Gibbs free energies for phases in the system Al2O3-SiO2-H2O have been calculated from reversed experiments in order to correct earlier values and to calculate a phase diagram consistent with more recent experiments. An internally consistent diagram could not be calculated that agreed with all published experiments, and choices of preferred data were made. The following Gibbs free energies, relative to the elements at STP (298.15 K, 1 bar), have been derived

The above values were calculated assuming literature values for corundum, quartz, and H2O (v).

Examination of available thermodynamic, experimental, and observational data on the aluminum hydroxides gibbsite, boehmite, bayerite, and nordstrandite suggests that these minerals are metastable with respect to diaspore + water at STP and at higher temperatures. Similarly, halloysite and dickite are metastable with respect to kaolinite at these conditions. The occurrence of these minerals in soils must therefore be ascribed to nonequilibrium processes, and the use of equilibrium phase diagrams to explain their occurrence is inappropriate.

Diagenetic clay minerals usually occur as heterogeneous assemblages of submicroscopic layers consisting of different structure types such as illite, smectite and chlorite, with variable composition within a given structure type, and with highly variable concentrations of imperfections. The dimensions of mixed-layering, the semi-coherent to coherent nature of the structures across the layering, and compositional heterogeneity occur at a scale well below that of an individual thermodynamic phase. These relations imply that most clays are not distinct minerals or phases, and that assemblages of clays in shales and mudstones are incompatible with the phase rule. Such relations are better evaluated in terms of the formation of metastable materials with each small unit having unique chemical properties, rather than as a small number of stable homogeneous phases. Consequently, treatment of most clay minerals in terms of equilibrium stability with either a thermodynamic or experimental approach is subject to error.

Chemical reactions involving most clay minerals are best understood with kinetic models. These involve a great variety of parameters such as time, fluid/rock ratio, deformation history, nature of starting materials and transformation mechanisms, as well as the variables, such as temperature, pressure and composition, that are commonly used to define equilibrium. Solubility experiments on the stabilities of clay minerals are unlikely to attain equilibrium at low temperatures. Moreover, the activity of soluble species may be controlled by surface equilibria, or by absorbed or exchangeable cations. Interpretations of available experiments on the solubility of illite vs. other mineral assemblages are in violation of Schreinemakers’ rules and indicate lack of equilibrium.

Predictable sequences of clay minerals as a function of temperature are best understood through the Ostwald step rule, in which clay mineral assemblages undergo reactions in response to kinetic factors that represent reaction progress rather than an approach to equilibrium. Currently used clay mineral thermometers (illite crystallinity, smectite/illite reaction, chlorite composition) are not based on equilibrium reactions. Such systems are not accurate thermometers and therefore have questionable utility.

A review of the models proffered to advance the notion of the metastability of illite shows that these models are not supported by the various data groups that have become available. Given that clay minerals are products of water–rock interactions, low-temperature hydrothermal experiments provide singular insights into their relative stabilities; such experiments with natural materials of contrasting pedigree (illites, sericites, muscovites, and chlorites) show that clay-mineral behaviors in low-temperature hydrothermal solutions are amenable to equilibrium thermodynamic conventions. The data from hydrothermal experiments coupled with data from geothermal fields indicate that muscovite is not a stable phase in the P-T-X range in which authigenic illite occurs; given that experimental data and field occurrence suggest that muscovite and illite have different P-T stability regimes, the continued use of muscovite as a proxy for illite in thermodynamic models is of questionable utility. Furthermore, morphometric studies of clays undergoing illitization show that crystal-size distributions exhibit log-normal patterns. Because log-normal distributions derive from maximum entropy effects, these crystal-size distributions may reflect the effects of entropy production during crystallization rather than kinetically driven Ostwald ripening of illitic phases; the small crystal size of clay minerals may derive from constraints imposed by the physicochemical conditions of their environments of formation. Presumably, irreversible thermodynamics provides the framework for a quantitative understanding of the evolution of complex clay minerals in space and time.

The first occurrence of Ni-rich stevensite found in the ophiolite complex of Othrys, Central Greece is described. The stevensite, which develops in cracks in a host serpentinite, formed at the expense of serpentine. Two varieties of stevensite have been described: a Mg-rich, Ni-poorer variety with 0.4–1.2 octahedral Ni atoms per half formula unit (p.h.f.u.) and a Ni-rich variety with >2 Ni atoms p.h.f.u. The layer charge in both varieties is −0.24 p.h.f.u.. Stevensite layers are completely separated when dispersed in dilute polyvinylpyrrolidone (PVP) solutions and begin to convert to talc after heating at 250°C for 90 min. Total conversion to talc is observed at 550°C. Formation of Ni-rich stevensite took place at ambient temperature during supergene processes. The scarcity of Ni-rich stevensite occurrences in nature is attributed to the metastability of smectite and to the analytical procedures used in previous studies. Stevensite is considered a phase containing domains with variable numbers of octahedral vacancies. A new experimental protocol is proposed for the determination of Ni-rich stevensite, based on a combination of XRD after solvation with various organic liquids and subsequent heating at 750°C.

