Chaotic Behavior of the Baker’s Map Under Quantization

Abstract

Chaotic systems, such as turbulent flow, are characterized by sensitivity towards initial conditions. Turbulent flow is caused by fluid pressure gradients, which cause “eddies” or vortices, and these gradients and vortices diminish in scale over time. If fluid flow simulations were run on a computer, eventually, the gradients would occur on such a small scale that they would no longer be recognizable by the computer, due to limited precision. While this may be an artifact of current technological limits, studying the effects of quantization on a relatively large scale (2^-52) may shed light on the effects of the real-world quantization of spacetime in a relatively simple and accessible manner. In this project, we study the effects of information loss, bit-shifting, and truncation on the discrete Baker’s map with 25-bit precision. We also observe the distinction between a “practical” attractor, which encodes the finite limitations of the system, as opposed to a “theoretical” attractor, which may fail on minute computational scales. Using similar methods, we may be able to find practical attractors for different systems under different quantization procedures. This insight provides a nontraditional, simplistic approach to finding attractors that hold true under the grainy nature of spacetime