \int\_{\left(\mathbb{R}^{\mathbf{J}} \times \mathbb{R}^{\mathbf{J}}\right)^{\mathbf{r}}} d\mathbf{x}\_{\mathbf{r}+1} \dots d\mathbf{x}\_{\mathbf{r}+n} \mathfrak{A}\_{\mathbf{1}+\mathbf{n}}(\mathbf{t}, \{\mathbf{Y}\}, \mathbf{X} \{\mathbf{Y}\}) \prod\_{i=1}^{\mathbf{r}+n} F\_1(\mathbf{t}, \mathbf{x}\_i), \quad \mathbf{s} \ge \mathbf{2}. \tag{14}$$

In series (14) we used the notations *Y* � ð Þ 1, … , *s* , *X* � ð Þ 1, … , *s* þ *n* , and the ð Þ *n* þ 1 *th*-order generating operator V<sup>1</sup>þ*<sup>n</sup>*ð Þ*t* , *n*≥0 is defined as follows [15]:

<sup>V</sup><sup>1</sup>þ*<sup>n</sup>*ð Þ *<sup>t</sup>*, f g *<sup>Y</sup>* ,*X*n*<sup>Y</sup>* <sup>≐</sup> <sup>X</sup>*<sup>n</sup> k*¼0 ð Þ �<sup>1</sup> *<sup>k</sup>* <sup>X</sup>*<sup>n</sup> m*1¼1 … *n*�*m*1� <sup>X</sup>… �*mk*�<sup>1</sup> *mk*¼1 *n*! ð Þ *n* � *m*<sup>1</sup> � … � *mk* ! � Ab<sup>1</sup>þ*n*�*m*1� … �*mk* ð Þ *t*, f g *Y* , *s* þ 1, … , *s* þ *n* � *m*<sup>1</sup> � … � *mk* Y *k j*¼1 X*mj k j* <sup>2</sup>¼0 …

*Kinetic Equations of Granular Media DOI: http://dx.doi.org/10.5772/intechopen.90027*

ð Þ¼ *B t*ð Þ, *F*ð Þ 0 ð Þ *B*ð Þ 0 , *F t*ð Þ ,

where the sequence *F t*ðÞ¼ ð Þ 1, *F*1ð Þ*t* , … , *Fs*ð Þ*t* , … is a solution of the Cauchy problem of the BBGKY hierarchy of hard spheres with inelastic collisions [14]. The last equality signifies the equivalence of two pictures of the description of the evolution of hard spheres by means of the BBGKY hierarchy [7] and the dual

Hereinafter we consider initial states of hard spheres specified by a one-particle

n*<sup>s</sup>*

� � is a characteristic function of allowed configura-

<sup>1</sup> <sup>∈</sup>*L*<sup>1</sup> <sup>3</sup> � <sup>3</sup> � �. Initial data (13) is intrinsic for

is the sequence of initial marginal distribution

*i*¼1

*n*! ð Þ *n* � *m*<sup>1</sup> � … � *mk* !

X*mj*

…

*k j* <sup>2</sup>¼0

Y *k*

*j*¼1

*F*1ð Þ *t*, *xi* , *s*≥2*:*

(14)

�

, *s*≥1, (13)

Y*s i*¼1 *F*0 <sup>1</sup> ð ÞX *xi* 3*<sup>s</sup>*

the kinetic description of many-particle systems because in this case all possible states are described by means of a one-particle marginal distribution function.

In the case of initial states (13), the dual picture of the evolution to the picture of the evolution by means of observables of a system of hard spheres with inelastic collisions governed by the dual BBGKY hierarchy (8) for marginal observables is the evolution of states described by means of the non-Markovian Enskog kinetic equation and a sequence of explicitly defined functionals of a solution of such

Indeed, in view of the fact that the initial state is completely specified by a oneparticle marginal distribution function on allowed configurations (13), for mean

functions (13) and the sequence *F t*ð Þ¼ j*F*1ð Þ*t* ð Þ 1, *F*1ð Þ*t* , *F*2ð Þ *t*j*F*1ð Þ*t* , … , *Fs*ð Þ *t*j*F*1ð Þ*t* is a sequence of the marginal functionals of the state *Fs t*, *x*1, … , *xs* ð Þ j*F*1ð Þ*t* represented by the series expansions over the products with respect to the one-particle marginal

In series (14) we used the notations *Y* � ð Þ 1, … , *s* , *X* � ð Þ 1, … , *s* þ *n* , and the ð Þ *n* þ 1 *th*-order generating operator V<sup>1</sup>þ*<sup>n</sup>*ð Þ*t* , *n*≥0 is defined as follows [15]:

…

*n*�*m*1�

¼ ð Þ *B*ð Þ 0 , *F t*ð Þ j*F*1ð Þ*t* ,

*dxs*þ<sup>1</sup> … *dxs*þ*<sup>n</sup>* <sup>V</sup><sup>1</sup>þ*<sup>n</sup>*ð Þ *<sup>t</sup>*, f g *<sup>Y</sup>* ,*X*n*<sup>Y</sup>* <sup>Y</sup>*<sup>s</sup>*þ*<sup>n</sup>*

<sup>X</sup>… �*mk*�<sup>1</sup>

*mk*¼1

BBGKY hierarchy (8).

*Progress in Fine Particle Plasmas*

where X 3*<sup>s</sup>*

kinetic equation.

where *<sup>F</sup>*ð Þ*<sup>c</sup>* <sup>¼</sup> 1, *<sup>F</sup>*ð Þ*<sup>c</sup>*

distribution function *F*1ð Þ*t* :

*Fs <sup>t</sup>*, *<sup>x</sup>*1, … , *xs* ð Þ <sup>j</sup>*F*1ð Þ*<sup>t</sup>* <sup>≐</sup> <sup>X</sup><sup>∞</sup>

**168**

tions <sup>3</sup>*<sup>s</sup>*

marginal distribution function, namely,

*F*ð Þ*<sup>c</sup>*

<sup>n</sup>*<sup>s</sup>* � X*<sup>s</sup> <sup>q</sup>*1, … , *qs*

<sup>n</sup>*<sup>s</sup>* of *<sup>s</sup>* hard spheres and *<sup>F</sup>*<sup>0</sup>

*<sup>s</sup>* ð Þ¼ *x*1, … , *xs*

**4. The non-Markovian Enskog kinetic equation**

value functional (12), the following representation holds [14, 15]:

*B t*ð Þ, *<sup>F</sup>*ð Þ*<sup>c</sup>* � �

*<sup>s</sup>* , …

3�<sup>3</sup> ð Þ*<sup>n</sup>*

ð Þ �<sup>1</sup> *<sup>k</sup>* <sup>X</sup>*<sup>n</sup>*

Ab<sup>1</sup>þ*n*�*m*1� … �*mk* ð Þ *t*, f g *Y* , *s* þ 1, … , *s* þ *n* � *m*<sup>1</sup> � … � *mk*

*m*1¼1

<sup>1</sup> , … , *F*ð Þ*<sup>c</sup>*

*n*¼0

<sup>V</sup><sup>1</sup>þ*<sup>n</sup>*ð Þ *<sup>t</sup>*, f g *<sup>Y</sup>* ,*X*n*<sup>Y</sup>* <sup>≐</sup> <sup>X</sup>*<sup>n</sup>*

1 *n*! ð

*k*¼0

� �

$$\begin{pmatrix} k\_{n-m\_1\ldots\ldots\ldots m\_j+s-1}^j & s+n-m\_1\ldots\ldots m\_j & \mathbf{1} \\ \sum\_{k\_{n-m\_1\ldots\ldots m\_j+s}^j=0} & \prod\_{i\_j=1}^j & \left(k\_{n-m\_1\ldots\ldots\ldots m\_j+s+1-i\_j}^j - k\_{n-m\_1\ldots\ldots\ldots m\_j+s+2-i\_j}^j\right)! \\ \widehat{\mathfrak{A}}\_{1\star k\_{n-m\_1\ldots\ldots m\_j+s+1-i\_j}^j - k\_{n-m\_1\ldots\ldots m\_j-s+2-i\_j}^j}^j \left(t, i\_j, s+n-m\_1-\ldots-m\_j+1+\right. \\ & k\_{s+n-m\_1\ldots\ldots m\_j+2-i\_j}^j, \ldots, s+n-m\_1-\ldots-m\_j+k\_{s+n-m\_1\ldots\ldots m\_j+1-i\_j}^j\right), \end{pmatrix}$$

where it means that *k <sup>j</sup>* <sup>1</sup> � *mj*, *<sup>k</sup> <sup>j</sup> <sup>n</sup>*�*m*1� … �*mj*þ*s*þ<sup>1</sup> � 0, and we denote the 1ð Þ þ *n th*order scattering cumulant by the operator Ab1þ*n*ð Þ*t* :

$$\widehat{\mathfrak{A}}\_{1+n}(t,\{Y\},X\backslash Y) \doteq \mathfrak{A}^\*\_{1+n}(t,\{Y\},X\backslash Y) \mathcal{X}\_{\mathbb{R}^{\mathcal{N}+n}\backslash \mathcal{W}\_{s+n}} \prod\_{i=1}^{s+n} \mathfrak{A}^\*\_1(t,i)^{-1},$$

and the operator A<sup>∗</sup> <sup>1</sup>þ*<sup>n</sup>*ð Þ*t* is the 1ð Þ þ *n th*-order cumulant of adjoint semigroups of hard spheres with inelastic collisions.

