EXTENSIVE RESEARCH SHOWS THAT PARTICLES IN A WIDE RANGE OF PLASMA ENVIRONMENTS THROUGHOUT THE HELIOSPHERE SOLAR CORONA AND MAGNETOSPHERE DEVIATE FROM TRADITIONALLY ASSUMED GAUSSIAN DISTRIBUTIONS. OBSERVED DISTRIBUTIONS SHOW LARGE SUPRATHERMAL COMPONENTS THAT MAY EXERT NON-NEGLIGIBLE INTERNAL PRESSURE WITHIN THE PLASMA. UNDERSTANDING THE RELATIONSHIPS BETWEEN SUPRATHERMAL ION POPULATIONS AND COMMONLY OCCURRING PHENOMENA SUCH AS WAVE-PARTICLE INTERACTIONS TURBULENCE MAGNETIC RECONNECTION SHOCK WAVES AND PLASMA COMPRESSIONS REMAINS A MISSING YET ESSENTIAL ELEMENT OF PLASMA DYNAMICS THAT APPLIES THROUGHOUT THE HELIOSPHERE MAGNETOSPHERE AND OTHER PLANETARY ENVIRONMENTS. UNDERSTANDING SUPRATHERMAL DISTRIBUTIONS IS REQUIRED TO MAKE SENSE OF: IBEX AND VOYAGER OBSERVATIONS IN THE HELIOSHEATH; OBSERVATIONS OF SUPRATHERMAL PARTICLES IN FAST AND SLOW SOLAR WIND AND UPSTREAM OF EARTH S BOW SHOCK. ESTABLISHING THE THEORETICAL UNDERPINNINGS OF SUPRATHERMAL DISTRIBUTIONS WILL BE NECESSARY FOR FUTURE OBSERVATIONS FROM SOLAR ORBITER (SO) SOLAR PROBE PLUS (SPP) AND THE INTERSTELLAR MAPPING AND ACCELERATION PROBE (IMAP). HERE OUR OBJECTIVE IS TO DETERMINE THE ORIGIN OF SUPRATHERMAL ION DISTRIBUTIONS AND THE PHYSICAL PROCESSES THAT CREATE THEM. THIS WORK ANSWERS FUNDAMENTAL SCIENCE QUESTIONS IN BASIC PLASMA PHYSICS AND PROVIDES THE MISSING LINK BETWEEN BULK PLASMA IN THERMAL EQUILIBRIUM AND NON-EQUILIBRIUM PROCESSES SUCH AS DIFFUSIVE SHOCK ACCELERATION WHICH CREATES HIGH-ENERGY PARTICLES. OUR WORK INVESTIGATES A COMPELLING PROBLEM ADDRESSING THE FOURTH OF THE HIGH-LEVEL SCIENCE GOALS FROM THE HELIOPHYSICS DECADAL SURVEY: [TO] DISCOVER AND CHARACTERIZE FUNDAMENTAL PROCESSES THAT OCCUR BOTH WITHIN THE HELIOSPHERE AND THROUGHOUT THE UNIVERSE. OUR LACK OF A DETAILED UNDERSTANDING OF THE ORIGIN OF SUPRATHERMAL ION POPULATIONS THWARTS PROGRESS IN HELIOPHYSICS. A BASIC FRAMEWORK FOR DESCRIBING SUPRATHERMAL ION DISTRIBUTIONS IS NEEDED TO SOLVE SIGNIFICANT PROBLEMS RANGING FROM THE ORIGIN OF THE IBEX RIBBON TO THE INJECTION OF SEED PARTICLES AT CORONAL AND INTERPLANETARY SHOCKS TO FORM SOLAR ENERGETIC PARTICLES. THEREFORE A DEDICATED INVESTIGATION IS REQUIRED TO ESTABLISH THE ORIGIN OF SUPRATHERMAL ION POPULATIONS ENCOMPASSING DIVERSE HELIOSPHERIC PLASMA REGIMES AND DOMAINS. WE ANSWER FOUR SCIENCE QUESTIONS: HOW DO TURBULENCE WAVES AND STOCHASTIC PROCESSES CREATE NON-MAXWELLIAN ION DISTRIBUTIONS? HOW DO SHOCKS COMPRESSION REGIONS AND MAGNETIC RECONNECTION PARTICIPATE IN ENERGIZING SUPRATHERMAL IONS AND HOW MIGHT THESE PARTICLES FILL THE HELIOSPHERE? WHAT ARE THE BASIC ANALYTICAL FRAMEWORKS IN WHICH NON-EQUILIBRIUM PROCESSES CAN BE DESCRIBED? AND WHAT OBSERVATIONAL CHARACTERISTICS CAN BE USED TO DISTINGUISH BETWEEN LOCAL NON-EQUILIBRIUM ION ACCELERATION PROCESSES AND NON-LOCAL TRANSPORT EFFECTS? WE PROPOSE A UNIFIED APPROACH IN WHICH WE TEST THEORIES THROUGH NUMERICAL SIMULATIONS COMPARE THE RESULTS WITH OBSERVATIONS IN DIFFERENT PARTS OF THE HELIOSPHERE AND MAKE PREDICTIONS FOR MEASUREMENTS BY FUTURE NASA MISSIONS. WE RELY ON HYBRID (ION SCALE) SIMULATIONS COMPLEMENTED BY SELECTED FULL KINETIC (ELECTRON SCALE) SIMULATIONS TO TEST HOW DIFFERENT PROCESSES CAN ACCELERATE SUPRATHERMAL POPULATIONS ON DIFFERENT SCALES. IN PARALLEL WE DEVELOP ANALYTICAL FRAMEWORKS TO UNDERSTAND HOW REALISTIC SUPRATHERMAL ACCELERATION MECHANISMS MODIFY THE SUBSEQUENT ACCELERATION TO 100S OF KEV OR ABOVE. THIS ALLOWS US TO DEVELOP A KINETIC TRANSPORT MODEL THAT IS COUPLED TO GLOBAL MHD SIMULATIONS. WE COMPARE RESULTS FROM THESE GLOBAL SIMULATIONS WITH OBSERVATIONS FROM NASA HELIOPHYSICS SYSTEM OBSERVATORY (HSO) AND MAKE PREDICTIONS FOR SPP SO AND IMAP.