A Numerical Model of Mixing
For much of physics, models of response to stimuli are most easily thought of in terms of a linear response, the output (acceleration) of a system being linearly proportional to the input (force) to the system. However, real physical systems only demonstrate linear response in relatively narrow windows of phase space, and a more complete understanding of physical systems can only be understood by examining the instabilities that lead to nonlinear or even chaotic response, usually as the input parameter is increased. Convection is an example of such instability, where the linear conduction of a fluid is no longer able to transport the heat that is input to a system. A local instability in the density of the fluid leads to mass transport, or convection. As the amount of heat is increased even further, the pattern itself can be subject to another instability, the regular convection rolls changing to a complex, chaotic pattern. Chaotic behavior is a dynamic response that can take essentially an infinite amount of time to repeat itself. As such, chaotic behavior is one way to study non-equilibrium thermodynamic systems, where the constraints of equilibrium thermostatistics may or may not be observed.