Methods Of Statistical Mechanics Research Articles

Classical microscopic theory of dispersion, emission and absorption of light in dielectrics

This paper is a continuation of a recent one in which, apparently for the first time, the existence of polaritons in ionic crystals was proven in a microscopic electrodynamic theory. This was obtained through an explicit computation of the dispersion curves. Here the main further contribution consists in studying electric susceptibility, from which the spectrum can be inferred. We show how susceptibility is obtained by the Green-Kubo methods of Hamiltonian statistical mechanics, and give for it a concrete expression in terms of time-correlation functions. As in the previous paper, here too we work in a completely classical framework, in which the electrodynamic forces acting on the charges are all taken into account, both the retarded forces and the radiation reaction ones. So, in order to apply the methods of statistical mechanics, the system has to be previously reduced to a Hamiltonian one. This is made possible in virtue of two global properties of classical electrodynamics, namely, the Wheeler-Feynman identity and the Ewald resummation properties, the proofs of which were already given for ordered system. The second contribution consists in formulating the theory in a completely general way, so that in principle it applies also to disordered systems such as glasses, or liquids or gases, provided the two general properties mentioned above continue to hold. A first step in this direction is made here by providing a completely general proof of the Wheeler-Feynman identity, which is shown to be the counterpart of a general causality property of classical electrodynamics. Finally it is shown how a line spectrum can appear at all in classical systems, as a counterpart of suitable stability properties of the motions, with a broadening due to a coexistence of chaoticity. The relevance of some recent results of the theory of dynamical systems in this connection is also pointed out.

‘Quantum hairs’ and entropy of the quantum isolated horizon from Chern–Simons theory

We articulate the fact that the loop quantum gravity (LQG) description of the quantum macrostates of black hole horizons, modeled as quantum isolated horizons (QIHs), is completely characterized in terms of two independent integer-valued ‘quantum hairs’, viz, the coupling constant of the quantum Chern–Simons (CS) theory describing QIH dynamics, and the number of punctures produced by the bulk spin network edges piercing the isolated horizon (which act as pointlike sources for the CS fields). We demonstrate that the microcanonical entropy of macroscopic (both parameters assuming very large values) QIHs can be obtained directly from the microstates of this CS theory using standard statistical mechanical methods, without having to additionally postulate the horizon as an ideal gas of punctures, or incorporate any additional classical or semiclassical input from general relativity vis-a-vis the functional dependence of the isolated horizon mass on its area, or indeed, without having to restrict to any special class of spins. Requiring the validity of the Bekenstein–Hawking area law relates these two parameters (as an equilibrium ‘equation of state’), and consequently allows the Barbero–Immirzi parameter to take any real and positive value depending on the value of . The logarithmic correction to the area law obtained a decade ago by R Kaul and one of us (PM), ensues straightforwardly, with precisely the coefficient , making it a signature of the LQG approach to black hole entropy.

Dimensionality and Design of Isotropic Interactions that Stabilize Honeycomb, Square, Simple Cubic, and Diamond Lattices

We use inverse methods of statistical mechanics and computer simulations to investigate whether an isotropic interaction designed to stabilize a given two-dimensional (2D) lattice will also favor an analogous three-dimensional (3D) structure, and vice versa. Specifically, we determine the 3D ordered lattices favored by isotropic potentials optimized to exhibit stable 2D honeycomb (or square) periodic structures, as well as the 2D ordered structures favored by isotropic interactions designed to stabilize 3D diamond (or simple cubic) lattices. We find a remarkable `transferability' of isotropic potentials designed to stabilize analogous morphologies in 2D and 3D, irrespective of the exact interaction form, and we discuss the basis of this cross-dimensional behavior. Our results suggest that the discovery of interactions that drive assembly into certain 3D periodic structures of interest can be assisted by less computationally intensive optimizations targeting the analogous 2D lattices.

