Techniques Of Statistical Mechanics Research Articles

Designer spin systems via inverse statistical mechanics. II. Ground-state enumeration and classification

In the first paper of this series (DiStasio, Jr., Marcotte, Car, Stillinger, and Torquato, Phys. Rev. B 88, 134104 (2013)), we applied inverse statistical-mechanical techniques to study the extent to which targeted spin configurations on the square lattice can be ground states of finite-ranged radial spin-spin interactions. In this sequel, we enumerate all of the spin configurations within a unit cell on the one-dimensional integer lattice and the two-dimensional square lattice up to some modest size under periodic boundary conditions. We then classify these spin configurations into those that can or cannot be unique classical ground states of the aforementioned radial pair spin interactions and found the relative occurrences of these ground-state solution classes for different system sizes. As a result, we also determined the minimal radial extent of the spin-spin interaction potentials required to stabilize those configurations whose ground states are either unique or degenerate (i.e., those sharing the same radial spin-spin correlation function). This enumeration study has established that unique ground states are not limited to simple target configurations. However, we also found that many simple target spin configurations cannot be unique ground states.

Designer spin systems via inverse statistical mechanics

In this work, we extend recent inverse statistical-mechanical methods developed for many-particle systems to the case of spin systems. For simplicity, we focus in this initial study on the two-state Ising model with radial spin-spin interactions of finite range (i.e., extending beyond nearest-neighbor sites) on the square lattice under periodic boundary conditions. Our interest herein is to find the optimal set of shortest-range pair interactions within this family of Hamiltonians, whose corresponding ground state is a targeted spin configuration such that the difference in energies between the energetically closest competitor and the target is maximized. For an exhaustive list of competitors, this optimization problem is solved exactly using linear programming. The possible outcomes for a given target configuration can be organized into the following three solution classes: unique (nondegenerate) ground state (class I), degenerate ground states (class II), and solutions not contained in the previous two classes (class III). We have chosen to study a general family of striped-phase spin configurations comprised of alternating parallel bands of up and down spins of varying thicknesses and a general family of rectangular block checkerboard spin configurations with variable block size, which is a generalization of the classic antiferromagnetic Ising model. Our findings demonstrate that the structurally anisotropic striped phases, in which the thicknesses of up- and down-spin bands are equal, are unique ground states for isotropic short-ranged interactions. By contrast, virtually all of the block checkerboard targets are either degenerate or fall within class III solutions. The degenerate class II spin configurations are identified up to a certain block size. We also consider other target spin configurations with different degrees of global symmetries and order. Our investigation reveals that the solution class to which a target belongs depends sensitively on the nature of the target radial spin-spin correlation function. In the future, it will be interesting to explore whether such inverse statistical-mechanical techniques could be employed to design materials with desired spin properties.

Statistical mechanics of dictionary learning

Finding a basis matrix (dictionary) by which objective signals are represented sparsely is of major relevance in various scientific and technological fields. We consider a problem to learn a dictionary from a set of training signals. We employ techniques of statistical mechanics of disordered systems to evaluate the size of the training set necessary to typically succeed in the dictionary learning. The results indicate that the necessary size is much smaller than previously estimated, which theoretically supports and/or encourages the use of dictionary learning in practical situations.

Entropy of two-phase flows in the mode of separation

The new-developed approach to analyzing two-phase flow in mineral separation and beneficiation is based on the physical analogy of the flows and the perfect gas kinetic theory. The analysis from these viewpoints allowed formulation and basis of the invariants of such flows, where the key invariant is entropy of the two-phase flow, and its derivatives are separation potential, chaotization factor and mobility. The mathematical modeling of the two-phase flow from the new points of view permits analysis of the flow using statistical mechanics techniques.

Modelling of phase transitions: do it yourself

We present the basics of a powerful contemporary statistical mechanical technique that can be used by students to explore first-order phase transitions by themselves and for models of their own construction. The technique is a generalization of the well-known Peierls argument and is applicable to various models on a lattice. We illustrate the technique with the help of two simple models that were recently used to simulate phase transitions on surfaces.

Enhancement of Nematic Order and Global Phase Diagram of a Lattice Model for Coupled Nematic Systems

We use an infinite-range Maier–Saupe model, with two sets of local quadrupolar variables and restricted orientations, to investigate the global phase diagram of a coupled system of two nematic subsystems. The free energy and the equations of state are exactly calculated by standard techniques of statistical mechanics. The nematic–isotropic transition temperature of system A increases with both the interaction energy among mesogens of system B, and the two-subsystem coupling J. This enhancement of the nematic phase is manifested in a global phase diagram in terms of the interaction parameters and the temperature T. We make some comments on the connections of these results with experimental findings for a system of diluted ferroelectric nanoparticles embedded in a nematic liquid-crystalline environment.

