Free energies for materials with memory in terms of state functionals

A peridynamic approach to computation of elastic and entropic interactions of inclusions on a lipid membrane

Energy-stable boundary conditions based on a quadratic form: Applications to outflow/open-boundary problems in incompressible flows

In general, materials with linear memory constitutive relations are characterized by a relaxation function. This leads to a situation where the free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. An alternative procedure is proposed which characterizes a material by the kernel of the rate of dissipation functional. Using some recent results, we find that a unique free energy and relaxation function may then be deduced. An example is given for discrete spectrum materials. Also, the new results are used to show that a previously derived general representation of rate of dissipation and free energy functionals is not complete, in the sense that there are valid functionals which cannot be described by this general formula.

The analysis of linear elastic structures having cable-like members, that is members which can take tension but not compression, is formulated as one in quadratic programming. The objective function is the total complementary energy function, a quadratic form in terms of the variables which are forces and force resultants, and the constraints are the equations of equilibrium of forces at the joints. A technique is indicated to reduce the size of the quadratic program obtained so that the number of variables in the reduced program is approximately the larger of the degree of indeterminacy and the number of the cable-like members. The formulation is performed entirely for a truss.

This report describes a general yield function for anisotropic elastoplastic materials written in a quadratic form in terms of the stress components. The function contains 28 independent coefficients that must be determined experimentally. A set of inequality constraint conditions imposed on these coefficients is derived from the convexity requirement for all yield functions. A number of previously derived yield functions are recognized here as special cases of the general quadratic yield function, and the convexity condition for each of them is also provided. These inequality conditions should aid experimental projects for material identification.

We consider two quadratic forms on a class of manifolds conformally equivalent to the euclidean space. We give necessary and sufficient conditions for the equivalence of these quadratic forms in terms of a weighted Hardy inequality.

Minkowski established a lower bound for the determinant D of a Minkowski-reduced quadratic form in terms of the product of its diagonal coefficients aii (i = 1, …n). Oppenheim and Barnes found, for n = 3 and n = 4 respectively, the precise minimum of D in terms of the aii; in each case the minimum is a polynomial in the aii. Here it is shown that no such result exists when n = 5; however a polynomial in all,…, a55 is determined which gives the minimum of D when a55 is sufficiently large.

In an attempt to improve automated gene prediction in the untranslated region of a gene, we completed an in-depth analysis of the minimum free energy for 8,689 sub-genetic DNA sequences. We expanded Zhang's classification model and classified each sub-genetic sequence into one of 27 possible motifs. We calculated the minimum free energy for each motif to explore statistical features that correlate to biologically relevant sub-genetic sequences. If biologically relevant sub-genetic sequences fall into distinct free energy quanta it may be possible to characterize a motif based on its minimum free energy. Proper characterization of motifs can lead to greater understanding in automated genefinding, gene variability and the role DNA structure plays in gene network regulation. Our analysis determined: (1) the average free energy value for exons, introns and other biologically relevant sub-genetic sequences, (2) that these subsequences do not exist in distinct energy quanta, (3) that introns exist however in a tightly coupled average minimum free energy quantum compared to all other biologically relevant sub-genetic sequence types, (4) that single exon genes demonstrate a higher stability than exons which span the entire coding sequence as part of a multi-exon gene and (5) that all motif types contain a free energy global minimum at approximately nucleotide position 1,000 before reaching a plateau. These results should be relevant to the biochemist and bioinformatician seeking to understand the relationship between sub-genetic sequences and the information behind them.

