Elastic interactions and complex patterns in binary systems

One of the many beauties of condensed matter physics is that it can reveal and illuminate how two systems or phenomena that at first appear to be quite different actually involve similar physics. This is an example of universality: for emergent phenomena, many details don't really matter. One example is the similarities between superconductivity and superfluidity. A consequence of universality is that the same concepts, techniques, toy models, and effective theories can be used to describe a wide range of systems.

The complex organometallic molecules, known by the misnomer "spin crossover" compounds, exhibit a rich range of phase transitions and types of spatial order. Key aspects of the physics are the following.

• Each transition metal ion can be in one of two possible states: low-spin or high-spin. 
• The size of each molecular complex depends on the spin state.
• Consequently, the molecules interact with their neighbours via elastic interactions.

A toy model that can describe this is expanding balls connected by springs. Various versions of this type of model are reviewed here. The simplest version is the chain model below.

It turns out there are other classes of systems described by similar models. As far as I am aware, this was first pointed out in Consequences of Lattice Mismatch for Phase Equilibrium in Heterostructured Solids Layne B. Frechette, Christoph Dellago, Phillip L. Geissler

That paper is motivated by experiments on the growth of semiconductor quantum dots, by ion exchange, such as when CdSe is bathed in an Ag-rich solution and Ag2Se is produced with heterostructures (i.e., patterns of Ag and Se ions) that are different from the bulk crystal.

They consider the balls and springs model above on a triangular lattice.

They also point out how similar physics is relevant to binary metal alloys, e.g, AgCu, citing 

Ising model for phase separation in alloys with anisotropic elastic interaction—I. Theory, P. Fratzl and O. Penrose

Those authors consider a square lattice with elastic interactions associated with bond stretching along the edges and diagonals of the squares and bending of the square angles.

Frechette et al. also mention experiments on thin films of  DNA modified metallic nanoparticles. Compared to atomic systems these can tolerate larger lattice-mismatch before the formation of defects due to lattice strain.

Other systems (not mentioned) described by similar Ising models are metal-hydrogen systems, where the Ising pseudospin signifies whether a hydrogen atom is present at a particular site in the metallic crystal.

Frechette et al. start with the ball and springs model and "integrate out" the springs to obtain an effective Hamiltonian, which is an Ising model.

The spatial range of the interaction between Ising spins is shown in the colour-shaded plot below.

The interaction has two components.

One is an infinite range "ferromagnetic" part, seen as the light blue below.

The second is a short-range interaction which is mostly "antiferromagnetic" (i.e., red), but extends over several lattice sites. (Note, this interaction will be frustrated on the triangular lattice).

Using this toy model, Frechette et al. can obtain complex patterns (heterostructures) similar to those seen in quantum dots grown by ion exchange.

There is some subtle (and confusing) physics associated with deriving the Ising model from the ball and springs model. 

Due to the long-range nature of elastic interactions, the boundary conditions matter. 

The infinite range part of the Ising interaction arises from dealing with the lattice constant for the crystal, depending on the net "magnetisation" of the "spins". But that is a story for another day.

Topological defects determine the strength and growth rate of crystals

 The quantum theory of solids developed in the 1920s provided a theoretical estimate of the ideal strength of crystals. The problem was that this estimate was a thousand times greater than the measured strength of metals. This paradox was resolved in 1934, when Egon Orowan, Michael Polanyi and G. I. Taylor, independently proposed that plastic deformation could be explained in terms of the theory of dislocations. Aside: this is an example of how macroscopic properties can be determined by structures at the mesoscale rather than microscopic properties.

By 1940 the accepted theory of crystal growth was that it occurred by nucleation of successive close-packed layers of the crystal and this provided algebraic expressions for growth rates that were consistent with experiment. However, around 1950 Keith Burton estimated the parameters in the theory and pointed out that it predicted a growth rate that was smaller than observed growth rates by a factor 10^1000, i.e., 1000 orders of magnitude!

This quantitative discrepancy was resolved by Burton, Nicolas Cabrera and Charles Frank in 1951 who showed the central role played by screw dislocations. A crystal does not grow by the independent nucleation of separate layers. Rather it grows from just one layer that heloicoidally overlapping itself. A signature of this growth mode is the presence of spiral steps on crystal surfaces and they were subsequently observed.

This history is beautifully recounted in the introduction to a review article on Snow Crystals published by Charles Frank in 1982. It was reprinted in 2009 with an introduction by Andrew Fisher.

Following the introduction Frank discusses how snow is an important example of crystal growth that is not attributable to the presence of screw dislocations.

In 2015, D.P. Woodruff wrote a commentary on the classic 1951 paper by Burton, Cabrera, and Frank.

Did Turing really "explain" pattern formation?

