Chapter 1

Introduction

1.1 Pattern Formation in Nonequilibrium Systems

Pattern formation is a ubiquitous phenomenon in nature, e.g., emergence of animal skin patterns, crystal growth from a seed, structure formation in galaxies and the universe, etc. The appearance of patterns is associated with the development of inhomogeneous structure on one or more length scales. In the context of physics, patterns usually characterize the nonequilibrium evolution of systems. Nonequilibrium systems exhibit fascinating spatio­temporal dynamics, and it is often found that vastly different systems, e.g., fluids, granular materials, lasers, biological tissues, etc., show similar dy­namical behaviors. One of the major challenges in this area is to identify the similarities between different systems and develop theories for a quantitative understanding of these universal phenomena [1].

1.1.1 Nonequilibrium Systems Let us first provide a working definition of nonequilibrium systems. An equilibrium system is characterized by the following features: (a) the macroscopic variables (e.g., pressure, temperature) do not change with time; (b) there are no macroscopic currents of quantities like heat, mass, charge, etc. Systems which only satisfy (a) (but not (b)) are referred to as steady-state systems.

A nonequilibrium system is then defined as a system which in not in equi­librium. Broadly speaking, there are four classes of nonequilibrium systems:

(i) Systems which are near-equilibrium and approaching equilibrium. This class of systems can be studied in the framework of linear response theory (LRT), which describes nonequilibrium behavior in terms of equilibrium cor­relation functions. (ii) Systems which are far-from-equilibrium and approaching equilibrium. One cannot apply LRT in this case because the evolution of the system is strongly nonlinear. (iii) Systems which are near steady-state and approaching a steady state. This class also includes steady-state systems. (iv) Systems which are far-from-steady-state and approaching a steady state.

Typically, rich examples of pattern formation are found in classes (ii) and (iv) above, as the system evolution involves strongly nonlinear and time­dependent processes. In this thesis, we will study a variety of such nonlinear processes, and quantitatively characterize the system evolution in each case. All the problems addressed in this thesis constitute examples of "phase or­dering dynamics", which we discuss in the next subsection.

1.1.2 Phase Ordering Dynamics

The term "phase ordering dynamics" refers to the evolution of a homoge­neous multi-phase system, which is rendered thermodynamically unstable by a rapid change of parameters, e.g., temperature, pressure, etc. There has been much interest in such problems in the general context of far-from­equilibrium statistical physics [2, 3]. An example of a two-phase mixture is a ferromagnet at high temperature and zero magnetic field, which consists of a homogeneous mixture of "up" and "down" spins. The relevant phase diagram for the ferromagnet is shown in Figure 1.1. Below the critical tem­perature (Te), the system prefers to be in a spontaneously-magnetized state, even if there is no external field. We consider a situation where the system at temperature Ti is rapidly quenched to T, < Te at time t = 0, as shown in Figure 1.1.

The evolution of the system from the unstable initial state is a complex nonlinear process. The system does not order immediately after the quench from the disordered phase to the ordered phase. The length scale of ordered regions grows with time as the different broken-symmetry phases compete to select the equilibrium state. The typical system evolution is depicted in Figure 1.2, which is obtained from a Monte Carlo (MC) simulation of a phase-ordering ferromagnet. The numbers labeling the evolution pictures denote times in terms of Monte Carlo steps. The domain growth process is a scaling phenomenon, i.e., the domain patterns at later times are statistically

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similar to those at earlier times, apart from a change of length scale. In the above example, at high temperature and zero magnetic field, the

spins are randomly oriented in both "up" and "down" directions with equal probability, so that the magnetization of the system is zero at any given time - this corresponds to the paramagnetic phase. However, below the critical temperature Te , the spins tend to align along a single direction in space, giving rise to a non-zero local magnetization M. This is referred to as the ferromagnetic phase. The onset of this behavior is a continuous phase transition - the local magnetization increases from zero as the temperature is reduced below Te. In this context, the spontaneous magnetization constitutes a convenient "order parameter" to describe the phase transition. In the time­dependent evolution of Figure 1.2, the order parameter depends upon space (f') and time (t), and we denote the order parameter as 'l/J(r, t).