Stochastic bursting synchronization in a population of subthreshold Izhikevich neurons

We consider a population of subthreshold Izhikevich neurons that cannot fire spontaneously without noise. As the coupling strength passes a threshold, individual neurons exhibit noise-induced burstings (i.e., discrete groups or bursts of noise-induced spikes). We investigate stochastic bursting synchronization by varying the noise intensity. Through competition between the constructive and the destructive roles of noise, collective coherence between noise-induced burstings is found to occur over a large range of intermediate noise intensities. This kind of stochastic bursting synchronization is well characterized by using the techniques of statistical mechanics and nonlinear dynamics, such as the order parameter, the raster plot of neural spikes, the time series of the ensemble-averaged global potential, and the phase portraits of limit cycles. In contrast to spiking neurons showing only spike synchronization (characterizing a temporal relationship between spikes), bursting neurons are found to exhibit both spike synchronization and burst synchronization (characterizing a temporal relationship between the onset times of the active phases of repetitive spikings). The degree of stochastic bursting synchronization is also measured in terms of a synchronization measure that reflects the resemblance of the global potential to the individual potential.

Thermodynamics of Lithium in TiO2(B) from First Principles

We use first-principles density functional theory (DFT) calculations combined with statistical mechanical techniques based on the cluster expansion method and Monte Carlo simulations to predict the lithium site occupancies, voltage curves, and phase diagram for TiO2(B), a candidate anode material for lithium ion batteries. We find that Li intercalation is thermodynamically favorable up to a Li/Ti ratio of 1.25, higher than the theoretical maximum usually assumed for TiO2. The calculated phase diagram at 300 K contains three first-order phase transformations corresponding to major changes in the favored intercalation sites at increasing Li concentrations. Calculations based on DFT predict the stability of a new Li site at high Li concentrations in TiO2(B) and the occurrence of a dramatic site-inversion as Li is added to the host.

Can persistent Epstein–Barr virus infection induce chronic fatigue syndrome as a Pavlov reflex of the immune response?

Chronic fatigue syndrome is a protracted illness condition (lasting even years) appearing with strong flu symptoms and systemic defiances by the immune system. Here, by means of statistical mechanics techniques, we study the most widely accepted picture for its genesis, namely a persistent acute mononucleosis infection, and we show how such infection may drive the immune system towards an out-of-equilibrium metastable state displaying chronic activation of both humoral and cellular responses (a state of full inflammation without a direct ‘causes–effect’ reason). By exploiting a bridge with a neural scenario, we mirror killer lymphocytes TK and B cells to neurons and helper lymphocytes and to synapses, hence showing that the immune system may experience the Pavlov conditional reflex phenomenon: if the exposition to a stimulus (Epstein–Barr virus antigens) lasts for too long, strong internal correlations among B,TK and TH may develop ultimately resulting in a persistent activation even though the stimulus itself is removed. These outcomes are corroborated by several experimental findings.

Statistical Mechanics of Reward-Modulated Learning in Decision-Making Networks

The neural substrates of decision making have been intensively studied using experimental and computational approaches. Alternative-choice tasks accompanying reinforcement have often been employed in investigations into decision making. Choice behavior has been empirically found in many experiments to follow Herrnstein's matching law. A number of theoretical studies have been done on explaining the mechanisms responsible for matching behavior. Various learning rules have been proved in these studies to achieve matching behavior as a steady state of learning processes. The models in the studies have consisted of a few parameters. However, a large number of neurons and synapses are expected to participate in decision making in the brain. We investigated learning behavior in simple but large-scale decision-making networks. We considered the covariance learning rule, which has been demonstrated to achieve matching behavior as a steady state (Loewenstein & Seung, 2006 ). We analyzed model behavior in a thermodynamic limit where the number of plastic synapses went to infinity. By means of techniques of the statistical mechanics, we can derive deterministic differential equations in this limit for the order parameters, which allow an exact calculation of the evolution of choice behavior. As a result, we found that matching behavior cannot be a steady state of learning when the fluctuations in input from individual sensory neurons are so large that they affect the net input to value-encoding neurons. This situation naturally arises when the synaptic strength is sufficiently strong and the excitatory input and the inhibitory input to the value-encoding neurons are balanced. The deviation from matching behavior is caused by increasing variance in the input potential due to the diffusion of synaptic efficacies. This effect causes an undermatching phenomenon, which has been often observed in behavioral experiments.