We study a phase-field variational model for the solvaiton of charged molecules with an implicit solvent. The solvation free-energy functional of all phase fields consists of the surface energy, solute excluded volume and solute-solvent van der Waals dispersion energy, and electrostatic free energy. The surface energy is defined by the van der Waals--Cahn--Hilliard functional with squared gradient and a double-well potential. The electrostatic part of free energy is defined through the electrostatic potential governed by the Poisson--Boltzmann equation in which the dielectric coefficient is defined through the underlying phase field. We prove the continuity of the electrostatics---its potential, free energy, and dielectric boundary force---with respect to the perturbation of dielectric boundary. We also prove the $\Gamma$-convergence of the phase-field free-energy functionals to their sharp-interface limit, and the equivalence of the convergence of total free energies to that of all individual parts of free energy. We finally prove the convergence of phase-field forces to their sharp-interface limit. Such forces are defined as the negative first variations of the free-energy functional; and arise from stress tensors. In particular, we obtain the force convergence for the van der Waals--Cahn--Hilliard functionals with minimal assumptions.

The minimum free energy (MFE) of ribonucleic acids (RNAs) increases at an apparent linear rate with sequence length. Simple indices, obtained by dividing the MFE by the number of nucleotides, have been used for a direct comparison of the folding stability of RNAs of various sizes. Although this normalization procedure has been used in several studies, the relationship between normalized MFE and length has not yet been investigated in detail. Here, we demonstrate that the variation of MFE with sequence length is not linear and is significantly biased by the mathematical formula used for the normalization procedure. For this reason, the normalized MFEs strongly decrease as hyperbolic functions of length and produce unreliable results when applied for the comparison of sequences with different sizes. We also propose a simple modification of the normalization formula that corrects the bias enabling the use of the normalized MFE for RNAs longer than 40 nt. Using the new corrected normalized index, we analyzed the folding free energies of different human RNA families showing that most of them present an average MFE density more negative than expected for a typical genomic sequence. Furthermore, we found that a well-defined and restricted range of MFE density characterizes each RNA family, suggesting the use of our corrected normalized index to improve RNA prediction algorithms. Finally, in coding and functional human RNAs the MFE density appears scarcely correlated with sequence length, consistent with a negligible role of thermodynamic stability demands in determining RNA size.

We develop a novel data-driven approach to the inverse problem of classical statistical mechanics: Given the experimental data on the collective motion of a classical many-body system, how does one characterize the free energy landscape of that system? By combining non-parametric Bayesian inference with physically motivated constraints, we develop an efficient learning algorithm that automates the construction of approximate free-energy functionals. In contrast to optimization-based machine learning approaches, which seek to minimize a cost function, the central idea of the proposed Bayesian inference is to propagate a set of prior assumptions through the model, derived from physical principles. The experimental data are used to probabilistically weigh the possible model predictions. This naturally leads to humanly interpretable algorithms with full uncertainty quantification of predictions. In our case, the output of the learning algorithm is a probability distribution over a family of free energy functionals, consistent with the observed particle data. We find that surprisingly small data samples contain sufficient information for inferring highly accurate analytic expressions of the underlying free-energy functionals, making our algorithm highly data efficient. In particular, we consider classical particle systems with excluded volume interactions, which are ubiquitous in nature, while being highly challenging in terms of free energy modeling. We validate our approach on the paradigmatic case of one-dimensional fluid and develop inference algorithms for the canonical and grand-canonical statistical-mechanical ensembles. Extensions to higher dimensional systems are conceptually straightforward, while standard coarse-graining techniques allow one to easily incorporate attractive interactions.

Expanding on an earlier work that relied on linear force-free magnetic fields, we self-consistently derive the instantaneous free magnetic energy and relative magnetic helicity budgets of an unknown three-dimensional nonlinear force-free magnetic structure extending above a single, known lower-boundary magnetic field vector. The proposed method does not rely on the detailed knowledge of the three-dimensional field configuration but is general enough to employ only a magnetic connectivity matrix on the lower boundary. The calculation yields a minimum free magnetic energy and a relative magnetic helicity consistent with this free magnetic energy. The method is directly applicable to photospheric or chromospheric vector magnetograms of solar active regions. Upon validation, it basically reproduces magnetic energies and helicities obtained by well-known, but computationally more intensive and non-unique, methods relying on the extrapolated three-dimensional magnetic field vector. We apply the method to three active regions, calculating the photospheric connectivity matrices by means of simulated annealing, rather than a model-dependent nonlinear force-free extrapolation. For two of these regions we correct for the inherent linear force-free overestimation in free energy and relative helicity that is larger for larger, more eruptive, active regions. In the third studied region, our calculation can lead to a physical interpretation of observed eruptive manifestations. We conclude that the proposed method, including the proposed inference of the magnetic connectivity matrix, is practical enough to contribute to a physical interpretation of the dynamical evolution of solar active regions.

