Active matter refers to systems driven out of thermal equilibrium by the uptake and dissipation of energy directly at the level of the individual constituents, which then undergo systematic movement in a direction decided by their own internal state. This category of nonequilibrium systems was defined as the physical model of motile, metabolizing matter, but the definition has a wider application. In this thesis we work with monolayer of macro-scopic granular particles lying on a vibrated surface and show that it provides a faithful realisation of active matter. The vibration feeds energy into the tilting vertical motion of the particles, which transduces it into a horizontal movement via frictional contact with the base in a direction determined by its orientation in the plane. We show that the dynamics of the particles can be easily controlled by manipulating their geometrical shapes. In the second part of the thesis, not addressing active matter, we do experiments on a soft condensed mat-ter system of viscoelastic surfactant gel formed of an entangled network of wormlike micelles and shows shear-thinning and is therefore non-Newtonian. These systems have relaxation times of the order of seconds and we have studied their non-equilibrium response properties when driven out of equilibrium externally by the gravitational sedimentation of objects and rising air-bubbles. Chapter 1 gives a general introduction to the term active matter and emphasize particularly on how these systems are internally driven and work far away from the equilibrium. We then explain in detail how a system of granular particles lying on a vibrating surface acts as active matter. We later give a brief introduction to the field of soft condensed matter and discuss the viscoelastic properties of surfactant solutions and their phase behaviour. We end this chapter by giving a brief introduction to flocking and non-equilibrium fluctuation relations which act as prerequisite to the following chapters. In Chapter 2 we discuss the experimental techniques used by us. We will first describe the shapes and dimensions of the granular particles used in the experiments. Next we introduce the shaker set-up and describe the experimental cell in which the particles are confined and variation in cell’s boundary. We show the dynamics of the particles in a quasi one-dimensional channel and then in two-dimensions. We give a brief account of image analysis and tracking algorithms employed and other data analyses techniques. In Chapter 3, we study the non-equilibrium fluctuations of a self-propelled polar particle moving through a background of non-motile spherical beads in the context of the Gallavotti-Cohen Fluctuation Relation (GCFR), which generalizes the second law of thermodynamics by quantifying the relative probabilities of the instantaneous events of entropy consumption and production. We find a fluctuation relation for a non-thermodynamic quantity, the velocity component along the long axis of the particle. We calculate the Large Deviation Function (LDF) of the velocity fluctuations and find the first experimental evidence for its theoretically predicted slope singularity at zero. We also propose an independent way to estimate the mean phase-space contraction rate. In Chapter 4 we expand the analysis done in Chapter 3 and study the two-dimensional velocity vector of the particle in the context of Isometric Fluctuation Relation (IFR) which measures the relative probability of current fluctuations in different directions in space of dimension >1. We first show that the dynamics of the particle is not isotropic and present a minimal model for its dynamics as a biased random walker, driven by a noise with anisotropic strength and construct an Anisotropic IFR (AIFR). We then show that the velocity statistics of the polar particle agree with the AIFR. We also confirm that the GCFR can be obtained as a special case of AIFR when the velocity vectors point in opposite directions. We calculate the LDF of particle’s velocity vector and find an extended kink in the velocity plane. In Chapter 5 we study the flocking phenomenon of a collection of polar particles when moving through a background of non-motile beads. We show that in the presence of bead medium, polar particles can flock at much lower concentrations, in contrast to the Vicsek model which predicts flocking at high concentrations. We show that the moving rods lead to a bead flow which in turn helps them to communicate their orientations and velocities at much greater distances. We provide a phase diagram in the parameter space of concentrations of beads and polar particles and show power-law spatial correlations as we approach the phase boundary. We also discuss the numerical simulations and theoretical model presented which support the experiments results. In Chapter 6 we experimentally study the angle dependence of the trapping of collection of active granular rods in a chevron shaped geometry. We show the particles undergo a trapping-detrapping transition at θ = 1150. On the contrary, this angle value is θ = 700 for a single rod. We find a substantial decrease in rotational noise for a collection of particles inside a trap as compared to a single rod which explains the increased value of θ for the trapping-detrapping transition. We also show that polar active particles which tend to change their direction of motion do not show the trapping phenomenon. In Chapter 7 we conduct experiments on falling balls and rising air bubbles through a non-Newtonian solution of surfactant CTAT in water, which forms a viscoelastic wormlike micellar gel. We show that the motion of the ball undergoes a transition from a steady state to oscillatory as the diameter of the ball is increased. The oscillations in velocity of the ball are non-sinusoidal, consisting of high-frequency bursts occurring periodically at intervals long compared to the period within the bursts. We present a theoretical model based on a slow relaxation mechanism owing to structural instabilities in the constituent micelles of the viscoelastic gel. For the case of air bubbles, we show that an air bubble rising in the viscoelastic gel shows a discontinuous jump in the velocity beyond a critical volume followed by a drastic change in its shape from a teardrop to almost spherical. We also observe shape oscillations for bigger bubbles with the tail swapping in and out periodically.

