Poisson

Siméon Denis Poisson (1781–1840) was a French mathematician and physicist whose work helped shape probability theory and mathematical physics. His research spanned mechanics, optics, electromagnetism, and the theory of partial differential equations. Poisson’s name is attached to several foundational concepts in mathematics and science, reflecting his broad impact on both theory and application.

Poisson distribution: A discrete probability distribution that models the number of events occurring in a fixed interval of time or space when events occur with a known constant mean rate and independently of the time since the last event. It is characterized by a single parameter λ, which is both the expected value and the variance.

Poisson process: A counting process with stationary, independent increments; it yields Poisson-distributed counts in any interval and models random events that occur continuously and independently over time, such as calls arriving at a switchboard or particles emitted from a source.

Poisson's equation: A partial differential equation, ∇^2 φ = f, arising in electrostatics, mechanical engineering, and theoretical physics. It extends Laplace's equation by including a source term and is central to potential theory.

Poisson ratio: In materials science, the negative ratio of transverse to axial strain in uniaxial loading. It quantifies how materials tend to expand or contract in directions perpendicular to the applied force.

Other concepts bearing Poisson's name include the Poisson bracket in Hamiltonian mechanics, the Poisson kernel in harmonic analysis, and the Poisson summation formula in Fourier analysis, all reflecting Poisson's broad influence on mathematics.

Poisson's work contributed to formalizing randomness, potential theory, and the mathematical underpinnings of physical phenomena in the 19th century.