Next, let us consider the case of a homogeneous binary (AB) alloy at high temperature. The homogeneous mixture prefers to be phase-separated, when it is quenched below the coexistence curve. The corresponding phase diagram is shown in Figure 1.3. In Figure 1.4, we depict the kinetics of the phase-separation process from an initially homogeneous state, after the system has been quenched below the critical temperature Te. For mixtures with equal amounts of A and B, the local difference in densities of the two components is zero at high temperatures. However, below Te , this quantity becomes nonzero. In this case, the relevant order parameter is the local difference in densities of A and B.

1.1.3 Modeling of Phase-Ordering Systems

In both ofthe above examples, viz., the two-state ferromagnet and the binary mixture, a reasonable model is the Ising Hamiltonian:

H {Si} = - L JijSiSj , Si = ±1, (ij)

(1.1 )

where Jij (> 0) is the exchange coupling strength between the spins at sites i and j. In the case of a ferromagnet, the spin variable Si refers to the spin state at site i. In the case of the binary alloy, the states Si = + 1 and -1 can be identified with species A and B, respectively.

The classical Ising model has no intrinsic dynamics associated with it. Therefore, we associate stochastic dynamics with the Ising model by plac­ing it in a heat bath. The simplest dynamics with a nonconserved order parameter (e.g., ordering of a ferromagnet) is the "spin-flip Glauber dynam­ics" [4] where, during the evolution process, any particular spin is flipped if it reduces the system energy. On the other hand, the simplest conserved

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h

(0,0) ....... -------II----L-~ T

Figure 1.1: Phase diagram of a two-state ferromagnet. At magnetic field h = 0, the magnetization is zero above the critical temperature Te. Below Te , the system is in spontaneously magnetized state. The heavy line from (T = 0, h = 0) to (T = Te , h = 0) denotes a line of first-order transitions. We consider temperature quenches from Ti to Tf at time t = O.

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t=OMCS t=50 MCS

t=IOO MCS t=500 MCS

-Figure 1.2: Evolution of a spin-1/2 ferromagnet from a random initial condi­tion on an N 2-lattice with N = 128. The system temperature was T = O.5Tc. Periodic boundary conditions were applied in both directions. The black re­gions denote domains of "up" spins, and the white regions denote domains of "down" spins. The snapshots are labeled by appropriate evolution times in units of Monte Carlo steps (MCS).

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Phase-separated mixture

T

T· -----------------1

Tf ------

Homogeneous mixture

Homogeneous mixture metastable

~----~~~--~--~~~--------~ CA Ca Co Cp

Figure 1.3: Phase diagram of a binary (AB) mixture. The parameters are concentration of A (denoted as CA ); and temperature T. The different re­gions of the phase diagram are labeled by the appropriate phase. We con­sider quenches from Ti to Tf at t = O. Below the coexistence curve (de­noted as a solid line), the mixture segregates into A-rich and A-poor regions, with A-concentrations C{3 and Co, respectively. The homogeneous mixture is metastable in the region between the dashed curve and the coexistence curve.

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t=OMCS t=100 MCS

t=1000MCS t=5000 MCS

Figure 1.4: Evolution of a homogeneous binary (AB) mixture from a homoge­neous initial condition on an N 2-lattice (with N = 128), after it is quenched to T = O.85Tc. Periodic boundary conditions were applied in both directions. The black and white regions denote domains of A and B, respectively. The snapshots are labeled by appropriate evolution times in units of Monte Carlo steps (MCS).

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dynamics (e.g., phase separation of a binary alloy) is the "spin-exchange Kawasaki dynamics" [5], where a pair of spins are interchanged if the system energy is reduced.

Recall that, in the case of the Ising model, the spins can align in either "up" or "down" directions. If one allows the spins to align in any direction in a plane, the relevant model is referred to as the XY model. If the spins are 3-component vectors, the corresponding model is referred to as the Heisenberg model. A planar ferromagnet (where the system can get magnetized in any direction in a plane) is describable by the XY model, whereas an isotropic ferromagnet (where the system can be magnetized in any direction in the 3-dimensional space) is describable by the Heisenberg model.

