 bookstore is by Professor Kim of KAIST on first passage dynamics of primary parties. Thank you. Thank you. Good afternoon. First of all, I'd like to thank professor Jeong Yeo Jo for all the effort, preparing this nice program. Today I'll talk about some recent work on the first passage dynamics. 이선셜 아이디어로 이동할 수 있을 것입니다. 너무 많은 디테일이 있습니다. 첫번째 문제는? 지난 5년에 코스프인 덱스에서 피해의 불쌍한 불쌍, 높은 흥미를 보였습니다. 최근에 흥미를 보였습니다. 그래서 한 몇 년 정도의 스톡을 보셔야 한다면, 오리지널의 비용을 알 수 있을 것입니다. 이 상황에서, 아낌올을 흘러내리기 위해 식물을 찾을 수 있을 것입니다. 이 시기에는, 간단한 모델을 설명할 수 있습니다. 이의 흡수의 흡수는 흡수의 흡수의 흡수의 흡수의 흡수의 흡수의 흡수의 흡수의 흡수는 어떤 가능성이 될지 모르고, 그 후에 그의 흡수는 좀 더 흡수한 것입니다. 물론, 비코스코픽에 우리의 이유로 인해 기술과 정보, 예를 들은 바퀴리오키모 택시, 특정 DNA 시퀴스타일, DNA 바퀴리오키모 proteins. 모든 이 문제는 다른 지역과 다른 길을 보입니다. 하지만, 사실은 스태트스크류 피지컬 프로듀스에 첫 번째 문제입니다. 이 문제입니다. 이 문제입니다. 우리는 previous slide, this is a universal problem, we experience this kind of problem every day, right? Then sometimes the searcher can be a person or can be an animal, microscopic organism, molecules, even mathematical index, such as a cospy index, right? In this business, one of the central question or key question is how long does it take for a searcher to encounter a target, which is usually referred to as a mean first passage time, right? And this is the central quantity of general interest and which we want to understand here. Okay, let me begin with the elementary stuff. First passage dynamics is the classical subject in stochastic theory, originally formulated by Smolochowski and Cramer's almost 100 years ago. This is a typical setup. We have a metastable potential like this, and a particle escapes from this metastable region to reach here, and here the potential suddenly changes into negative infinity, right? So this position can be considered as the location of a target, right? And we want to estimate or calculate the mean first passage time or mean exit time, the time required to from this metastable region, right? Okay, it can be calculated using focal plane application in general. For example, for diffusing particle, the transition probability to find a particle position at r' at time t, given initial condition at r, satisfies this focal plane application, right? And this is the survival probability that the particle is not observed until time t, and usually slowly decaying function of time, and given by the positional integral of the transition probability. It is a first passage probability that the particle absorption or barrier crossing occurs for the first time at time t, right? Then the mean first passage time, the original quantity of our interest can be calculated by average time with a rate of this first passage probability, right? Then simply integrating by part, you can see that this mean first passage time is given by the time integral of the survival probability, right? And also given by the spatial temporal integral of the transition probability, right? And then the transition probability is expanded using the eigenfunction and eigenvalue lambda of the focal plane co-operator, right? Then this is the mean first passage time according to the previous definition. Then applying the backward focal plane co-operator on both sides, and you can easily see that this mean first passage time satisfies the backward or joint focal plane co-occasion. This is the second differential occasion. So given boundary condition, it can be calculated, right? As a simple example, consider a one-dimensional simple diffusion without any external potential, right? And we put reflecting boundary here and observing boundary on the other side and integrating this equation twice with this boundary condition. Then you can calculate the mean first passage time as a function of the initial position of particle, right? In this simple case, everything just as expected. The mean first passage time is given by the time scale to diffuse over the length scale of the system with the diffusion constant D, right? Very simple. Okay, this is the picture I'd like to show when explaining a random in the classroom. This is a load in our campus called endless load, Y. It leads to a marketplace, nearby marketplace with many bars and puffs, right? So after drinking a lot at night, you have to take this load to get back to your dormitory, right? But sometimes you drink too much with too much of C2H50H, you become like this, right? And you feel the load is winding and never ends, right? So next morning, you don't remember anything sometime, but interestingly, you always find yourself in your room, right? How is this possible? You can actually prove it mathematically. I can show it. Okay, this is the starting point called linear relation. User starting point of first passage problem in many cases. So this is the previous occupation probability in discretized version, right? So this is the probability to arrive site X after n step. Given started at X0. F is the previous first passage probability to arrive site X for the first time in j-step, right? It's decomposed like this, which means that you arrive at site in the previous j-step, right? And for the remaining n-j-step, you just simply make a returning work, right? So this occupation probability can be expressed in the form of a convoluted summation. Therefore, it is useful to introduce a generating function or discrete Laplace transform, right? Then this p can be expressed in the form of a convoluted integral. So simply product of respective generating functions, right? Okay, then now focus on the long-time asymptotic behavior. Then this discrete summation is replaced with continuous time integral and also sharp cut-off approximation can be used. Which means that we neglect this integral after this upper bound of the time scale. So the long-time behavior, long-time asymptotic behavior of the function can be studied by considering the limit where this z approaches 1, right? Then this t star goes to the infinity, right? And then now suppose that we have a simple isotropic diffusion, right? Starting at the origin, then we know that the PDF is given by the simple Gaussian distribution and then the long-time behavior of this function can be considered from the generating function with z approaching 1, right? So given like this. And due to this linear relation, then generating function of f is related to the occupation priority, right? Also this define the