Emergence, Universality and Renormalization

In the realm of “low-energy” physics, the basic building blocks of nature and their interactions were defined almost a century ago. As articulated (mischievously) by Robert Laughlin during his Nobel lecture, in the terrestrial world, physical scientists have the “Theory of Everything” [11]. Matter is made up of electrons and ions, and these particles interact through electromagnetic forces. By fixing stoichiometry and temperature, the “fundamental theory” –encapsulated in the quantum mechanical Schrodinger equation– can describe everything; air, water, rocks, etc…! After a moment of reflection, it quickly becomes evident that such a fundamental theory isn’t a practical theory of anything. To quote Philip Anderson [10], “…the reductionist hypothesis does not… imply a ‘constructionist’ one: The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe.” In his influential essay, “More is Different”, written almost 50 years ago, Anderson goes on to say, “The constructionist hypothesis breaks down when confronted with the twin difficulties of scale and complexity. The behaviour of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles. Instead, at each level of complexity entirely new properties appear…”. Although now somewhat clichéd, these insights seem particularly prescient as biology enters an era in which the “fundamental building blocks” – the genes are gene products- and their interactions are becoming resolved. The triumph of 19th and 20th century physics was to understand that complexities at the microscopic – or nano – scale translate to (often unexpected) emergent phenomena at the mesoscale that cannot be predicted, or even conceived, from the behaviour of two or three “elemental” particles [10,11]. For example, when tuned by pressure or temperature, interactions between atoms or molecules in a liquid can drive a transition into an ordered crystalline phase in which fundamentally new collective excitations –sound waves– emerge. Similarly, when electrons or atoms condense into the same quantum state, there emerge new collective phenomena in the form of a “super-flow” involving the dissipationless transport of current. Importantly, these emergent behaviours are often encapsulated through “coarse-grained”, or hydrodynamic, theories involving few composite variables, themselves complex and usually unknown functions of the fundamental or microscopic parameters. But why should it be that details at a microscopic scale can be surrendered without losing information on the dynamics at the macroscale? Crucially, when systems are poised at the transition point between phases, statistical fluctuations can become length-scale independent. In such critical states, hydrodynamic theories can be systematically derived by successive coarse-graining of the microscopic degrees of freedom, a process known as “renormalization” [12] in which, at the largest scales, the properties of different microscopic models converge or “flow” to those of the same hydrodynamic theory. In this way, the phase behaviour of entirely different physical systems, such as magnets or liquids, obtain equivalent statistical dependences defined by the same theory. In the language of statistical physics such “attractor theories” constitute universality classes. In physics, much of the focus has been on the equilibrium – or near-equilibrium – phase behaviour of (often complex) ensembles of inanimate particles or compounds. These days, the question of whether and how collective phenomena emerge in driven non-equilibrium systems has evolved as a major frontier of statistical physics [13], embracing phenomena such as jamming in particulate matter [14], swarming and flocking of active systems [15,16], epidemics [17], voting patterns [18], risk management and financial markets [19], to mention just a few. In many such cases, it has been found empirically that systems positioned far from equilibrium may be driven towards critical states by collective dynamics, even without fine-tuning of parameters. Therefore, as in critical equilibrium states, the large-scale statistical properties non-equilibrium systems are defined by a limited number of “universal” theories obtained as the renormalization “fixed points” of whole classes of distinct microscopic models (See figure).

Figure 1. Schematic showing the principle of renormalization group flow. Systems undergoing continuous phase transitions can be prepared in a state where statistical fluctuations become scale free. Systems far from thermal equilibrium, like biological systems, can approach such states through collective dynamics (grey line). Mathematically, critical systems form a subset (dotted area) of all possible systems described by a set of parameters p1, p2, …. Successive coarse graining (renormalization) drives a broad range of critical systems into the same “hydrodynamic” attractor theory encapsulating the basic symmetries of the microscopic system.

Under these conditions, probability distributions of critical states often converge to self-similar “scaling” forms at long times, such that their behaviour is entirely defined by a single, time-dependent scale. Can such concepts of emergence and universality provide insight into the behaviour of living systems, where the constant flux of energy from the environment leaves them far from thermal equilibrium?

The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation

What's new

The Riemann zeta function $latex {zeta(s)}&fg=000000$ is defined in the region $latex {hbox{Re}(s)>1}&fg=000000$ by the absolutely convergent series

$latex displaystyle zeta(s) = sum_{n=1}^infty frac{1}{n^s} = 1 + frac{1}{2^s} + frac{1}{3^s} + ldots. (1)&fg=000000$

Thus, for instance, it is known that $latex {zeta(2)=pi^2/6}&fg=000000$, and thus

$latex displaystyle sum_{n=1}^infty frac{1}{n^2} = 1 + frac{1}{4} + frac{1}{9} + ldots = frac{pi^2}{6}. (2)&fg=000000$