The starting point of most analytical approaches in phase ordering dy­namics is a stochastic partial differential equation obtained from the coarse­grained version of the Hamiltonian. Let us consider the case of the two-state ferromagnet. In this case, the coarse-grained free-energy functional has the form (in dimensionless units):

F[?j;(i, t)] = J di [~(V?j;)2 + ~Sgn(T - Tc )7/i + l?j;4] , (1.2)

which is referred to as the ?j;4-free energy functional. In equilibrium, the functional derivative of the free energy F[?j;(i, t)] satisfies

<5F[?j;] = 0 8?j; . (1.3)

When the system is slightly out of equilibrium, it is assumed that the rate at which the system relaxes to equilibrium is proportional to the generalized force. This assumption leads to the following equation for the rate of change of the order parameter in the overdamped limit

8?j;(i, t) _ _ r8F[?j;(i, t)] . 8t - 8?j;(i, t) + nOlse, (1.4)

where r is a phenomenological parameter. This equation is usually referred to as the time-dependent Ginzburg-Landau (TDGL) equation [2, 6]. Using the expression for the free energy in Eq. (1.2), we obtain

8?j;(i, t) 01.( .... ) 201.( .... ) ( .... )3 8t = 'f/ r, t + \7 'f/ r, t -?j; r, t + noise. (1.5)

A class of model equations, referred to as Ginzburg-Landau models because of their similarity with the TDGL equation, has the general form:

8?j;~, t) = (a + ib)\72?j;(i, t) + f(I?j;(i, t)12)?j;(i, t), (1.6)

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where f(= II + ih) is an arbitrary complex function of I~(f', t)12. The dynamics of the phase separation process, is assumed to be gov­

erned by diffusion in a chemical-potential gradient. Conservation of material implies the equation of continuity

a~ (f', t) + 0 . 7( .... t) = 0 at v J r, , (1.7)

where J(i, t) is the phenomenological current, given by

J(f', t) -MV /1(f', t) + noise

.... [6F[~(f', t)]] . -M\7 6~(i, t) + nOIse. (1.8)

Here M is a phenomenological constant which plays the role of mobility. Following this assumption, one obtains

a~(f', t) _ M,,2 [6F[~(i, t)]] . at - v 6~(i, t) + nOIse. (1.9)

Eq. (1.9) is referred to as the Cahn-Hilliard-Cook (CHC) equation [7]. The deterministic version of this equation (without noise) is referred to as the Cahn-Hilliard (CH) equation.

1.1.4 Topological Defects

Defects playa very important role in the dynamics of domain growth. In the case of two-component systems, the relevant defects are the domain walls between dissimilar phases. Domain walls are the simplest form of topologi­cal defects and occur in systems described by scalar (one-component) order parameters. They are formally defined as surfaces on which the order param­eter vanishes, i.e., ~ = O. Local changes in the order parameter can move the wall but cannot destroy it. Thus, a domain wall is topologically stable.

For vector order parameters, stable topological defects can be constructed in analogy with the scalar case. Thus, in d-dimensional space, all n compo­nents of the order parameter must vanish on a surface of (d - n) dimension. The physical existence of such defects requires that n ~ d. If n < d, the defects are spatially extended. In that case, growth occurs by a straighten­ing out of the defect. This reduces the total area of domain walls or length of a vortex line (e.g., XY model in d = 3). For n = d, i.e., in the case of point defects, coarsening occurs due to the annihilation of defect-antidefect pairs (XY model in d = 2). Figure 1.5 demonstrates a few typical defects in phase-ordering systems.

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(.) I 1

(c)

Figure 1.5: A few topological defects in 2- and 3- spatial dimensions. (a) Domain wall (d = 2, n = 1). (b) Vortex (d = 2, n = 2). (c) Vortex line (d=3,n=2).

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1.1.5 Growth Exponents The primary mechanism for domain coarsening is the curvature-driven anni­hilation of interfaces (or defects), which results in the system energy being lowered. In many phase-ordering systems, the asymptotic domain length scale exhibits a power-law behavior L(t) I'V t¢, where 4> is referred to as the growth exponent. The value of this exponent depends on the relevant conservation laws, and whether or not there are hydrodynamic effects.

Let us consider the growth laws which arise for pure and isotropic sys­tems. In the case of a ferromagnet undergoing an ordering transition (see Figure 1.2), where the order-parameter (spontaneous magnetization) is not a conserved quantity, we have 4> = 1/2. This growth law is referred to as the Lifshitz-Cahn-Allen (LeA) law [8]. On the other hand, for a phase-separating binary mixture (see Figure 1.4), where the order parameter (local difference in densities of A and B) is conserved, the growth exponent is 4> = 1/3 in the absence of hydrodynamic effects. This growth law is referred to as the Lifshitz-Slyozov (LS) law [9]. When hydrodynamic effects are relevant, e.g., phase separation of a binary fluid, there are a variety of exponents - depend­ing upon the time-regime and dimensionality [10].