survival probability. The probability that particle is not observed until time t, right? And following this scheme, you can actually calculate the long-time asymptotic behavior of s and you can find that this s has interesting dependence on the spatial dimensionality, right? In the low-dimensional system, this s becomes always zero but in high-dimensional system, it remains finite, right? And this long-time behavior, long-time behavior of survival probability depends only on the spatial dimensionality, not on the details of random law, right? Therefore, the probability to find your home with random steps is always one in two-dimensional system. So even if you are completely drunk, you don't have to worry, you always find your way back to home. But this is not the case if you are wandering in three-dimensional system, okay? Okay, this dimensionality dependence can be understood by a simple scaling argument. So consider a d-dimensional random walk, then the radius explored by this random walk is given by the time to the 1 over walk dimension. And the total number of sites within this volume explored by the random walk is determined by the fractal dimension. And the total number of visited sites by this volume within this random walk will be proportional to the time. Therefore, the fraction of visited sites in this volume will be given by the id time with the power of 1 minus d over 2. Therefore, in dimensions higher than 3, this fraction of visited sites becomes vanishingly small, right? And walks become very transparent, okay? Right. So this classical subject of target search or first path dynamics has regained a growing interest recently. This problem consists of four elements, first searcher, target, searching domain, and underlying dynamics, right? And these four elements have been extended in various directions. For example, you can consider a target search problem in complex environment such as scaling the network or crowded intracellular medium, right? And also, you can think of different dynamics other than simple diffusive motion such as leviflight or fractional Brownian motion. You can also bring up with many interesting ideas such as the motility of searcher's target, okay? Right. So these problem has been studied widely and extended in various direction. However, most of them considered a single particle, single searcher, right? So it is natural to extend to consider the presence of multiple searchers competing for the same target, right? Suppose that we have n searchers instead of 1. Then in this case, one interesting question will be is your search time still given in this way? I mean in terms of the single particle, search time renormalized by the number of searchers. If so, what will this exponent alpha? Is it 1 or any other value, right? Okay. When you have searchers more than 1, then the problem becomes already quite complicated even in the case of non-interacting particles. Why? Suppose that you have n non-interacting particles, right? With different initial positions, then they will record respective first passage time, right? And from the point of view of first passage dynamics, only minimum value among n first passage time should be counted, which means that we need an order statistics, right? So this can be done only through a full distribution, first passage time distribution, right? So this is the problem we considered. We considered n non-interacting or independent searchers in a finite domain of volume v with dimension d, right? And I'm going to skip the detail rather focus on more recent works, but you can actually solve this problem analytically. This is possible because for non-interacting systems, the trajectories of particles are statistically independent, right? So therefore the probability distribution function of n particle can be decomposed into the product of a single particle distribution function, right? So that's the reason. And also assuming a uniform distribution of searcher, then you can show that when properly rescale, this search time or mean first passage time shows a universal behavior. When the number of searcher is small, then the gain in the search time is proportional to the number of searcher. But when n is large, then the gain is more enhanced. So it is proportional to one of n squared, right? And using extensive numerical simulation, we confirm our theoretical prediction and the simulation here are performed for different systems with different sizes, different number of particles in different dimensionality, right? And as you can see here, all the data points nicely collapse on a single curve as predicted by our theory. Okay, in the previous example, we considered uniform initial distribution of searcher, right? But you can think of different strategies employing different initial distribution, of course, right? And as I said, important question in this business is to optimize the system parameter to reduce the mean first passage time, right? Then in that case, one important question is what is the optimal distribution of searcher, right? That minimizes the mean first passage time, right? Okay, so this is the question we would like to answer here. Now, the target position is given in the form of a probability distribution function. What does it mean? Suppose that we have a missing child who was lastly seen an hour ago here, right? Then you may guess the current position of the child as a Gaussian distribution centered around here, right? Like this. Then how should we deploy the searching squad? You can think of different strategies, right? row the distribution or narrow wall or the same as the target distribution, right? What will be the best one? So in that case, the formulation almost remains the same as before except that the target position is given by the probability distribution function. And then you want to minimize this mean first passage time as varying this probability distribution function of searcher position, okay? Okay, let me consider a simple case to get some idea. Here we have a target position as a truncated Gaussian in a finite domain, right? And the target is most probable at the center and we have a single searcher. Then what will be your choice for the starting position of search? Natural choice will be the center, right? And this is actually calculated mean first passage