A model of the radiation-induced bystander effect based on an analogy with ferromagnets. Application to modelling tissue response in a uniform field

We propose a model of the radiation-induced bystander effect based on an analogy with magnetic systems. The main benefit of this approach is that it allowed us to apply powerful methods of statistical mechanics. The model exploits the similarity between how spin–spin interactions result in correlations of spin states in ferromagnets, and how signalling from a damaged cell reduces chances of survival of neighbour cells, resulting in correlated cell states. At the root of the model is a classical Hamiltonian, similar to that of an Ising ferromagnet with long-range interactions. The formalism is developed in the framework of the Mean Field Theory. It is applied to modelling tissue response in a uniform radiation field. In this case the results are remarkably simple and at the same time nontrivial. They include cell survival curves, expressions for the tumour control probability and effects of fractionation. The model extends beyond of what is normally considered as bystander effects. It offers an insight into low-dose hypersensitivity and into mechanisms behind threshold doses for deterministic effects.

Minimum vertex cover problems on random hypergraphs: replica symmetric solution and a leaf removal algorithm.

The minimum vertex-cover problems on random α-uniform hypergraphs are studied using two different approaches, a replica method in statistical mechanics of random systems and a leaf removal algorithm. It is found that there exists a phase transition at the critical average degree e/(α-1), below which a replica symmetric ansatz in the replica method holds and the algorithm estimates exactly the same solution of the problem as that by the replica method. In contrast, above the critical degree, the replica symmetric solution becomes unstable and the leaf-removal algorithm fails to estimate the optimal solution because of the emergence of a large size core. These results strongly suggest a close relation between the replica symmetry and the performance of an approximation algorithm. Critical properties of the core percolation are also examined numerically by a finite-size scaling.

The D(+) + H2 reaction: differential and integral cross sections at low energy and rate constants at low temperature.

The D(+) + H2 reaction is investigated by means of a time independent quantum mechanical (TIQM) and statistical quantum mechanical (SQM) methods. Differential cross sections and product rotational distributions obtained with these two theoretical approaches for collision energies between 1 meV and 0.1 eV are compared to analyze the dynamics of the process. The agreement observed between the TIQM differential cross sections and the SQM predictions as the energy increases revealed the role played by the complex-forming mechanism. The importance of a good description of the asymptotic regions is also investigated by calculating rate constants for the title reaction at low temperature.

Online Learning through Moving Ensemble Teachers — An Exact Solution of a Linear Model —

Since a model in which a student learns from two or more teachers who themselves are learning has a certain similarity with actual human society, the analysis of such a model is interesting. In this paper, a model composed of a true teacher, multiple moving ensemble teachers existing around the true teacher, and a student, which are all linear perceptions, is analyzed using the statistical–mechanical method in the framework of on-line learning. The dependences of the generalization performance on the ensemble teachers’ learning rate, the student’s learning rate, and the number of ensemble teachers are clarified. Furthermore, it is shown that the generalization error can be reduced to the lower bound in the case of moving ensemble teachers, while there are unattainable generalization errors in the case of stationary ensemble teachers. These results show that it is important for teachers to continue learning in order to educate students.

Structure and Reactivity of Sub-Monolayer Pt on CeO2 Surface from First Principles Thermodynamics

Transition metal catalysts supported on metal-oxide surfaces are promising materials for usage in heterogeneous catalysis due to strong metal-support interaction, which affects the intrinsic activity at the surface, and results in enhanced dispersion of metal atoms in some cases, thus promoting stability. One of the most widely investigated such systems are Pt/CeO2 systems, which have been found to be highly active toward the water gas shift reaction [1]. Experimental and theoretical studies point to electron and oxygen transfer between adsorbed Pt atoms and CeO2 support as the reason for the enhanced activity in such systems [2, 3]. The effect of this metal-support interaction, however, is far from clear in context of the redox reactions that occur in DMFCs and PEMFCs. Especially, the role of CeO2, well known for its oxygen storage capacity, remains yet to be investigated in co-catalysis. To this end, understanding and characterizing the dispersion of Pt on the CeO2 surface is of utmost importance. Stability and activity of a few Pt layers on rutile metal-oxides has been investigated computationally, yielding insights into the geometric and electronic influences of the support on the catalyst [4]. Sub-monolayer deposition of catalyst atoms opens up the support surface to direct engagement with the reactants and is well worth investigating. In this study, first principles DFT calculations are used in combination with rigorous statistical mechanics methods to find stable phases of Pt atoms dispersed on the CeO2 (111) surface as a function of their chemical potential. Employing the Cluster Expansion formalism in tandem with the Metropolis Monte Carlo Method to account for the configurational contribution to the system’s free energy [5], the important interactions dictating the surface morphology are identified. In agreement with experiments [2], a remarkable transfer of oxygen from CeO2 to the Pt atoms is observed, and a tendency of Pt atoms to aggregate at low surface coverage is also seen. Further, using a linear Gibbs energy relation, the energetics of various reaction steps in a 2 e- oxygen reduction (ORR) process of the Pt/CeO2 system are computed for several instances of oxygen co-adsorption and are compared to those of a model Pt (111) system as a function of the electrode potential. Additionally, the effect of support on the dissolution potential of Pt atoms is also evaluated to understand the stabilizing effect of the support, which has been observed experimentally [6]. Enhancement in ORR kinetics for Pt/CeO2 systems has been observed experimentally, but such studies are few [7]. Two mechanisms are proposed in this study to explain such a behavior: (1) the preferential adsorption of OH on only Pt in contrast to the co-adsorption of O on both Pt and CeO2, and (2) the activation of O-O bond breaking due to the oxygen starved CeO2 surface. In principle, such systems with the possibility for the support material to take part in the reaction along with the catalyst atoms can open up new channels that facilitate ORR.