1.1.6 Correlation Functions and Structure Factors Two standard tools for the quantitative characterization of the evolving mor­phology in pattern-forming systems are the equal-time correlation function C(i, t); and the structure factor s(f, t), which is the Fourier transform of the correlation function. The quantity S(k, t) is directly measurable from small­angle scattering experiments with probes of the appropriate wavelength, e.g., light, X-rays, neutrons, etc. For an n-component order-parameter field ~(i, t), the correlation function is defined as

C(i, t) = ~ J dR { (~(R, t) . ~(R + i, t)) - (~(R, t)) . (~(R + i, t)) } , (1.10)

where the angular brackets refer to an averaging over initial conditions and noise ensembles; and V is the system volume.

The corresponding definition for the structure factor is

S(k, t) ! dieik.rC(i, t)

(~(k, t) . ~(-k, t)), (1.11)

where the angular brackets have the same meaning as that in the correlation function.

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For isotropic systems, we have C(i, t) = C(r, t) and S(k, t) = S(k, t). Furthermore, an examination of Figures 1.2 and 1.4 suggests that the statis­tical properties of the morphology are unchanged in time upto a scale factor, which is merely the growing length scale. This self-similar behavior of the evolving system is reflected in the scaling behavior of C(r, t) and S(k, t) [11], VIZ.,

C(r, t) - f (L~t)) , (1.12)

and S(k, t) L(t)dg(kL(t)). (1.13)

Though the generality of this scaling hypothesis has not been tested from first principles (except in some simple cases), it is supported by a large body of experimental and numerical results [2].

1.2 Overview of Numerical Techniques In the present context, numerical simulations playa crucial role in statistical physics. In this section, we would like to briefly describe some important techniques which are used extensively in non equilibrium statistical physics: (a) Simulations of continuum dynamical equations; (b) Monte Carlo simulations; (c) Molecular dynamics simulations. We have developed and used these techniques for our investigations described in this thesis, and will describe them in greater detail at appropriate places.

1.2.1 Simulations of Continuum Dynamical Equations In this approach, nonlinear partial differential equation models are solved, often by using simple Euler-discretization techniques. Let us illustrate the Euler-discretization scheme in the context of the TDGL equation [12], i.e.,

ov;~, t) = v;(r, t) + V2v;(i, t) - v;(i, t)3. (1.14)

We will consider the simulation of the TDGL equation on a square lattice in dimensionality d = 2. Using the Euler formula, we can write the approximate forms of the derivatives as

ov;(i, t) ot

o2V;(i, t) ox?

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v;(r, t + ~t) - v;(r, t) ~t

V;(Xi + ~x, t) + V;(Xi - ~x, t) - 2'IjJ(Xi' t) ) rv (~X)2 i = 1, 2{1.15

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where 6.x and 6.t are discretization mesh sizes. The accuracy of these dis­crete derivatives could be improved by incorporating intermediate points also. Using this discretization scheme, one obtains the field at some later time (t + 6.t), from the field at time t as

'ljJ(f, t + 6.t) = 'ljJ(i, t) + 6.t ['ljJ(f, t) - 'ljJ(f, t)3] + aV~'ljJ(i, t), (1.16)

where a = 6.t/(6.x)2 and Vb is the discrete Laplacian. We should stress that we are not particularly interested in obtaining pre­

cise numerical solutions of the nonlinear equation of interest. Rather, we expect the numerical solution to provide the same physics as that of the partial differential equation. The solution should be stable to small fluctu­ations around the equilibrium values. In the case of the TDGL equation, once the domains saturate out to their equilibrium values 'ljJ = ±1, the bulk is stable against fluctuations around these values. Let us examine the sta­bility property for the discretized version of the TDGL equation by setting 'ljJ(i, t) = 1 + 6'ljJ(i, t). After linearizing the resulting equation in 6'ljJ(f, t), we obtain

6'ljJ(f, t + 6.t) = (1 - 26.t)6'ljJ(f, t) + aVb6'ljJ(i, t). (1.17) We can take the Fourier transform of this equation to obtain