 Exactly seventy years ago, Alan Turing published a seminal article, in which he proposed a simple reaction-diffusion model for pattern formation in biological systems. The basic idea is that there are two molecules (morphogens) that react with one another chemically and also diffuse through the system.

The potential relevance of the model can be seen by comparing the lower panels below. The left panel is a real fish and the right panel shows the results of a simulation. The figure above is taken from a beautiful review article published a decade ago.

Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation  Shigeru Kondo and Takashi Miura

The authors state that the model is not accepted by many experimental biologists and hope their review will lead to a greater engagement with it. Some of the reasons are related to issues in the philosophy of science and how to model complex systems. What is an explanation? What is the role of simple models for complex systems that ignore so many details?

Kondo and Miura point out the universality of the reaction-diffusion model in the sense that a similar model can be derived where the "molecules" are instead a circuit of cellular signals. Diffusion can be replaced by a relay of signals between cells. Alfred Gierer and  Hans Meinhardt in 1972 showed that all that is required is a network with "a short-range positive feedback [competing] with a long-range negative feedback."

A short video from the Sante Fe Institute also provides a helpful introduction including some simulations.

 

There is another problem with Turing's model that is succinctly described in the opening paragraph of a recent PRL. In a system with two molecular species, patterns only form when there is a large difference between the diffusivity of the two molecules. However, this seems unrealistic because one expects the molecules to have comparable diffusivities.

Turing’s Diffusive Threshold in Random Reaction-Diffusion Systems 

Pierre A. Haas and Raymond E. Goldstein 

 In 1952, Turing described the pattern-forming instability that now bears his name [1]: diffusion can destabilize a fixed point of a system of reactions that is stable in well-mixed conditions. Nigh on threescore and ten years on, the contribution of Turing’s mechanism to chemical and biological morphogenesis remains debated, not least because of the diffusive threshold inherent in the mechanism: chemical species in reaction systems are expected to have roughly equal diffusivities, yet Turing instabilities cannot arise at equal diffusivities [2,3]. It remains an open problem to determine the diffusivity difference required for generic systems to undergo this instability, yet this diffusive threshold has been recognized at least since reduced models of the Belousov–Zhabotinsky reaction [4,5] only produced Turing patterns at unphysically large diffusivity differences.

I first became aware of this paper through a commentary by Changbong Hyeon, at the Journal Club for Condensed Matter. It is also helpful because it explains the simple mathematics behind the threshold value of the model parameters for pattern formation. 

Self-organisation in complex fluids

 I am at the beach this week and so a lot of time is spent staring at waves, clouds, sunsets, and patterns in the sand. There is a lot of beauty and a lot of beautiful science, most of which I know only a little about. For example, what is the essential physics and simplest theory that can explain the patterns below?

To start understanding the beautiful patterns seen in natural systems I have found helpful the two-page Quick Study in Physics Today

The universe in a cup of coffee by John Wettlaufer

Your morning java or tea is a rotating, cooling laboratory that reflects the physics of such large-scale phenomena as stellar dynamics and energy transport in Earth’s atmosphere and oceans. 

A nice demonstration is to put the hot liquid in a glass jar and then just add a few drops of cold milk and see the beautiful patterns that emerge.

The key idea is there is a balance between thermal bouyancy (hot air rises) and viscous stresses. This balance can lead to symmetry breaking and self-organisation. In planetary systems rotation can play a significant role, particularly when there is a balance of viscous forces and the coriolis force. This can lead to the formation of vortices. The Quick Study includes snapshops from a video that is worth watching,  supplementary material from this PRL.

The article also discusses the importance of Rayleigh-Benard convection in many geophysical phenomena. Something interesting I learnt is that this is actually a misnomer, as is often the case in science. According to Wikipedia, 

This pattern of convection, whose effects are due solely to a temperature gradient, was first successfully analyzed in 1916 by Lord Rayleigh (1842–1919).^[16] Rayleigh assumed boundary conditions in which the vertical velocity component and temperature disturbance vanish at the top and bottom boundaries (perfect thermal conduction). Those assumptions resulted in the analysis losing any connection with Henri Bénard's experiment. This resulted in discrepancies between theoretical and experimental results until 1958, when John Pearson (1930– ) reworked the problem based on surface tension.^[9] This is what was originally observed by Bénard.

Sociological insights from statistical physics

Condensed matter physics and sociology are both about emergence. Phenomena in sociology that are intellectually fascinating and important for public policy often involve qualitative change, tipping points, and collective effects. One example is how social networks influence individual choices, such as whether or not to get vaccinated. In my previous post, I briefly introduced some Ising-type models that allow the investigation of fundamental questions in sociology. The main idea is to include heterogeneities and interactions in models of decision. 