Thermodynamic formalism for a modified shift map

We introduce a transfer operator and use it to prove some theorems of a classical flavor from thermodynamic formalism (including existence and uniqueness of appropriately defined Gibbs states and equilibrium states for potential functions satisfying Dini's condition and stochastic laws for Hölder continuous potential and observable functions) in a novel setting: the "alphabet" E is a compact metric space equipped with an a priori probability measure ν and an endomorphism T. The "modified shift map" S is defined on the product space Eℕ by the rule (x1x2x3…) ↦ (T(x2)x3…). The greatest novelty is found in the variational principle, where a term must be added to the entropy to reflect the transformation of the first coordinate by T after shifting. Our motivation is that this system, in its full generality, cannot be treated by the existing methods of either rigorous statistical mechanics of lattice gases (where only the true shift action is used) or dynamical systems theory (where the a priori measure is always implicitly taken to be the counting measure).

Fluctuating observation time ensembles in the thermodynamics of trajectories

The dynamics of stochastic systems, both classical and quantum, can be studied by analysing the statistical properties of dynamical trajectories. The properties of ensembles of such trajectories for long, but fixed, times are described by large-deviation (LD) rate functions. These LD functions play the role of dynamical free energies: they are cumulant generating functions for time-integrated observables, and their analytic structure encodes dynamical phase behaviour. This ‘thermodynamics of trajectories’ approach is to trajectories and dynamics what the equilibrium ensemble method of statistical mechanics is to configurations and statics. Here we show that, just like in the static case, there are a variety of alternative ensembles of trajectories, each defined by their global constraints, with that of trajectories of fixed total time being just one of these. We show how the LD functions that describe an ensemble of trajectories where some time-extensive quantity is constant (and large) but where total observation time fluctuates can be mapped to those of the fixed-time ensemble. We discuss how the correspondence between generalized ensembles can be exploited in path sampling schemes for generating rare dynamical trajectories.

Pinning of steps near equilibrium without impurities, adsorbates, or dislocations

Discontinuous surface tension of a vicinal surface is calculated numerically based on a statistical mechanical method. The microscopic model is a restricted solid-on-solid (RSOS) model with a point-contact type step–step attraction (p-RSOS model). The discontinuity in the surface tension leads to step faceting at low temperatures near equilibrium, where macrosteps with smooth side surfaces are formed, resulting in two-surface coexistence. Using the Monte Carlo method with the Metropolis algorithm, we show the formation of the merged steps (macrosteps) and the slow velocity of the merged steps. We demonstrate that steps on a vicinal surface are pinned without impurities, adsorbates, or dislocations. We also propose simple ideas to avoid non-uniform growing of a crystal caused by the discontinuous surface tension of the crystal.

Protein Transport Across Nanopores: A Statistical Mechanical Perspective From Coarse-Grained Modeling and Approaches

We survey the transport of proteins across nanopores in the framework of coarse-grained modeling. The advantage of a reduced complexity with respect to full-atomistic techniques lies in the possibility of massive sampling of events, thus allowing a statistical mechanical description of translocation in terms of ensemble averages. Often, protein transport through narrow channels tightly couples with unfolding pathways causing a richer phenomenology compared to unstructured polymer translocation. This reflects into a process controlled by the presence of protein-specific free-energy barriers which can be conveniently estimated by statistical mechanical methods implemented in coarse-grained simulations. We illustrate how protein transport dynamics can be characterized by the statistical properties of trajectories and sometimes interpreted as driven diffusion of a single collective coordinate over a free-energy landscape. We also discuss, through selected examples, the connection between reduced-model simulations and recent experimental results.

