Statistical Physics of Design by Andrei A. Klishin A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in The University of Michigan 2020 Doctoral Committee: Assistant Professor Greg van Anders, Queen’s University, Co- Chair Professor Mark E. Newman, Co-Chair Professor Sharon C. Glotzer Professor Roberto D. Merlin Associate Professor David J. Singer Andrei A. Klishin [email protected] ORCID iD: 0000-0002-5740-8520 © Andrei A. Klishin 2020 ACKNOWLEDGEMENTS I went to graduate school hoping to become a complex systems scientist. That more or less happened. However, the path towards that result has been incredibly winding and very different from what I could project going in. Most importantly, I met an incredible cast of people who in many ways helped me to make it through. The work in this dissertation is chiefly a collaboration with my advisor, Greg van Anders. With Greg’s guidance, I understood how to efficiently expand the boundaries of what we call physics, or science in general, and how to sell that understanding to broader community. Greg extensively drilled me with writing and re-writing my texts, drawing and re-drawing my figures in the interest of distilling the message to its clearest form. I am grateful for his comprehensive program of training the next generation of scientists. I am also deeply grateful to him for morally and financially supporting my outreach activities during graduate school, even though they did not directly contribute to this research. I am also fortunate to have deep connections with the Michigan faculty who formed my dissertation committee. Mark Newman taught me everything I know about networks, and provided formal support in a crucial moment by becoming a co-chair of my dissertation. Dave Singer taught me everything I know about Naval Engineering, and on many occasions defended my work more valiantly than I could have done myself. Sharon Glotzer drove a lot of my thinking about self- assembly, served as a great role model in how to weave together scientific narratives, ii and taught me the truth about entropy. Roberto Merlin provided an important perspective on graduate school and the academic world that was largely agnostic of my research subject and thus not tainted by it. I would also like to thank Michael Brenner for giving me an opportunity to spend two years at Harvard School of Engineering and Applied Sciences, a rich academic environment complementary to that of Michigan. Michael taught me a lot about self-assembly (again), and Chapter VII of this dissertation largely builds on the work developed in his group. I am grateful to the Brenner group for providing me with an intellectual home during this time and exposing me to many diverse scientific ideas. I would like to thank my collaborators on projects presented here. Colin Shields and Conner Goodrum filled in a lot of Naval Engineering and design context in my work. Alec Kirkley assisted with computations in Chapter V. Adam Jermyn introduced me to the tensor network methods and provided the backend software package PyTNR for tensor network computations. Norman Mackay helped with the TenZ package that wraps physics around the PyTNR core. I would like to thank James Sethna and Nigel Goldenfeld for useful discussions on the history and philosophy of certain statistical mechanics concepts. I would like to thank the Office of Naval Research and specifically Kelly Cooper for funding this research. During my studies I enjoyed an ability to travel to a variety of professional events, and I don’t take that opportunity for granted. I would like to thank people who made my time at Michigan much more pleasant. I was attracted to Michigan by the Center for Studies of Complex Systems, led by Charlie Doering and his staff, who do a superb job at attracting resident faculty and guest speakers, as well as providing systematic complex systems training to iii students of different levels. On the Physics Department side, I was supported multiple times by Bradford Orr during his time as department chair. Socially, I greatly enjoyed the company of Harry Liu, Steve Novakov, Yuri Popov, and Markus Wallerberger over the years. I would like to thank the teachers who initially put me on the path towards becoming a physicist during my years in high school and early years of university in Belarus. While the things they taught me are