We emphasize that in fact functionals (14) characterize the correlations generated by dynamics of a hard sphere system with inelastic collisions.

The second element of the sequence *F t*ð Þ j*F*1ð Þ*t* , i.e., the one-particle marginal distribution function *F*1ð Þ*t* , is determined by the following series expansion:

$$F\_1(t, \mathbf{x}\_1) = \sum\_{n=0}^{\infty} \frac{1}{n!} \int\_{\left(\mathbb{R}^3 \times \mathbb{R}^3\right)^n} d\mathbf{x}\_2 \dots d\mathbf{x}\_{n+1} \mathfrak{A}\_{1+n}^\*(t) \mathcal{X}\_{\mathbb{R}^{3(1+n)} \backslash W\_{1+n}} \prod\_{i=1}^{n+1} F\_1^0(\mathbf{x}\_i),\tag{15}$$

where the generating operator A<sup>∗</sup> <sup>1</sup>þ*<sup>n</sup>*ðÞ� *<sup>t</sup>* <sup>A</sup><sup>∗</sup> <sup>1</sup>þ*<sup>n</sup>*ð Þ *<sup>t</sup>*, 1, … , *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> is the 1ð Þ <sup>þ</sup> *<sup>n</sup> th*order cumulant of adjoint semigroups of hard spheres with inelastic collisions.

For *t* ≥0 the one-particle marginal distribution function (15) is a solution of the following Cauchy problem of the non-Markovian Enskog kinetic equation [14, 15]:

$$\begin{split} \frac{\partial}{\partial t} F\_1(t, q\_1, p\_1) &= - \left\langle p\_1, \frac{\partial}{\partial q\_1} \right\rangle F\_1(t, q\_1, p\_1) \\ &+ \sigma^2 \int\_{\mathbb{R}^3 \times \mathbb{S}\_+^2} dp\_2 d\eta \langle \eta, (p\_1 - p\_2) \rangle \left( \frac{1}{(1 - 2\varepsilon)^2} F\_2(t, q\_1, p\_1^\diamond, q\_1 - \sigma \eta, p\_2^\diamond) F\_1(t) \right) \\ &= - \left\langle p\_1, \frac{1}{(1 - \varepsilon)^2} \right\rangle \end{split} \tag{17.14}$$

�*F*<sup>2</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*1, *<sup>q</sup>*<sup>1</sup> <sup>þ</sup> *ση*, *<sup>p</sup>*2j*F*1ð Þ*<sup>t</sup>* � �Þ, (16)

$$\left.F\_1(t)\right|\_{t=0} = F\_1^0,\tag{17}$$

where the collision integral is determined by the marginal functional of the state (14) in the case of *<sup>s</sup>* <sup>¼</sup> 2, and the expressions *<sup>p</sup>*<sup>⋄</sup> <sup>1</sup> and *p*<sup>⋄</sup> <sup>2</sup> are the pre-collision momenta of hard spheres with inelastic collisions (7), i.e., solutions of Eq. (4).

We note that the structure of collision integral of the non-Markovian Enskog equation for granular gases (16) is such that the first term of its expansion is the collision integral of the Boltzmann–Enskog kinetic equation and the next terms describe all possible correlations which are created by hard sphere dynamics with inelastic collisions and by the propagation of initial correlations connected with the forbidden configurations.

We remark also that based on the non-Markovian Enskog equation (16), we can formulate the Markovian Enskog kinetic equation with inelastic collisions [14].

For the abstract Cauchy problem of the non-Markovian Enskog kinetic equation (16), (17) in the space of integrable functions , the following statement is true [14]. A global in time solution of the Cauchy problem of the non-Markovian Enskog

equation (16) is determined by function (15). For small densities and *F*0 <sup>1</sup> ∈ *L*<sup>1</sup> <sup>0</sup> <sup>3</sup> � <sup>3</sup> � �, function (15) is a strong solution, and for an arbitrary initial data *F*0 <sup>1</sup> <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> <sup>3</sup> � <sup>3</sup> � �, it is a weak solution.

where the momenta *p*<sup>⋄</sup>

*Kinetic Equations of Granular Media*

*DOI: http://dx.doi.org/10.5772/intechopen.90027*

<sup>w</sup> � lim<sup>ϵ</sup>!<sup>0</sup>

gation of the initial chaos).

*∂ ∂t*

ð<sup>∞</sup> 0

ð<sup>∞</sup> 0

**171**

**6. One-dimensional granular gases**

inelastic collisions (7).

<sup>1</sup> and *p*<sup>⋄</sup>

tionals of the state (14), the following statement holds:

ϵ2*s*

<sup>2</sup> are pre-collision momenta of hard spheres with

*f* <sup>1</sup> *t*, *xj*

*j*¼1

As noted above, all possible correlations of a system of hard spheres with inelastic collisions are described by marginal functionals of the state (14). Taking into consideration the fact of the existence of the Boltzmann–Grad scaling limit of a solution of the non-Markovian Enskog kinetic equation (16), for marginal func-

*Fs <sup>t</sup>*, *<sup>x</sup>*1, … , *xs* <sup>ð</sup> <sup>j</sup>*F*1ð Þ*<sup>t</sup>* Þ � <sup>Y</sup>*<sup>s</sup>*

where the limit one-particle distribution function *f* <sup>1</sup>ð Þ*t* is governed by the Boltzmann kinetic equation for granular gases (18). This property of marginal functionals of the state (14) means the propagation of the initial chaos [7].

It should be emphasized that the Boltzmann–Grad asymptotics of a solution of the non-Markovian Enskog equation (16) in a multidimensional space are analogous of the Boltzmann–Grad asymptotic behavior of a hard sphere system with the elastic collisions [10]. Such asymptotic behavior is governed by the Boltzmann equation for a granular gas (18), and the asymptotics of marginal functionals of the state (14) are the product of its solution (this property is interpreted as the propa-

As is known, the evolution of a one-dimensional system of hard spheres with elastic collisions is trivial (free motion or Knudsen flow) in the Boltzmann–Grad scaling limit [7], but, as it was taken notice in paper [16], in this approximation the kinetics of inelastically interacting hard spheres (rods) is not trivial, and it is governed by the Boltzmann kinetic equation for one-dimensional granular gases [16–19]. Below the approach to the rigorous derivation of Boltzmann-type equation for one-dimensional granular gases will be outlined. It should be emphasized that a system of many hard rods with inelastic collisions displays the basic properties of granular gases inasmuch as under the inelastic collisions only the normal compo-

In case of a one-dimensional granular gas for *t*≥0 in dimensionless form, the

*F*<sup>1</sup> *t*, *q*1, *p*<sup>1</sup> � �<sup>þ</sup>

*<sup>F</sup>*<sup>2</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*1, *<sup>q</sup>*<sup>1</sup> <sup>þ</sup> <sup>ϵ</sup>, *<sup>p</sup>*<sup>1</sup> � *<sup>P</sup>*j*F*1ð Þ*<sup>t</sup>* � ��

*<sup>F</sup>*1ð Þj *<sup>t</sup> <sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> *<sup>F</sup>*<sup>ϵ</sup>,0

<sup>1</sup> *<sup>p</sup>*1, *<sup>P</sup>* � �, *<sup>q</sup>*<sup>1</sup> � <sup>ϵ</sup>, *<sup>p</sup>*<sup>⋄</sup> <sup>2</sup> *<sup>p</sup>*1, *<sup>P</sup>* � �j*F*1ð Þ*<sup>t</sup>* � ��

<sup>1</sup> *<sup>p</sup>*1, *<sup>P</sup>* � �, *<sup>q</sup>*<sup>1</sup> <sup>þ</sup> <sup>ϵ</sup>, *<sup>p</sup>*~<sup>⋄</sup> <sup>2</sup> *<sup>p</sup>*1, *<sup>P</sup>* � �j*F*1ð Þ*<sup>t</sup>* � ��

þ (20)

<sup>1</sup> , (21)

,

nent of relative velocities dissipates in a multidimensional case.