For $latex {hbox{Re}(s) leq 1}&fg=000000$, the series on the right-hand side of (1) is no longer absolutely convergent, or even conditionally convergent. Nevertheless, the $latex {zeta}&fg=000000$ function can be extended to this region (with a pole at $latex {s=1}&fg=000000$) by analytic continuation. For instance, it can be shown that after analytic continuation, one has $latex {zeta(0) = -1/2}&fg=000000$, $latex {zeta(-1) = -1/12}&fg=000000$, and $latex {zeta(-2)=0}&fg=000000$, and more generally

$latex displaystyle zeta(-s) = – frac{B_{s+1}}{s+1} (3)&fg=000000$

for $latex {s=1,2,ldots}&fg=000000$, where $latex {B_n}&fg=000000$ are the Bernoulli numbers. If one formally applies

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Becoming a Scientist

Giddings, Morgan C. “On the Process of Becoming a Great Scientist.” PLOS Computational Biology 4, no. 2 (2008): e33.

Erren, Thomas C., et al. “Ten Simple Rules for Doing Your Best Research, According to Hamming.” PLOS Computational Biology 3, no. 10 (2007): e213.

Writing a Fellowship Proposal

Guidelines & Advice on Applying to Graduate Fellowships (PDF)
(Courtesy of Diana Chien and John Casey, Biological Engineering Communication Lab. Used with permission.)

Choosing a Research Problem

Alon, Uri. “How to Choose a Good Scientific Problem.”Molecular Cell 35, no. 6 (2009): 726–8

Writing a Paper

Whitesides, George M. “Whitesides’ Group: Writing a Paper.” Advanced Materials 16, no. 15 (2004): 1375–7.

Doerr, Allison. “How to Write a Cover Letter.” Nature Methods (2013).

Reading Effectively

This resource may not render correctly in a screen reader.How to Navigate a Scientific Paper with Time Constraints: A Graphic Approach (PDF)


Drubin, David G. “Any Jackass can Trash a Manuscript, but it Takes Good Scholarship to Create One (How MBoC Promotes Civil and Constructive Peer Review).” Molecular Biology of the Cell 22, no. 5 (2011): 525–7.

Making Figures

Wong, Bang. “Points of View: Gestalt Principles (Part 1).” Nature Methods 7, no. 11 (2010): 863.

———. “Points of View: Gestalt Principles (Part 2).”Nature Methods 7, no. 12 (2010): 941.

Giving a Talk

McConnell, Susan. “Designing Effective Scientific Presentations.” iBiology.org.

———. “The Importance of Giving a Good Talk.” iBiology.org.

Scientific Ethics

Glass, Bentley. “The Ethical Basis of Science.” Science150, no. 3701 (1965): 1254–61.

Buy at Amazon Engineering, and Public Policy Committee on Science, Institute of Medicine, et al. On Being a Scientist: A Guide to Responsible Conduct in Research. National Academies Press, 2009. ISBN: 9780309119702. [Preview with Google Books]

Curriculum Vitae

Resumes, CVs, Cover Letters, and LinkedIn

Life after the Ph.D.

Austin, Jim, and Bruce Alberts. “Planning Career Paths for Ph.D.s.” Science 337, no. 6099 (2012): 1149.

English Communication for Scientists

  1. http://www.nature.com/scitable/ebooks/english-communication-for-scientists-14053993/communicating-as-a-scientist-14238273
  2. http://www.americanscientist.org/issues/pub/the-science-of-scientific-writing
  3. http://www.cmu.edu/student-org/pcr/media-files/pcr-sessions/swan_handout.pdf
  4. https://www.brown.edu/academics/science-center/sites/brown.edu.academics.science-center/files/uploads/Quick_Guide_to_Science_Communication_0.pdf

Reaction Diffusion Systems/Phenomena

Taeuber U.C.-Critical Dynamics_ A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior-CUP (2014)

Generic reaction-diffusion models are in fact utilized to describe a multitude of phenomena in various disciplines, ranging from population dynamics in ecology, competition of bacterial colonies in microbiology, dynamics of magnetic monopoles in the early Universe
in cosmology, equity trading on the stock market in economy, opinion exchange
in sociology, etc. More concrete physical applications systems encompass excitons kinetics in organic semconductors, domain wall interactions in magnets, and
interface dynamics in growth models. Yet most of our current knowledge in this
area stems from extensive computer simulations, and actual experimental realizations allowing accurate quantitative analysis are still deplorably rare

Brownian Dynamics Simulation

LAMMPS and GROMACS are popular simulation packages for molecular dynamics (MD) simulations. Both can be used to simulate Brownian Dynamics by using Langevin dynamics. See “fix langevin” in LAMMPS or “bd integrator” in GROMACS.

LAMMPS: lammps.sandia.gov

GROMACS: http://www.gromacs.org

GROMACS Tutorial: http://www.gromacs.org/WIKI-import/Main_Page/Tutorials

SMOG: Structure-based Models for Biomolecules: http://smog-server.org/


BDpack is a Brownian dynamics package developed recently. It is an open source code and is written in parallel with high computational efficiency. It can be found at: http://amir-saadat.github.io/BDpack/