In this paper, we propose new Metropolis–Hastings and simulated annealing algorithms on a finite state space via modifying the energy landscape. The core idea of landscape modification rests on introducing a parameter c, such that the landscape is modified once the algorithm is above this threshold parameter to encourage exploration, while the original landscape is utilized when the algorithm is below the threshold for exploitation purposes. We illustrate the power and benefits of landscape modification by investigating its effect on the classical Curie–Weiss model with Glauber dynamics and external magnetic field in the subcritical regime. This leads to a landscape-modified mean-field equation, and with appropriate choice of c the free energy landscape can be transformed from a double-well into a single-well landscape, while the location of the global minimum is preserved on the modified landscape. Consequently, running algorithms on the modified landscape can improve the convergence to the ground state in the Curie–Weiss model. In the setting of simulated annealing, we demonstrate that landscape modification can yield improved or even subexponential mean tunnelling time between global minima in the low-temperature regime by appropriate choice of c, and we give a convergence guarantee using an improved logarithmic cooling schedule with reduced critical height. We also discuss connections between landscape modification and other acceleration techniques, such as Catoni’s energy transformation algorithm, preconditioning, importance sampling, and quantum annealing. The technique developed in this paper is not limited to simulated annealing, but is broadly applicable to any difference-based discrete optimization algorithm by a change of landscape.

This article examines large-time behaviour of finite-state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) of the time required for convergence of the empirical measure process of the N-particle system to its invariant measure; we show that when time is of the order $\exp\{N\Lambda\}$ for a suitable constant $\Lambda > 0$ , the process has mixed well and it is close to its invariant measure. We then obtain large-N asymptotics of the second-largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as $\exp\{{-}N\Lambda\}$ . The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-N limit. As an application of the study of large-time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain ‘entropy’ function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.

In this doctoral thesis, we show how the bounded functional interpretation of F. Ferreira and P. Oliva can be used and contribute to the Proof Mining program, a program which aims to extract computational information from mathematical theorems using proof-theoretic techniques. We present a method for the elimination of sequential weak compactness arguments from the quantitative analysis of certain mathematical results. This method works as a “macro” and allowed us to obtain quantitative versions of important results of F. E. Browder, R. Wittmann, and H. H. Bauschke in fixed point theory in Hilbert spaces. Although the theorems of Browder and Wittmann were previously analyzed by U. Kohlenbach using the monotone functional interpretation, it was not clear why such analyses did not require the use of functionals defined by bar recursion. This phenomenon is now fully understood by a theoretical justification for the elimination of sequential weak compactness in the context of the bounded functional interpretation. Bauschke’s theorem is an important generalization of Wittmann’s theorem and its original proof is also analyzed here. The analyses of these results also require a quantitative version of a projection argument which turned out to be simpler when guided by the bounded functional interpretation than when using the monotone functional interpretation. In the context of the theory of monotone operators, results due to Boikanyo/Moroşanu and Xu for the strong convergence of variants of the proximal point algorithm are analyzed and bounds on the metastablility property of these iterations are obtained. These results are the first applications of the bounded functional interpretation to the proof mining of concrete mathematical results.

Abstract prepared by Pedro Pinto.

E-mail: pinto@mathematik.tu-darmstadt.de

URL: http://hdl.handle.net/10451/42661

We extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong convergence for Cesàro means of non-expansive maps on Banach spaces.

Detailed balance of a chemical reaction network can be defined in several different ways. Here we investigate the relationship among four types of detailed balance conditions: deterministic, stochastic, local, and zero-order local detailed balance. We show that the four types of detailed balance are equivalent when different reactions lead to different species changes and are not equivalent when some different reactions lead to the same species change. Under the condition of local detailed balance, we further show that the system has a global potential defined over the whole space, which plays a central role in the large deviation theory and the Freidlin–Wentzell-type metastability theory of chemical reaction networks. Finally, we provide a new sufficient condition for stochastic detailed balance, which is applied to construct a class of high-dimensional chemical reaction networks that both satisfies stochastic detailed balance and displays multistability.

