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We formulate a thermodynamic theory applicable to both classical and quantum systems. These systems are depicted as thermodynamic system–bath models capable of handling isothermal, isentropic, thermostatic, and entropic processes. Our approach is based on the use of a dimensionless thermodynamic potential expressed as a function of the intensive and extensive thermodynamic variables. Using the principles of dimensionless minimum work and dimensionless maximum entropy derived from quasi-static changes of external perturbations and temperature, we obtain the Massieu–Planck potentials as entropic potentials and the Helmholtz–Gibbs potentials as free energy. These potentials can be interconverted through time-dependent Legendre transformations. Our results are verified numerically for an anharmonic Brownian system described in phase space using the low-temperature quantum Fokker–Planck equations in the quantum case and the Kramers equation in the classical case, both developed for the thermodynamic system–bath model. Thus, we clarify the conditions for thermodynamics to be valid even for small systems described by Hamiltonians and establish a basis for extending thermodynamics to non-equilibrium conditions. [ABSTRACT FROM AUTHOR]

Nonlinear optical methods, such as vibrational sum frequency generation (vSFG) and second harmonic generation (SHG), are powerful techniques to study elusive structures at charged buried interfaces. However, for the separation and determination of the Stern and diffuse layer spectra at these charged interfaces, complex vSFG spectra and, hence, the absolute phase need to be retrieved. The maximum entropy method is a useful tool for the retrieval of complex spectra from the intensity spectra; however, one caveat is that an understanding of the error phase is required. Here, for the first time, we provide a physically motivated understanding of the error phase. Determining the error phase from simulated spectra of oscillators with a spectral overlap, we show that for broadband vSFG spectra, such as for the silica/water interface, the diffuse and Stern layers' spectral overlap within the O–H stretching window results in a correlation between the error phase and the phase shift between the responses of these layers. This correlation makes the error phase sensitive to changes in Debye length from varying the ionic strength among other variations at the interface. Furthermore, the change in the magnitude of the error phase can be related to the absolute SHG phase, permitting the use of an error phase model that can utilize the SHG phase to predict the error phase and, hence, the complex vSFG spectra. Finally, we highlight limitations of this model for vSFG spectra with a poor overlap between the diffuse and Stern layer spectra (silica/HOD in D2O system). [ABSTRACT FROM AUTHOR]

Empirical force fields employed in molecular dynamics simulations of complex systems are often optimized to reproduce experimentally determined structural and thermodynamic properties. In contrast, experimental knowledge about the interconversion rates between metastable states in such systems is hardly ever incorporated in a force field due to a lack of an efficient approach. Here, we introduce such a framework based on the relationship between dynamical observables, such as rate constants, and the underlying molecular model parameters using the statistical mechanics of trajectories. Given a prior ensemble of molecular dynamics trajectories produced with imperfect force field parameters, the approach allows for the optimal adaption of these parameters such that the imposed constraint of equally predicted and experimental rate constant is obeyed. To do so, the method combines the continuum path ensemble maximum caliber approach with path reweighting methods for stochastic dynamics. When multiple solutions are found, the method selects automatically the combination that corresponds to the smallest perturbation of the entire path ensemble, as required by the maximum entropy principle. To show the validity of the approach, we illustrate the method on simple test systems undergoing rare event dynamics. Next to simple 2D potentials, we explore particle models representing molecular isomerization reactions and protein–ligand unbinding. Besides optimal interaction parameters, the methodology gives physical insights into what parts of the model are most sensitive to the kinetics. We discuss the generality and broad implications of the methodology. [ABSTRACT FROM AUTHOR]

A novel method combining the maximum entropy principle, the Bayesian-inference of ensembles approach, and the optimization of empirical forward models is presented. Here, we focus on the Karplus parameters for RNA systems, which relate the dihedral angles of γ, β, and the dihedrals in the sugar ring to the corresponding 3J-coupling signal between coupling protons. Extensive molecular simulations are performed on a set of RNA tetramers and hexamers and combined with available nucleic-magnetic-resonance data. Within the new framework, the sampled structural dynamics can be reweighted to match experimental data while the error arising from inaccuracies in the forward models can be corrected simultaneously and consequently does not leak into the reweighted ensemble. Carefully crafted cross-validation procedure and regularization terms enable obtaining transferable Karplus parameters. Our approach identifies the optimal regularization strength and new sets of Karplus parameters balancing good agreement between simulations and experiments with minimal changes to the original ensemble. [ABSTRACT FROM AUTHOR]