What follows is drawn from Sections 2 and 3 of the following paper from the Journal of Statistical Physics. 

Crises and Collective Socio-Economic Phenomena: Simple Models and Challenges by Jean-Philippe Bouchaud

Bouchaud first considers a homogeneous population which reaches an equilibrium state. This is then described by an Ising model with an interaction (between agents) J, in an external field, F that describes the incentive for the agents to make one of the choices. The state of the model (in the mean-field approximation) is then found by solving the Curie-Weiss equation. In the sociological context, this was first derived by Weidlich and in the economic context re-derived by Brock and Durlauf.  (Aside: The latter paper is in one of the "top-five" economic journals, was published five years after submission, and has been cited more than 2000 times.)

As first noted by Weidlich, a spontaneous “polarization” of the population occurs in the low noise regime β>β c , i.e. [the average equilibrium value of S_z] ϕ ∗≠1/2 even in the absence of any individually preferred choice (i.e. F=0). When F≠0, one of the two equilibria is exponentially more probable than the other, and in principle the population should be locked into the most likely one: ϕ ∗>1/2 whenever F>0 and ϕ ∗<1/2 whenever F<0.

Unfortunately, the equilibrium analysis is not sufficient to draw such an optimistic conclusion. A more detailed analysis of the dynamics is needed, which reveals that the time needed to reach equilibrium is exponentially large in the number of agents, and as noted by Keynes, "in the long run, we are all dead." This situation is well-known to physicists, but is perhaps not so well appreciated in other circles—for example, it is not discussed by Brock and Durlauf.

Bouchaud then discusses the meta-stability associated with the two possible polarisations, as occurs in a first-order phase transition. From a non-equilibrium dynamical analysis, based on a Langevin equation, 

one finds that the time τ needed for the system, starting around ϕ=0, to reach ϕ ∗≈1 is given by: 𝜏 ∝ exp[𝐴𝑁(1−𝐹/𝐽)], where A is a numerical factor. This means that whenever 0<F<J, the system should really be in the socially good minimum ϕ ∗≈1, but the time to reach it is exponentially large in the population size.  The important point about this formula is the presence of the factor N(1−F/J) in the exponential.

In other words, it has no chance of ever getting there on its own for large populations. Only when F reaches J, i.e. when the adoption cost C becomes zero will the population be convinced to shift to the socially optimal equilibrium...

This is very different from the standard model of innovation diffusion, based on a simple differential equation proposed by Bass in 1969 [cited more than 10,000 times].

In physics, the existence of mutually inaccessible minima with different potentials is a pathology of mean-field models that disappears when the interaction is short-ranged. In this case, the transition proceeds through “nucleation”, i.e. droplets of the good minimum appear in space and then grow by flipping spins at the boundaries. 

This suggests an interesting policy solution when social pressure resists the adoption of a beneficial practice or product: subsidize the cost locally, or make the change compulsory there, so that adoption takes place in localized spots from which it will invade the whole population. The very same social pressure that was preventing the change will make it happen as soon as it is initiated somewhere.

This analysis provides concepts to understand wicked problems. Societies get "trapped" in situations that are not for the common good and outside interventions, such as providing incentives for individuals to make better choices, have little impact.

In the next post, I hope to discuss the role of heterogeneity (i.e. the role of a random field in the Ising model). A seminal paper published in the American Journal of Sociology in 1978 is Threshold models of collective behavior  by Mark Granovetter. It has been cited more than 6000 times. The central idea is how changes in heterogeneity can induce a transition between two different collective states.

Aside: The famous Keynes quote was in his 1923 publication, The Tract on Monetary Reform. The fuller quote is “But this long run is a misleading guide to current affairs. In the long run we are all dead. Economists set themselves too easy, too useless a task, if in tempestuous seasons they can only tell us, that when the storm is long past, the ocean is flat again.”

Phase diagram of snowflakes

I like "collecting" interesting phase diagrams, partly because they are fun to show students when teaching introductory thermodynamics. I recently discovered the one below that I feel I really should have known about. It shows the morphology of different snow crystals as a function of temperature and water supersaturation (relative to ice).
It should be pointed out that this is a non-equilibrium phase diagram as it involves supercooled liquid water.

The figure below is taken from the beautiful review
The physics of snow crystals 
Kenneth G Libbrecht

This diagram was originally constructed by Ukichiro Nakaya in the 1930's. The physics behind it is still poorly understood.

I came across the diagram while browsing through the Forces of Nature book by Brian Cox and Andrew Cohen.

While on the subject here is a nice video.

Are power laws good for anything?

It is rather amazing that many complex systems, ranging from proteins to stock markets to cities, exhibit power laws, sometimes over many decades.
A critical review is here, which contains the figure below.