6'ljJ(k, t + 6.t) [1 - 26.t - 2a {2 - cos (kx6.x) - cos (ky6.y)}]6'ljJ(k, t) A(k)6'ljJ(k, t). (1.18)

The eigenvalue A(k) determines the temporal behavior of the fluctuation for the mode with wavevector k. The fluctuation will decay if IA(k)1 < 1; and grow if IA(k)1 > 1. The growth of the fluctuation is unphysical and our stability condition is obtained by requiring IA(k)1 < 1. In our case, notice that A(k) < 1 for all k; but A(k) > -1, if

(6.x )2 (6.x)2 + 4 ~ 6.t. (1.19)

Therefore,the numerical simulation will be stable if 6.x and 6.t are chosen so that the condition in Eq. (1.19) is satisfied. Of course, this condition does not address the issue of numerical accuracy. As we mentioned earlier, it suffices that the discrete models replicate the correct physical behavior.

1.2.2 Monte Carlo Simulations The Monte Carlo (MC) technique [13] is primarily used for systems defined on a discrete lattice. We will discuss it in the context of the Ising model Hamiltonian, presented in Eq. (1.1).

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We will discuss different steps of the MC procedure in the context of non­conserved order parameter. Step 1: A random initial configuration Si of the spins is chosen to implement the far-from-equilibrium state of the system. Recall that we are interested at a temperature regime which is below the order-disorder transition tem­perature. In this regime, the equilibrium state is the situation where all the spins are aligned in the same direction. Step 2: Flip a randomly chosen spin Si to S~. Step 3: Compute the probability W for the transition from Si to S~ as

W e-oH/T if bH > 0 , , 1, if bH ~ 0, (1.20)

where bH is the change in energy due to the spin-flip. Step 4: Generate a random number p between 0 and 1 and compare it with W. If W > p, take S: as the new configuration of the system. If W < p, take Si as the new configuration of the system. Step 5: Repeat the steps 2 to 4 again and again. In case of conserved dynamics (binary mixture AB), spins are not flipped. In this case two spins are interchanged to meet the conservation property satisfied by the system. For detailed discussion of MC techniques and its application in statistical physics, see Ref. [14].

1.2.3 Molecular Dynamics Simulations In the molecular dynamics (MD) method, the Newtonian dynamical equa­tions are solved for a large number of particles with relevant interactions. Two major types of algorithm are used to implement this on computer. 1. Event driven algorithm: This algorithm is useful only when there is no ex­ternal force (e.g., gravitational force). In the absence of any external forces, particles move in straight lines with constant velocities during the time inter­val between two successive collisions. Once the collision has taken place, the new velocities (post collision) of the colliding particles are calculated keeping velocities of all the other particles unchanged.

Let us consider the example of a collection of large number of identical hard particles having unit mass, in the absence of external force. In this case, after the collision between the ith and jth particles having velocities Vi and Vj respectively, the new velocities are given as

...., v· J

.... l+e( .... (.... ....)) .... V· - -- n· V· - V· n t 2 t J ,

.... l+e( .... (.... ....)) .... Vj + -2- n· Vi - Vj n,

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(1.21)

where e is the coefficient of restitution and fi is the unit vector parallel to the relative position of the particles in the time of collision. 2. Time step driven algorithm: This algorithm is used for the simulation when there is external force so that the particles no longer can move in straight lines between two collisions. In this case, the particles are allowed to move in straight lines only for a fixed size of time which is taken to be very small to minimize error.

There are two subclasses of this algorithm, viz., predictor corrector algo­rithm and verlet algorithm. But we will not discuss them here. Interested readers are referred to the book by Allen and Tildesley [15].

1.3 Overview of Thesis In this section, we would like to give a brief overview of the problems we have studied in this thesis. In subsequent chapters we will describe the problems in greater detail. The Fortran code for relevant computer programs is also appended at the end of each chapter.

1.3.1 Nonequilibrium Dynamics in the Complex Ginzburg-Landau Equation

The complex Ginzburg-Landau (CGL) equation arises in diverse contexts (see the review by Cross and Hohenberg [1]), e.g., chemical oscillations, thermal conduction in binary fluids, multimode lasers, etc. In its general form, the CGL equation can be written as a special case of Eq. (1.6), viz.,

a1/J~, t) = 1/J(i, t) + (1 + ia)\121/J(i, t) - (1 + i,6)I1/J(i, t)121/J(i, t), (1.22)

where 1/J(i, t) is a complex order-parameter field; and a, (3 are real parameters. The CGL equation exhibits a rich range of dynamical behavior with the variation of parameters a and (3. In a large range of parameter space, the emergence and interaction of spiral and antispiral defects plays an important role in determining the evolution and morphology.

