PY722

When the universe began, it was very, very hot.  The temperature was much greater than the ionization energy of hydrogen (then, as now, the most common form of baryonic matter), so protons and electrons were dissociated and electromagnetic radiation could propagate freely.  As the universe expanded, it cooled, and eventually the temperature dropped below the ionization energy.  Protons and electrons combined to make neutral hydrogen, which is opaque to electromagnetic radiation.  So began what in astrophysics is called the "Dark Ages".  As the universe cooled further, though, the first stars formed and began to pump out prodigious amounts of energy--enough to reionize the hydrogen of the universe and make it transparent to light once again.  Though there are no current methods for studying the first phase transition (light to dark), we do have ways to probe the second phase transition of the universe (dark to light).

This paper by an entire page of authors, submitted to Nature, discusses one such attempt.  Gamma-ray bursts are the most luminous events in the known universe, second only to the Big Bang itself.  They are bright enough, in fact, that we expect to be able to observe them all the way back to reionization, which began at a redshift of z = 11 (meaning the universe was 11 times smaller than it is at present, at an age of just 400 million years).  The gamma-ray burst 090423 currently holds the record for most distant object ever observed, at a redshift of 8.2.  This is during the reionization period, and comparison with other very distant gamma-ray bursts, particularly in the UV spectrum (which is what 13.6eV-photons fall into), should offer a lot of information about the phase state of the universe as it changed from opaque to transparent.

As far as I'm aware, this is the most significant phase change in astrophysics/cosmology, because understanding the properties of the reionization constrains the formation rate of stars and galaxies in the early universe.  In addition, the presence or absence of gamma-ray bursts impacts our models of the evolution of vary large, metal-poor (i.e. almost entirely hydrogen and helium) stars--do they age and die with a bang, or with something else altogether?

If you were planning on reading the paper, please don't be scared off by the size.   Of the 41 pages in the PDF, the first three are authors and affiliations, and the actual paper ends at page 9.  And it's double-spaced.

When rapidly cooled, liquids  can experience a glassy transition, where we have seen particles can be trapped in a potential.  A similar effect happens to dense granular materials, save the potential is termed a 'cage'.  If a grain is going to leave the cage, it requires cooperative motion from its nearest neighbors to create a void space for it to relocate. 

The caging effect is readily visible from the Mean Square Displacement <delta r^2>, from the upturn in this curve at the end of the plateau when plotted against time.   At large times the motion is diffusive but the coefficient is much reduced.  The group used a correlation function C_cage defined as the fraction of particles with the same neighbors between t and t+delta t.    The group also endeavored to measure the cage size. 

 Breaking cages is analogous to taking a random walk, and techniques from random walks can be used to unambiguously estimate the fraction of particles involved in these uncaging events at any time.   These reaarangements occur more frequently in regions of higher disorder (measured from bond angles).

This paper does not give definitive results of any particular parameter at which this transition occurs.  However, it does give tools in which soft condensed matter physicists can use to search for the glassy transition.

Link

First-order phase transitions in a 2D random field Ising Model with conflicting dynamics is a recent paper by Nuno Crokidakis from the Federal University in Rio, Brazil published in the Journal of Statistical Mechanics.

 Just like the Ising model is a generic model of a continuous phase transition in an equilibrium system, the random field Ising Model wants to be your friendly-neighborhood generic non-equilibrium model.  The physical intuition behind the idea is that impurities and diffusion in the domains of a magnet produce random magnetic fields of their own whose effect may dominate the behavior of the system.  Indeed, previous results in 1D Ising Models seem to suggest that the behavior of the magnetization is intimately related with the details of the impurities/diffusion of domains, rather than being universal and independent of microscopic details.  As you can imagine, such a suggestion would be ripe for debate.

 The cool thing/big claim about this paper is that it shows that a random field ising model can be tuned from a continuous (second order) to an abrupt (first-order) phase transition without changing wild parameters like dimensionality or boundary conditions (which can also tune you into different types of transitions).  No, sir. This paper shows that by changing a simple physical parameter (the magnitude of the random field), you can go from observing the traditional continuous phase transition of the magnetization, to one with a discrete/clear jump in the magnetization.