Complexity theory makes much of these power laws.

But, sometimes I wonder what the power laws really tell us, and particularly whether for social and economic issues they are good for anything.
Recently, I learnt of a fascinating case. Admittedly, it does not rely on the exact mathematical details (e.g. the value of the power law exponent!).

The case is described in an article by Dudley Herschbach,
Understanding the outstanding: Zipf's law and positive deviance
and in the book Aid at the Edge of Chaos, by Ben Ramalingam.

Here is the basic idea. Suppose that you have a system of many weakly interacting (random) components. Based on the central limit theorem one would expect that a particular random variable would obey a normal (Gaussian) distribution. This means that large deviations from the mean are extremely unlikely. However, now suppose that the system is "complex" and the components are strongly interacting. Then the probability distribution of the variable may obey a power law. In particular, this means that large deviations from the mean can have a probability that is orders of magnitude larger than they would be if the distribution was "normal".

Now, lets make this concrete. Suppose one goes to a poor country and looks at the weight of young children. One will find that the average weight is significantly smaller than in an affluent country, and most importantly the average less than is healthy for brain and physical development. These low weights arise from a complex range of factors related to poverty: limited money to buy food, lack of diversity of diet, ignorance about healthy diet and nutrition, famines, giving more food to working members of the family, ...
However, if the weights of children obeys a power law, rather than a normal, distribution one might be hopeful that one could find some children who have a healthy weight and investigate what factors contribute to that. This leads to the following.

Positive Deviance (PD) is based on the observation that in every community there are certain individuals or groups (the positive deviants), whose uncommon but successful behaviors or strategies enable them to find better solutions to a problem than their peers. These individuals or groups have access to exactly the same resources and face the same challenges and obstacles as their peers. 

The PD approach is a strength-based, problem-solving approach for behavior and social change. The approach enables the community to discover existing solutions to complex problems within the community. 

The PD approach thus differs from traditional "needs based" or problem-solving approaches in that it does not focus primarily on identification of needs and the external inputs necessary to meet those needs or solve problems. A unique process invites the community to identify and optimize existing, sustainable solutions from within the community, which speeds up innovation. 

The PD approach has been used to address issues as diverse as childhood malnutrition, neo-natal mortality, girl trafficking, school drop-out, female genital cutting (FGC), hospital acquired infections (HAI) and HIV/AIDS.

A central result of non-equilibrium statistical physics

Here is a helpful quote from William Bialek. It is a footnote in a nice article, Perspectives on theory at the interface of physics and biology.

The Boltzmann distribution is the maximum entropy distribution consistent with knowing the mean energy, and this sometimes leads to confusion about maximum entropy methods as being equivalent to some sort of equilibrium assumption (which would be obviously wrong). But we can build maximum entropy models that hold many different expectation values fixed, and it is only when we fix the expectation value of the Hamiltonian that we are describing thermal equilibrium. What is useful is that maximum entropy models are equivalent to the Boltzmann distribution for some hypothetical system, and often this is a source of both intuition and calculational tools.

This type of approach features in the statistical mechanics of income distributions.

Examples where Bialek has applied this includes voting patterns of the USA Supreme Court, flocking of birds, and antibody diversity.

For a gentler introduction to this profound idea [which I still struggle with] see
*James Sethna's textbook, Entropy, Order parameters, and Complexity.
* review articles on large deviation theory by Hugo Touchette, such as this and this.
I thank David Limmer for bringing the latter to my attention.

Irreversibility is an emergent property

Time has a direction. Macroscopic processes are irreversible. Mixing is a simple example. The second law of thermodynamics encodes universal property of nature.
Yet the microscopic laws of nature [Newton's equations or Schrodinger's equation] are time reversal invariant. There is no arrow of time in these equations. So, where does macroscopic irreversibility come from?

It is helpful to think of irreversibility [broken time-reversal symmetry] as an emergent property. It only exists in the thermodynamic limit. Strictly speaking for a finite number of particles there is a "recurrence time" [whereby the system can return to close to its initial state]. However, for even as few as a thousand particles this becomes much longer than any experimental time scale.
There is a nice analogy to spontaneously broken symmetry in phase transitions.  Strictly speaking for a finite number of particles there is no broken symmetry as the system can tunnel backwards and forwards between different states. However, in reality for even a small macroscopic system the time scale for this is ridiculously long.

Deriving irreversibility from microscopic equations is a major theoretical challenge. The first substantial contribution was that of Boltzmann's H-theorem. There are many subtleties associated with why it is not the final answer, but my understanding is superficial...




This post was stimulated by some questions from students when I recently visited Vidyasagar University.

Deconstructing noise in organic charge transfer salts

There are several things that I used to find very puzzling about electrical noise measurements on the metallic phase of organic charge transfer salts.