The quantum harmonic oscillator is the fundamental building block to compute thermal properties of virtually any dielectric crystal at low temperatures in terms of phonons, extended further to cases with anharmonic couplings, or even disordered solids. In general, Path Integral Monte Carlo or Path Integral Molecular Dynamics methods are powerful tools to determine stochastically thermodynamic quantities without systematic bias, not relying on perturbative schemes. Addressing transport properties, for instance calculating thermal conductivity from PIMC, however, is substantially more difficult. Although correlation functions of current operators can be determined by PIMC from analytic continuation on the imaginary time axis, Bayesian methods are usually employed for the numerical inversion back to real time response functions. This task not only strongly relies on the accuracy of the PIMC data but also introduces noticeable dependence on the model used for the inversion. Here, we address both difficulties with care. In particular, we first devise improved estimators for current correlations, which substantially reduce the variance of the PIMC data. Next, we provide a neat statistical approach to the inversion problem, blending into a fresh workflow the classical stochastic maximum entropy method together with recent notions borrowed from statistical learning theory. We test our ideas on a single harmonic oscillator and a collection of oscillators with a continuous distribution of frequencies and provide indications of the performance of our method in the case of a particle in a double well potential. This work establishes solid grounds for an unbiased, fully quantum mechanical calculation of transport properties in solids. [ABSTRACT FROM AUTHOR]

Three-dimensional (3D) organization of the human genome plays an essential role in all DNA-templated processes, including gene transcription, gene regulation, and DNA replication. Computational modeling can be an effective way of building high-resolution genome structures and improving our understanding of these molecular processes. However, it faces significant challenges as the human genome consists of over 6 × 109 base pairs, a system size that exceeds the capacity of traditional modeling approaches. In this perspective, we review the progress that has been made in modeling the human genome. Coarse-grained models parameterized to reproduce experimental data via the maximum entropy optimization algorithm serve as effective means to study genome organization at various length scales. They have provided insight into the principles of whole-genome organization and enabled de novo predictions of chromosome structures from epigenetic modifications. Applications of these models at a near-atomistic resolution further revealed physicochemical interactions that drive the phase separation of disordered proteins and dictate chromatin stability in situ. We conclude with an outlook on the opportunities and challenges in studying chromosome dynamics. [ABSTRACT FROM AUTHOR]

Machine learning-based interatomic potentials are currently garnering a lot of attention as they strive to achieve the accuracy of electronic structure methods at the computational cost of empirical potentials. Given their generic functional forms, the transferability of these potentials is highly dependent on the quality of the training set, the generation of which can be highly labor-intensive. Good training sets should at once contain a very diverse set of configurations while avoiding redundancies that incur cost without providing benefits. We formalize these requirements in a local entropy-maximization framework and propose an automated sampling scheme to sample from this objective function. We show that this approach generates much more diverse training sets than unbiased sampling and is competitive with hand-crafted training sets. [ABSTRACT FROM AUTHOR]

We address the calculation of microcanonical reaction rates for processes involving significant nuclear quantum effects using ring-polymer molecular dynamics (RPMD), both with and without electronically non-adiabatic transitions. After illustrating the shortcomings of the naive free-particle direct-shooting method, in which the temperature of the internal ring-polymer modes is set to the translational energy scale, we investigate alternative strategies based on the expression for the microcanonical rate in terms of the inverse Laplace transform of the thermal reaction rate. It is shown that simple application of the stationary-phase approximation (SPA) dramatically improves the performance of the microcanonical rates using RPMD, particularly in the low-energy region where tunneling dominates. Using the SPA as a Bayesian prior, numerically exact RPMD microcanonical rates are then obtained using maximum entropy inversion of the thermal reaction rates for both electronically adiabatic and non-adiabatic model systems. Finally, the direct-shooting method is revisited using the SPA-determined temperature for the internal ring-polymer modes, leading to a simple, direct-simulation method with improved accuracy in the tunneling regime. This work suggests a general strategy for the extraction of microcanonical dynamical quantities from RPMD (or other approximate thermal) simulations. [ABSTRACT FROM AUTHOR]

We show that the generalized Boltzmann distribution is the only distribution for which the Gibbs-Shannon entropy equals the thermodynamic entropy. This result means that the thermodynamic entropy and the Gibbs-Shannon entropy are not generally equal, but rather the equality holds only in the special case where a system is in equilibrium with a reservoir. [ABSTRACT FROM AUTHOR]