To do detailed quantitative analysis one can look at the shape of the susceptibility to confirm that it is indeed a first-order transition.

To understand why the system behave the way it does, we should talk about avalanches and their stability... which will be my final project. 

This article simulates ferroelectric nanoparticles of transition metal oxides and relates the various phases available to the particles as a function of temperature, particle size and particle shape. Simulations were done on disks, where the diameter was greater than the height, and rods, where the diameter was less than the height. The electronic structure phases that resulted were highly dependent on the shape of the nanoparticle. Two phases were modeled. The A phase had a z component in the toroidal moment; the B phase had an x and y phase to the torroidal moment. A toroidal moment can be thought of as a colection of dipoles which form a ring in the plane of the moment. In other words, a z direction toroidal moment is a ring in the z constant plane. The net macroscopic field of a toroidal moment is zero, but the local feild and the interaction energy with an external field is nonzero. Additionally, toroidal phases may still have a dipole moment. Upon cooling in and electric field, the A phase was favored for a nanoparticle only when the diamater was above a certain critical diameter. In other words, nanodisks favored toroidal moments in the z direction. The B phase was favored by the nanorods. The applications here mainly envolve memory storage and the A phase. While the B phase is not very useful, the A phase has two states, in the +z direction or the -z direction. This means that the A phase can have counterclockwise or clockwise toroidal structures. The material could be made to switch phases from an applied electromagnetic field, and magnetic component is resonsible for the flipping. Unlike bulk ferroelectric memory, which changes phase based on an applied static field, this type of memory may be easier to read and write because no physical contacts would be required.

The paper that I chose was "From complex structures to complex processes: Percolation theory applied to the formation of a city", by Agnieszka Bitner, Robert Ho?yst, and Marcin Fia?kowski published in Physical Review E (2009), and it modeled the formation of urban centers in large cities.  The authors studies how the average size of land parcels varied with distance from the city center.   They used this information to decide if a region was in the urban phase or not.  Initially a region starts out as a rural region with large parcels of land.  As time goes on, roads are built and the parcels are divided and sub-divided.  The size of the parcels continues to shrink until a final size is reached which is approximately 1000 square meters.  Once a region has reached this average parcel size it is stable and the region is considered to be in the urban phase.  The authors examined 29 cities from around the world and found that they all showed the same general trends.   The average area of a parcel would increase as a power law from the center of the city except in the center where the average size of 1000 square meters had already been reached and stabilized.  I thought that this was an interesting article because it suggests that large tracts of land are an unstable state around a city, and that the natural tendency is for the land to be subdivide until some average size is reached.  Also, one the cities that was studied was Raleigh, NC so I thought that was neat.

    This article was interesting because it talked about a small aspect of the phase transition of crystallization of water; exploring the dipole interactions of the molecules with an internal electric field.  At the surface of water/vapor/ice there is a double electric layer.  Crystallization/condensation can be described as molecules going through this layer and adhering to the surface of a condensate.  Molecules with dipoles entering this layer will want to align with the field created by the layer, emitting EM radiation as they do so.  For water, this radiation has a frequency of ~150kHz.  Detection of this radiation may add useful information to scientists attempting to research weather patterns and water/ice formation.
    The electric fields in this layer are non-uniform in places.  Non-uniformity at the corners and tops of ice clusters acts as an attractor to the incoming molecule, causing it to orient and place itself in a particular location on the surface (such as to the stops of cylindrical columns or along corners of ice clusters).  The addition of an external electric field can also help grow ice columns in specific directions.  
    I found this article interesting because this effect helps explain the diversity and formation of snowflakes.  I like that this article explores an aspect of phase transitions (microscopic electromagnetic interactions of the surface molecules) in which very little energy is lost (compared to latent heat) and finds that it provides important information for the formation of complex crystal designs (like snowflakes).