A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit metastable mixing with modes$S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min\!\big(\Phi_{\beta}\big(S_{\beta}^{(1)}\big), \ldots, \Phi_{\beta}\big(S_{\beta}^{(k)}\big)\big)$ of its modes for all large values of $\beta$. We give simple sufficient conditions for a family of Markov chains to exhibit metastability in this sense, and verify that these conditions hold for a prototypical Metropolis–Hastings chain targeting a mixture distribution. The existing metastability literature is large, and our present work is aimed at filling the following small gap: finding sufficient conditions for metastability that are easy to verify for typical examples from statistics using well-studied methods, while at the same time giving an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp). Our bounds from this paper are used in a companion paper (O. Mangoubi, N. S. Pillai, and A. Smith, arXiv:1808.03230) to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions.

Classical alloy design strategies often aim to benefit from metastability. Examples are numerous: metastable transformation- and twinning-induced plasticity steels, cobalt or titanium based alloys, age hardenable aluminum alloys, and severe plastic deformed nanostructured copper. In each of these cases, superior engineering property combinations are achieved by exploring limits of stability. For the case of high-entropy alloys (HEAs), on the other hand, majority of present research efforts focus on exploring compositions that would yield stable single-phase structures. HEA metastability and its effects on microstructure and property development constitute only a relatively small fraction of ongoing work. To help motivate and guide a corresponding shift in HEA research efforts, here in this paper, we provide an overview of the research activities on metastability in HEAs. To this end, we categorize the past research on the topic into two groups based on their focus, namely, compositional and structural stability, and discuss the most relevant and exciting findings.

Experimental breakdown of staurolite + inclusions of albite and muscovite (representing an aluminous Fe-rich metapelitic composition) occurs within the KNFMASH system at 800°C and 0.1 GPa under fluid-present conditions over periods of 2 days to 8 weeks. The overall melt-producing reaction is staurolite + muscovite + plagioclase + vapour = hercynite + corundum + mullite + quartz + liquid, and involves three reactions; staurolite + vapour = hercynite + corundum + quartz + liquid, albite + vapour = mullite + liquid, and muscovite + vapour = mullite + liquid. Melt (glass) compositions are siliceous (77.2–64.5 wt.% SiO2), peraluminous (A/CNK = 1.5–5.5), and have granite-trondjhemite compositions in terms of normative Ab-Ab-Or contents. Textures indicate that quartz formed later than corundum. The corundum + quartz + hercynite assemblage after staurolite is considered to be metastable and only likely to be found in nature where staurolite-bearing rocks have undergone rapid heating and melting followed by quenching.

We construct Cheeger-type bounds for the second eigenvalue of a substochastic transition probability matrix in terms of the Markov chain's conductance and metastability (and vice versa) with respect to its quasistationary distribution, extending classical results for stochastic transition matrices.

Many complex systems occurring in various application share the property that theunderlying Markov process remains in certain regions of the state space for long times,and that transitions between such metastable sets occur only rarely.Often the dynamics within each metastable set is of minor importance, but the transitions between these sets are crucial for the behavior and the understanding of the system.Since simulations of the original process are usually prohibitively expensive, the effective dynamics of the system, i.e. the switching between metastable sets, has to be approximated in a reliable way.This is usually done by computing the dominant eigenvectors and eigenvalues of the transferoperator associated to the Markov process. In many real applications, however, the matrix representing the spatially discretized transfer operator can be extremely large, such that approximating eigenvectors and eigenvaluesis a computationally critical problem.In this article we present a novel method to determine the effective dynamics via the transfer operator without computing its dominant spectral elements. The main idea is that a time series of the process allows to approximate the sampling kernel of the process, which is an integral kernel closely related to the transition function of the transfer operator.Metastability is taken into account by representing the approximative sampling kernel by a linear combination of kernels each of which represents the process on one of the metastable sets.The effect of the approximation error on the dynamics of the system is discussed,and the potential of the new approach is illustrated by numerical examples.

We consider a dynamical system in $\mathbb{R}$ driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Lévy noise of small intensity and such that the heaviest tail of its Lévy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tailnature of the random perturbation, the results differ strongly from the well studied purely Gaussian case.

This article reviews recent advances in understanding amorphous glassy states in dense colloidal suspensions with or without short-range interparticle attractions. Experiments, theory, and simulation show that two kinds of glassy states are possible, dominated respectively by repulsion and attraction. Under suitable conditions, a small change in the interparticle potential can lead to a transition between these two kinds of colloidal glasses that entails sharp changes in material properties such as the shear modulus. This may provide novel routes for fine-tuning the properties of industrial pastes and slurries.