The  measured noise spectrum is close to (but not exactly) 1/f.

The disparity of time/energy scales.
What is the relationship (if any) between the noise (which is sometimes measured on time scales as long as one thousand seconds (mHz)) and microscopics (which one might calculate with quantum chemistry and/or Hubbard models, but typically involves energies larger than meV or frequencies that can be ten orders of magnitude larger)?

Obscure trends.
If one looks at the actual exponent alpha of the noise, 1/f^alpha. It varies in a non-monotonic way as the temperature T varies. This looks rather "random" to me (i.e. I found it hard to believe there was any systematics involved).

However, Jens Muller and collaborators have used a model due to Dutta, Dimon, and Horn (DDH) to nicely elucidate what is going on in a series of papers such as this one.

Origin of the glass-like dynamics in molecular metals κ-(BEDT-TTF)2X: implications from fluctuation spectroscopy and ab initio calculations 
Jens Müller, Benedikt Hartmann, Robert Rommel, Jens Brandenburg, Stephen M Winter, and John A Schlueter

Here are the basic ideas of the DDH model.
There is a distribution of relaxation times tau, which arise because there are a distribution of activation energies E for relaxation.

tau0 is a typical "attempt frequency"/molecular vibration frequency for something like a conformational change of a molecule.

One assumes that for a specific tau that the noise is simply Lorentzian. But one then averages over D(E), the distribution of activation energies.

One can then show that at a given temperature the noise has a 1/f^alpha form with an exponent given by,

A specific consistency test of the model is to then compare the measured alpha to that calculated from the above expression using the observed temperature dependence of the noise spectrum. This comparison is shown in the figure above. 

One can also invert the equation above to extract D(E), giving the result in the figure below.

These two points give a better understanding of where the temperature dependence of alpha comes from; it has a reasonable explanation in terms of the distribution of activation energies.

Furthermore, the origin of the low frequency noise is the relatively large value of the activation energies. This leads to conformational transitions being extremely rare. In particular, I find it amazing that the noise at the Hz scale is detecting the fact that in the macroscopic crystal about every one second a single molecule (yes, just one undergoes a conformational change)!

Note that the activation energy distribution D(E) is peaked around 230 meV. This is the same energy that is deduced from studies of the activation energy for the glassy behaviour seen in NMR, specific heat, and thermal expansion. Moreover it is also the energy barrier calculated from quantum chemistry for the transition between the two conformations of the ethylene end groups (staggered vs. eclipsed) that I discussed in a recent post.

The reference given above also gives an explanation using ab initio calculations as to why the presence of the glass transition depends on the chemical identity of the anion X in kappa-(BEDT-TTF)2X. It relates to the relative strength of the bonding between X and the ethylene end groups of the BEDT-TTF molecules.

One thing that is not clear is what determines the width of the distribution D(E).

There are subtleties that I have glossed over here and other interesting things but the aim of this post is to focus on the big picture and some of my basic puzzles.

I thank Jens Muller for a very helpful discussion about his work.

The mysterious origin of resistivity in Fermi liquids

It is hard to believe that we really don't understand the basic issues that I am going to discuss.

Resitivity occurs in a metal because scattering causes decay of charge currents. This means that the total momentum of the electrons in the presence of an electric field decays.
However, in a Fermi liquid metal with strong electron-electron interactions the main scattering of electrons is due to electron-electron scattering. But, in such collisions the total momentum of the two electrons is the same before and after the collision.
One can calculate the life time of the quasi-particles and it is inversely proportional to the temperature squared. The quasi-particle scattering rate ~ T^2.
Suppose one makes the relaxation time approximation in the Boltzmann equation or equivalently, neglects vertex corrections in the corresponding current-current correlation function associated with the Kubo formula for the conductivity. Then the resistivity is proportional to the quasi-particle scattering rate and one has resistivity ~ T^2. We say the transport lifetime is the same as the quasi-particle lifetime.
However, these are approximations, and strictly speaking there is no decay of the total electron momentum (or current) by electron-electron scattering and so the resistivity should be zero!
One way to save the situation is when there is Umklapp scattering. However, this requires a special relation between the shape of the Fermi surface and the Brillouin zone, as illustrated below.

These issues and puzzles are highlighted in a beautiful paper

Scalable T^2 resistivity in a small single-component Fermi surface 
Xiao Lin, Benoît Fauqué, Kamran Behnia

By chemical doping they tune the charge density and Fermi energy by several orders of magnitude, with the size of the Fermi surface increasing from some very small fraction of the Brilloiun zone.
In all cases the resistivity equals A T^2, characteristic of electron-electron scattering.
The figure below shows how the proportionality factor A scales with the density.
They also find A scales with the inverse of the effective mass squared as one expects from the Kadowaki-Woods ratio.