ð Þ <sup>1</sup> � <sup>2</sup>*<sup>ε</sup>* <sup>2</sup> *<sup>F</sup>*<sup>2</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*<sup>⋄</sup>

*<sup>F</sup>*<sup>2</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*1, *<sup>q</sup>*<sup>1</sup> � <sup>ϵ</sup>, *<sup>p</sup>*<sup>1</sup> <sup>þ</sup> *<sup>P</sup>*j*F*1ð Þ*<sup>t</sup>* � ��

ð Þ <sup>1</sup> � <sup>2</sup>*<sup>ε</sup>* <sup>2</sup> *<sup>F</sup>*<sup>2</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*~<sup>⋄</sup>

*∂ ∂q*1

Cauchy problem (16),(17) takes the form [20]:

� � ¼ �*p*<sup>1</sup>

*F*<sup>1</sup> *t*, *q*1, *p*<sup>1</sup>

*dPP* <sup>1</sup>

*dPP* <sup>1</sup>

� � ! <sup>¼</sup> 0,

Thus, if initial state is specified by a one-particle marginal distribution function on allowed configurations, then the evolution, describing by marginal observables governed by the dual BBGKY hierarchy (8), can be also described by means of the non-Markovian kinetic equation (16) and a sequence of marginal functionals of the state (14). In other words, for mentioned initial states, the evolution of all possible states of a hard sphere system with inelastic collisions at arbitrary moment of time can be described by means of a one-particle distribution function without any approximations.

## **5. The Boltzmann kinetic equation for granular gases**

It is known [7, 8] the Boltzmann kinetic equation describes the evolution of many hard spheres in the Boltzmann–Grad (or low-density) approximation. In this section the possible approaches to the rigorous derivation of the Boltzmann kinetic equation from dynamics of hard spheres with inelastic collisions are outlined.

One approach to deriving the Boltzmann kinetic equation for hard spheres with inelastic collisions, which was developed in [10] for a system of hard spheres with elastic collisions, is based on constructing the Boltzmann–Grad asymptotic behavior of marginal observables governed by the dual BBGKY hierarchy (8). A such scaling limit is governed by the set of recurrence evolution equations, namely, by the dual Boltzmann hierarchy for hard spheres with inelastic collisions [14]. Then for initial states specified by a one-particle distribution function (13), the evolution of additive-type marginal observables governed by the dual Boltzmann hierarchy is equivalent to a solution of the Boltzmann kinetic equation for granular gases [12], and the evolution of nonadditive-type marginal observables is equivalent to the property of the propagation of initial chaos for states [10].

One more approach to the description of the kinetic evolution of hard spheres with inelastic collisions is based on the non-Markovian generalization of the Enskog equation (16).

Let the dimensionless one-particle distribution function *F*ϵ,0 <sup>1</sup> , specifying initial state (13), satisfy the condition, ∣*F*<sup>ϵ</sup>,0 <sup>1</sup> ð Þ *<sup>x</sup>*<sup>1</sup> ∣ ≤ *ce*�*<sup>β</sup>* 2*p*2 <sup>1</sup> , where ϵ>0 is a scaling parameter (the ratio of the diameter *σ* > 0 to the mean free path of hard spheres), *β* >0 is a parameter, and *c*< ∞ is some constant, and there exists the following limit in the sense of a weak convergence: w � lim <sup>ϵ</sup>!<sup>0</sup> <sup>ϵ</sup><sup>2</sup>*F*<sup>ϵ</sup>,0 <sup>1</sup> ð Þ� *x*<sup>1</sup> *f* 0 <sup>1</sup> ð Þ *x*<sup>1</sup> � � <sup>¼</sup> 0. Then for finite time interval the Boltzmann–Grad limit of dimensionless solution (15) of the Cauchy problem of the non-Markovian Enskog kinetic equation (16) and (17) exists in the same sense, namely, w � lim<sup>ϵ</sup>!<sup>0</sup> <sup>ϵ</sup><sup>2</sup>*F*1ð Þ� *<sup>t</sup>*, *<sup>x</sup>*<sup>1</sup> *<sup>f</sup>* <sup>1</sup>ð Þ *<sup>t</sup>*, *<sup>x</sup>*<sup>1</sup> � � <sup>¼</sup> 0, where the limit

one-particle distribution function is a weak solution of the Cauchy problem of the Boltzmann kinetic equation for granular gases [6, 12]:

$$\begin{split} \frac{\partial}{\partial t} f\_1(t, \mathbf{x}\_1) &= - \left\langle p\_1, \frac{\partial}{\partial q\_1} \right\rangle f\_1(t, \mathbf{x}\_1) \\ &+ \int\_{\mathbb{R}^3 \times \mathbb{S}^1\_+} dp\_2 d\eta \langle \eta, (p\_1 - p\_2) \rangle \left( \frac{1}{\left(1 - 2\varepsilon\right)^2} f\_1(t, q\_1, p\_1^\*) f\_1(t, q\_1, p\_2^\*) - f\_1(t, \mathbf{x}\_1) f\_1(t, q\_1, p\_2) \right), \end{split} \tag{18}$$

$$\left.f\_1(t,\varkappa\_1)\right|\_{t=0} = f\_1^0(\varkappa\_1),\tag{19}$$

*Kinetic Equations of Granular Media DOI: http://dx.doi.org/10.5772/intechopen.90027*

equation (16) is determined by function (15). For small densities and

**5. The Boltzmann kinetic equation for granular gases**

property of the propagation of initial chaos for states [10].

Let the dimensionless one-particle distribution function *F*ϵ,0

<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> <sup>3</sup> � <sup>3</sup> � �, it is a weak solution.

*Progress in Fine Particle Plasmas*

<sup>0</sup> <sup>3</sup> � <sup>3</sup> � �, function (15) is a strong solution, and for an arbitrary initial data

Thus, if initial state is specified by a one-particle marginal distribution function on allowed configurations, then the evolution, describing by marginal observables governed by the dual BBGKY hierarchy (8), can be also described by means of the non-Markovian kinetic equation (16) and a sequence of marginal functionals of the state (14). In other words, for mentioned initial states, the evolution of all possible states of a hard sphere system with inelastic collisions at arbitrary moment of time can be described by means of a one-particle distribution function without any

It is known [7, 8] the Boltzmann kinetic equation describes the evolution of many hard spheres in the Boltzmann–Grad (or low-density) approximation. In this section the possible approaches to the rigorous derivation of the Boltzmann kinetic equation from dynamics of hard spheres with inelastic collisions are outlined.

One approach to deriving the Boltzmann kinetic equation for hard spheres with inelastic collisions, which was developed in [10] for a system of hard spheres with elastic collisions, is based on constructing the Boltzmann–Grad asymptotic behavior of marginal observables governed by the dual BBGKY hierarchy (8). A such scaling limit is governed by the set of recurrence evolution equations, namely, by the dual Boltzmann hierarchy for hard spheres with inelastic collisions [14]. Then for initial states specified by a one-particle distribution function (13), the evolution of additive-type marginal observables governed by the dual Boltzmann hierarchy is equivalent to a solution of the Boltzmann kinetic equation for granular gases [12], and the evolution of nonadditive-type marginal observables is equivalent to the

One more approach to the description of the kinetic evolution of hard spheres with inelastic collisions is based on the non-Markovian generalization of the Enskog

<sup>1</sup> ð Þ *<sup>x</sup>*<sup>1</sup> ∣ ≤ *ce*�*<sup>β</sup>*

(the ratio of the diameter *σ* > 0 to the mean free path of hard spheres), *β* >0 is a parameter, and *c*< ∞ is some constant, and there exists the following limit in the

time interval the Boltzmann–Grad limit of dimensionless solution (15) of the Cauchy problem of the non-Markovian Enskog kinetic equation (16) and (17) exists

one-particle distribution function is a weak solution of the Cauchy problem of the

ð Þ <sup>1</sup> � <sup>2</sup>*<sup>ε</sup>* <sup>2</sup> *<sup>f</sup>* <sup>1</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*<sup>⋄</sup>

2*p*2

<sup>ϵ</sup><sup>2</sup>*F*1ð Þ� *<sup>t</sup>*, *<sup>x</sup>*<sup>1</sup> *<sup>f</sup>* <sup>1</sup>ð Þ *<sup>t</sup>*, *<sup>x</sup>*<sup>1</sup>

<sup>1</sup> ð Þ� *x*<sup>1</sup> *f*

1 � �*<sup>f</sup>* <sup>1</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*<sup>⋄</sup>

0 <sup>1</sup> ð Þ *x*<sup>1</sup> � � <sup>¼</sup> 0. Then for finite

� � <sup>¼</sup> 0, where the limit

2

<sup>1</sup> ð Þ *x*<sup>1</sup> , (19)

� � !

� � � *<sup>f</sup>* <sup>1</sup>ð Þ *<sup>t</sup>*, *<sup>x</sup>*<sup>1</sup> *<sup>f</sup>* <sup>1</sup> *<sup>t</sup>*, *<sup>q</sup>*1, *<sup>p</sup>*<sup>2</sup>

,

(18)

<sup>1</sup> , specifying initial

<sup>1</sup> , where ϵ>0 is a scaling parameter

*F*0 <sup>1</sup> ∈ *L*<sup>1</sup>

*F*0

approximations.

equation (16).