time as a function of initial position of searcher, right? And as expected, it is minimized when you start your search at the center. Then what do you expect if you have two particles, two searchers? Still is the most efficient to put two searchers together at the center where the target is most probable or not? The answer is no. Actually, the efficient way is to put two searchers separated from each other with a certain distance. Why? This can be understood by collective searching efficiency. If initial positions are identical or too close, then searching domain tend to be largely overlapped, right? Lowering the searching efficiency, okay? So anyway, the exact answer of optimal searcher distribution is a highly collate problem and it's quite difficult to obtain general answer. So in this sense, we propose an alternative approach which makes sense when, for the case of large number of searchers, when you have large number of searchers, like a thousand or 10,000 particles, then it's not so practical to deploy each of them precisely in a highly correlated manner even if we know the exact answer of the optimal distribution, right? Instead, we consider independent deployment, placing particles according to a prescribed independent distribution, which is u here, right? Then our question becomes things like that. So what will be the optimal distribution of u that minimizes the mean first passage time, right? Then, okay, so this is the question we would like to answer and it's equivalent to taking the functional derivative of the previous quantity with respect to u. And after some calculation, we can find that this optimal distribution is given by one-third of the target distribution, little bit broader of the original target distribution, right? So using the Langebaum dynamic simulations, we checked our theoretical validity of our theoretical prediction. Here, for example, we considered Gaussian target distribution, which means that we generate target position according to a Gaussian distribution. Then also generate many searchers according to this distribution with different value of gamma, right? And this is inverse mean first passage time renormalized by the searching time when gamma is equal to 3, right? So as you can see here, these solid lines are our theoretical prediction and the symbols are numerical result clearly you see that the searching time is minimized when the gamma is equal to 3 as consistent with our theoretical prediction, okay? Okay, let me skip this. So far, we have considered the target search by multiple searchers, but they are still not interacting, right? However, in many cases, rather usual cases, some form of interaction certainly exist between searchers, right? For example, think of predator such as wolves or whales. They communicate with each other to locate or hunt down a prey, right? More importantly, the first passage dynamics was originally proposed to describe chemical reaction of molecules, and we know that molecule experience various kind of interaction, right? Renaissance potential or DLVO potential. Anyway, so this is the question we would like to answer or understand here. How does the mean first passage time depend on the interaction, right? And actually in many of the system, rigorous consideration of interaction is always non-trivial and has been one of the fundamental questions in theoretical study, right? For example, let me remind you of how we describe the effect of interaction on the equilibrium pressure of real interacting gas system in graduate statistical mechanics course, right? So we know that for real gas, it is given in the form of a variable expansion, right? So in order to derive that simplification, what we have done like this, we consider canonical partition function for n-particle system where the Hamiltonian is given like this. p is a momentum, u is an interaction potential. Then the momentum integral and positional integral can be done separately. This is the configurational integral, which is further expressed using moment expansion and u0 here indicates the average with respect to non-interacting system, right? Then to calculate the thermodynamic potential or thermodynamic function, log of partition function should be considered which is expressed in terms of cumulant expansion, right? All right, but this cumulant expansion or moment expansion, still, perturbative approach, which means that they are valid only for weakly interacting system, not so practical for strongly interacting system, such as a system with hardcore potential, right? So in order to consider general form of interaction, you have to do some kind of a partial resummation, right? Summation of all two point clusters, for example, in low density limit. Then you can see that all reducible clusters subtracted out in the variance and only irreducible clusters remains in the final expression in the variable form, right? And this kind of expansion is a little bit complicated to do in a canonical picture, so usually we consider this cluster expansion in a grand canonical system, right? That's what we learned from the textbook, right? And at the end, then you can show that the equilibrium pressure of interacting gas system is given in the expansion in powers of density, right? Okay. Then, so this is a kind of a brief review how to deal with the interacting system. Now, let me get back to our original problem. So, we have an interacting Brownian particle searching for a single target in a finite domain. So this is a dynamical problem, right? So in order to estimate or calculate the mean first passage time, then you have to solve the focal length occasion for this n-interacting particle, right? Which is, of course, a nontrivial task, right? So our problem corresponds to the generalization of the previous example of equilibrium pressure to the n-body dynamical problem, right? So I'm going to I'm going to skip the details of mathematical derivation. Actually, we have tried, actually we solved this problem using three different methods, right? And then we obtained every time identical result. So anyway, we found that kind of a partial resumption similar to the previous example is also possible for the solution of this focal length occasion, right? And as a result, the searching time is given in this way. So the correction to the