“Would you believe it, Ariadne?—said Theseus.—The Minotaur barely defended himself.”
Jorge Luis Borges, “The house of Asterion”
It takes hard work to connect detailed molecular motions with macroscopic properties of materials. Perhaps for this reason, thermodynamics is still often taught following the empirical route first taken by engineers some two hundred years ago, not the constructive, but often perilous, path taken by the likes of Boltzmann. Already evident in studying the gas phase, the complexity of the task becomes even greater for understanding condensed phases, where intimate features of the intermolecular interactions come into prominence. Ever practical, the condensed matter curriculum in many departments relies on two disjoint pillars: theory of liquids and solid state physics. The former directly relates observed properties, such as pairwise correlations or heat capacity, to detailed inter-particle potentials. The latter assumes at the onset that solids are stable and allows one to express the response of the material—both mechanical and optoelectronic—as that of modestly interacting quasi-particles, viz., phonons and electrons. Recent advances in computation have allowed us to obtain the phonon and electronic spectra from scratch using quantum mechanics, but often at the expense of clarity. To quote P. W. Anderson's Nobel speech, “It can be a disadvantage rather than an advantage to be able to compute or to measure too accurately, since often what one measures or computes is irrelevant in terms of mechanism. After all, the perfect computation simply reproduces Nature, does not explain her.” Despite these shortcomings, molecular simulation is a powerful tool that has helped us meet recurring technological challenges; it is here to stay. Accordingly, modern instruction of statistical mechanics must bridge the seemingly disparate styles of mathematical elegance and detailed computer modeling. How do we teach student to perform time-consuming computer simulations not for their own sake but to answer questions that have intellectual merit? In the first place, how do we approach phenomena whose computational complexity is still too great for the modern-day technology, such as the dynamics of realistic glassy liquids on technologically relevant time scales?
One expects Stillinger's text to meet today's challenges as well as one could: Having studied with the (underappreciated) genius John G. Kirkwood, Stillinger is as comfortable with paper-and-pencil calculations as he is with computer simulations, a field he himself helped establish decades ago. Stillinger's opus is a collection of graduate-level mini-courses covering many topics under the broad rubric of statistical mechanics; the notion of energy landscapes is the obligato theme, as the book's title suggests.
Quite appropriately, the word “landscape” evokes images of mountain ranges separating picturesque valleys. In Stillinger's language, the valley bottoms are called inherent structures. The inherent structures correspond to energy minima. As such, they are zero-temperature configurations, be they stable or metastable. (Important examples of inherent structures are the cis- and trans-isomers of retinal within the protein rhodopsin.) In equilibrium, the system must sample not just the minima but also a huge number of other configurations such as the vibrations around the minima and the vastly more numerous transition-state configurations, all according to the Gibbs-Boltzmann distribution.
The author starts by briefly reiterating basic notions of quantum mechanics and statistical mechanics before continuing onto the main topics. These cover structural aspects of solid state physics, the theory of liquids, the basics of supercooled liquids and glasses, select nanoscopic systems such as molecular clusters, the physics of helium, an extensive set of lectures on water and ice, and, finally, the statistical mechanics of polymer solutions and protein folding. The book is clearly not intended as a primary text on statistical mechanics. Rather, the covered topics are meant to complement existing texts, such as McQuarrie's “Statistical Mechanics,” and can be used as standalone mini-courses at the graduate level. The narrative is pedagogical and suitable for the theoretically and experimentally inclined student alike.
Although understandably brief, the review of the basics of quantum and statistical mechanics is not generic. For instance, the reader will benefit from the focused discussion of inverse-power potentials and the Gaussian core model. A Gaussian core crystal exhibits the interesting phenomenon of a pressure-driven FCC to BCC transition, something that was shown elegantly by the author in the 70s. Within the solid-state section, the discussion of the order-disorder transition of crystalline defects is noteworthy; it lends itself well to a discussion within the inherent-structure framework and, at the same time, forms a great platform to discuss mean field theory, a traditionally under-taught topic. The discussion of ionic liquids and liquid interfaces, as part of the theory of liquids, is timely with regard to modern-day applications such as energy storage. The reader will also find a brief discussion of transport phenomena, which naturally leads to a chapter on supercooled liquids and glasses. The chapter on molecular clusters touches on the charming topic of fractals; it also gives one a flavor of how one may attempt to bookkeep the myriad inherent structures, a topic covered much more extensively in Wales's comprehensive monograph “Energy Landscapes.” The liquid Helium chapter is a welcome foray into real quantum mechanics that helps one appreciate Feynman's early work on the subject.
The lectures on water and ice are a high point of the text; they are heartily recommended to everyone. The author firmly guides the reader through this important topic, which has seen a resurgence of controversy concerning phase transitions within the liquid state; many more controversies are sure to follow in the future. The authoritative yet pedagogical tone of the chapter is perfectly flanked by reference to important old work, such as that of Pauling on the residual entropy of ice. Even very simplified ways to count inherent structures go a long way here.
Although warranted on a few occasions, the obligato recurrence of the inherent-structure theme often ends up being a distraction. For instance, the attractive portions of intermolecular interactions must be sampled during the vapor-to-liquid transition of a pure substance. Yet the actual minima—either of the individual pairwise potentials or of the total potential energy—are barely sampled, contrary to what the book seems to imply. This is brought home by the notion that even the much lower temperature, liquid-to-crystal transition is largely entropically driven, not really an energetic phenomenon at its heart. This was understood already in the 1930s by Bernal, and by Mott and Gurney: Except for substances with very directional bonding, such as water, the fusion entropy largely stems from the confinement of particles to specific sites in the periodic solid. This entropy is well approximated by the entropy of an ideal gas of particles confined to a free volume , minus the entropy of particles each confined to a fixed cell of volume : , roughly consistent with the experimental value of or so per particle. Likewise, the discussion of glassy liquids in the framework of inherent structures produces no quantitative predictions and is, frankly, dated. Much more powerful quantitative theories now exist, including the classical density functional theory and the replica symmetry-breaking framework. The author brings up energy landscapes in the context of hard sphere liquids, even though the energy of hard spheres is either identically zero or infinity, a singular behavior the author himself acknowledges later in the text. The landscape of a hard sphere liquid is much like a classic 2D labyrinth; it is purely entropic. The landscape of a realistic liquid must contain both energetic and entropic contributions; it is a free energy landscape. By overlooking the entropic contribution, one obtains a microscopic picture that represents not a true landscape but, rather, a still life, to borrow terminology from art. Already the ancient Greeks emphasized the presence of thermal motions. Recent calculations put those early, prescient notions on a firm formal basis by showing that the type of cooperativity observed in zero-temperature systems, such as colloidal suspensions or sand dunes, pertains to the portion of the phase space separated from the glassy liquid states by a set of thermodynamic and mechanical instabilities.
To summarize, Stillinger's volume is a well-written and referenced source that will enrich the library of a graduate instructor and student alike, especially in traditional topics of statistical mechanics. The author brings to the effort extensive expertise in both analytical and computer calculations. The book gives a hint of the maze-like complexity of the landscapes of glassy liquids and proteins, but to navigate that maze, the reader will need an Ariadne's thread offered by more modern sources that discuss the progress of the last twenty five years.
Vassiliy Lubchenko is Associate Professor of Chemistry and Physics at the University of Houston. His research interests include the structural glass transition, protein aggregation, and solid-state inorganic chemistry.