The free energy functions for electron transfer reactions in solution are explored using a previously developed microscopic simulation approach that provides a clear definition of these functions and the variable (the reaction coordinate) used as their argument. The issue of whether the curvatures of the two functions (which correspond to states with nonpolar and polar solutes) are different is given special attention. It is found, in contrast to some previous suggestions, that the curvatures of the two functions are quite similar, even when one would expect differences due to dielectric saturation effects, and that Marcus’ approximation (and, in fact, the linear response theory inherent in this approximation) provides a valid description of the solvent’s role in electron transfer reactions over a wide range of conditions. The present study demonstrates that direct simulations of the reactant and product states do not provide the data needed for determination of the free energy functions in high energy regions. On the other hand, a free energy perturbation approach developed in our early studies does allow one to obtain statistically significant data for these regions of high free energy. Inclusion of the solvent electronic polarizabilities in the potential results in a marked lowering of the solvent reorganization energy, but does not change the ratio of the curvatures of the free energy functions for the nonpolar and polar states. Finally, the effect of solvent tunneling is evaluated with the dispersed polaron method and shown to be rather small, and it is argued that the observed deviations from the classical Marcus theory in the inverted region are due to quantum mechanical tunneling associated with the solute vibrations, rather than to differences in the curvatures of the solvent free energy functions.

Protein molecules in solution have a broad distribution of enthalpy states. A good approximation to the distribution function for enthalpy states can be calculated, using the maximum-entropy method, from the moments of the distribution that, in turn, are obtained from the experimental temperature dependence of the heat capacity. In the present paper, we show that the enthalpy probability distribution can then be formulated in terms of a free energy function that gives the free energy of the protein corresponding to a particular value of the enthalpy. By the location of the minima in this function, the free energy distribution graphically indicates the most probable values of the enthalpy for the protein. We find that the behavior of the free energy functions for proteins falls somewhere between two different cases: a two-state like function with two minima, the relative levels of the two states changing with temperature; and, a single-minimum function where the position of the minimum shifts to higher enthalpy values as the temperature is increased. We show that the temperature dependence of the free energy function can be expressed in terms of a central free energy distribution for a given, fixed temperature (which is most conveniently chosen as the temperature of the maximum in the heat capacity). The nature of this central free energy function for a given protein thus yields all of the thermodynamic behavior of that protein over the temperature range of the denaturation process.

A material with memory typically has a set of many free energy functionals associated with it, all members of which yield the same constitutive relations. An alternative interpretation of this set is explored in the present work. Explicit formulae are derived for the free energy and total dissipation of an arbitrary material in the cases of step function and sinusoidal/exponential histories. Expressions for the fraction of stored and dissipated energy are deduced. Also, various formulae are given for discrete spectrum materials. For materials with relaxation function containing one decaying exponential, the associated Day functional is the physical free energy. For more general materials, we seek a best fit of the relaxation function with one decaying exponential to that chosen for the general case. The free energy, total dissipation and fractions of stored and dissipated energies relating to the Day material are derived for the various histories. Similar data, in the case of the general material, are explored for the minimum and maximum free energies and also for a centrally located free energy given in the literature. Various plots of aspects of this data, including comparisons between the behaviour for general and Day materials, are presented and discussed.