Session L53: Frontiers of Statistical Physics

Lars Onsager Prize Lecture: A Random Walk Through Theoretical Physics

A historical account will be given of my efforts to apply conformal field theory techniques to experimentally relevant models of condensed matter. This began with a so far unsuccessful attempt to find the exact critical exponents for the localization transition in the integer quantum Hall effect, using techniques developed by field/string theorists. It was followed by a program to classify critical behavior of Heisenberg antiferromagnetic spin chains of arbitrary spin magnitude. It eventually led to a general theory of the low energy behavior of quantum impurity models including exact solutions for non-Fermi liquid critical points.   

Dannie Heineman Prize for Mathematical Physics Lecture: Understanding Nonequilibrium via Rare Fluctuations

Irreversible processes are a hot subject in statistical mechanics. During the last decade through the effort of several people, including the recipient of the prize and his collaborators, a progress in understanding stationary nonequilibrium states has been achieved. The key has been the study of rare fluctuations. The talk will review some basic ideas, results and perspectives.   

Experimental confirmation of Landauer's principle

Landauer's principle, formulated in 1961, postulates that irreversible logical or computational operation such as memory erasure must dissipate heat, no matter how slowly they are performed. For example, to ``reset to one'' a memory that can be in state 0 or 1 requires at least kT ln2 of work, which is dissipated as heat. In 1982, Bennett pointed out a link to Maxwell's Demon: Were Landauer's principle to fail, it would be possible to repeatedly extract work from a heat bath. We report the first confirmation of Landauer's principle in an experimental system, where a virtual double-well potential is created via a feedback loop. We observe the position of a charged, fluorescent, colloidal particle in water and calculate and then apply a force = -grad U(x,t) via an electric field. In a first experiment, the probability of ``erasure'' (resetting to one) is unity, and at long cycle times, we observe that the work is compatible with kT ln2. In a second, the probability of erasure is zero; the system may end up in two states; and, at long cycle times, the measured work tends to zero.   

The Basis of the Second Law of Thermodynamics in Quantum Field Thoery

We derive the quantum Boltzmann equation for a closed system with a two-particle collision process on the basis of quantum field theory. In the thermodynamic limit, the system evolves deterministically and irreversibly towards equilibrium, on the time scale of the scattering time of the particles. This irreversibility is related to the loss of information which comes from the vanishing off-diagonal phase coherence in the system. By calculating the time evolution of the off-diagonal elements of the generalized density matrix, we show that these terms decay rapidly due to the interaction. In the case of Bose-Einstein condensates, all phase coherence is not lost. We deduce the onset of phase coherence in a Bose-Einstein condensate, which gives rise to macroscopic wavelike behaviors of Bose systems. We also derive the H-theorem by combining our results with standard definitions of entropy.   

The Stochastic Geometry of non-Gaussian Fields

Gaussian random fields pervade various areas of physics and have distinctive and well understood stochastic properties. Here we study the stochastic geometry of a random surface, whose height is given by a nonlinear function of a Gaussian field. We find that, as a result of the non-Gaussianity, the density of maxima and minima no longer match and calculate the relative imbalance between the two. We perform similar calculations for the density of umbilical points, which are topological defects of the lines of curvature. Our results apply to the analysis of speckle patterns generated by nonlinear random waves and more generally to detect and quantify non-Gaussianities present in any scalar field that can be represented as a smooth two-dimensional surface.   

Differential geometry of the space of Ising models

We use information geometry to understand the emergence of simple effective theories, using an Ising model perturbed with terms coupling non-nearest-neighbor spins as an example. The Fisher information is a natural metric of distinguishability for a parameterized space of probability distributions, applicable to models in statistical physics. Near critical points both the metric components (four-point susceptibilities) and the scalar curvature diverge with corresponding critical exponents. However, connections to Renormalization Group (RG) ideas have remained elusive. Here, rather than looking at RG flows of parameters, we consider the reparameterization-invariant flow of the manifold itself. To do this we numerically calculate the metric in the original parameters, taking care to use only information available after coarse-graining. We show that under coarse-graining the metric contracts very anisotropically, leading to a ``sloppy'' spectrum with the metric's Eigenvalues spanning many orders of magnitude. Our results give a qualitative explanation for the success of simple models: most directions in parameter space become fundamentally indistinguishable after coarse-graining.   

Growth Inside a Corner: Limiting Interface Shape

We investigate a simple model for crystal growth in which elemental cubes are stochastically deposited onto the inside of a three-dimensional corner. The interface of this crystal evolves into a deterministic limiting shape in the long-time limit. We incorporate known results from the corresponding two-dimensional system and use geometrical symmetries of the three-dimensional problem to conjecture an equation of motion for the interface profile which we solve analytically. The agreement between the result of the calculation and simulations of the growth process is excellent. We also present a generalization of our equation of interface motion to arbitrary spatial dimension.   

The magnetisation distribution of the Ising model for $d\ge 5$

The magnetisation distribution of the Ising model for $d>4$ is excellently fitted by a generalised binomial distribution. We have computed exactly an ansatz expression which can be fitted to the distribution near $T_c$. Though the ansatz is long and complicated it only has three parameters, besides the number of vertices, which then provides us with details about the distribution. This method also provided us with an estimate of $T_c$ for $d=6$. For extremely dense regular graphs, such as a complete bipartite graph, we can show what the parameter values are and that these values give asymptotically correct behaviour of eg the susceptibility and the free energy. Also a possible approach for $d\le 4$ will be briefly discussed as well as boundary effects for $d=5$.   

A Topological Phase Transition in Models of River Networks

The classical Scheidegger model of river network formation and evolution is investigated on non-Euclidean geometries, which model the effects of regions of convergent and divergent flows - as seen around lakes and drainage off mountains, respectively. These new models may be differentiated by the number of basins formed. Using the divergence as an order parameter, we see a phase transition in the number of distinct basins at the point of a flat landscape. This is a surprising property of the statistics of river networks and suggests significantly different properties for riverine networks in uneven topography and vascular networks of arteries versus those of veins among others.   

Finite size scaling of the dynamical free-energy in the interfacial regime of a kinetically constrained model

Glassy phenomena have proven difficult to understand: they present a variety of features --~slow dynamics, ageing, dynamical heterogeneity, frustration~-- which make their study arduous from a theoretical point of view. Kinetically Constrained Models (KCMs) are a simple class of lattice gas whose dynamics present features similar to those of glassy phenomena, with the advantage that no disorder is present in the model --~making them easier to study. A dynamical approach has been recently proposed: it consists in determining the large deviation function associated to the probability distribution function of time-integrated observables quantifying the ``activity'' of histories followed by the system. We determine the finite size corrections to the large deviation function of the activity in a KCM (the Fredrickson-Anderson model in one dimension), in the regime of dynamical phase coexistence. Numerical results agree with an effective model where the boundary between active and inactive regions is described by a Brownian interface. We show that the scalings of this physical picture are reflected in the finite size scaling of the dynamical free energy of the model. We expect the same picture to hold in other kinetically constrained models where the particle numberis not conserved.   

Effective Temperature Dynamics of Radiation Induced Amorphization

Materials exposed to radiation suffer structural changes over time. Typically, after exposure to radiation, a crystal will gradually lose its periodic structure and become amorphous. A theory of radiation amorphization should provide a description of the structural evolution. We study radiation amorphization in a simple molecular dynamics model and show that one can describe the amorphous steady-state using a structural effective temperature (different from the thermal bath temperature). We derive a theory that predicts the value of the steady-state effective-temperature as a function of the thermal bath temperature for a constant intensity of radiation. The theory agrees well with simulations results.