IIT Bombay Online is the worldwide online access to free official and certification courses from India's leading research institute, IIT Bombay.
For more info visit : www.cdeep.iitb.ac.in/ Facebook : facebook.com/IITBOnlineCDEEP/ Instagram : instagram.com/iitbombayonline/ Twitter : twitter.com/IITBombayOnline
great, i live in jakarta, indonesia, i have been following your teaching , quite clear, i graduate master in biochemistry, when i was 50, it was 15 years ago
# Lecture on Random Walks and Diffusion ## Chapter 1: Introduction to Random Walks ****0:00** - **2:17**** - Governing equation for the probability of a random walk to be at location R. - Introduction of the Discrete Fourier Transform \( G_n(k) \). - Evaluation of \( G_n(k) \) leading to binomial distribution for the probability \( P_n(r) \). ## Chapter 2: Discrete Fourier Transform and Its Implications ****2:17** - **6:29**** - Detailed derivation of \( G_n(k) \) and the inverse Fourier Transform. - Explanation of the integration limits from \(-\pi\) to \(\pi\) due to the discrete nature of space. - Comparison with the continuous case and the concept of Kronecker Delta. ## Chapter 3: Generating Functions ****6:29** - **10:12**** - Introduction of the generating function \( \Gamma(r, z) \) as a discrete Laplace transform. - Relationship between generating function and probability \( P_n(r) \). - Calculation and significance of higher-order derivatives of the generating function. ## Chapter 4: Recurrence in Random Walks ****10:12** - **14:29**** - Concept of recurrence and the probability of a random walker returning to the origin. - Calculation of the mean number of returns to the origin using the generating function. ## Chapter 5: Fourier Transform of the Generating Function ****14:29** - **18:00**** - Calculation of the Fourier transform of the generating function. - Specific case for symmetric random walks (\( p = q = 0.5 \)). - Evaluation of the integral for the generating function. ## Chapter 6: Recurrence and Transience in Different Dimensions ****18:00** - **23:06**** - Definition of recurrence and transience. - Explanation that 1D and 2D random walks are recurrent, while 3D random walks are transient. ## Chapter 7: Continuous Limit and Diffusion Equation ****23:06** - **32:04**** - Transition from discrete random walks to continuous random walks. - Derivation of the diffusion equation from the discrete random walk equations. - Discussion of symmetric and asymmetric random walks and their impact on the diffusion equation. ## Chapter 8: Solving the Diffusion Equation ****32:04** - **39:00**** - Solving the diffusion equation using Fourier transforms. - Calculation of the probability distribution and its normalization. - Implications for the mean displacement and second moment. ## Chapter 9: Scaling and Universality in Diffusion ****39:00** - **45:04**** - Discussion on the scaling behavior of the second moment (\( \langle x^2 angle \)) with time. - Introduction to anomalous diffusion, including superdiffusion and subdiffusion. ## Chapter 10: Random Walk Polymers ****45:04** - **53:39**** - Concept of random walks in space as models for polymers. - Calculation of the end-to-end distance for 1D random walk polymers. - Transition to higher dimensions and the resulting probability distributions. ## Chapter 11: Ideal Polymers and Gaussian Distributions ****53:39** - End** - Discussion on ideal polymers and their statistical properties. - Calculation of the radius of gyration as a better measure of polymer size. - Connection between the theoretical models and physical properties of polymers.
# Lecture Summary: Random Walks in Biology ## Chapter 1: Introduction to Random Walks in Biology (0:00-1:20) - Overview of E. coli diffusion as a model system. - Recommendation to read "Random Walks in Biology" by Howard Berg. - Importance of understanding bacterial motility and stochastic variables. ## Chapter 2: Model System: E. coli (1:20-8:04) - Description of E. coli as a model organism. - Comparison between prokaryotic and eukaryotic cells. - Explanation of E. coli's structure: size, pili, and flagella. - Role of flagella in bacterial locomotion. - The crowded environment inside cells affecting molecular movement. ## Chapter 3: Flagella and Bacterial Locomotion (8:04-16:07) - Detailed structure of flagella and its motor. - Motion types: counterclockwise (run) and clockwise (tumble). - Description of run-and-tumble behavior. - Measurement and analysis of tumbling angles and intervals. - Introduction of rotational diffusion during the run phase. ## Chapter 4: Chemotaxis and Bacterial Motion (16:07-29:02) - Chemotaxis: bacterial movement in response to chemical gradients. - Experiment with Helicobacter pylori demonstrating chemotaxis. - Explanation of chemoattractants and chemorepellents. - Role of chemo receptors in sensing chemical gradients. - Experiment demonstrating different bacterial responses based on genetic mutations. ## Chapter 5: Signal Detection and Multiple Signals (29:02-42:32) - Discussion on detecting multiple signals using different receptors. - Experiments showing responses to urea (chemoattractant) and HCL (chemorepellent). - Conclusion that bacteria have multiple receptors for different chemicals. - Impact of knocking out specific receptors on bacterial motion. ## Chapter 6: Physical and Chemical Modulation of Swimming Strategies (42:32-45:25) - How physical confinement and chemical gradients affect swimming strategies. - Changes in running distribution from exponential to power law under confinement. - Impact on tumbling angles and overall bacterial movement. ## Chapter 7: Mathematical Modeling of Random Walks (45:25-1:11:01) - Introduction to the mathematical framework of random walks. - One-dimensional random walk with discrete time and space. - Derivation of recurrence relation for probability distribution. - Calculation of characteristic function and its role in determining moments. - Solution for probability distribution using binomial distribution. - Connection to central limit theorem and Gaussian distribution for large steps. ## Chapter 8: Recurrence and Transients in Random Walks (1:11:01-end) - Introduction to the concept of recurrence in random walks. - Discussion on the probability of returning to the origin. - Plan to take the continuous space-time limit and connection to diffusion equations. --- This summary captures the main points and themes of the lecture, dividing them into clear chapters with timestamp ranges for easy reference.
Hello Sir. Your lectures are really good. Can you make the assignments, lecture plan and notes open source, so that the whole RF Community can get benefit of this.
Thank you so much for the lecture, I searched everywhere for something that could help me compare both models for a practical class and this absolutely made my day!
I have studied at two IITs, and Dr V Balki is being humble here, there are "NO" experts in any IIT, especially in Physics department, I know many professors who would understand nothing out of this too .. they are just sitting there awestruck by the fluency of Dr Balki and wish if they were like him or knew things from such basics as him. I have seen people from Cambridge and Oxford saying that they hardly have anyone as good as him when it comes to teaching Physics, forget IITs, Basic sciences still have no standard in IITs, just fancy engineering colleges
Imagine you're on a bungee trampoline, connected from above to a ceiling and below to the ground. If you're in the middle, you can bounce freely within the full range. But if we adjust the trampoline's height closer to the ceiling, you can't bounce up as high because you hit the ceiling. Similarly, if it's closer to the ground, you can't bounce down as low because you hit the ground. This is like an electrical signal in a circuit. If you bias the circuit closer to the power supply, the signal can't swing up to its full potential. If it's biased closer to the ground, it can't swing down fully. Also, if you increase the signal's strength, it's like pulling harder on the bungee cord, causing the person to hit both the ground and the ceiling more forcefully which will led to ot going to the full range up and down and this is significantly the clipping that we are seeing in the signal when we increase the input signal swing range 😀
