# Mathematics AA HL IA – Probability distribution of the distance between random points in a square

이 IA 는 math AA HL IA 이며 주제는 second law of thermodynamics 를 수학적으로 증명하는 내용입니다. 방에 공기 분자들이 왜 골고루 퍼지는지, heat energy는 왜 골고루 퍼지는지, 등 일상생활에 존재하는 물리법칙을 확률을 통해 증명하였습니다. IB 에서 요구하는 창의적인 접근방법, 그리고 새로운 주제를 탐구하여 최종점수로 19/20 를 받았습니다. 만점에 가까운 점수를 받았으므로 고득점을 위한 요소들을 알기 위해 큰 도움이 될것으로 예상됩니다.

[본문내용]

### Introduction

When I was learning thermodynamics, I was fascinated by the fact that every natural tendency is determined by probability. In nature, it is observable that energy or particles would be distributed evenly around the space, also referred to as “maximizing disorder” of the system (“The Second Law of Thermodynamics”). For instance, air molecules will evenly spread across the room; a hot cup of coffee will cool down in a cold room. The reason for such natural tendencies is solely due to the higher probability of the system being in a disordered state of even distribution. I personally wanted to investigate the reason why nature tends towards “maximum disorder” by mathematically showing that moderate separation is more probable than uneven distribution. In this investigation, the distance between particles in the 2 dimensional plane will be obtained using Python, then plotted on a probability distribution graph. The probability distribution graphs of 2, 3, and 4 particles will be compared to answer the research question: “why do air molecules spread evenly around a room?”

### Background information

A continuous probability distribution is defined as a probability distribution where the random variable X could be any number within a range of values, having infinitely many values that X could be. For example, distance is a continuous variable that could have any value within the given range.
For graphical display of graphs and shapes, an online graphing tool “Desmos” was used, along with Google Documents “Drawing” tool. And for Python coding, “Google Colaboratory” was used.
Maths AA topics related to this investigation are continuous random variables, and integration. The concept used in these topics will be discussed further in the Methodology section…

• 총 페이지수: 27 pages
• 과목명: Mathematics
• 주제: Probability distribution of the distance between random points in a square
• The file is in PDF format.