I found this paper to be really interesting.  It is a review paper of the types of optical storage (using lasers) which currently are used in single write media (CDs DVDs, etc), but aim to compete with hard drive technologies in the future.  In phase-change materials, lasers can induce the change of the material  both back and forth between ordered and amorphous phases repeatably.  The crystalline order is obtained by just heating the sample to just above the glass transition temperature, which anneals into a crystal as it slowly cools, while the amorphous state is obtained by heating and actively cooling rapidly, quenching the disordered phase.  The materials that work well for this type of storage must have different optical properties in their two states, allowing a lower power laser can read the state easily, not exciting a change.  Further applications I found interesting include the 3D storage of data.  They have dual layer optical disks now, but it's really interesting to think about using multiple lasers or highly focused laser to just deliver the power to the region of interest in the bulk of a material.

This paper seems to be inspired by the economic crisis. The main issue here is the collective bankruptcies measured by the the default rates in the a economy. The authors are trying to simulate what happened in reality by a rating dynamic with short memory, in which the rate change of each firm only depends on the last step of the market. With the major part of the paper emphasizing on the effect of the interaction between  nodes, namely the competition or coordination depending on the sign, it is shown that three phases  are found in different regions of parameters, resembling the  paramagnetic, ferromagnetic and spin glass states respectively. Firms' individual effort is explored in some simplified cases which suggests that it could help to stabilize the system for weak interaction.

 As admitted in the paper, the major issues to be improved in the model are that the firms could have different size or market share, leading to effects such as asymmetric influence on each other, size distribution, etc, and new firms with lower rating should show up after the old ones went bankrupt.  

This paper is an attempt to model the crystallization process of a vapor. The paper uses a unique approach, that the “droplets” of vapor first nucleate into liquid particles and then to crystals. It was really interesting that they found these crystals form from the inside of the liquid droplet with at least a monolayer of vapor around the crystal core formation. The authors used many concepts we talked about in class recently, like the Lennard-Jones Potential, surface tension, metastable states and the changing chemical potentials between phases. The authors use a Monte-Carlo simulation with the Lennard-Jones pair potential to simulate pairs of vapor droplets just below the triple point on the phase diagram of the substance. The simulation leads to coagulation and evolution of the vapor droplets. The system goes to this intermediate liquid phase because the liquid-vapor interface required less energy than the crystal-vapor interface. The authors use their simulations to determine approximate nucleation rates of these transitions. To simulate real world conditions of constant vapor pressure the simulations are run in the grand canonical ensemble, so the total number of particles can fluctuate but the amount in the gas phase remains relatively constant. This article has many interesting and wide ranged applications, to such things as ice formation, protein crystallization, and the fabrication of nanocrystals. The wide range of applications was the basis for my initial interest in the article.

This paper from Nature is an attempt to use the fluctuation-dissipation theorem we talked about in class away from equilibrium in a granular system.  Their work suggests that it is possible to do so! You may find it surprising since aside from being a system far from equilibrium, driven and with damping forces, the system is also spatially non-homogeneous and anisotropic.   

 The details of the experiment are similar in spirit to Eli's experiment if you've ever seen it.  They put a type of sensor in a granular pack that is being shaken at a given frequency.  They measure the magnitude of the vibration using an accelerometer, and the use the the sensor (which is a torsional oscillator) to do two experiments. (1) measure the power spectrum of the sensor moving under no external torque and (2) measure the susceptibility of the granular medium under an external torque.  In both of these experiments they were able to show that an equation "like" the fluctuation-dissipation theorem checks out to first order.    The defined two parameters, an effective temperature related to the power spectrum of the sensor under no external force, and an effective friction relation to the susceptibility.

 The idea that non-equilibrium systems might to "first-order" be described by relations at equilibrium is just wild.  Good for them that it worked, because I would not have bet any money on it working. 