We undertake a detailed analytical and numerical study of nonequilibrium dynamics of the CGL equation in both two and three dimensions [16, 17, 18]. In particular, we characterize evolution morphologies using spiral defects. In Chapter 2, we use Hagan's [19] single-spiral solution of the CGL equation to calculate the correlation function C (r) of a single spiral defect of size L, and undertake its asymptotic analysis in the limit rlL --+ 0 with r/~ » 1, where ~ is the size of the defect core. We find that there is a sequence of

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singularities in this limit, which are reminiscent of singularities for defects with O(n) symmetry, where n is even [3]. However, the dominant singularity as r / L --+ 0 corresponds to the case of vortex defects in the XY model [3, 20].

Chapter 2 also investigates the validity of the Gaussian auxiliary field (GAF) ansatz [3] in the context of multispiral morphologies. For early times, the GAF ansatz is reasonable when the domain growth in the CGL equation is analogous to that for the XY model. For late times, we find that the GAF ansatz is not reasonable, as it is unable to account for order-parameter modulations at the defect-defect boundaries.

In Chapter 3, we present detailed numerical results from simulations of the CGL equation in d = 2,3. In particular, we compare our numerical results with the analytical predictions discussed in Chapter 2.

1.3.2 Dynamics of Phase Separation in Multi-Component Mixtures

Multi-component mixtures have considerable technological and scientific im­portance, especially in the context of materials science and metallurgy. Ac­cording to LS theory, the time-dependent length scale of the growing domains in phase-separating binary mixtures exhibits the growth law L( t) rv t1/3 . It is now clear from experimental, theoretical and numerical studies that the LS growth law is rather universal - independent of spatial dimensionality; material parameters; and details of the interaction potential. However, much less is known about the effect of the number of components on the growth law.

In Chapter 4, we study the dynamics of phase separation in multi-compon­ent mixtures, modeled by the q-state Potts model [21] with conserved kinet­ics [22]. We use the Monte Carlo renormalization-group (MCRG) method [23, 24, 25] to investigate the asymptotic behavior. The asymptotic domain growth law is found to be consistent with the LS law, regardless of the value of q. We also present results for the scaled correlation function and domain-size distribution function for q = 3,4,5,10.

1.3.3 Inhomogeneous Cooling in Granular Fluids Consider a granular gas evolving under the effect of interparticle collisions. Due to the inelastic nature of collisions, two colliding particles lose a fraction of their kinetic energy, c (= 1 - e2 ), where c is the degree of inelasticity and e is the coefficient of normal restitution. Therefore unlike an ordinary gas, the "granular gas" loses its kinetic energy due to inelastic collisions, if there

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is no continuous energy input from outside. Particles become progressively more parallel after every collision, and their random motion decreases.

During the initial cooling process, the system stays spatially homogeneous and is said to be in the homogeneous cooling state (RCS). In the RCS, the system energy decreases as

(1.23)

which is known as the Raff [26] cooling law. In Eq. (1.23) /0 = f.ld, where d is the spatial dimensionality; and T is the average number of collisions suffered by a particle. After a certain time Te, there is a crossover from the RCS to a state where the particles start forming clusters which grow with time. This is referred to as the inhomogeneous cooling state (ICS) [27] and the Raff cooling law is no longer valid in this state [28]. The crossover time Te depends upon f. and the density of particles p. The clustering dynamics is reminiscent of the phase-separation process, and can be discussed in a similar context. We undertake a detailed study of the kinetics of the RCS and ICS states and the relevant crossovers [29]. A detailed description of this study is provided in Chapter 5 of this thesis.

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[14] Monte Carlo Methods in Statistical Physics, ed. K. Binder, Springer·· Verlag, Berlin (1986); K. Binder and D.W. Herrmann, Monte Carlo Sim·· ulations in Statistical Physics: An Introduction, Springer-Verlag, Berlin (1988); The Monte Carlo Method in Condensed Matter Physics, ed. K Binder, Springer-Verlag, Berlin (1992).

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[29] S.K. Das and S. Puri, in preparation.

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