mean first passage time for the real gas system is given in the form of a virial expansion, right? And as expressed with the virial coefficient this is a universal result valid for any kind of potential. And we have demonstrated this universality by performing a Langebaum Dynamics simulation considering various potentials such as exponential, hardcore, Sutherland potential, Renaissance potential of different number of particle interaction strengths and system size, right? And as you can see here all the data points numerical results nicely collapsed on a straight line as predicted by our theory, okay? Right, so this is summary. We have considered target search by multiple searcher and when they are non-interacting and initially uniformly distributed then we have universal scaling behavior with respect to number of particle and when you have a degree of freedom to adjust the initial distribution then the optimal distribution is given by one third of the target distribution finally when they are interacting this dynamical problem still can be solved giving the answer in the form of a virial expansion. Okay, most important part this work was done by my former PhD students Dr. Seung-Hwan Lu currently at MIT as a postdoc and in collaboration with Professor Ju Hyun-hee sitting over there in Busan National University so I'd like to specially thank these collaborators and of course I'd like to thank you all for listening. This is the last result for the search time in the Interactive Particle Number 3 Is this for 3 dimensions, right? Actually, the dimension doesn't matter Actually, the dimension doesn't matter It doesn't matter It's a general result for arbitrary dimensions because it doesn't matter Okay, thank you What if the searchers are obstructing each other or what if the searchers are obstructing each other Suppose this is a treasure hunt and physical interaction or However, you want to phrase the interaction so that one searcher wants to get to the target before the others That is already included because we have to do some kind of order statistics The first one is meaningful So physical interaction these kind of order statistics among the N first passage times are already taken into account in this study Thanks In the summary slide, you presented three results Results No. 1 and 3 Can they be connected? For instance You showed that time decreases with a number of searches without interaction and you presented the results for the case of interaction So for instance, in the weak interaction limit can you reproduce No. 1 from No. 3? But you have to be careful because it is the video expansion so that makes sense only in the dialogue limit So in order to check this kind of crossover or universal scaling behavior with respect to N then you have to take into account the next order term or actually infinite order term in terms of the density So maybe another approach may be useful or will be better I don't know the exact answer how to extend to consider this large N system So every assumption applied to the equilibrium pressure calculation in the textbook should be also applied here except that this is a dynamical problem So can you apply this kind of idea to real situation like small child trying to find many people people are intending each other so can you calculate B2 kind of thing and you can give a consultation to the police So that's an interesting point but you have to be careful because it is based on still smaller chopstick occasion or focal plant occasion so underlying dynamics is very random of but in our scale search we make a kind of directed motion and we have also long distance communication method so in real life search maybe you can apply that to the some kind of searching algorithm for drawn something like that but not for human I guess I have a question for your number 2 Can you intuitively understand why the linear distribution should be broader than target distribution Actually that's partly can be understood by this collective searching efficiency so it is a diffusive motion it is highly there is a high non-negligible probability that the searching domain can be overlapped so on the one hand it is better to minimize this searching domain and on the other side you want to put your resources at the target position where the target is mostly expected so this is kind of a balance between the 2 different opposite behaviors so the exact number of 3 I don't know the precise argument but the qualitative argument is just like that I also have a short question So this are expected time for search condition no success in higher dimensions there is always a probability of failure Is that correct? in more than 2 dimensions 3 and above there is always a finite probability of failure that you don't find the target If you are working in the infinite space without any boundary in finite system Thank you for the clarification So just a technical question So this video coefficient So this result was obtained for without any boundary condition just free space in 3 and 2 and 1 dimension Actually we have in order to get some numerical simulation you have to put boundary otherwise there is a certain probability up to infinite time Right So then the video coefficient itself is actually boundary condition dependent it is known that if you assign some boundary then it's like you are applying some sort of potential field so your whole particle is actually not only interacting actually interacting particle actually feel the potential to each other but if you set the boundary potential field such that you have to reformulate your VR coefficient P2 this is known for the engineers as far as I can tell Yeah, but actually we are looking at very dilute limit and small number of particle in very large system so thermodynamically this boundary condition shouldn't be so relevant in cases otherwise an isotropic space then it might be true very long cylinder something like that or some space between the plate then boundary condition can be relevant but if you think of very isotropic space then I think it will be very difficult to detect any boundary effect in the video coefficient so that is usually the thermodynamic limit we consider so volume and number of particle both of them go to the infinity with a finite density Okay, let's thank Professor Kim again Thank you