perhaps too elementary to be cited in this dissertation, I still firmly believe that I would not be the scientist I am today if all of my education happened in the United States. Belarus might be in political turmoil now, but I still believe that it has a huge reservoir of untapped human talent that will make itself known. Жыве Беларусь! I would like to thank my parents who let me go out into the world and make my own decisions, but supported all of my decisions nonetheless. Lastly, I would like to thank Chrisy Xiyu Du, my partner in all things, who has been by my side for almost the whole time in graduate school. iv TABLE OF CONTENTS ACKNOWLEDGEMENTS ::::::::::::::::::::::::::::::::::: ii LIST OF FIGURES ::::::::::::::::::::::::::::::::::::::: viii ABSTRACT :::::::::::::::::::::::::::::::::::::::::::: xvii CHAPTER I. Introduction ....................................... 1 1.1 What is design? . 1 1.2 Central problems . 4 1.3 Overview of the dissertation . 6 II. Design is not an Optimization Problem ...................... 9 2.1 Wicked problems in design . 9 2.2 Design approaches . 15 2.3 Design as a collective phenomenon . 19 2.4 Design for self-assembly . 24 2.5 Why is design physics? . 28 2.5.1 Model–Compute–Learn loop . 28 2.5.2 Model: the spaces of design . 30 2.5.3 Compute: forward, inverse, and in between . 32 2.5.4 Learn: physics knowledge structures . 35 III. Methods .......................................... 38 3.1 Statistical Mechanics . 38 3.1.1 Maximal entropy perspective . 38 3.1.2 Statistical mechanics and statistical physics . 41 3.1.3 Extensive and intensive parameters . 44 3.1.4 Bath perspective . 46 3.1.5 Computing observables . 49 3.1.6 Coarse graining . 52 3.1.7 Formal definitions of design space . 55 3.1.8 Statistical mechanics computations . 57 3.2 Tensor Networks . 61 3.2.1 Basic notions . 61 3.2.2 Applications . 63 3.2.3 Software solutions . 69 3.2.4 TenZ package . 71 3.3 Statistical mechanics for self-assembly . 73 3.3.1 Scope of theory needed . 73 v 3.3.2 Why don’t we have a theory yet? . 75 3.3.3 Conceptual questions . 78 3.3.4 Implementation questions . 80 IV. Phase Transitions in Design ............................. 83 4.1 Introduction . 83 4.2 Systems Physics Framework . 86 4.3 Arrangement Problem Model . 87 4.4 Results . 92 4.4.1 Case 1, Homogeneous Embeddings: Cost/Design Freedom Trade-off 93 4.4.2 Case 2, Inhomogeneous Embeddings: Cost/Design Freedom/Per- formance Trade-offs . 97 4.5 Conclusion . 103 4.6 Supplementary Methods . 105 4.6.1 Evaluation of Outcomes and Variability . 105 4.6.2 Case 1: Computation Details . 106 4.6.3 Case 2: Computation Details . 107 4.7 Supplementary Results . 108 V. Robust Design ...................................... 112 5.1 Introduction . 112 5.2 Robust Design from Statistical Physics . 116 5.2.1 General Approach . 117 5.2.2 Systems Physics . 124 5.2.3 Architecture Classes . 125 5.2.4 Quantifying Robustness . 126 5.2.5 Beyond Ultimate Stress . 129 5.3 Example System: Model, Results, and Discussion . 132 5.3.1 Model System . 132 5.3.2 Results and Discussion . 135 5.4 Conclusion . 145 VI. Interplay of Logical and Physical Architecture ................. 147 6.1 Introduction . 147 6.2 Systems Physics Framework . 149 6.2.1 Motivation: Design Challenges Lurk Between Logical and Physical Architectures . 149 6.2.2 Statistical Physics Approach . 152 6.2.3 Example Model System . 155 6.3 Results . 158 6.3.1 Avoidance . 158 6.3.2 Adjacency . 164 6.3.3 Association . 169 6.4 Discussion . 174 6.5 Methods . 178 6.5.1 Tensor Network Construction . 178 6.5.2 Move 1: External Legs . 179 6.5.3 Move 2: Anchors . 180 6.5.4 Move 3: Modified Coupling . 181 6.5.5 Bond Diagrams . 181 6.5.6 Numerical Aspects of Computation . 182 vi 6.6 Supplementary Results . 184 6.6.1 Avoidance: Excess Density . 184 6.6.2 Adjacency: All Bond Diagrams . 186 VII. Topological Design in Heterogeneous Self-Assembly .............. 189 7.1 Introduction . 189 7.2 Statistical mechanics of self-assembly . 192 7.2.1 Self-assembly design space . 193 7.2.2 Partition functions . 195 7.2.3 Experimental control and yields . 199 7.2.4 Polymer entropy and loop penalty . 203 7.3 Chain sequence combinatorics . 205 7.3.1 Hierarchical structures . 206 7.3.2 The transfer matrix . 207 7.3.3 Partition functions . 207 7.3.4 Spectral analysis . 209 7.3.5 Divergence analysis . 212 7.3.6 Computing yields . ..