Inverse statistical problems: from the inverse Ising problem to data science

Inverse problems in statistical physics are motivated by the challenges of ‘big data’ in different fields, in particular high-throughput experiments in biology. In inverse problems, the usual procedure of statistical physics needs to be reversed: Instead of calculating observables on the basis of model parameters, we seek to infer parameters of a model based on observations. In this review, we focus on the inverse Ising problem and closely related problems, namely how to infer the coupling strengths between spins given observed spin correlations, magnetizations, or other data. We review applications of the inverse Ising problem, including the reconstruction of neural connections, protein structure determination, and the inference of gene regulatory networks. For the inverse Ising problem in equilibrium, a number of controlled and uncontrolled approximate solutions have been developed in the statistical mechanics community. A particularly strong method, pseudolikelihood, stems from statistics. We also review the inverse Ising problem in the non-equilibrium case, where the model parameters must be reconstructed based on non-equilibrium statistics.     

Generalized Brownian motion and elasticity

We introduce a family of stochastic processes which are a natural extension of Brownian motion to a tensor form. This allows us to solve a Dirichlet problem of linear elasticity obeying Lamé's equation, [1–(d– 1)]²V(x)+ [·V(x)]=0.     

Potential-induced structure transitions in self-assembled monolayers: ethanethiol on Au(1 0 0)

We have characterized the structural behaviour of ethanethiol self-assembled monolayers (SAMs) on Au(1 0 0) in 0.1 M H₂SO₄ as a function of electrode potential, using in-situ scanning tunneling microscopy (STM). After modification of the Au(1 0 0) electrode in an ethanolic solution of ethanethiol, STM images in air reveal a disordered thiol adlayer and a surface that is covered by 25% of monoatomic high gold islands, which originate from lifting of the (hex) reconstruction during thiol adsorption. In contrast to alkanethiol SAMs on Au(1 1 1), no vacancy islands are seen on the Au(1 0 0) surface. After contact of the SAM-covered Au(1 0 0) electrode with 0.1 M H₂SO₄ under potential control, two different structures are observed, depending on the potential range positive or negative of +0.3 V vs. SCE. In both cases the emerging ordered structures are quadratic, their unit cells being rotated by 45° with respect to the main crystallographic axes of the substrate. However, the ordered structure at negative potentials is more densely packed than the one at positive potentials, and in addition the surface reveals an almost 50% coverage of monoatomic high gold islands. The structure of the SAM changes reversibly with the electrode potential, the long range order gradually decreasing with each transition. Concomittant with this structure transition monoatomic deep holes are created when the potential is stepped from the cathodic to the anodic region. The experimental observations are rationalized by a high mobility of the gold thiolate moiety, causing the surface density of the SAM-covered gold to change drastically with potential.     

A cluster Monte Carlo algorithm for 2-dimensional spin glasses

A new Monte Carlo algorithm for 2-dimensional spin glasses is presented. The use of clusters makes possible global updates and leads to a gain in speed of several orders of magnitude. As an example, we study the 2-dimensional ±J Edwards-Anderson model. The new algorithm allows us to equilibrate systems of size 100² down to temperature T = 0.1. Our main result is that the correlation length diverges as an exponential ( ξ∼e ^2βJ) and not as a power law as T↦T _c = 0. Received 10 January 2001 and Received in final form 29 May 2001     

The one-body and two-body density matrices of finite nuclei with an appropriate treatment of the center-of-mass motion⋆

The one-body and two-body density matrices in coordinate space and their Fourier transforms in momentum space are studied for a nucleus (a nonrelativistic, self-bound finite system). Unlike the usual procedure, suitable for infinite or externally bound systems, they are determined as expectation values of appropriate intrinsic operators, dependent on the relative coordinates and momenta (Jacobi variables) and acting on intrinsic wave functions of nuclear states. Thus, translational invariance (TI) is respected. When handling such intrinsic quantities, we use an algebraic technique based upon the Cartesian representation, in which the coordinate and momentum operators are linear combinations of the creation and annihilation operators and for oscillator quanta. Each of the relevant multiplicative operators can then be reduced to the form: one exponential of the set { } times another exponential of the set { }. In the course of such a normal-ordering procedure we offer a fresh look at the appearance of “Tassie-Barker” factors, and point out other model-independent results. The intrinsic wave function of the nucleus in its ground state is constructed from a nontranslationally-invariant (nTI) one via existing projection techniques. As an illustration, the one-body and two-body momentum distributions (MDs) for the ⁴He nucleus are calculated with the Slater determinant of the harmonic-oscillator model as the trial, nTI wave function. We find that the TI introduces quite important effects in the MDs.     

