Statistical Mechanical Approximation Research Articles

Scaling limits of bosonic ground states, from many-body to non-linear Schrödinger

How and why could an interacting system of many particles be described as if all particles were independent and identically distributed? This question is at least as old as statistical mechanics itself. Its quantum version has been rejuvenated by the birth of cold atoms physics. In particular, the experimental creation of Bose–Einstein condensates leads to the following variant: why and how can a large assembly of very cold interacting bosons (quantum particles deprived of the Pauli exclusion principle) all populate the same quantum state? In this text I review the various mathematical techniques allowing to prove that the lowest energy state of a bosonic system forms, in a reasonable macroscopic limit of large particle number, a Bose–Einstein condensate. This means that indeed in the relevant limit all particles approximately behave as if independent and identically distributed, according to a law determined by minimizing a non-linear Schrödinger energy functional. This is a particular instance of the justification of the mean-field approximation in statistical mechanics, starting from the basic many-body Schrödinger Hamiltonian.

Tensor networks contraction and the belief propagation algorithm

Belief Propagation is a well-studied message-passing algorithm that runs over graphical models and can be used for approximate inference and approximation of local marginals. The resulting approximations are equivalent to the Bethe-Peierls approximation of statistical mechanics. Here we show how this algorithm can be adapted to the world of PEPS tensor networks and used as an approximate contraction scheme. We further show that the resultant approximation is equivalent to the ``mean field'' approximation that is used in the Simple-Update algorithm, thereby showing that the latter is a essentially the Bethe-Peierls approximation. This shows that one of the simplest approximate contraction algorithms for tensor networks is equivalent to one of the simplest schemes for approximating marginals in graphical models in general, and paves the way for using improvements of BP as tensor networks algorithms.

Statistical Mechanical Approximations to More Efficiently Determine Polymorph Free Energy Differences for Small Organic Molecules.

Methods to efficiently determine the relative stability of polymorphs of organic crystals are highly desired in crystal structure predictions (CSPs). Current methodologies include calculating the free energy of static lattice phonons, quasi-harmonic approximations (QHA), and computing the full thermodynamic cycle using replica exchange molecular dynamics (REMD). We found that 13 out of the 29 systems minimized from experimental crystal structures restructured to a lower energy minimum when heated and annealed using REMD, a phenomenon that QHA alone cannot capture. Here, we present a series of methods that are intermediate in accuracy and expense between QHA and computing the full thermodynamic cycle, which can save 42-80% of the computational cost and introduces, on this benchmark, a relatively small (0.16 ± 0.04 kcal/mol) error relative to the full thermodynamic cycle. In particular, a method that Boltzmann weights harmonic free energies from along the trajectory of REMD replica exchange appears to be an appropriate intermediate between QHA and the full thermodynamic cycle using MD when screening crystal polymorph stability.

Microscopic theory of onset of decaging and bond-breaking activated dynamics in ultradense fluids with strong short-range attractions.

We theoretically study thermally activated "in cage" elementary dynamical processes that precede full structural relaxation in ultradense particle liquids interacting via strong short-range attractive forces. The analysis is based on a microscopic theory formulated at the particle trajectory level built on the dynamic free energy concept and an explicit treatment of how attractive forces control the formation and lifetime of physical bonds. Mean time scales for bond breaking, the early stage of cage escape, and non-Fickian displacement by a fixed amount are analyzed in the repulsive glass, bonded repulsive (attractive) glass, fluid, and dense gel regimes. The theory predicts a strong length-scale-dependent growth of these time scales with attractive force strength at fixed packing fraction, a much weaker slowing down with density at fixed attraction strength, and a strong decoupling of the shorter bond-breaking time with the other two time scales that are controlled mainly by perturbed steric caging. All results are in good accord with simulations, and additional testable predictions are made. The classic statistical mechanical projection approximation of replacing all bare attractive and repulsive forces with a single effective force determined by pair structure incurs major errors for describing processes associated with thermally activated escape from transiently localized states.

Statistical analysis of loopy belief propagation in random fields.