Yet for small densities (and Fermi surfaces) it is just not clear how one can have electron-electron scattering since Umklapp scattering is not relevant.

This major puzzle awaits an explanation.

I thank David Cavanagh, Jure Kokalj, Jernez Mravlje, and Peter Prevlosek for stimulating discussions about this topic.

Note added. The theoretical issues are nicely reviewed in
Resistivity of non-Galilean-invariant Fermi- and non-Fermi liquids 
 H. K. Pal, V. I. Yudson, D. L. Maslov

Universal distributions for wealth distribution from physical ideas

I finally read most of an interesting Colloquium article in Reviews of Modern Physics
Statistical mechanics of money, wealth, and income 
Victor M. Yakovenko and J. Barkley Rosser, Jr.

[I mentioned the review 2 years ago in a post about the science of economic inequality].

It reviews the history and concept of econophysics, pointing out how some of the founders of statistical mechanics actually had a vision for its application to economics and sociology. Most of the review is about analogues with statistical mechanics that use the notion of money as a conserved quantity that is exchanged by individuals, leading to Boltzmann type distributions for wealth and income.
I found the article a nice accessible introduction to the field.

What is impressive is that the simple exponential distribution (Boltzmann) does describe empirical data over two orders of magnitude. Furthermore, the analysis gives some insight into economic inequality. This is summarised in the following sentences from the abstract and the figure below showing data from the USA.

Data analysis of the empirical distributions of wealth and income reveals a two-class distribution. The majority of the population belongs to the lower class, characterized by the exponential (“thermal”) distribution, whereas a small fraction of the population in the upper class is characterized by the power-law (“superthermal”) distribution. The lower part is very stable, stationary in time, whereas the upper part is highly dynamical and out of equilibrium.

Another result that is interesting is the income of spouses seems to be uncorrelated leading to the distribution shown below for total household income. The solid line is the simple functional form following from two uncorrelated Boltzmann distributions.

Clouds, climate change, and emergence

On tuesday the Telluride Science Research Center hosted a nice public lecture, "Clouds in a bowl of soup," by Graham Feingold, an atmospheric scientist.

He emphasised how the atmosphere is a complex system that exhibits emergent phenomena, particularly pattern formation and synchronisation. He discussed how one sees these phenomena in other systems, such as soup (Rayleigh-Benard convection cells) and fire flies.
Aside: in a similar vein there is a nice Physics Today, Quick Study, The universe in a cup of coffee by John Wettlaufer.

Emergent behaviour results from simple rules. The four rules for clouds are

1. Drops form on aerosols (suspended particles) and grow by vapour diffusion.

2. Drop coalescence generates rain. (Aerosols can influence rain).

3. Drops fall and evaporate.

4. Continuity of air flow.

Some of the work he described is in this paper, which includes the figure below.

Some key physics relevant to climate change is that clouds generally reflect sunlight and have a cooling effect. The big question is whether global warming then increases or decreases cloud formation. Is the feedback positive or negative? It seems people aren't really sure. Intuitively, you might think that the increased water in the atmosphere means more clouds but (as is often the case) it turns out to be more complicated than that.

The challenge of non-equilibrium thermodynamics

This week I am in Telluride at the bi-annual workshop on Condensed Phase Dynamics. I really enjoyed the talks today. A common topic was that of non-equilibrium thermodynamics, particularly in nanoscale systems.

Abe Nitzan began his talk mentioning a recent PRL, Quantum Thermodynamics: A Nonequilibrium Green’s Function Approach, which unfortunately, is not valid because the expressions it gives do not give the correct result in the equilibrium limit. This is shown in

Quantum thermodynamics of the driven resonant level model 
 Anton Bruch, Mark Thomas, Silvia Viola Kusminskiy, Felix von Oppen, and Abraham Nitzan

What is striking to me about both papers is that they consider a non-interacting model, i.e. the Hamiltonian is quadratic in fermion operators and exactly soluble.
This shows just how far we are from any sort of theory of a realistic system, i.e. one with interactions and which is not integrable.

Phil Geissler gave a nice introduction to different theorems for fluctuations in the dissipation (defined as the difference between the entropy change and heat/temperature). The most general theorem is that due to Gavin Crooks and implies the Jarzynski inequality, the fluctuation theorem, and the second law of thermodynamics.
A key question is what sorts of non-equilibrium processes (protocols) minimise the dissipation and whether the distribution is Gaussian (it often is).
He then described near optimal protocols to invert the magnetisation in a two-dimensional Ising model.