Differential Entropy and Dynamics of Uncertainty

We analyze the functioning of Gibbs-type entropy functionals in the time domain, with emphasis on Shannon and Kullback-Leibler entropies of time-dependent continuous probability distributions. The Shannon entropy validity is extended to probability distributions inferred from L ²(R ⁿ ) quantum wave packets. In contrast to the von Neumann entropy which simply vanishes on pure states, the differential entropy quantifies the degree of probability (de)localization and its time development. The associated dynamics of the Fisher information functional quantifies nontrivial power transfer processes in the mean, both in dissipative and quantum mechanical cases. PACS NUMBERS: 05.45.+b, 02.50.-r, 03.65.Ta, 03.67.-a     

A Pseudo-Stokes Mesh Motion Algorithm

This work presents a moving mesh methodology based on the solution of a pseudo flow problem. The mesh motion is modeled as a pseudo Stokes problem solved by an explicit finite element projection method. The mesh quality requirements are satisfied by employing a null divergent velocity condition. This methodology is applied to triangular unstructured meshes and compared to well known approaches such as the ones based on diffusion and pseudo structural problems. One of the test cases is an airfoil with a fully meshed domain. A specific rotation velocity is imposed as the airfoil boundary condition. The other test is a set of two cylinders that move toward each other. A mesh quality criterion is employed to identify critically distorted elements and to evaluate the performance of each mesh motion approach. The results obtained for each test case show that the pseudo-flow methodology produces satisfactory meshes during the moving process.     

Integration of large chemical kinetic mechanisms via exponential methods with Krylov approximations to Jacobian matrix functions

Recent trends in hydrocarbon fuel research indicate that the number of species and reactions in chemical kinetic mechanisms is rapidly increasing in an effort to provide predictive capabilities for fuels of practical interest. In order to cope with the computational cost associated with the time integration of stiff, large chemical systems, a novel approach is proposed. The approach combines an exponential integrator and Krylov subspace approximations to the exponential function of the Jacobian matrix. The components of the approach are described in detail and applied to the ignition of stoichiometric methane–air and iso-octane–air mixtures, here described by two widely adopted chemical kinetic mechanisms. The approach is found to be robust even at relatively large time steps and the global error displays a nominal third-order convergence. The performance of the approach is improved by utilising an adaptive algorithm for the selection of the Krylov subspace size, which guarantees an approximation to the matrix exponential within user-defined error tolerance. The Krylov projection of the Jacobian matrix onto a low-dimensional space is interpreted as a local model reduction with a well-defined error control strategy. Finally, the performance of the approach is discussed with regard to the optimal selection of the parameters governing the accuracy of its individual components.     

Fundamental relationships between deformation-induced surface roughness,critical strain localisation and failure in AA5754-O

This research employs two approaches to characterise the apparent structure observed in localised strain maps constructed from surface topography data acquired from AA5754-O sheet stock that was deformed in three in-plane stretching modes. The first uses a conventional two-point autocorrelation function (ACF), while the second uses the combination of the eigenvalue spectrum associated with each map and information theory. The results from the ACF analysis are inconclusive, implying that this technique lacks the sensitivity necessary to quantify the relationships between multi-point clustering and strain localisation. The information theory-based approach reveals that the relative spectral entropy increases monotonically, attains a maximum and then decreases sharply to the failure strain. This behaviour occurs in all three strain modes and results from two competing processes: one where the formation of structure is favourable and one where it is not. The crossover point is a clear indicator of the onset of critical strain localisation and, therefore, can be regarded as a precursor to failure because once the dominant process shifts, additional strain results in the precipitate formation of a critical strain localisation event.     

Study on the denoising method for the electromagnetic ultrasonic echoes from multiple interfaces

The electromagnetic ultrasound is used in the detection of interfaces of the adhesive multilayer structures to solve the unstable coupling problem in ultrasonic testing by traditional piezoelectric transducers. Based on the analysis of the transforming mechanism of electromag-netic ultrasound energy and the resultant dead zone from mutual inductance of the transducer, the wavelet filtering by soft-thresholding and adaptive noise canceling methods are used simul-taneously to the detected electromagnetic ultrasonic signals to overcome the drawbacks of the low signal to noise ratio （SNR） and the wide intrinsic dead zone of the transducer. Processed results in the interface detection of a three layered adhesive sample of steel and rubber materials demonstrate that the wavelet filtering enhances the SNR about 12dB while the adaptive noise canceling narrows the dead zone effectively.