Loopy belief propagation (LBP), which is equivalent to the Bethe approximation in statistical mechanics, is a message-passing-type inference method that is widely used to analyze systems based on Markov random fields (MRFs). In this paper, we propose a message-passing-type method to analytically evaluate the quenched average of LBP in random fields by using the replica cluster variation method. The proposed analytical method is applicable to general pairwise MRFs with random fields whose distributions differ from each other and can give the quenched averages of the Bethe free energies over random fields, which are consistent with numerical results. The order of its computational cost is equivalent to that of standard LBP. In the latter part of this paper, we describe the application of the proposed method to Bayesian image restoration, in which we observed that our theoretical results are in good agreement with the numerical results for natural images.

SHADOW DARK MATTER AS A MANIFESTATION OF i ↔ -i SYMMETRY IN PRE-QUANTUM TRACE DYNAMICS

We propose an alternate version of the "shadow world" hypothesis for the origin of dark matter. Instead of postulating that the shadow world is a "mirror" world under parity or charge-parity reflection, we suggest that the existence of a shadow world arises from the fact that i or –i can be the imaginary unit in complex quantum mechanics. This mechanism is a natural consequence of "trace dynamics" pre-quantum theory, from which quantum mechanics over the complex number field emerges as a statistical mechanical approximation. Because the pre-quantum dynamics does not pick out a preferred imaginary unit, the emergent quantum dynamics contains two sectors, one based on i and the other on –i, with both sectors coupled to gravity.

Optimized atomic statistical potentials: assessment of protein interfaces and loops.

Statistical potentials have been widely used for modeling whole proteins and their parts (e.g. sidechains and loops) as well as interactions between proteins, nucleic acids and small molecules. Here, we formulate the statistical potentials entirely within a statistical framework, avoiding questionable statistical mechanical assumptions and approximations, including a definition of the reference state. We derive a general Bayesian framework for inferring statistically optimized atomic potentials (SOAP) in which the reference state is replaced with data-driven 'recovery' functions. Moreover, we restrain the relative orientation between two covalent bonds instead of a simple distance between two atoms, in an effort to capture orientation-dependent interactions such as hydrogen bonds. To demonstrate this general approach, we computed statistical potentials for protein-protein docking (SOAP-PP) and loop modeling (SOAP-Loop). For docking, a near-native model is within the top 10 scoring models in 40% of the PatchDock benchmark cases, compared with 23 and 27% for the state-of-the-art ZDOCK and FireDock scoring functions, respectively. Similarly, for modeling 12-residue loops in the PLOP benchmark, the average main-chain root mean square deviation of the best scored conformations by SOAP-Loop is 1.5 Å, close to the average root mean square deviation of the best sampled conformations (1.2 Å) and significantly better than that selected by Rosetta (2.1 Å), DFIRE (2.3 Å), DOPE (2.5 Å) and PLOP scoring functions (3.0 Å). Our Bayesian framework may also result in more accurate statistical potentials for additional modeling applications, thus affording better leverage of the experimentally determined protein structures. SOAP-PP and SOAP-Loop are available as part of MODELLER (http://salilab.org/modeller).

SOME INEQUALITIES FOR THEORETICAL SPATIAL ECOLOGY

AbstractInequalities for spatial competition verify the pair approximation of statistical mechanics introduced to theoretical ecology by Matsuda, Satō and Iwasa, among others. Spatially continuous moment equations were introduced by Bolker and Pacala and use a similar assumption in derivation. In the present article, I prove upper bounds for the$k\mathrm{th} $central moment of occupied sites in the contact process of a single spatial dimension. This result shows why such moment closures are effective in spatial ecology.

Quasicontinuous approximation in classical statistical mechanics

Within the framework of classical statistical mechanics, we consider infinite continuous systems of point particles with strong superstable interaction. A family of approximate correlation functions is defined to take into account solely the configurations of particles in the space $$ {{\mathbb R}^d} $$ that contain at most one particle in each cube of a given partition of the space $$ {{\mathbb R}^d} $$ into disjoint hypercubes of volume a d : It is shown that the approximations of correlation functions thus defined are pointwise convergent to the exact correlation functions of the system if the parameter of approximation a approaches zero for any positive values of the inverse temperature β and fugacity z: This result is obtained both for two-body interaction potentials and for many-body interaction potentials.