Due to their diversity, soft matter systems provide a convenient platform to study a variety of physical phenomena like phase transitions and collective motion. Encompassing a wide range of equilibrium and non-equilibrium systems, they often provide significant insight into the statistical mechanics of different kinds of many-body systems. Though large scale properties of such systems are of fundamental interest in their own accord, since most experimental realizations of soft matter systems are finite sized, there is a growing need to understand the effects of confinement or boundary conditions on the collective behaviour of such systems. The primary purpose of this thesis is to study the effects of boundary conditions or confinement on both equilibrium and non-equilibrium soft matter systems via theoretical modelling. For equilibrium systems we have studied a system of colloidal particles in harmonic confinement, and for non-equilibrium systems we consider a system of self-propelled rods in both harmonic and hard wall confinement. In Chapter 1 we first lay down some basic concepts of stochastic dynamics and Brownian motion, before discussing some of the recent results on confinement effects on colloidal systems, showing how the properties of a finite sized colloidal system can be very different from those of large, un confined systems. Thereafter turning to non-equilibrium active systems, we discuss various fundamental problems posed by these systems due to their unique ability to generate and dissipate energy on their own. We also point out some instances of observed confinement effects in such systems, such as boundary aggregation and transient hedgehog-like clusters near the boundary. Chapter 2 deals with the effect of harmonic confinement on a finite sized colloidal assembly, where we show that such finite size effects coupled with a confining potential can give rise to special features like initial position dependent expulsion of dopant particles. First we model experimentally studied small two-dimensional colloidal assemblies trapped by a defocussed laser beam by Langevin dynamics simulations in the presence of harmonic confinement and demonstrate how the system shows a crossover from liquid state to crystalline state as a function of the stiffness of the confinement. We also show that in the crystalline state the system can be effectively modelled as a rigid body under small force perturbations. Notably, while studying the dynamics of a defect particle inside these crystallites, we found evidence for the occurrence of self purification by the crystallites. In this process, a dopant is spontaneously expelled out of the crystallite. Surprisingly, this phenomena has a strong dependence on the initial position of the dopant, which turns out to be the consequence of the non monotonic spatial variation of the free energy of the system as a function of the dopant position. This is caused by a difference in the rate of change of internal energy and entropy with the dopant position, with the entropy decreasing faster when the dopant is closer to the centre. This can be attributed to the amount of disruption of crystalline order in the assembly due to the incommensurate dimensions of the defect particle. In order to put these results in a general perspective, we verify in the last part of this chapter that the presence of this free energy barrier is independent of the exact functional forms of the confining potential and the interaction of a defect particle with the host particles, as well as the shape and size of the defect particle. Moving to non-equilibrium systems, we consider, in Chapter 3, the effect of harmonic and hard wall confinement on a two-dimensional system of self-propelled rods (SPRs). Though there have been very limited studies of confinement effects on such systems, existing studies are adequate to show that their behaviour near a boundary wall can be very different, e.g. formation of hedgehog like clusters near a boundary wall. First we show that for harmonic confinement small systems show polar order, which decays with system size, eventually going away for large systems. But the effect of hard wall confinement turns out to be rather different, where the system shows isotropic and clustered states depending on the values of activity and density. We construct a complete activity-density phase diagram showing four distinct phases. For high density and high activity, the rods spontaneously arrange themselves into a stable vortex structure in which the rods exhibit global radial polar order. Surprisingly this order does not decay with system size: the radial orientation of the rods exhibit strong spatial correlation even in large systems, ruling out the possibility that the radial order is a finite-size effect. Using other geometrical shapes of the hard wall boundary, we confirm this phase to be independent of the shape of the boundary. We also demonstrate how small modifications of the boundary conditions at the hard wall can collapse the clustered and vortex phases to a global flocking phase similar to that found in earlier studies of hydrodynamic active particles under confinement. Based on these observations, we conclude that the bulk of the system is strongly affected by the subjected boundary condition, which is rather unusual for large systems. In Chapter 4 this thesis concludes with a summary of the main results and suggestions for future work along similar lines