Cavity-based robustness analysis of interdependent networks: Influences of intranetwork and internetwork degree-degree correlations

We develop a methodology for analyzing the percolation phenomena of two mutually coupled (interdependent) networks based on the cavity method of statistical mechanics. In particular, we take into account the influence of degree-degree correlations inside and between the networks on the network robustness against targeted (random degree-dependent) attacks and random failures. We show that the developed methodology is reduced to the well-known generating function formalism in the absence of degree-degree correlations. The validity of the developed methodology is confirmed by a comparison with the results of numerical experiments. Our analytical results indicate that the robustness of the interdependent networks depends on both the intranetwork and internetwork degree-degree correlations in a nontrivial way for both cases of random failures and targeted attacks.

Landau-Silin kinetic equation in the theory of normal Fermi liquid

In the frame of the Kadanoff-Baym version of the Green’s function method in quantum statistical mechanics it is proved that the Landau-Silin kinetic equation for a normal quantum Fermi liquid is valid in the case when the widths of oneparticle energy levels are taken into account exactly.

Species abundance distributions, statistical mechanics and the priors of MaxEnt

The methods of Maximum Entropy have been deployed for some years to address the problem of species abundance distributions. In this approach, it is important to identify the correct weighting factors, or priors, to be applied before maximising the entropy function subject to constraints. The forms of such priors depend not only on the exact problem but can also depend on the way it is set up; priors are determined by the underlying dynamics of the complex system under consideration. The problem is one of statistical mechanics and it is the properties of the system that yield the correct MaxEnt priors, appropriate to the way the problem is framed. Here I calculate, in several different ways, the species abundance distribution resulting when individuals in a community are born and die independently. In the usual formulation the prior distribution for the number of species over the number of individuals is 1/n; the problem can be reformulated in terms of the distribution of individuals over species classes, with a uniform prior. Results are obtained using master equations for the dynamics and separately through the combinatoric methods of elementary statistical mechanics; the MaxEnt priors then emerge a posteriori. The first object is to establish the log series species abundance distribution as the outcome of per capita guild dynamics. The second is to clarify the true nature and origin of priors in the language of MaxEnt. Finally, I consider how it may come about that the distribution is similar to log series in the event that filled niches dominate species abundance. For the general ecologist, there are two messages. First, that species abundance distributions are determined largely by population sorting through fractional processes (resulting in the 1/n factor) and secondly that useful information is likely to be found only in departures from the log series. For the MaxEnt practitioner, the message is that the prior with respect to which the entropy is to be maximised is determined by the nature of the problem and the way in which it is formulated.

Probing the limitations of isotropic pair potentials to produce ground-state structural extremes via inverse statistical mechanics

Inverse statistical-mechanical methods have recently been employed to design optimized short-range radial (isotropic) pair potentials that robustly produce novel targeted classical ground-state many-particle configurations. The target structures considered in those studies were low-coordinated crystals with a high degree of symmetry. In this paper, we further test the fundamental limitations of radial pair potentials by targeting crystal structures with appreciably less symmetry, including those in which the particles have different local structural environments. These challenging target configurations demanded that we modify previous inverse optimization techniques. In particular, we first find local minima of a candidate enthalpy surface and determine the enthalpy difference ΔH between such inherent structures and the target structure. Then we determine the lowest positive eigenvalue λ(0) of the Hessian matrix of the enthalpy surface at the target configuration. Finally, we maximize λ(0)ΔH so that the target structure is both locally stable and globally stable with respect to the inherent structures. Using this modified optimization technique, we have designed short-range radial pair potentials that stabilize the two-dimensional kagome crystal, the rectangular kagome crystal, and rectangular lattices, as well as the three-dimensional structure of the CaF(2) crystal inhabited by a single-particle species. We verify our results by cooling liquid configurations to absolute zero temperature via simulated annealing and ensuring that such states have stable phonon spectra. Except for the rectangular kagome structure, all of the target structures can be stabilized with monotonic repulsive potentials. Our work demonstrates that single-component systems with short-range radial pair potentials can counterintuitively self-assemble into crystal ground states with low symmetry and different local structural environments. Finally, we present general principles that offer guidance in determining whether certain target structures can be achieved as ground states by radial pair potentials.