Statistical performance analysis by loopy belief propagation in Bayesian image modeling

The mathematical structures of loopy belief propagation are reviewed for Bayesian image modeling from the standpoint of statistical mechanical informatics. We propose some schemes for evaluating the statistical performance of probabilistic binary image restoration. The schemes are constructed by means of the LBP, which is known as the Bethe approximation in statistical mechanics. We show some results of numerical experiments obtained by using the LBP algorithm as well as the statistical performance analysis for the probabilistic image restorations.

Thermodynamic mechanism for solution phase chiral amplification via a lattice model

We formulate a two-dimensional lattice model to study the equilibrium phase behavior of a ternary mixture composed of two enantiomeric forms of a chiral molecule and a nonchiral liquid solvent. Numerical solution of the model invoking the mean-field approximation of statistical mechanics allows the calculation of a ternary phase diagram. A prominent feature of the phase diagram is the appearance of a mirror-image pair of triple points involving coexistence of a liquid phase enriched in one of the enantiomers with two solid phases: a racemic and an enantiopure crystal. Thus, over broad ranges of initial composition, including liquid mixtures containing almost equal amounts of the two enantiomers, the equilibrium state of the system produces liquid-phase chiral amplification. The calculations predict that chiral amplification is favored at low temperatures, and by strengthening those molecular interactions that stabilize the racemic crystal. The phase behavior that we obtain is qualitatively identical to that reported in a recent experimental study of solutions of amino acids, aimed at probing the liquid-phase control of asymmetric catalysis. Those results, and the present calculations that provide molecular-level insight into their underlying causes, suggest a possible thermodynamic scenario for the liquid-phase emergence of chiral imbalance in a prebiotic and presumably nearly racemic world.

Methods for calculating the desorption rate of molecules from a surface at non-zero coverage: Water on MgO(0 0 1)

We present calculations of the desorption rate of water molecules from MgO(0 0 1) at a range of coverages θ and temperatures T. Our aim is to demonstrate that this can be done without making uncontrollable statistical mechanical approximations, and we achieve this by using the potential of mean force method reported previously. As in our earlier work on desorption of isolated molecules, we use a classical interaction model. We find that correlations between adsorbed molecules greatly increase the simulation time needed to obtain good statistical accuracy, compared with the isolated molecule. The activation energy for desorption varies significantly with coverage. The calculations also yield the chemical potential of adsorbed molecules as a function of θ and T, from which we can deduce the critical temperature and coverage for phase separation of adsorbed molecules.

Spectral density of complex networks with a finite mean degree

In order to clarify the statistical features of complex networks, the spectral density of adjacency matrices has often been investigated. Adopting a static model introduced by Goh, Kahng and Kim, we analyse the spectral density of complex scale free networks. For this purpose, we utilize the replica method and effective medium approximation (EMA) in statistical mechanics. As a result, we identify a new integral equation which determines the asymptotic spectral density of scale free networks with a finite mean degree p. In the limit p → ∞, known asymptotic formulae are rederived. Moreover, the 1/p corrections to known results are analytically calculated by a perturbative method.

Quantum mechanics as an approximation of statistical mechanics for classical fields

We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the term of the second order. To escape technical difficulties related to the infinite dimension of phase space for quantum mechanics, we start with a detailed presentation of our approach for the finite-dimensional quantum mechanics. We also separate real and complex cases, because the reproduction of the complex structure of quantum mechanics is a special problem which is not related to approximation of classical averages. In our approach quantum mechanics is an approximative theory. It predicts statistical averages only with some precision. In principle, there might be found deviations of averages calculated within the quantum formalism from experimental averages (which are supposed to be equal to classical averages given by our model).