*∂ ∂t*

**170**

state (13), satisfy the condition, ∣*F*<sup>ϵ</sup>,0

in the same sense, namely, w � lim<sup>ϵ</sup>!<sup>0</sup>

*∂q*1 � �

*<sup>f</sup>* <sup>1</sup>ð Þ¼� *<sup>t</sup>*, *<sup>x</sup>*<sup>1</sup> *<sup>p</sup>*1, *<sup>∂</sup>*

þ ð 3�<sup>2</sup> þ

sense of a weak convergence: w � lim <sup>ϵ</sup>!<sup>0</sup> <sup>ϵ</sup><sup>2</sup>*F*<sup>ϵ</sup>,0

Boltzmann kinetic equation for granular gases [6, 12]:

*f* <sup>1</sup>ð Þ *t*, *x*<sup>1</sup>

*dp*<sup>2</sup> *dη η*, *p*<sup>1</sup> � *p*<sup>2</sup>

� � � � 1

*f* <sup>1</sup>ð Þ *t*, *x*<sup>1</sup> � � *<sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> *<sup>f</sup>* 0

where the momenta *p*<sup>⋄</sup> <sup>1</sup> and *p*<sup>⋄</sup> <sup>2</sup> are pre-collision momenta of hard spheres with inelastic collisions (7).

As noted above, all possible correlations of a system of hard spheres with inelastic collisions are described by marginal functionals of the state (14). Taking into consideration the fact of the existence of the Boltzmann–Grad scaling limit of a solution of the non-Markovian Enskog kinetic equation (16), for marginal functionals of the state (14), the following statement holds:

$$\mathbf{w} - \lim\_{\epsilon \to 0} \left( \epsilon^2 F\_s(t, \varkappa\_1, \dots, \varkappa\_s | F\_1(t)) - \prod\_{j=1}^s f\_1(t, \varkappa\_j) \right) = \mathbf{0},$$

where the limit one-particle distribution function *f* <sup>1</sup>ð Þ*t* is governed by the Boltzmann kinetic equation for granular gases (18). This property of marginal functionals of the state (14) means the propagation of the initial chaos [7].

It should be emphasized that the Boltzmann–Grad asymptotics of a solution of the non-Markovian Enskog equation (16) in a multidimensional space are analogous of the Boltzmann–Grad asymptotic behavior of a hard sphere system with the elastic collisions [10]. Such asymptotic behavior is governed by the Boltzmann equation for a granular gas (18), and the asymptotics of marginal functionals of the state (14) are the product of its solution (this property is interpreted as the propagation of the initial chaos).

## **6. One-dimensional granular gases**

As is known, the evolution of a one-dimensional system of hard spheres with elastic collisions is trivial (free motion or Knudsen flow) in the Boltzmann–Grad scaling limit [7], but, as it was taken notice in paper [16], in this approximation the kinetics of inelastically interacting hard spheres (rods) is not trivial, and it is governed by the Boltzmann kinetic equation for one-dimensional granular gases [16–19]. Below the approach to the rigorous derivation of Boltzmann-type equation for one-dimensional granular gases will be outlined. It should be emphasized that a system of many hard rods with inelastic collisions displays the basic properties of granular gases inasmuch as under the inelastic collisions only the normal component of relative velocities dissipates in a multidimensional case.

In case of a one-dimensional granular gas for *t*≥0 in dimensionless form, the Cauchy problem (16),(17) takes the form [20]:

$$\begin{aligned} \frac{\partial}{\partial t} F\_1(t, q\_1, p\_1) &= -p\_1 \frac{\partial}{\partial q\_1} F\_1(t, q\_1, p\_1) + \\\\ \int\_0^\infty dP P \Big( \frac{1}{\left(1 - 2\varepsilon\right)^2} F\_2(t, q\_1, p\_1^\diamond(p\_1, P), q\_1 - \epsilon, p\_2^\diamond(p\_1, P) | F\_1(t)) - \\\\ F\_2(t, q\_1, p\_1, q\_1 - \epsilon, p\_1 + P | F\_1(t)) \Big) + \\\\ \int\_0^\infty dP P \Big( \frac{1}{\left(1 - 2\varepsilon\right)^2} F\_2(t, q\_1, \bar{p}\_1^\diamond(p\_1, P), q\_1 + \epsilon, \bar{p}\_2^\diamond(p\_1, P) | F\_1(t)) - \\\\ F\_2(t, q\_1, p\_1, q\_1 + \epsilon, p\_1 - P | F\_1(t)) \Big), \\\\ F\_1(t)|\_{t = 0} &= F\_1^{\diamond, 0}, \end{aligned} \tag{21}$$

where ϵ> 0 is a scaling parameter (the ratio of a hard sphere diameter (the length) *σ* > 0 to the mean free path), the collision integral is determined by marginal functional (14) of the state *F*1ð Þ*t* in the case of *s* ¼ 2, and the expressions:

where *x*<sup>⋄</sup>

*∂ ∂t*

> Ið Þ *n* <sup>0</sup> � <sup>1</sup> *n*! ð<sup>∞</sup> 0

space:

**173**

following expressions:

*f* <sup>1</sup>ð Þ¼� *t*, *q*, *p p*

*<sup>j</sup>* � *qj*

*Kinetic Equations of Granular Media*

, *p*<sup>⋄</sup> *j*

*DOI: http://dx.doi.org/10.5772/intechopen.90027*

point particles with inelastic collisions [20]

1

In kinetic equation (26) the remainder P<sup>∞</sup>

ð Þ � *<sup>n</sup>*

ð Þ � *<sup>n</sup>*

*S* b 0

series over the density so that the terms <sup>I</sup>ð Þ *<sup>n</sup>*

*∂ ∂q*

�

determined by the expressions

� *<sup>f</sup>* <sup>1</sup> *<sup>t</sup>*, *<sup>q</sup>*, *<sup>p</sup>*<sup>⋄</sup>

� *<sup>f</sup>* <sup>1</sup> *<sup>t</sup>*, *<sup>q</sup>*, *<sup>p</sup>*~<sup>⋄</sup>

*dPP*<sup>ð</sup>

þ ð<sup>∞</sup> 0

*dPP*<sup>ð</sup>

� � and the pre-collision momenta *<sup>p</sup>*<sup>⋄</sup>

ðþ<sup>∞</sup> �∞

ð Þ <sup>1</sup> � <sup>2</sup>*<sup>ε</sup>* <sup>2</sup> *<sup>f</sup>* <sup>1</sup> *<sup>t</sup>*, *<sup>q</sup>*, *<sup>p</sup>*<sup>⋄</sup> ð Þ*<sup>f</sup>* <sup>1</sup>ð*t*, *<sup>q</sup>*, *<sup>p</sup>*<sup>⋄</sup>

<sup>2</sup>ð Þ *<sup>p</sup>*, *<sup>P</sup>* � � � *<sup>f</sup>* <sup>1</sup>ð Þ *<sup>t</sup>*, *<sup>q</sup>*, *<sup>p</sup> <sup>f</sup>* <sup>1</sup>ð ÞÞ *<sup>t</sup>*, *<sup>q</sup>*, *<sup>p</sup>* <sup>þ</sup> *<sup>P</sup>* <sup>Y</sup>*<sup>n</sup>*þ<sup>2</sup>

<sup>2</sup>ð Þ *<sup>p</sup>*, *<sup>P</sup>* � � � *<sup>f</sup>* <sup>1</sup>ð Þ *<sup>t</sup>*, *<sup>q</sup>*, *<sup>p</sup> <sup>f</sup>* <sup>1</sup>ð ÞÞ *<sup>t</sup>*, *<sup>q</sup>*, *<sup>p</sup>* � *<sup>P</sup>* <sup>Y</sup>*<sup>n</sup>*þ<sup>2</sup>

*<sup>n</sup>*ð Þ *<sup>t</sup>*, 1, … , *<sup>n</sup>* <sup>≐</sup> *<sup>S</sup>* <sup>∗</sup> ,0

integral for one-dimensional granular gases stated in [17, 21].

*p*⋄ *j* 1 ¼ *pj* 2 <sup>þ</sup> *<sup>ε</sup>* <sup>2</sup>*<sup>ε</sup>* � <sup>1</sup> *pj*

*p*⋄ *j* 2 ¼ *pj* 1 � *<sup>ε</sup>* <sup>2</sup>*<sup>ε</sup>* � <sup>1</sup> *pj*

*f* <sup>1</sup>ð Þþ *t*, *q*, *p*

*j* 1 , *p*<sup>⋄</sup> *j* 2

<sup>1</sup> Þ � *f* <sup>1</sup>ð*t*, *q*, *p*Þ*f* <sup>1</sup>ð*t*, *q*, *p*1Þ

<sup>0</sup> of the collision integral is

ð Þ <sup>1</sup> � <sup>2</sup>*<sup>ε</sup>* <sup>2</sup> *<sup>f</sup>* <sup>1</sup> *<sup>t</sup>*, *<sup>q</sup>*, *<sup>p</sup>*<sup>⋄</sup> <sup>1</sup> ð Þ *<sup>p</sup>*, *<sup>P</sup>* � �

> , *pi* � �

ð Þ <sup>1</sup> � <sup>2</sup>*<sup>ε</sup>* <sup>2</sup> *<sup>f</sup>* <sup>1</sup> *<sup>t</sup>*, *<sup>q</sup>*, *<sup>p</sup>*~<sup>⋄</sup> <sup>1</sup> ð Þ *<sup>p</sup>*, *<sup>P</sup>* � �

> , *pi* � �,

<sup>0</sup> , *n*≥ 1, of the collision integral in kinetic

*:* (26)

1

*f* <sup>1</sup> *t*, *qi*

1

*F*<sup>1</sup> *t*, *qi*

*i*¼3

*i*¼3

Y*n i*¼1

*S* <sup>∗</sup> ,0 <sup>1</sup> ð Þ *<sup>t</sup>*, *<sup>i</sup>* �<sup>1</sup>

1 � *pj* 2 � �,

1 � *pj* 2 � �*:*

For *t* ≥0 the limit one-particle distribution function represented by series (25) is a weak solution of the Cauchy problem of the Boltzmann-type kinetic equation of

*dp*<sup>1</sup> ∣*p* � *p*1∣

*dq*3*dp*<sup>3</sup> … *dqn*þ<sup>2</sup>*dpn*þ<sup>2</sup>V<sup>1</sup>þ*<sup>n</sup>*ð Þ*<sup>t</sup>*

*dq*3*dp*<sup>3</sup> … *dqn*þ<sup>2</sup>*dpn*þ<sup>2</sup>V<sup>1</sup>þ*<sup>n</sup>*ð Þ*<sup>t</sup>*

where the generating operators V<sup>1</sup>þ*<sup>n</sup>*ðÞ� *t* V<sup>1</sup>þ*<sup>n</sup>*ð Þ *t*, 1, 2 f g, 3, … , *n* þ 2 , *n*≥ 0, are represented by expansions (15) with respect to the cumulants of semigroups of scattering operators of point hard rods with inelastic collisions in a one-dimensional

*<sup>n</sup>* ð Þ *t*, 1, … , *s*

In fact, the series expansions for the collision integral of the non-Markovian Enskog equation for a granular gas or solution (23) are represented as the power

equation (18) are corrections with respect to the density to the Boltzmann collision

Since the scattering operator of point hard rods is an identity operator in the approximation of elastic collisions, namely, in the limit *ε* ! 0, the collision integral of the Boltzmann kinetic equation (26) in a one-dimensional space is identical to zero. In the quasi-elastic limit [21], the limit one-particle distribution function (25)

> 0 ð Þ *t*, *q*, *v* ,

lim*<sup>ε</sup>*!<sup>0</sup> *<sup>ε</sup><sup>f</sup>* <sup>1</sup>ð Þ¼ *<sup>t</sup>*, *<sup>q</sup>*, *<sup>v</sup> <sup>f</sup>*

!

*<sup>n</sup>*¼<sup>1</sup>Ið Þ *<sup>n</sup>*

are determined by the

þX<sup>∞</sup> *n*¼1

Ið Þ *n* <sup>0</sup> *:*

(25)

$$\begin{aligned} p\_1^\circ(p\_1, P) &= p\_1 - P + \frac{\varepsilon}{2\varepsilon - 1} P, \\ p\_2^\circ(p\_1, P) &= p\_1 - \frac{\varepsilon}{2\varepsilon - 1} P \end{aligned}$$

and

$$\begin{aligned} \bar{p}\_1^\circ(p\_1, P) &= p\_1 + P - \frac{\varepsilon}{2\varepsilon - 1} P, \\ \bar{p}\_2^\circ(p\_1, P) &= p\_1 + \frac{\varepsilon}{2\varepsilon - 1} P, \end{aligned}$$

are transformed pre-collision momenta in a one-dimensional space.

If initial one-particle marginal distribution functions satisfy the following condition: ∣*F*ϵ,0 <sup>1</sup> ð Þ *<sup>x</sup>*<sup>1</sup> ∣ ≤*Ce*�*<sup>β</sup>* 2*p*2 <sup>1</sup> , where *β* >0 is a parameter, *C*< ∞ is some constant, then every term of the series

$$F\_1^{\varepsilon}(t, \mathbf{x}\_1) = \sum\_{n=0}^{\infty} \frac{1}{n!} \int\_{(\mathbb{R} \times \mathbb{R})^n} d\mathbf{x}\_2 \dots d\mathbf{x}\_{n+1} \mathfrak{A}\_{1+n}^\*(t) \prod\_{i=1}^{n+1} F\_1^{\varepsilon, 0}(\mathbf{x}\_i) \mathcal{X}\_{\mathbb{R}^{(1+n)} \backslash \mathbb{W}\_{1+n}},\tag{22}$$

exists, for finite time interval function (23) is the uniformly convergent series with respect to *x*<sup>1</sup> from arbitrary compact, and it is determined a weak solution of the Cauchy problem of the non-Markovian Enskog equation (20), (22). Let in the sense of a weak convergence there exist the following limit:

$$\mathbf{w} - \lim\_{\epsilon \to 0} \left( F\_1^{\epsilon, 0}(\varkappa\_1) - f\_1^0(\varkappa\_1) \right) = \mathbf{0}\_\* $$

then for finite time interval there exists the Boltzmann–Grad limit of solution (23) of the Cauchy problem of the non-Markovian Enskog equation for onedimensional granular gas (20) in the sense of a weak convergence:

$$\mathbf{w} - \lim\_{\epsilon \to 0} \left( F\_1^{\epsilon}(t, \varkappa\_1) - f\_1(t, \varkappa\_1) \right) = \mathbf{0},\tag{23}$$

where the limit one-particle marginal distribution function is defined by uniformly convergent arbitrary compact set series:

$$f\_1(t, \mathbf{x}\_1) = \sum\_{n=0}^{\infty} \frac{1}{n!} \int\_{(\mathbb{R} \times \mathbb{R})^n} d\mathbf{x}\_2 \dots d\mathbf{x}\_{n+1} \mathfrak{A}\_{1+n}^0(t) \prod\_{i=1}^{n+1} f\_1^0(\mathbf{x}\_i),\tag{24}$$

and the generating operator A<sup>0</sup> <sup>1</sup>þ*<sup>n</sup>*ðÞ� *<sup>t</sup>* <sup>A</sup><sup>0</sup> <sup>1</sup>þ*<sup>n</sup>*ð Þ *<sup>t</sup>*, 1, … , *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> is the ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *th*-order cumulant of adjoint semigroups *S* <sup>∗</sup> ,0 *<sup>n</sup>* ð Þ*t* of point particles with inelastic collisions. An infinitesimal generator of the semigroup of operators *S* <sup>∗</sup> ,0 *<sup>n</sup>* ð Þ*t* is defined as the operator:

$$\begin{split} \left(\mathcal{L}\_{n}^{\*,\ast}f\_{n}\right)(\mathbf{x}\_{1},\ldots,\mathbf{x}\_{n}) &= -\sum\_{j=1}^{n}p\_{j}\frac{\partial}{\partial q\_{j}^{\ast}}f\_{n}(\mathbf{x}\_{1},\ldots,\mathbf{x}\_{n}) \\ &+\sum\_{j\_{1}$$

**172**

*Kinetic Equations of Granular Media DOI: http://dx.doi.org/10.5772/intechopen.90027*

where ϵ> 0 is a scaling parameter (the ratio of a hard sphere diameter (the length) *σ* > 0 to the mean free path), the collision integral is determined by marginal

<sup>1</sup> *<sup>p</sup>*1, *<sup>P</sup>* � � <sup>¼</sup> *<sup>p</sup>*<sup>1</sup> � *<sup>P</sup>* <sup>þ</sup> *<sup>ε</sup>*

<sup>1</sup> *<sup>p</sup>*1, *<sup>P</sup>* � � <sup>¼</sup> *<sup>p</sup>*<sup>1</sup> <sup>þ</sup> *<sup>P</sup>* � *<sup>ε</sup>*

<sup>2</sup> *<sup>p</sup>*1, *<sup>P</sup>* � � <sup>¼</sup> *<sup>p</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ε</sup>*

are transformed pre-collision momenta in a one-dimensional space.