Relaxation dynamics of an elastic string in random media is a cool article about simulations of a elastic string in a heat bath.  I was interested in it initially because this research is essentially modeling idealized polymer folding or melting.  Rather than just look at the properties of the equilibrated string, they actually model how it behaves as a funcion of time from being completely straight to equilibrium.  It turns out there are a few regions of different behavior depending on the disorder initially present in the string.  At first the displacement from one end to the other (in the transverse direction) follows a non-universal power law (ie with no single, universal exponent) transitioning to a region where it follows a logarithmic law.  The cool difference they see is between starting in a perfectly ordered straight state and a state with quenched disorder, where they claim pinning centers of disorder may lead to the "nonuniversal" power law behavior, before the correlation length becomes greater than the distance between these areas of quenched disorder, leading to a transition into the usual logarithmic regime.  In the power law regime, the polymer acts as if it is overcoming energy barriers which scale as l^(1/3) where l is the length of the part of the polymer that is changing.  This paper uses a lot of topics from this course, and is pretty interesting, although I wish there were more discussions of the physical interpretation of what is going on.  I think a video of the behavior would help quite a bit in the interpretation of these kinds of results.  What is the behavior looking like in the transition between power-law and logarithmic behavior?

The radial distribution function is generally not known for most systems, however approximations have been made to attempt to analytically approximate simple systems. In this paper, the authors review several approximations that have been attempted for g(r), and put forward a new term.

The pair distribution function we used in class was close to g(r)~exp[-ßu(r)], which is not a terrible approximation. However, it can be written in terms of two new functions, so that g(r)=exp[-ßu(r) + y(r) +b(r)]. This paper defines y(r) as the difference between the total correlation function and the direct correlation function, it's called the indirect correlation function. The third function, b(r) is approximated in many different ways. The simplest way is b(r)=-y(r)+ln(1+y(r)), but the inclusion of non-linear terms of y(r) into b(r) will give a better result.

This paper proposes a b(r) approximation given by b(r)=-y(r)+ln(1+S(r)+aS(r)^2), where S(r) is given by S(r)=(exp[y(r)(1-exp(-hr)]-1)/(1-exp(-hr), where h is an adjustable parameter. This give b(r) in terms of two adjustable parameters, a and h.

There are many approximations for b(r) used in the literature. This paper compares the above approximation to b(r)=-y(r)+ln(1+S(r)), the first method is named ERY and the second RY. Each function was used to model a simulation for the Yukawa potential.

The ERY method did a slightly better job than the RY method at calculating g(r) and b(r), but from the plots shown I don't see much of a great leap forward. Indeed, seeing as the ERY method has one more parameter, we would expect a better fit. The paper does not go into detail on how the parameters are calculated, saying only that they come from thermodynamic consistency. I'm not sure if they treated h and a as fit parameters to model their simulations, or if h and a can be given by some unnamed thermodynamic relation. The paper allows h to be adjusted, but says that a is found from thermodynamic consistency. I'm not sure what "thermodynamic consistency" means when defining an arbitrary parameter; the paper needs to give more detail on this point. If a is merely another adjustable parameter, than of course the ERY approximation will work better. The h parameter is shared by both methods, so the a parameter is one more freedom in the fit. In this case, the paper really can't conclude anything. If however, a is given by some unnamed thermodynamic relation, then I think the paper is worthwhile. As it stands, the paper leaves out this information, making it difficult to see what the ultimate conclusion will be.

Thermodynamically consistent integral equation for soft repulsive spheres
Mauricio D. Carbajal-Tinoco, J. Chem. Phys. 128, 184507 (2008), DOI:10.1063/1.2918495

This paper gives a quantitative way to show the relationship between different parts of the same laser beam.  This method allows you to study a laser beam that has N different polarization zones.  The correlation function is used to determine the amount of correlation between two points in the cross section of the laser beam. This correlation function can be used to help determine the total polarization of the beam. The beam can be partially polarized, locally polarized, locally coherent, or randomly polarized. The correlation function can tell the experimenter how much of the beam is polarized or unpolarized and approximately how much of each polarization exists.

The paper does make a few generalizations. They assume that the laser beam has a uniform intensity, which is not usually the case, it usually has a Gaussian distribution of intensity centered at the center of the beam. The authors do take this into account when doing an experiment buy using the middle of the beam, but they do mention that this only captures about 95% of the beams intensity.  Ultimately the paper finds that the degree of polarization of the beam is inversely proportional to the square root of N, where N is the amount of original polarization zones in the incident laser beam. 