Ab initio statistical mechanics of surface adsorption and desorption. I. H2O on MgO (001) at low coverage

We present a general computational scheme based on molecular dynamics (MD) simulation for calculating the chemical potential of adsorbed molecules in thermal equilibrium on the surface of a material. The scheme is based on the calculation of the mean force in MD simulations in which the height of a chosen molecule above the surface is constrained and subsequent integration of the mean force to obtain the potential of mean force and hence the chemical potential. The scheme is valid at any coverage and temperature, so that in principle it allows the calculation of the chemical potential as a function of coverage and temperature. It avoids all statistical mechanical approximations, except for the use of classical statistical mechanics for the nuclei, and assumes nothing in advance about the adsorption sites. From the chemical potential, the absolute desorption rate of the molecules can be computed, provided that the equilibration rate on the surface is faster than the desorption rate. We apply the theory by ab initio MD simulation to the case of H2O on MgO (001) in the low-coverage limit, using the Perdew-Burke-Ernzerhof (PBE) form of exchange correlation. The calculations yield an ab initio value of the Polanyi-Wigner frequency prefactor, which is more than two orders of magnitude greater than the value of 10(13) s(-1) often assumed in the past. Provisional comparison with experiment suggests that the PBE adsorption energy may be too low, but the extension of the calculations to higher coverages is needed before firm conclusions can be drawn. The possibility of including quantum nuclear effects by using path-integral simulations is noted.

Quantum mechanics as the quadratic Taylor approximation of classical mechanics: The finite-dimensional case

We show that in contrast to a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. We consider an approximation based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the second-order term. To escape technical difficulties related to the infinite dimensionality of the phase space for quantum mechanics, we consider finite-dimensional quantum mechanics. On one hand, this is a simple example with high pedagogical value. On the other hand, quantum information operates in a finite-dimensional state space. Therefore, our investigation can be considered a construction of a classical statistical model for quantum information.

Accuracy of the Bethe approximation for hyperparameter estimation in probabilistic image processing

We investigate the accuracy of statistical-mechanical approximations for the estimation of hyperparameters from observable data in probabilistic image processing, which is based on Bayesian statistics and maximum likelihood estimation. Hyperparameters in statistical science correspond to interactions or external fields in the statistical-mechanics context. In this paper, hyperparameters in the probabilistic model are determined so as to maximize a marginal likelihood. A practical algorithm is described for grey-level image restoration based on a Gaussian graphical model and the Bethe approximation. The algorithm corresponds to loopy belief propagation in artificial intelligence. We examine the accuracy of hyperparameter estimation when we use the Bethe approximation. It is well known that a practical algorithm for probabilistic image processing can be prescribed analytically when a Gaussian graphical model is adopted as a prior probabilistic model in Bayes' formula. We are therefore able to compare, in a numerical study, results obtained through mean-field-type approximations with those based on exact calculation.

04/00542 Peas in a pod: quasi-one-dimensional C 60 molecules in a nanotube : Calbi, M. M. et al. Los Alamos National Laboratory, Preprint Archive, Condensed Matter, 2003, 1–12.

We evaluate the equation of state of the quasi-one-dimensional (1D) phase of C60 molecules in small carbon nanotubes, nicknamed ``peas in a pod''. The chemical potential and 1D pressure are evaluated as functions of the temperature and density, initially with the approximation of nearest neighbor interactions and classical statistical mechanics. Quantum corrections and long-range interaction corrections are discussed, as are the effects of interactions with neighboring peapods. Transition phenomena involving the 3D coupling are evaluated.

Statistical mechanics of interacting peapods

We evaluate the equation of state of the quasi-one-dimensional (1D) phase of ${\mathrm{C}}_{60}$ molecules in small carbon nanotubes, nicknamed peapods. The chemical potential and 1D pressure are evaluated as functions of the temperature and density, initially with the approximation of nearest-neighbor interactions and classical statistical mechanics. Quantum corrections and long-range interaction corrections are discussed, as are the effects of interactions with neighboring peapods. Transition phenomena involving the 3D coupling are evaluated.

Globally accurate theory of structure and thermodynamics for soft-matter liquids

Standard statistical mechanical approximations (e.g. mean-field approximations) for pair-correlation functions of strongly interacting systems that yield adequate thermodynamics away from critical points typically break down badly in critical regions. The self-consistent Ornstein–Zernike approximation (SCOZA) transcends this difficulty, yielding globally accurate structure and thermodynamics. The SCOZA has been applied successfully to a variety of Hamiltonian models and the result will be briefly summarized. We end with a progress report on the applications of the SCOZA to some soft-matter systems.