*dx*<sup>2</sup> … *dxn*þ<sup>1</sup> <sup>A</sup><sup>∗</sup>

*F*ϵ,0 <sup>1</sup> ð Þ� *x*<sup>1</sup> *f*

(23) of the Cauchy problem of the non-Markovian Enskog equation for one-

dimensional granular gas (20) in the sense of a weak convergence:

*F*ϵ

ð Þ � *<sup>n</sup>*

<sup>1</sup>þ*<sup>n</sup>*ðÞ� *<sup>t</sup>* <sup>A</sup><sup>0</sup>

*f <sup>n</sup>*ð ÞÞ *x*1, … , *xn δ qj*

2*ε* � 1

*P*

2*ε* � 1

*P*,

<sup>1</sup> , where *β* >0 is a parameter, *C*< ∞ is some constant, then

*F*ϵ,0

� � <sup>¼</sup> 0, (23)

<sup>1</sup>þ*<sup>n</sup>*ð Þ*t*

*<sup>n</sup>* ð Þ*t* of point particles with inelastic collisions. An

ð Þ <sup>1</sup> � <sup>2</sup>*<sup>ε</sup>* <sup>2</sup> *<sup>f</sup> <sup>n</sup> <sup>x</sup>*1, … , *<sup>x</sup>*<sup>⋄</sup>

,

Y*n*þ1 *i*¼1 *f* 0

<sup>1</sup>þ*<sup>n</sup>*ð Þ *<sup>t</sup>*, 1, … , *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> is the ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *th*-order

<sup>1</sup> ð Þ *xi* , (24)

*<sup>n</sup>* ð Þ*t* is defined as the operator:

� �

�

*j* 1 , … , *x*<sup>⋄</sup> *j* 2 , … , *xn*

<sup>1</sup> ð ÞX *xi* ð Þ <sup>1</sup>þ*<sup>n</sup>* <sup>n</sup>1þ*<sup>n</sup>*

, (22)

Y*n*þ1 *i*¼1

2*ε* � 1

2*ε* � 1

<sup>1</sup>þ*<sup>n</sup>*ð Þ*t*

0 <sup>1</sup> ð Þ *x*<sup>1</sup> � � <sup>¼</sup> 0,

If initial one-particle marginal distribution functions satisfy the following con-

exists, for finite time interval function (23) is the uniformly convergent series with respect to *x*<sup>1</sup> from arbitrary compact, and it is determined a weak solution of the Cauchy problem of the non-Markovian Enskog equation (20), (22). Let in the

then for finite time interval there exists the Boltzmann–Grad limit of solution

<sup>1</sup>ð Þ� *t*, *x*<sup>1</sup> *f* <sup>1</sup>ð Þ *t*, *x*<sup>1</sup>

*dx*<sup>2</sup> … *dxn*þ<sup>1</sup> <sup>A</sup><sup>0</sup>

*f <sup>n</sup>*ð Þ *x*1, … , *xn*

<sup>∣</sup> <sup>1</sup>

1 � *qj* 2 � �

where the limit one-particle marginal distribution function is defined by uni-

*P*,

*P*,

functional (14) of the state *F*1ð Þ*t* in the case of *s* ¼ 2, and the expressions:

<sup>2</sup> *<sup>p</sup>*1, *<sup>P</sup>* � � <sup>¼</sup> *<sup>p</sup>*<sup>1</sup> � *<sup>ε</sup>*

*p*⋄

*p*⋄

*p*~⋄

*p*~⋄

and

dition: ∣*F*ϵ,0

*F*ϵ

<sup>1</sup> ð Þ *<sup>x</sup>*<sup>1</sup> ∣ ≤*Ce*�*<sup>β</sup>*

*Progress in Fine Particle Plasmas*

X∞ *n*¼0

1 *n*! ð

ð Þ � *<sup>n</sup>*

sense of a weak convergence there exist the following limit:

<sup>w</sup> � lim<sup>ϵ</sup>!<sup>0</sup>

<sup>w</sup> � lim<sup>ϵ</sup>!<sup>0</sup>

X∞ *n*¼0

infinitesimal generator of the semigroup of operators *S* <sup>∗</sup> ,0

<sup>þ</sup> <sup>X</sup>*<sup>n</sup> j* <sup>1</sup> <*j* <sup>2</sup>¼1 ∣*pj* 2 � *pj* 1

X*n j*¼1 *pj ∂ ∂qj*

1 *n*! ð

formly convergent arbitrary compact set series:

*f* <sup>1</sup>ð Þ¼ *t*, *x*<sup>1</sup>

and the generating operator A<sup>0</sup>

cumulant of adjoint semigroups *S* <sup>∗</sup> ,0

� �ð Þ¼� *<sup>x</sup>*1, … , *xn*

<sup>L</sup><sup>∗</sup> ,0 *<sup>n</sup> f <sup>n</sup>*

**172**

every term of the series

<sup>1</sup>ð Þ¼ *t*, *x*<sup>1</sup>

2*p*2

where *x*<sup>⋄</sup> *<sup>j</sup>* � *qj* , *p*<sup>⋄</sup> *j* � � and the pre-collision momenta *<sup>p</sup>*<sup>⋄</sup> *j* 1 , *p*<sup>⋄</sup> *j* 2 are determined by the following expressions:

$$\begin{aligned} p\_{j\_1}^\bullet &= p\_{j\_1} + \frac{\varepsilon}{2\varepsilon - 1} \left( p\_{j\_1} - p\_{j\_2} \right), \\ p\_{j\_2}^\bullet &= p\_{j\_1} - \frac{\varepsilon}{2\varepsilon - 1} \left( p\_{j\_1} - p\_{j\_2} \right). \end{aligned}$$

For *t* ≥0 the limit one-particle distribution function represented by series (25) is a weak solution of the Cauchy problem of the Boltzmann-type kinetic equation of point particles with inelastic collisions [20]

$$\begin{split} \frac{\partial}{\partial t} f\_1(t, q, p) &= -p \frac{\partial}{\partial q} f\_1(t, q, p) + \int\_{-\infty}^{+\infty} dp\_1 |p - p\_1| \\ &\times \left( \frac{1}{(1 - 2\varepsilon)^2} f\_1(t, q, p^\*) f\_1(t, q, p\_1^\*) - f\_1(t, q, p) f\_1(t, q, p\_1) \right) + \sum\_{n=1}^{\infty} \mathcal{I}\_0^{(n)}. \end{split} \tag{25}$$

In kinetic equation (26) the remainder P<sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup>Ið Þ *<sup>n</sup>* <sup>0</sup> of the collision integral is determined by the expressions

$$\begin{split} \mathcal{I}\_{0}^{(n)} & \equiv \frac{1}{n!} \Big[ \mathop{\rm dP}{\mathop{\rm dP}} \Big[ \mathop{\rm }\_{\left(\mathbb{R} \times \mathbb{R}\right)^{\rm s}} \limits( \mathop{\rm d}{\mathop{\rm p}}\_{\left(\mathbb{R} \times \mathbb{R}\right)^{\rm s}} \boldsymbol{d}q\_{3} \boldsymbol{d}p\_{3} \ldots \boldsymbol{d}q\_{n+2} \boldsymbol{d}p\_{n+2} \mathfrak{D}\_{1+n}(t) \bigg( \frac{1}{(1-2\epsilon)^{2}} f\_{1}(t,q,p\_{1}^{\boldsymbol{s}}(p,P)) \Big) \\ & \times f\_{1}(t,q,p\_{2}^{\boldsymbol{s}}(p,P)) \, -f\_{1}(t,q,p) f\_{1}(t,q,p+P) \bigg) \prod\_{i=3}^{n+2} f\_{1}(t,q\_{i},p\_{i}) \\ & \quad + \Big[ \mathop{\rm d}{\mathop{\rm d}}\_{\left(\mathbb{R} \times \mathbb{R}\right)^{\rm s}} \boldsymbol{d}q\_{3} \boldsymbol{d}p\_{3} \ldots \boldsymbol{d}q\_{n+2} \boldsymbol{d}p\_{n+2} \mathfrak{D}\_{1+n}(t) \bigg( \frac{1}{(1-2\epsilon)^{2}} f\_{1}(t,q,p\_{1}^{\boldsymbol{s}}(p,P)) \\ & \times f\_{1}(t,q,p\_{2}^{\boldsymbol{s}}(p,P)) \, -f\_{1}(t,q,p) f\_{1}(t,q,p-P) \bigg) \prod\_{i=3}^{n+2} F\_{1}(t,q\_{i},p\_{i}) \,, \end{split}$$

where the generating operators V<sup>1</sup>þ*<sup>n</sup>*ðÞ� *t* V<sup>1</sup>þ*<sup>n</sup>*ð Þ *t*, 1, 2 f g, 3, … , *n* þ 2 , *n*≥ 0, are represented by expansions (15) with respect to the cumulants of semigroups of scattering operators of point hard rods with inelastic collisions in a one-dimensional space:

$$\hat{S}\_n^0(t, \mathbf{1}, \dots, n) \doteq \mathcal{S}\_n^{\*,0}(t, \mathbf{1}, \dots, s) \prod\_{i=1}^n \mathcal{S}\_1^{\*,0}(t, i)^{-1}. \tag{26}$$

In fact, the series expansions for the collision integral of the non-Markovian Enskog equation for a granular gas or solution (23) are represented as the power series over the density so that the terms <sup>I</sup>ð Þ *<sup>n</sup>* <sup>0</sup> , *n*≥ 1, of the collision integral in kinetic equation (18) are corrections with respect to the density to the Boltzmann collision integral for one-dimensional granular gases stated in [17, 21].