Influence of Synaptic Depression on Memory Storage Capacity

Synaptic efficacy between neurons is known to change within a short time scale dynamically. Neurophysiological experiments show that high-frequency presynaptic inputs decrease synaptic efficacy between neurons. This phenomenon is called synaptic depression, a short term synaptic plasticity. Many researchers have investigated how the synaptic depression affects the memory storage capacity. However, the noise has not been taken into consideration in their analysis. By introducing "temperature", which controls the level of the noise, into an update rule of neurons, we investigate the effects of synaptic depression on the memory storage capacity in the presence of the noise. We analytically compute the storage capacity by using a statistical mechanics technique called Self Consistent Signal to Noise Analysis (SCSNA). We find that the synaptic depression decreases the storage capacity in the case of finite temperature in contrast to the case of the low temperature limit, where the storage capacity does not change.

Elastic Maier-Saupe-Zwanzig model and some properties of nematic elastomers

We introduce a simple mean-field lattice model to describe the behavior of nematic elastomers. This model combines the Maier-Saupe-Zwanzig approach to liquid crystals and an extension to lattice systems of the Warner-Terentjev theory of elasticity, with the addition of quenched random fields. We use standard techniques of statistical mechanics to obtain analytic solutions for the full range of parameters. Among other results, we show the existence of a stress-strain coexistence curve below a freezing temperature, analogous to the P-V diagram of a simple fluid, with the disorder strength playing the role of temperature. Below a critical value of disorder, the tie lines in this diagram resemble the experimental stress-strain plateau and may be interpreted as signatures of the characteristic polydomain-monodomain transition. Also, in the monodomain case, we show that random fields may soften the first-order transition between nematic and isotropic phases, provided the samples are formed in the nematic state.

Unusual ground states via monotonic convex pair potentials

We have previously shown that inverse statistical-mechanical techniques allow the determination of optimized isotropic pair interactions that self-assemble into low-coordinated crystal configurations in the d-dimensional Euclidean space R(d). In some of these studies, pair interactions with multiple extrema were optimized. In the present work, we attempt to find pair potentials that might be easier to realize experimentally by requiring them to be monotonic and convex. Encoding information in monotonic convex potentials to yield low-coordinated ground-state configurations in Euclidean spaces is highly nontrivial. We adapt a linear programming method and apply it to optimize two repulsive monotonic convex pair potentials, whose classical ground states are counterintuitively the square and honeycomb crystals in R(2). We demonstrate that our optimized pair potentials belong to two wide classes of monotonic convex potentials whose ground states are also the square and honeycomb crystal. We show that these unexpected ground states are stable over a nonzero number density range by checking their (i) phonon spectra, (ii) defect energies and (iii) self assembly by numerically annealing liquid-state configurations to their zero-temperature ground states.

STATISTICAL MECHANICS OF COLLISIONLESS ORBITS. II. STRUCTURE OF HALOS

In this paper, we present the density, \rho, velocity dispersion, \sigma, and \rho/\sigma^3 profiles of isotropic systems which have the energy distribution, N(E)\propto[\exp(\phi_0-E)-1], derived in Paper I. This distribution, dubbed DARKexp, is the most probable final state of a collisionless self-gravitating system, which is relaxed in terms of particle energies, but not necessarily in terms of angular momentum. We compare the DARKexp predictions with the results obtained using the extended secondary infall model (ESIM). The ESIM numerical scheme is optimally suited for the purpose because (1) it relaxes only through energy redistribution, leaving shell/particle angular momenta unaltered, and (2) being a shell code with radially increasing shell thickness it has very good mass resolution in the inner halo, where the various theoretical treatments give different predictions. The ESIM halo properties, and especially their energy distributions, are very well fit by DARKexp, implying that the techniques of statistical mechanics can be used to explain the structure of relaxed self-gravitating systems.

Exactly solvable model for helix-coil-sheet transitions in protein systems

In view of the important role helix-sheet transitions play in protein aggregation, we introduce a simple model to study secondary structural transitions of helix-coil-sheet systems using a Potts model starting with an effective Hamiltonian. This energy function depends on four parameters that approximately describe entropic and enthalpic contributions to the stability of a polypeptide in helical and sheet conformations. The sheet structures involve long-range interactions between residues which are far in sequence, but are in contact in real space. Such contacts are included in the Hamiltonian. Using standard statistical mechanical techniques, the partition function is solved exactly using transfer matrices. Based on this model, we study thermodynamic properties of polypeptides, including phase transitions between helix, sheet, and coil structures.

