classical field theory
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Classical field theory
The mathematical discipline that studies the behavior of distributions of matter and energy when their discrete nature can be ignored; also known as continuum physics or continuum mechanics. The discrete nature of matter refers to its molecular nature, and that of energy to the quantum nature of force fields and of the mechanical vibrations that exist in any sample of matter. The theory is normally valid when the sample is of laboratory size or larger, and when the number of quanta present is also very large. See Phonon, Photon, Quantum mechanics
Classical field theories can be formulated by the molecular approach, which seeks to derive the macroscopic (bulk) properties by taking local averages of microscopic quantities, or by the phenomenological approach, which ignores the microscopic nature of the sample and uses properties directly measurable with laboratory equipment. Although the microscopic treatment sometimes yields profounder insights, the phenomenological approach can use partial differential equations since neglecting the microscopic structure allows quantities such as density and pressure to be expressed by continuously varying numbers.
Examples of classical field theories include the deformation of solids, flow of fluids, heat transfer, electromagnetism, and gravitation. Solving the equations has produced a vast body of mathematics. Computers have aided in special calculations, but many mathematicians are working on the analytical theory of partial differential equations, and new results continue to be produced.