Combining Solvent Thermodynamic Profiles with Functionality Maps of the Hsp90 Binding Site to Predict the Displacement of Water Molecules

Intermolecular interactions in the aqueous phase must compete with the interactions between the two binding partners and their solvating water molecules. In biological systems, water molecules in protein binding sites cluster at well-defined hydration sites and can form strong hydrogen-bonding interactions with backbone and side-chain atoms. Displacement of such water molecules is only favorable when the ligand can form strong compensating hydrogen bonds. Conversely, water molecules in hydrophobic regions of protein binding sites make only weak interactions, and the requirements for favorable displacement are less stringent. The propensity of water molecules for displacement can be identified using inhomogeneous fluid solvation theory (IFST), a statistical mechanical method that decomposes the solvation free energy of a solute into the contributions from different spatial regions and identifies potential binding hotspots. In this study, we employed IFST to study the displacement of water molecules from the ATP binding site of Hsp90, using a test set of 103 ligands. The predicted contribution of a hydration site to the hydration free energy was found to correlate well with the observed displacement. Additionally, we investigated if this correlation could be improved by using the energetic scores of favorable probe groups binding at the location of hydration sites, derived from a multiple copy simultaneous search (MCSS) method. The probe binding scores were not highly predictive of the observed displacement and did not improve the predictivity when used in combination with IFST-based hydration free energies. The results show that IFST alone can be used to reliably predict the observed displacement of water molecules in Hsp90. However, MCSS can augment IFST calculations by suggesting which functional groups should be used to replace highly displaceable water molecules. Such an approach could be very useful in improving the hit-to-lead process for new drug targets.

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.

Coupled systems of two-dimensional turbulence

AbstractOrdinary two-dimensional turbulence corresponds to a Hamiltonian dynamics that conserves energy and the vorticity on fluid particles. This paper considers coupled systems of two-dimensional turbulence with three distinct governing dynamics. One is a Hamiltonian dynamics that conserves the vorticity on fluid particles and a quantity analogous to the energy that causes the system members to develop a strong correlation in velocity. The other two dynamics considered are non-Hamiltonian. One conserves the vorticity on particles but has no conservation law analogous to energy conservation; the other conserves energy and enstrophy but it does not conserve the vorticity on fluid particles. The coupled Hamiltonian system behaves like two-dimensional turbulence, even to the extent of forming isolated coherent vortices. The other two dynamics behave very differently, but the behaviours of all four dynamics are accurately predicted by the methods of equilibrium statistical mechanics.

Dynamics of the D+ + H2 → HD + H+ reaction at the low energy regime by means of a statistical quantum method

The D(+) +H2(v = 0, j = 0, 1) → HD+H(+) reaction has been investigated at the low energy regime by means of a statistical quantum mechanical (SQM) method. Reaction probabilities and integral cross sections (ICSs) between a collisional energy of 10(-4) eV and 0.1 eV have been calculated and compared with previously reported results of a time independent quantum mechanical (TIQM) approach. The TIQM results exhibit a dense profile with numerous narrow resonances down to Ec ~ 10(-2) eV and for the case of H2(v = 0, j = 0) a prominent peak is found at ~2.5 × 10(-4) eV. The analysis at the state-to-state level reveals that this feature is originated in those processes which yield the formation of rotationally excited HD(v' = 0, j' > 0). The statistical predictions reproduce reasonably well the overall behaviour of the TIQM ICSs at the larger energy range (Ec ≥ 10(-3) eV). Thermal rate constants are in qualitative agreement for the whole range of temperatures investigated in this work, 10-100 K, although the SQM values remain above the TIQM results for both initial H2 rotational states, j = 0 and 1. The enlargement of the asymptotic region for the statistical approach is crucial for a proper description at low energies. In particular, we find that the SQM method leads to rate coefficients in terms of the energy in perfect agreement with previously reported measurements if the maximum distance at which the calculation is performed increases noticeably with respect to the value employed to reproduce the TIQM results.