Random walk and emergent properties

Reference: [Set21] Chap. 2

Summary

• Random walk model is the simplest non-trivial example that we can track how the idea of statsitics is useful. Especially it demonstrate the idea of scale invariance and the idea of universality class.

• The macroscopic effective theory of the random walk model is governed by the diffusion equation. The connection between the microsopci physics and the macroscpoic physics is explicitly in this example.

• The central limit theorem shows the distribution function for the sum of random variables will converge to a gaussian function. The result is generally applicable to random variables with well defined mean and variance. It is another example of the concept of universality class described in mathematical langrage.

• The fourier method and the Green’s function method can solve the diffusion equation exactly. We found the solution of the diffusion equation is closely related to the central limit theorem.

Before we enter the discussion of statistical mechanics theory, let’s start with the simplest non-trivial example to learn what statistics could tell us. We will start from the one-dimensional random walk problem and the two-dimensional random drunkard’s walk. From the two examples, we introduce the idea of scale invariance and universality class. The simple calculation is also a warm-up for the basic calculations of statistical properties. Then, we will discuss the idea of irrelevant microscopic details and the relevant details using our simple examples. Once we understand the microscopic models in detail, we will try to develop an effective theory to describe the macroscopic property of the system. We will solve the macroscopic theory and bridge the macroscopic picture and the microscopic understandings.