Hello, I am applying to physics program with a focus on complex systems. This is my statement of purpose. Thanks for your any advice!

Dear Admissions Committee,

My interest in complex systems

I am applying to Northwestern University’ doctoral program in physics to pursue a career in academic research. I was attracted to physics world by the concise and universal theories. However, with further study, I found myself more enchanted by complex behavior arising from underlying, simple rules, particularly when I enrolled in an introductory course on nonlinear physics, learning about so many intriguing phenomena, such as period-doubling bifurcation and recursive, infinite self-similarity. It was my curiosity about complex systems that encouraged me to conduct academic research on this field.

The first research experience

Inspired by the fascinating collective motion of bird flocks and synchronized patterns, I started a project involving looking for collective behavior and self-organizing phenomena with Prof. Zhu at Nankai University. In this project, I designed a Hamiltonian system comprised of coupled particles which have short-range interactions with each other and each in a two-dimensional sinusoidal potential field. The motion of the whole system is described by a set of coupled ordinary differential equations and simulated in Python. Different Hamiltonian of this system obtained by adjusting potential field strength and initial velocity could give rise to distinctive motion patterns, which is a very engrossing part to analyze through motion visualization. To better comprehend the change of motion patterns, I define an order parameter to indicate the degree of order of the system. When interaction strength and potential field strength are of the same order, and initial velocity increases to certain critical point, the parameter would sharply jump up, which represents that phase transition occurs where ordered collective motion becomes disordered. That the macroscopic collective behavior and dynamical evolution result from certain microscopic interactions could provide insight into the way of engineering one system with desired behavior by controlling microscopic interactions.

The second research experience

My revealed interest in complex systems led me towards the research on real-world complex networks. In 2018, I got sponsored by Nankai University to carry out a summer research program in Germany. Under the guidance of Prof. Timme, I worked on a project which was aiming at developing a nodal vulnerability index for oscillatory networks, for instance, power grids, in the presence of stochastic perturbation as constrained by a given power spectral density. This index can help identify the most unstable nodes in the networks. Based on linear response theory for network dynamics, Fourier analysis for perturbation signals, we analytically derived the formula for nodal response under perturbation. However, the major difficulty was to gain a time-independent index in order to identify which nodes respond most violently. My major personal contribution was to find the vulnerability index, a topology-specific measure quantifying each grid’s response intensity and derive that the nodal response follows a Gaussian distribution by employing the central limit theory in probability theory. Hence we not only obtained the maximum nodal response from the index but from the distribution, we could also know how likely the nodes exceed the stable limit. From this research experience, I discovered that I especially enjoy carefully thinking about the highly mathematical parts of the problem, such as dealing with the time-dependent parts and deriving the probability distribution. Besides, I found it so stimulating and worthwhile to research complex networks. Deep understanding of such networks can have practical importance in real-world complex network design and control, such as maintaining the normal operation of power grids with fluctuating power input from renewable energy.

Some courses and my understanding for future development of this field.

In addition to giving me a general understanding of complex systems, my past work has also provided me with the skills needed for graduate school. To build my technical toolbox, I have taken classes in complex systems, such as introduction to nonlinear physics, Computational thinking, Fundamental of computational physics and software science, in all of which I scored above 90. From these courses and my research work, I have learned useful complex systems theories, including stability analysis, network analysis and basic graph theory. Furthermore, the experience increased my confidence in computer simulation and theoretical analysis. Apart from my previous research on collective behavior and network dynamics, however, there are still so many mysteries of complex systems attracting me: how complex networks form, evolve or even fall apart and die; how the behavior at one level gives rise to that of the next level, for example, how the phenomena we call “intelligence” and “consciousness” emerge from nonintelligent, nonconscious material. These secrets deserve further research since they may shed light on the origin of consciousness and life.

Why I am interested in NWU

To continue my pursuit of academic research on complex systems, I believe Northwestern University is the best place for me because of the favorable atmosphere of interdisciplinary research and the promising projects on complex systems here. In particular, I am interested in Prof. Mot’s research, for example, the project on “network control”. It is so exciting to learn from this project that instead of avoiding undesirable effects due to noise, we can even exploit noise to drive the networks into desirable states. While high-dimensional structure, nonlinear interaction and the physical constraints may pose challenges to generalizing this approach, I am confident that my extensive coursework and field preparation will equip me with the sufficient basic skills to start this project and the training of Northwestern’s program will provide me with novel insights into future research. I am ready to devote myself to exploring mysterious territories of complexity in Northwestern with both perseverance and alacrity. I am looking forward to seeing you on campus next summer.