Understanding The Drunkard's Walk: A Mathematical Model for Sobriety Checks and Heat Diffusion

Police departments worldwide employ a simple yet effective method to identify intoxication: the ability to walk a straight line. A sober individual, with a stride length of 'L', would cover a distance of 'D=NL' after 'N' steps. However, this linear progression dramatically shifts under the influence of alcohol. Despite maintaining a stride length of 'L', the direction of each step becomes unpredictable, leading to a random walk often referred to as the "drunkard's walk."

Decoding the Drunkard's Walk: Reading Signs of Intoxication

In the drunkard's walk, the person under examination meanders randomly, creating a zigzag pattern rather than a straight line. Interestingly, the total distance covered by the individual after 'N' steps becomes 'D=√NL'. That implies that while a sober person takes 'N' steps to cover a distance 'D' in a straight line, an intoxicated individual would require 'N^2' steps for the same distance. This phenomenon illustrates that if a 100-meter distance needs to be covered with a 1-meter stride, a sober person would require 100 steps, whereas an inebriated individual would need as high as 10,000 to cover the same distance.

Molecular Migration: The Drunkard's Walk in Action

This model of a series of random steps mirrors the behavior of molecules migrating from a high-temperature area to a cooler one. The molecules, moving at a faster pace due to their increased temperature, collide and push off other molecules randomly, mimicking the unpredictable steps of the drunkard's walk. Consequently, it takes considerable time for a newly heated room's effects to reach us, as the 'hot' molecules stagger into the room, akin to how a drunkard walks.

Heat Diffusion: The Random Walk Principle

In the context of heat diffusion, the 'random walk' principle provides an excellent model. When roasting a turkey, for instance, heat migrates from the external surface to the interior in a series of steps of length 'L', with the direction of each step being random. As a result, the time it takes for heat to travel between two points separated by 'NL' is proportionate to 'N^2'. 

Let's consider a spherical turkey for a moment (despite how amusing it sounds). Assuming the turkey has a uniform density, its volume and weight 'P' would be proportional to the cube of its radius 'R^3'. The time 'T' required for the heat to permeate from the periphery to the center would be proportional to 'R^2'. Therefore, the cooking time 'T' is proportional to 'P^(2/3)'. As a general rule, the cube of the cooking time should increase proportionately to the square of the turkey's weight. This intriguing principle offers a practical guide for gauging cooking times.