CLUSTERS OF GALAXIES HOLD A UNIQUE POSITION IN HIERARCHICAL STRUCTURE FORMATION THEY ARE BOTH POWERFUL COSMOLOGICAL PROBES AND EXCELLENT ASTROPHYSICAL LABORATORIES. ACCURATE MODELING OF THE CLUSTER PROPERTIES IS CRUCIAL FOR REDUCING SYSTEMATIC UNCERTAINTIES IN CLUSTER COSMOLOGY. HOWEVER THEORETICAL MODELING OF THE INTRACLUSTER MEDIUM (ICM) HAS LONG SUFFERED FROM THE ``COOLING-FLOW PROBLEM CLUSTERS WITH SHORT CENTRAL TIMES OR COOL CORES (CCS) ARE PREDICTED TO HOST MASSIVE INFLOWS OF GAS THAT ARE NOT OBSERVED. FEEDBACK FROM ACTIVE GALACTIC NUCLEI (AGN) IS BY FAR THE MOST PROMISING HEATING MECHANISM TO COUNTERACT RADIATIVE COOLING. RECENT HYDRODYNAMIC SIMULATIONS HAVE MADE REMARKABLE PROGRESS REPRODUCING PROPERTIES OF THE CCS. HOWEVER THERE REMAIN TWO MAJOR QUESTIONS THAT CANNOT BE PROBED USING PURELY HYDRODYNAMIC MODELS: (1) WHAT ARE THE ROLES OF COSMIC RAYS (CRS)? (2) HOW IS THE EXISTING PICTURE ALTERED WHEN THE ICM IS MODELED AS WEAKLY COLLISIONAL PLASMA? WE PROPOSE TO MOVE BEYOND LIMITATIONS OF PURE HYDRODYNAMICS AND PROGRESS TOWARD A COMPLETE UNDERSTANDING OF HOW AGN JET-INFLATED BUBBLES INTERACT WITH THEIR SURROUNDINGS AND PROVIDE HEAT TO THE ICM. OUR OBJECTIVES INCLUDE: (1) UNDERSTAND HOW CRDOMINATED BUBBLES HEAT THE ICM; (2) UNDERSTAND BUBBLE EVOLUTION AND SOUND-WAVE DISSIPATION IN THE ICM WITH DIFFERENT ASSUMPTIONS OF PLASMA PROPERTIES E.G. COLLISIONALITY OF THE ICM WITH OR WITHOUT ANISOTROPIC TRANSPORT PROCESSES; (3) DEVELOP A SUB-GRID MODEL OF AGN HEATING THAT CAN BE ADOPTED IN COSMOLOGICAL SIMULATIONS BASED ON STATE-OF-THE-ART ISOLATED SIMULATIONS. WE WILL USE A COMBINATION OF ANALYTICAL CALCULATIONS AND IDEALIZED SIMULATIONS TO ADVANCE OUR UNDERSTANDING OF EACH INDIVIDUAL PHYSICAL PROCESS. WE WILL THEN PERFORM THE FIRST THREE-DIMENSIONAL (3D) MAGNETO-HYDRODYNAMIC (MHD) SIMULATIONS OF SELF-REGULATED AGN FEEDBACK WITH RELEVANT CR AND ANISOTROPIC TRANSPORT PROCESSES IN ORDER TO QUANTIFY THE AMOUNT AND DISTRIBUTION OF HEATING FROM THE AGN. OUR PROPOSED WORK WILL ELUCIDATE THE POORLY UNDERSTOOD CR AND ANISOTROPIC TRANSPORT PROCESSES IN THE WEAKLY COLLISIONAL ICM AND SHED LIGHT ON THE LONG-STANDING MYSTERY OF AGN HEATING IN CC CLUSTERS. OUR INVESTIGATION WHICH INCORPORATES PLASMA EFFECTS INTO FLUID MODELS AND PROVIDES PHYSICAL FOUNDATION FOR COSMOLOGICAL SIMULATIONS WILL SERVE AS AN IMPORTANT BRIDGE BETWEEN PHYSICS ON BOTH MICRO AND MACRO SCALES. THIS STUDY WILL ENABLE ROBUST MODELING OF THE RADIO-MODE FEEDBACK OF AGN IN COSMOLOGICAL SIMULATIONS OF CLUSTER AND GALAXY FORMATION. IT WILL ALSO DIRECTLY IMPACT OBSERVATIONAL STUDIES OF CLUSTERS INCLUDING NASA MISSIONS SUCH AS CHANDRA XMM-NEWTON ASTRO-H/HITOMI FERMI HST AND PLANCK.