Since the scattering operator of point hard rods is an identity operator in the approximation of elastic collisions, namely, in the limit *ε* ! 0, the collision integral of the Boltzmann kinetic equation (26) in a one-dimensional space is identical to zero. In the quasi-elastic limit [21], the limit one-particle distribution function (25)

$$\lim\_{v \to 0} \epsilon f\_1(t, q, v) = f^0(t, q, v),$$

satisfies the nonlinear friction kinetic equation for granular gases of the following form [16, 21]:

$$\frac{\partial}{\partial t}f^0(t,q,v) = -v\frac{\partial}{\partial q}f^0(t,q,v) + \frac{\partial}{\partial v}\int\_{-\infty}^{\infty}dv\_1|v\_1 - v|(v\_1 - v)f^0(t,q,v\_1)f^0(t,q,v).$$

Taking into consideration result (24) on the Boltzmann–Grad asymptotic behavior of the non-Markovian Enskog equation (16), for marginal functionals of the state (14) in a one-dimensional space, the following statement is true [20]:

$$\mathbf{w} - \lim\_{\varepsilon \to 0} \left( F\_{\varepsilon} \left( t, \mathbf{x}\_1, \dots, \mathbf{x}\_{\varepsilon} | F\_1^{\varepsilon}(t) \right) - f\_{\varepsilon} \left( t, \mathbf{x}\_1, \dots, \mathbf{x}\_{\varepsilon} | f\_1(t) \right) \right) = \mathbf{0}, \quad \varepsilon \ge 2,$$

where the limit marginal functionals *fs <sup>t</sup>*<sup>j</sup> *<sup>f</sup>* <sup>1</sup>ð Þ*<sup>t</sup>* � �, *<sup>s</sup>*≥2, with respect to limit oneparticle distribution function (25) are determined by the series expansions with the structure similar to series (14) and the generating operators represented by expansions (15) over the cumulants of semigroups of scattering operators (27) of point hard rods with inelastic collisions in a one-dimensional space.

As mentioned above, in the case of a system of hard rods with elastic collisions, the limit marginal functionals of the state are the product of the limit one-particle distribution functions, describing the free motion of point hard rods.

Thus, the Boltzmann–Grad asymptotic behavior of solution (23) of the non-Markovian Enskog equation (20) is governed by the Boltzmann kinetic equation for a one-dimensional granular gas (18). Moreover, the limit marginal functionals of the state are represented by the appropriate series with respect to limit one-particle distribution function (25) that describe the propagation of initial chaos in onedimensional granular gases.

## **7. Conclusions**

In this chapter the origin of the kinetic description of the evolution of observables of a system of hard spheres with inelastic collisions was considered.

It was established that for initial states (13) specified by a one-particle distribution function, solution (10) of the Cauchy problem of the dual BBGKY hierarchy (8) and (9) and a solution of the Cauchy problem of the non-Markovian Enskog equation (16) and (17) together with marginal functionals of the state (14), give two equivalent approaches to the description of the evolution of states of a hard sphere system with inelastic collisions. In fact, the rigorous justification of the Enskog kinetic equation for granular gases (16) is a consequence of the validity of equality (14).

We note that the developed approach is also related to the problem of a rigorous derivation of the non-Markovian kinetic-type equations from underlying manyparticle dynamics which make it possible to describe the memory effects of granular gases.

**Author details**

**175**

Viktor Gerasimenko

*Kinetic Equations of Granular Media*

*DOI: http://dx.doi.org/10.5772/intechopen.90027*

Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: gerasym@imath.kiev.ua

provided the original work is properly cited.

One more advantage also is that the considered approach gives the possibility to construct the kinetic equations in scaling limits, involving correlations at initial time which can characterize the condensed states of a hard sphere system with inelastic collisions [10].

Finally, it should be emphasized that the developed approach to the derivation of the Boltzmann equation for granular gases from the dynamics governed by the non-Markovian Enskog kinetic equation (16) also allows us to construct higherorder corrections to the collision integral compared to the Boltzmann–Grad approximation.

*Kinetic Equations of Granular Media DOI: http://dx.doi.org/10.5772/intechopen.90027*

satisfies the nonlinear friction kinetic equation for granular gases of the follow-

*dv*<sup>1</sup> ∣*v*<sup>1</sup> � *v*∣ð Þ *v*<sup>1</sup> � *v f*

<sup>0</sup>ð Þ *<sup>t</sup>*, *<sup>q</sup>*, *<sup>v</sup>*<sup>1</sup> *<sup>f</sup>*

0 ð Þ *t*, *q*, *v :*

ing form [16, 21]:

<sup>0</sup>ð Þ¼� *<sup>t</sup>*, *<sup>q</sup>*, *<sup>v</sup> <sup>v</sup>*

*Progress in Fine Particle Plasmas*

<sup>w</sup> � lim<sup>ϵ</sup>!<sup>0</sup>

dimensional granular gases.

**7. Conclusions**

gases.

collisions [10].

approximation.

**174**

*∂ ∂q f* 0

ð Þþ *t*, *q*, *v*

hard rods with inelastic collisions in a one-dimensional space.

distribution functions, describing the free motion of point hard rods.

*Fs <sup>t</sup>*, *<sup>x</sup>*1, … , *xs*j*F*<sup>ϵ</sup>

*∂ ∂v* ð<sup>∞</sup> �∞

Taking into consideration result (24) on the Boltzmann–Grad asymptotic behavior of the non-Markovian Enskog equation (16), for marginal functionals of the state (14) in a one-dimensional space, the following statement is true [20]:

<sup>1</sup>ð Þ*<sup>t</sup>* � � � *fs <sup>t</sup>*, *<sup>x</sup>*1, … , *xs*<sup>j</sup> *<sup>f</sup>* <sup>1</sup>ð Þ*<sup>t</sup>* � � � � <sup>¼</sup> 0, *<sup>s</sup>*≥2,

where the limit marginal functionals *fs <sup>t</sup>*<sup>j</sup> *<sup>f</sup>* <sup>1</sup>ð Þ*<sup>t</sup>* � �, *<sup>s</sup>*≥2, with respect to limit oneparticle distribution function (25) are determined by the series expansions with the structure similar to series (14) and the generating operators represented by expansions (15) over the cumulants of semigroups of scattering operators (27) of point

As mentioned above, in the case of a system of hard rods with elastic collisions, the limit marginal functionals of the state are the product of the limit one-particle

Thus, the Boltzmann–Grad asymptotic behavior of solution (23) of the non-Markovian Enskog equation (20) is governed by the Boltzmann kinetic equation for a one-dimensional granular gas (18). Moreover, the limit marginal functionals of the state are represented by the appropriate series with respect to limit one-particle distribution function (25) that describe the propagation of initial chaos in one-

In this chapter the origin of the kinetic description of the evolution of observ-

It was established that for initial states (13) specified by a one-particle distribution function, solution (10) of the Cauchy problem of the dual BBGKY hierarchy (8) and (9) and a solution of the Cauchy problem of the non-Markovian Enskog equation (16) and (17) together with marginal functionals of the state (14), give two equivalent approaches to the description of the evolution of states of a hard sphere system with inelastic collisions. In fact, the rigorous justification of the Enskog kinetic equa-

We note that the developed approach is also related to the problem of a rigorous derivation of the non-Markovian kinetic-type equations from underlying manyparticle dynamics which make it possible to describe the memory effects of granular

One more advantage also is that the considered approach gives the possibility to construct the kinetic equations in scaling limits, involving correlations at initial time which can characterize the condensed states of a hard sphere system with inelastic

Finally, it should be emphasized that the developed approach to the derivation of the Boltzmann equation for granular gases from the dynamics governed by the non-Markovian Enskog kinetic equation (16) also allows us to construct higherorder corrections to the collision integral compared to the Boltzmann–Grad

ables of a system of hard spheres with inelastic collisions was considered.

tion for granular gases (16) is a consequence of the validity of equality (14).

*∂ ∂t f*

## **Author details**

Viktor Gerasimenko Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine

\*Address all correspondence to: gerasym@imath.kiev.ua

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

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[2] Goldhirsch I. Scales and kinetics of granular flows. Chaos. 1999;**9**:659-672. DOI: 10.1063/1.166440

[3] Mehta A. Granular Physics. Cambridge, UK: Cambridge University Press; 2007. 362 p. DOI: 10.1017/ CBO9780511535314

[4] Pöschel T, Brilliantov NV, editors. Granular gas dynamics. In: Lecture Notes in Phys. 624. Berlin, Heidelberg: Springer-Verlag; 2003. 369 p. ISBN 978-3-540-20110-6

[5] Brilliantov NV, Pöschel T. Kinetic Theory of Granular Gases. Oxford: Oxford University Press; 2004. 329 p. ISBN-13 978-0199588138

[6] Brey JJ, Dufty JW, Santos A. Dissipative dynamics for hard spheres. Journal of Statistical Physics. 1997;**87** (5-6):1051-1066. DOI: 10.1007/ BF02181270

[7] Cercignani C, Gerasimenko VI, Petrina DY. Many-Particle Dynamics and Kinetic Equations. The Netherlands: Springer; 2012. 247 p. ISBN 978-94- 010-6342-5

[8] Gallagher I, Saint-Raymond L, Texier B. From Newton to Boltzmann: Hard Spheres and Short-range Potentials. Zürich Lectures in Advanced Mathematics: EMS Publ House; 2014. 146 p. ISBN-10: 3037191295

[9] Pulvirenti M, Simonella S. The Boltzmann–Grad limit of a hard sphere system: Analysis of the correlation error. Inventiones Mathematicae. 2017;**207**(3): 1135-1237. DOI: 10.1007/s00222-016- 0682-4

Modeling and Simulation in Science, Engineering and Technology. Boston: Birkhäuser; 2000. 433 p. ISBN 978-1-