Suri Vaikuntanathan talked about coupled (classical) master equation models for biomolecular networks that have mathematical similarities to an electronic Su-Schrieffer-Heeger model which is an one-dimensional example of a topological insulator.
The work is described  in a preprint with A. Murugan,  "Topologically protected modes in non-equilibrium stochastic systems".
This is potentially  important because it may provide  "a framework for how biochemical systems can use non equilibrium driving to achieve robust function."

David Limmer gave a nice talk which considered thermodynamics as a large deviation theory and how that can even have meaning out of equilibrium and there is a notion of an entropy, a "free energy" and a "temperature". His slides are here.
A key notion is to focus on ensembles of trajectories rather than a probability distribution function. There are two alternative computational strategies: transition path sampling and diffusion Monte Carlo (the cloning algorithm).
He considered several concrete examples, such as thermal conductivity in carbon nanotubes, and electrochemical processes at electrode-water interfaces.

Do you really need to use Keldysh Green's functions?

Or are you cutting butter with a chainsaw? Or using a helicopter to cross the road?

Previously I posted that Green's functions are just a technique.
Here I have a new point. When using them one has to make a choice about how one treats the time variable in the complex plane. This is rather technical, subtle, and confusing. The choices available include imaginary time (Matsubara), retarded and advanced, Kadanoff-Baym, Keldysh, ....

Matsubara Green's functions can only describe equilibrium properties. They reflect a beautiful and profound connection between imaginary time and temperature. However, due to the very powerful fluctuation-dissipation relation (embodied in the Kubo formula), they can also be used to calculate transport properties in the linear response regime. They are also the easiest to use, both analytically and computationally. Although for some numerical methods such as Quantum Monte Carlo the analytic continuation to real frequencies can be a can of worms.

Kadanoff-Baym and Keldysh are constructed to allow description of non-equilibrium states. A beautiful and profound application of them is the derivation of Boltzmann-type transport equations. I think when Kadanoff and Baym did this it was a major conceptual advance that further buttressed Fermi liquid theory.
A nice accessible introduction to Keldysh is the review by Rammer and Smith. Nevertheless, keeping track of the contours and the relation between Keldysh, retarded and advanced components of the Greens functions can quickly become overwhelming.

There is no doubt that far from equilibrium, Keldysh is necessary and good. However, my concern is that sometimes people do a lot of formalism with Keldysh and then in the end they just do some linear response calculation that they could have just done with Matsubara. I conjecture this applies to some of the examples considered by Rammer and Smith.

In my Ph.D I used Keldysh to consider the non-linear interaction of zero sound with order parameter collective modes in superfluid 3He-B. Much later I realised that I could probably have done the same calculations with Matsubara.

Probing non-equilibrium dynamics in a quantum many-body system

This week at UQ there was a fascinating Quantum Science Seminar by Jorg Schmiedmayer, describing some beautiful ultra cold atom experiments.  Much of the talk is nicely discussed in a book chapter Does an isolated quantum system relax? [Answer is yes].
Some of the most recent results are in a Science paper, Experimental observation of a generalised Gibbs ensemble that appeared this month.

The experiments involve a one-dimensional Bose gas that can be described as a Luttinger liquid. This means that many properties, even non-equilibrium ones, can be calculated analytically and compared to experiment. This is a theorists dream!

Here are a few things that stood out.

Relaxation from a non-equilibrium state to the thermal equilibrium state occurs on several different time scales. First there is rapid relaxation to a "quasi-steady state" described by a Generalised Gibbs ensemble [This idea goes back to Jaynes] that involves an effective temperature [that one can even theoretically calculate in terms of the bose gas interaction strength].

There is a characteristic length scale associated with the relaxation.

One can even directly measure higher order correlation functions (4th, 6th and 10th order!), as seen below. Furthermore, in a Luttinger liquid [a non-interacting boson field theory] these should factorise in terms of the 2nd order correlation function [reflecting Wick's theorem]. One can even test this experimentally.

One can also simulate the sine Gordon theory and vary the coupling constant and so move through the associated phase transition.

Broken symmetry, rigidity, and dissipative structures

On monday I am giving the opening talk at the Australasian Workshop on Emergent Quantum Matter. Since it is a broad audience with a range of backgrounds I am going to give a tutorial talk, building on the UQ colloquium I gave earlier this year.  Later I will post my draft slides.

One concept I want to expand on is the concept of rigidity, associated with broken symmetry. To do this I am reading a nice article "Some general thoughts about broken symmetry," written by Phil Anderson in 1981. It is reprinted in A Career in Theoretical Physics, and here is a scanned copy of the article. It contains the figure above.

What is the connection between the "rigidity" of  solids and broken symmetry? A liquid is invariant under continuous translations and rotations. When it becomes a solid it is only invariant under discrete rotations and translations. Symmetry is broken. Unlike a liquid, a solid can "sustain/resist" a shear stress. Solids are rigid.

