Thank you very much! That's exactly what I wanted to hear.Ibix said:As kuruman says, it's helpful to provide links to things you want to ask questions about. We're helping out just for fun, and you're more likely to get help if we only have to click on one link rather than hunt through search engines for references. It isn't always possible to provide links, of course, but the Feynman Lectures are all online on CalTech's website - this is chapter 6.
He's proposing a game: toss a coin and if you get heads take a step to the left, if you get tails take a step to the right. If you keep repeating this game, how far will you be from your starting point? The answer is the difference between the number of heads you've got so far and the number of tails you've got so far. This is ##D##. He does algebra to get different expressions for ##D## in terms of the total number of tosses and the number of heads. This is useful because ##N## is not a random number and nor is 2, so the randomness of ##D## is dictated by the randomness of ##N_H##, and he already did the maths for that.
6.11 is just a rearrangement of ##D=2N_H-N##, stated in the paragraph above. 6.12 expects you to use 6.11 to see that the left hand side is the same as half the rms value of ##D## and then 6.10 to get the right hand side.
If that doesn't make sense, say which bits you don't follow and we'll see what we can do.
Especially when one is using a smartphone to search.Ibix said:We're helping out just for fun, and you're more likely to get help if we only have to click on one link rather than hunt through search engines for references.
Say what?AlexisBlackwell said:Formulas (6.11) and (6.12) in Feynman's lectures on physics explain how the energy of a system changes if it is divided into two parts. This is called a change in the internal energy of the system. Formula (6.11) says that the change in internal energy depends on the change in the volume of the system and pressure. And formula (6.12) shows that the change in internal energy also depends on the heat received or lost by the system. Thus, these formulas help to understand how the energy of a system changes under different conditions.
A random walk is a mathematical formalism that describes a path consisting of a succession of random steps. In the context of physics, it often models phenomena such as diffusion, stock market fluctuations, and particle movements in a fluid. The random walk concept helps in understanding how systems evolve over time when influenced by random processes.
The random walk is a fundamental model for diffusion, as it illustrates how particles spread out over time due to random motion. In a random walk, each step taken by a particle is independent and can be in any direction. Over time, this leads to a predictable pattern of diffusion, where the mean squared displacement of the particles increases linearly with time, in accordance with Fick's laws of diffusion.
Random walk theory has a wide range of applications across various fields. In physics, it is used to model diffusion processes, such as the spread of heat or the movement of molecules in a gas. In finance, it helps in understanding stock price movements, where prices are often assumed to follow a random walk. Additionally, it is applied in ecology to study animal foraging patterns and in computer science for algorithms related to search processes.
The Feynman Lectures on Physics introduce random walks in a clear and intuitive manner, emphasizing their significance in statistical mechanics and thermodynamics. Feynman illustrates the concept through thought experiments and simple mathematical models, making it accessible to students. He discusses the implications of random walks in understanding larger physical systems and their emergent properties.
The central limit theorem (CLT) plays a crucial role in random walks, as it states that the sum of a large number of independent random variables will tend to follow a normal distribution, regardless of the original distribution of the variables. In the context of random walks, this means that as the number of steps increases, the distribution of the position of the walker approaches a Gaussian distribution. This result is fundamental in understanding the statistical behavior of random walks and their convergence to normality over time.