*DOI: http://dx.doi.org/10.5772/intechopen.90027*

*Kinetic Equations of Granular Media*

[18] Williams DRM, MacKintosh FC. Driven granular media in one

dimension: Correlations and equation of states. Physical Review E. 1996;**R9**(R): 54. DOI: 10.1103/PhysRevE.54.R9

[19] Toscani G. One-dimensional kinetic models with dissipative collisions. Mathematical Modelling and Numerical Analysis. 2000;**34**(6):1277-1291. DOI:

Borovchenkova MS. The Boltzmann– Grad limit of the Enskog equation of one-dimensional granular gases. Reports of the NAS of Ukraine. 2013;**10**:11-17

hydrodynamic models of nearly elastic granular flows. Monatschefte für Mathematik. 2004;**142**:179-192. DOI:

10.1051/m2an:2000127

[20] Gerasimenko VI,

[21] Toscani G. Kinetic and

10.1007/s00605-004-0241-8

**177**

4612-6797-3

[10] Gerasimenko VI, Gapyak IV. Lowdensity asymptotic behavior of observables of hard sphere fluids. Advances in Mathematical Physics. 2018;**2018**:6252919. DOI: 10.1155/2018/ 6252919

[11] Pareschi L, Russo G, Toscani G, editors. Modelling and Numerics of Kinetic Dissipative Systems. N.Y.: Nova Science Publ. Inc.; 2006. 220 p. ISBN-13 978-1594545030

[12] Villani C. Mathematics of granular materials. Journal of Statistical Physics. 2006;**124**(2-4):781-822. DOI: 10.1007/ s10955-006-9038-6

[13] Capriz G, Mariano PM, Giovine P, editors. Mathematical Models of Granular Matter. (Lecture Notes in Math. 1937). Berlin Heidelberg: Springer-Verlag; 2008. 228 p. ISBN 978-3-540-78276-6

[14] Gerasimenko VI, Borovchenkova MS. On the non-Markovian Enskog equation for granular gases. Journal of Physics A: Mathematical and Theoretical. 2014; **47**(3):035001. DOI: 10.1088/1751-8113/ 47/3/035001

[15] Gerasimenko VI, Gapyak IV. Hard sphere dynamics and the Enskog equation. Kinetic and Related Models. 2012;**5**(3):459-484. DOI: 10.3934/ krm.2012.5.459

[16] Mac Namara S, Young WR. Kinetics of a one-dimensional granular medium in the quasielastic limit. Physics of Fluids A. 1993;**5**(1):34-45. DOI: 10.1063/ 1.858896

[17] Bellomo N, Pulvirenti M, editors. Modeling in applied sciences. In:

*Kinetic Equations of Granular Media DOI: http://dx.doi.org/10.5772/intechopen.90027*

Modeling and Simulation in Science, Engineering and Technology. Boston: Birkhäuser; 2000. 433 p. ISBN 978-1- 4612-6797-3

**References**

[1] Cercignani C. The Boltzmann

*Progress in Fine Particle Plasmas*

granular material. Philosophical Transactions of the Royal Society of London. 2002;**360**(1792):407-414. DOI:

10.1098/rsta.2001.0939

DOI: 10.1063/1.166440

CBO9780511535314

978-3-540-20110-6

BF02181270

010-6342-5

**176**

ISBN-13 978-0199588138

[6] Brey JJ, Dufty JW, Santos A. Dissipative dynamics for hard spheres. Journal of Statistical Physics. 1997;**87** (5-6):1051-1066. DOI: 10.1007/

[7] Cercignani C, Gerasimenko VI, Petrina DY. Many-Particle Dynamics and Kinetic Equations. The Netherlands: Springer; 2012. 247 p. ISBN 978-94-

[8] Gallagher I, Saint-Raymond L, Texier B. From Newton to Boltzmann:

[9] Pulvirenti M, Simonella S. The Boltzmann–Grad limit of a hard sphere system: Analysis of the correlation error. Inventiones Mathematicae. 2017;**207**(3):

Potentials. Zürich Lectures in Advanced Mathematics: EMS Publ House; 2014.

Hard Spheres and Short-range

146 p. ISBN-10: 3037191295

[3] Mehta A. Granular Physics.

equation approach to the shear flow of a

1135-1237. DOI: 10.1007/s00222-016-

[10] Gerasimenko VI, Gapyak IV. Low-

[11] Pareschi L, Russo G, Toscani G, editors. Modelling and Numerics of Kinetic Dissipative Systems. N.Y.: Nova Science Publ. Inc.; 2006. 220 p. ISBN-13

[12] Villani C. Mathematics of granular materials. Journal of Statistical Physics. 2006;**124**(2-4):781-822. DOI: 10.1007/

[13] Capriz G, Mariano PM, Giovine P, editors. Mathematical Models of Granular Matter. (Lecture Notes in Math. 1937). Berlin Heidelberg: Springer-Verlag; 2008. 228 p. ISBN

density asymptotic behavior of observables of hard sphere fluids. Advances in Mathematical Physics. 2018;**2018**:6252919. DOI: 10.1155/2018/

0682-4

6252919

978-1594545030

s10955-006-9038-6

978-3-540-78276-6

47/3/035001

krm.2012.5.459

1.858896

[14] Gerasimenko VI,

Borovchenkova MS. On the non-Markovian Enskog equation for granular gases. Journal of Physics A: Mathematical and Theoretical. 2014; **47**(3):035001. DOI: 10.1088/1751-8113/

[15] Gerasimenko VI, Gapyak IV. Hard sphere dynamics and the Enskog equation. Kinetic and Related Models. 2012;**5**(3):459-484. DOI: 10.3934/

[16] Mac Namara S, Young WR. Kinetics of a one-dimensional granular medium in the quasielastic limit. Physics of Fluids A. 1993;**5**(1):34-45. DOI: 10.1063/

[17] Bellomo N, Pulvirenti M, editors. Modeling in applied sciences. In:

[2] Goldhirsch I. Scales and kinetics of granular flows. Chaos. 1999;**9**:659-672.

Cambridge, UK: Cambridge University Press; 2007. 362 p. DOI: 10.1017/

[4] Pöschel T, Brilliantov NV, editors. Granular gas dynamics. In: Lecture Notes in Phys. 624. Berlin, Heidelberg: Springer-Verlag; 2003. 369 p. ISBN

[5] Brilliantov NV, Pöschel T. Kinetic Theory of Granular Gases. Oxford: Oxford University Press; 2004. 329 p. [18] Williams DRM, MacKintosh FC. Driven granular media in one dimension: Correlations and equation of states. Physical Review E. 1996;**R9**(R): 54. DOI: 10.1103/PhysRevE.54.R9

[19] Toscani G. One-dimensional kinetic models with dissipative collisions. Mathematical Modelling and Numerical Analysis. 2000;**34**(6):1277-1291. DOI: 10.1051/m2an:2000127

[20] Gerasimenko VI, Borovchenkova MS. The Boltzmann– Grad limit of the Enskog equation of one-dimensional granular gases. Reports of the NAS of Ukraine. 2013;**10**:11-17

[21] Toscani G. Kinetic and hydrodynamic models of nearly elastic granular flows. Monatschefte für Mathematik. 2004;**142**:179-192. DOI: 10.1007/s00605-004-0241-8

**Chapter 11**

**Abstract**

smaller than 100.

**1. Introduction**

process.

**179**

Comparison of Concentration

Transport Process

*Junsheng Zeng and Heng Li*

Transport Approach and MP-PIC

Method for Simulating Proppant

In this work, proppant transport process is studied based on two popular numerical methods: multiphase particle-in-cell method (MP-PIC) and concentration transport method. Derivations of governing equations in these two frameworks are reviewed, and then similarities and differences between these two methods are fully discussed. Several cases are designed to study the particle settling and conveying processes at different fluid Reynolds number. Simulation results indicate that two physical mechanisms become significant in the high Reynolds number cases, which leads to big differences between the simulation results of the two methods. One is the gravity convection effect in the early stage and the other is the particle packing, which determines the shape of sandbank. Above all, the MP-PIC method performs better than the concentration transport approach because more physical mechanisms are considered in the former framework. Besides, assumptions of ignoring unsteady terms and transient terms for the fluid governing equations in the concentration transport approach are only reasonable when Reynolds number is

**Keywords:** proppant transport, two-phase flow, gravity convection, multiphase

In unconventional oil and gas industry, there exists a significant granular flow process, which is known as the proppant transport [1]. It is necessary to pump highstrength granular materials such as ceramic particles and sand into the stimulated fracture networks with carrying fluid. Eventually after the flow-back of fluid, the granular materials remain in the fractures and fracture networks are efficiently propped, which contributes to a high conductivity for gas/oil exploitation. Therefore, it is important to reveal the physical mechanisms in the proppant transport

Essentially, proppant transport process is a two-phase flow problem constrained

in a channel with various widths. In previous works, concentration transport approach was very popular for simulating the proppant transport. In the approach, proppant is considered as a continuum, and is quantitatively described using

particle-in-cell method, concentration transport approach

## **Chapter 11**