Statistical mechanics of learning orthogonal signals for general covariance models

Statistical mechanics techniques have proved to be useful tools in quantifying the accuracywith which signal vectors are extracted from experimental data. However, analysis haspreviously been limited to specific model forms for the population covarianceC, which may be inappropriate for real world data sets. In this paper we obtainnew statistical mechanical results for a general population covariance matrixC. For data setsconsisting of p sample points in we use the replica method to study the accuracy of orthogonal signal vectors estimatedfrom the sample data. In the asymptotic limit of at fixed α = p/N, we derive analytical results for the signal direction learning curves. In the asymptoticlimit the learning curves follow a single universal form, each displaying a retarded learningtransition. An explicit formula for the location of the retarded learning transition isobtained and we find marked variation in the location of the retarded learning transitiondependent on the distribution of population covariance eigenvalues. The results of thereplica analysis are confirmed against simulation.

Gravitational backreaction in cosmological spacetimes

We develop a new formalism for the treatment of gravitational backreaction in the cosmological setting. The approach is inspired by projective tec hniques in non-equilibrium statistical mechanics. We employ group-averaging with respect to the action of the isotropy group of homogeneous and isotropic spacetimes (rather than spatial averaging), in order to define effective FRW variables for a generic spacetime. Using the Hamiltonian formalism for gravitating perfect fluids, we obtain a set of equations for the evolution of the effective variables; these equations incorporate the effects of backreaction by the inhomogeneities. Specializing to dust-filled spacetimes, we find regimes that lead to a closed set of backreaction equat ions, which we solve for small inhomogeneities. We then study the case of large inhomogeneities in relation to the proposal that backreaction can lead to accelerated expansion. In particular, we identify regions of the gravitational state space that correspond to effective cosmic a cceleration. Necessary conditions are (i) a strong expansion of the congruences corresponding to comoving observers, and (ii) a large negative value of a dissipation variable that appears in the effective equations (i.e, an effective ”anti-dissipation”).

How could the replica method improve accuracy of performance assessment of channel coding?

We explore the relation between the techniques of statistical mechanics and information theory for assessing the performance of channel coding. We base our study on a framework developed by Gallager in IEEE Trans. Inform. Theory IT-11, 3 (1965), where the minimum decoding error probability is upper-bounded by an average of a generalized Chernoff's bound over a code ensemble. We show that the resulting bound in the framework can be directly assessed by the replica method, which has been developed in statistical mechanics of disordered systems, whereas in Gallager's original methodology further replacement by another bound utilizing Jensen's inequality is necessary. Our approach associates a seemingly ad hoc restriction with respect to an adjustable parameter for optimizing the bound with a phase transition between two replica symmetric solutions, and can improve the accuracy of performance assessments of general code ensembles including low density parity check codes, although its mathematical justification is still open.

Hydrogen bond ordering in ice V and the transition to ice XIII

The proton ordered version of ice V, ice XIII, was recently identified using Raman spectroscopy and neutron diffraction techniques. The transformation, between 108 and 117 K, only occurred in the presence of a small amount of dopant, similar to the proton ordering transition of ice Ih/XI. In this work, we investigate the hydrogen bond fluctuations in ice V and XIII with statistical mechanical techniques that use results from periodic electronic density functional theory calculations as input. We find a number of low-lying hydrogen bond configurations, approximately 20 within 10 K/water above the ground state state configuration, the structure of which agrees with fully ordered ice XIII. Using an analytic theory, graph invariants, we developed effective spin-lattice Hamiltonians governing hydrogen bond fluctuations to perform statistical mechanical calculations for a large simulation cell containing 6048 water molecules. Two models were constructed, one more elaborate than the first, to gauge the robustness of our methods when the unit cell is very complex and a large number of configurations lie close in energy to the ground state. The predicted proton ordering transitions, 62 and 72 K for the two models, are in qualitative agreement with experiment. Occupation probabilities, obtained from our simulations, compare well with values from recent neutron diffraction experiments and help verify our effective Hamiltonians. In both models, we find that a second order phase transition intervenes about 10 K above the transition to ice XIII, but its effect is negligible on the behavior of thermodynamic functions near the transition to ice XIII.

Mean field and cavity analysis for coupled oscillator networks

We study coupled oscillator spin systems on sparse, random graphs. In particular, we examine the recent conjecture of Ichinomiya on the equivalence of a sparsely connected oscillator network with ferromagnetic interactions to a fully connected network with disordered (i.e., randomly quenched) interactions. By restricting our investigation to a Hamiltonian case we can use the techniques of equilibrium statistical mechanics to compare these two models analytically including phase diagrams and the calculation of order parameters in the ordered phase. We complete our investigation by performing some Monte Carlo simulations to compare our theoretical predictions against.