The techniques and ideas discussed in the paper have a wide range of applications and may prove to be of high importance. The main application I can see is that the analysis can be done before and after a beam passes through a medium, the correlation and polarization can be compared.  The medium suggested here is in liquid crystals used for television displays, and I would imagine as their characteristics become more know they will be used in other applications.

This paper includes all of the topics: pair correlation function, fluctuations, and disspation in Granular media.  The experimental technique used to locate the grains was very interesting: index match the grains with the fluid, and sweep a laser through the material.   They exponentially compact the material by heating/cooling cycles [Figure 2] (the container and grains have different thermal expansion coefficients).

On to the facinating Statistical Mechanics of this experiment:  

The correlations are between the preferred direction and the actual direction of motion of a grains.  The Preferred direction is the vector u [centroid of the voronoi cell  - center of grain].  theta is the angle between these vectors.  The conference paper investigates this correlation discriminating if the grains are in contact or not.

The group finds that this angle, theta, is indeed correlated to the motion of the particle.  Additionally, they tell us that it is independent of the volume of the voronoi cell  and the magnitude of the grain displacement.   In the 2009 paper, the groups shows that grains that are not in contact have a nearly uniformly distribution of thetas.  

This group emphasizes the geometric constaints on a grainular media.  Individual grain motion is highly correlated to the position neighbors in contact.  But, this contact correlation does not rule out the influence of force chains as a key driver in the local behavior of grains.  However, further research must distinguish these models. 

Note:

voronoi cells --> this is a geometric method of location the space around a grain that is nearest to it using planes that are perpendicular bisectors of the distance between the edges of the grains.

Note2:

[This paper also includes a small plot of the theoretical pair correlation function for a random packing of spheres shown in Fig 3 peaks at 1 grain radii, with another wider peak at 2 radii and quickly levels off to g(r)=1.]

Question:  

Does anyone know any good references on the "string-like motion in cooled liquids"?

Slotterback, Goff, Toiya, Douglas, Losert (both papers)

van Hecke (2009)

2008 PRE 

2009 Conference Proceedings 

Time for an oldie-but-goodie.  The thesis of Albert Bosma, defended in 1978, was a landmark publication in galactic physics.

Previously, it was assumed that the orbital speeds of stars in galaxies were Keplerian -- stable circular orbits about a massive central bulge and an extended massive disk out to the radius of orbit.  The orbits of stars in these galaxies would rise linearly with radius until outside the central bulge, then fall off as 1/sqrt(r) in the disk (this is derivable from basic orbital mechanics).  In 1959, the first example of non-Keplerian orbits was found.  Bosma's thesis extended this "anomaly" to many other galaxies: the majority of spiral galaxies in the sky are non-Keplerian in their velocity distribution.  What was being observed was the Keplerian linear rise out to a certain distance, but instead of the r^(-1/2) falloff expected, stars had constant orbital velocities outside of the central bulge.  This meant that one of two assumptions was incorrect: either Keplerian orbital mechanics (and thus Newtonian physics and general relativity) was inaccurate, or there was much more matter present that could not be seen by telescopes of any sort.  Given the previous successes of Newtonian physics and GR, the less outlandish choice was the existence of invisible matter around galaxies.  Observations in later decades, as well as numerous simulations, would support the hypothesis of dark matter as a dominating influence in the Universe.

I chose this paper because of its importance to cosmology and astrophysics, but also because it offers a good example of deriving radial distributions from experimental data.  Instead of scattering angle being the measured parameter, as we discussed in class, Bosma and others found velocities of hydrogen nebulae in the galaxies.  The paper is noteworthy as well because it is a great example of widely-accepted theories being found very wanting in the face of new data; these velocity curves necessitated radically different radial distributions of matter than could be seen.  As a result, our picture of the Universe changed dramatically, and we're still trying today to figure out what the actual reason is for the difference.