I also want to say something about non-equilibrium. Anderson has something critical to say about "dissipative structures" such as Bernard cells associated with self-organisation and hydrodynamic instabilities.

Hard questions about glasses

A recent book Dynamical heterogeneities in glasses, colloids, and granular media contains a fascinating chapter where four experts [Jorge Kurchan, James Langer, Thomas Witten, and Peter Wolynes] give their answers to the questions below.

I think we need more of these kind of frank discussions about scientific topics. I am slowly working through the answers. The most fascinating bit so far is Peter Wolynes inspiring response to Q9, including "I believe a young physicist who wants to work on any challenging problem in physics will eventually have to learn about glasses."

Q1) In your view, what are the most important aspects of the experimental data on the glass transition that any consistent theory explain? Is dynamical heterogeneity one of these core aspects?  

Q2) Why should we expect anything universal in the behavior of glass-forming liquids? Is the glass-transition problem well defined?  

Q3) In spin-glasses, the existence of a true spin-glass phase transition has been well established by simulations and experiments. Do you believe that a similar result will ever be demonstrated for molecular glasses? 

Q4) Why are there so many different theories of glasses? What kind of decisive experiments do you suggest to perform to rule out at least some of them? 

Q5) Can you briefly explain, and justify, why you believe your pet theory fares better than others? What, deep inside, are you worried about, that could jeopardize your theoretical construction?  

Q6) In the hypothesis that Random First-Order Theory [RFOT] forms a correct skeleton of the theory of glasses, what is missing in the theoretical construction that would convince the community?  

 Q7) Exactly solvable mean-field glass models exhibit an extraordinary complexity requiring impressive mathematical tools to solve them. 

 Q8) In your view, do the recent ideas and experimental developments concerning jamming in granular media and colloids contribute to our understanding of molecular glasses, or are they essentially complementary?  

Q9) If a young physicist asked you whether he or she should work on the glass problem in the next few years, would you encourage him or her and if so, which aspect of the glass problem would you recommend him or her to tackle?  

 Q10) In twenty years from now, what concepts, ideas or results obtained on the glass transition in the last twenty years will be remembered?  

Q11) If you met an omniscient God and were allowed one single question on glasses, what would it be?

I thank Peter Wolynes for bringing this to my attention.

Stokes-Einstein relation between viscosity and diffusion in liquids

The Stokes-Einstein equation

relates the diffusion constant D of a macroscopic particle of radius r undergoing a Brownian motion to the viscosity eta of the fluid in which it is immersed.
It is a beautiful and simple example of a fluctuation-dissipation relation.

But suppose now we think about one of the individual atoms or molecules in the fluid. It also undergoes Brownian motion and one can define a self-diffusion constant.
It is amazing to me that the Stokes-Einstein relation still holds for a wide range of liquids, temperatures, and pressures with r being of the order of the molecular radius.

The figure and table below are taken from this paper.

Can this relation be derived from microscopic theory?
Zwanzig gave a heuristic justification here.
Rah and Eu gave a derivation from stat. mech. here.

The Stokes-Einstein relation does break down as one approaches the glass temperature in a supercooled liquid, as for example shown here. The origin of that breakdown is controversial, as is many phenomena involving glasses.

A key concept in glasses: the entropy crisis

The figure below introduces the idea of an "entropy crisis" and the Kauzmann temperature in glasses. It also leads to profound and controversial questions about the intimate connection between thermodynamics and kinetics in glasses.

Each solid curve shows the temperature dependence of the entropy of a supercooled liquid, relative to that of the crystal, above T_g, the glass transition temperature. T_m is the melting temperature of the crystal. The dashed curves are entropy in the glassy state.

The figure is taken from a very helpful review and adapted from Walter Kauzmann's classic 1948 paper.

What is going on?
The entropy of a liquid is greater than a solid [think latent heat of melting] so Delta S is positive. But, the specific heat capacity of a liquid is also greater than that of a solid [the vibrational, translational, and rotational degrees of freedom are all "softer" and less constrained]. Hence, the slope of Delta S vs. T must be positive.
Now, suppose that the liquid is supercooled so incredibly slowly that the glass does not form and you keep lowering the temperature, then at some temperature Delta S becomes negative. This extrapolated temperature [see the light blue straight line] is known as the Kauzmann temperature.

Why does this matter?
By the third law of thermodynamics, the entropy of the crystal goes to zero as the temperature goes to zero. Thus the supercooled liquid, could have negative entropy, which is physically nonsense.
Formation of the glass prevents this possibility. But, formation of the glass involves kinetics. So is there some deep connection between thermodynamics and kinetics? The review  discusses some possible connections. The extent of that connection is one of the controversial questions in glasses.