Population balance equations (PBEs) have been introduced in several branches of modern science, mainly in branches with particulate entities. This includes topics like crystallization

Crystallization

Crystallization is the process of formation of solid crystals precipitating from a solution, melt or more rarely deposited directly from a gas. Crystallization is also a chemical solid–liquid separation technique, in which mass transfer of a solute from the liquid solution to a pure solid...

, liquid-liquid extraction, gas-liquid dispersions, liquid-liquid reactions, communition, aerosol engineering, biology

Biology

Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

 (where the separate entities are cells), polymerization

Polymerization

In polymer chemistry, polymerization is a process of reacting monomer molecules together in a chemical reaction to form three-dimensional networks or polymer chains...

, etc. PBEs define how populations of separate entities develop in specific properties over time. They are a set of Integro-partial differential equations which gives the behavior of a population of particles from the analysis of behavior of single particle in local conditions..
Particulate systems are characterized by the birth and death of particles.For example, consider precipitation

Precipitation (chemistry)

Precipitation is the formation of a solid in a solution or inside anothersolid during a chemical reaction or by diffusion in a solid. When the reaction occurs in a liquid, the solid formed is called the precipitate, or when compacted by a centrifuge, a pellet. The liquid remaining above the solid...

 process(formation of solid from liquid solution) which has the subprocesses nucleation

Nucleation

Nucleation is the extremely localized budding of a distinct thermodynamic phase. Some examples of phases that may form by way of nucleation in liquids are gaseous bubbles, crystals or glassy regions. Creation of liquid droplets in saturated vapor is also characterized by nucleation...

, agglomeration, breakage, etc., that result in the increase or decrease of the number of particles of a particular radius

Radius

In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

(assuming formation of spherical particles). Population balance is nothing but a balance on the number of particles of a particular state(in this example, size).

Solution to PBE

Monte Carlo method

Monte Carlo method

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

s and discretization methods are mainly used to solve these equations. The choice depends on the application and computing infrastructure.