Coarse-grained (CG) models allow efficient molecular simulation by reducing the degrees of freedom in the system. To recapitulate important physical properties, including many-body correlations at the CG resolution, an appropriate mapping from the atomistic to CG level is needed. Symmetry exhibited by molecules, especially when aspherical, can be lost upon coarse-graining due to the use of spherically symmetric CG effective potentials. This mismatch can be efficiently amended by imposing symmetry using virtual CG sites. However, there has been no rigorous bottom-up approach for constructing a many-body potential of mean force that governs the distribution of virtual CG sites. Herein, we demonstrate a statistical mechanical framework that extends a mapping scheme of CG systems involving virtual sites to provide a thermodynamically consistent CG model in the spirit of the principle of maximum entropy. Utilizing the extended framework, this work defines a center of symmetry (COS) mapping and applies it to benzene and toluene systems such that the planar symmetry of the aromatic ring is preserved by constructing two virtual sites along a normal vector. Compared to typical center of mass (COM) CG models, COS CG models correctly recapitulate radial and higher order correlations, e.g., orientational and three-body correlations. Moreover, we find that COS CG interactions from bulk phases are transferable to mixture phases, whereas conventional COM models deviate between the two states. This result suggests a systematic approach to construct more transferable CG models by conserving molecular symmetry, and the new protocol is further expected to capture other many-body correlations by utilizing virtual sites. [ABSTRACT FROM AUTHOR]

We formulate a nonequilibrium thermodynamic description for open chemical reaction networks (CRNs) described by a chemical master equation. The topological properties of the CRN and its conservation laws are shown to play a crucial role. They are used to decompose the entropy production into a potential change and two work contributions, the first due to time dependent changes in the externally controlled chemostats concentrations and the second due to flows maintained across the system by nonconservative forces. These two works jointly satisfy a Jarzynski and Crooks fluctuation theorem. In the absence of work, the potential is minimized by the dynamics as the system relaxes to equilibrium and its equilibrium value coincides with the maximum entropy principle. A generalized Landauer's principle also holds: the minimal work needed to create a nonequilibrium state is the relative entropy of that state to its equilibrium value reached in the absence of any work. [ABSTRACT FROM AUTHOR]

Entropy maximization methods that update a probability distribution P0(x) to a new distribution P(x) with the use of externally known, averaged constraints find use in diverse areas. Jaynes developed a Maximum Entropy Procedure (MEP) that is an objective approach to incorporate external data to update P0(x) to P(x). In this work, we consider the MEP in the context of external data known from a probability distribution versus that from a mean and a few higher moments. An immediate problem is that the conventional iterative Lagrange multiplier method, which relies on inverting a certain covariance matrix, is not applicable here because the covariance matrix is not invertible. We introduce an indicator function method that does not suffer from this problem. It leads to an analytic solution to this version of a MEP. As an example, a previously generated ensemble of peptide conformations used to characterize an intrinsically disordered protein is analyzed. The external constraint is on the radius of gyration probability distribution, p(RG), of this peptide. Ensemble observables such as geometric, shape characteristics, the residue end-to-end distance distribution, the all atom-pair distribution function related to the scattering intensity, the polyproline II content, and NMR 3JHNHα three bond couplings are evaluated with the initial and updated ensembles. Some observables are found to be insensitive and others sensitive to the external information. An example of a 24-residue peptide, histatin 5, where an experimentally derived p(RG) is available, is also analyzed. [ABSTRACT FROM AUTHOR]

Finding reduced models of spatially distributed chemical reaction networks requires an estimation of which effective dynamics are relevant. We propose a machine learning approach to this coarse graining problem, where a maximum entropy approximation is constructed that evolves slowly in time. The dynamical model governing the approximation is expressed as a functional, allowing a general treatment of spatial interactions. In contrast to typical machine learning approaches which estimate the interaction parameters of a graphical model, we derive Boltzmann-machine like learning algorithms to estimate directly the functionals dictating the time evolution of these parameters. By incorporating analytic solutions from simple reaction motifs, an efficient simulation method is demonstrated for systems ranging from toy problems to basic biologically relevant networks. The broadly applicable nature of our approach to learning spatial dynamics suggests promising applications to multiscale methods for spatial networks, as well as to further problems in machine learning. [ABSTRACT FROM AUTHOR]

We review here Maximum Caliber (Max Cal), a general variational principle for inferring distributions of paths in dynamical processes and networks. Max Cal is to dynamical trajectories what the principle of maximum entropy is to equilibrium states or stationary populations. In Max Cal, you maximize a path entropy over all possible pathways, subject to dynamical constraints, in order to predict relative path weights. Many well-known relationships of non-equilibrium statistical physics--such as the Green-Kubo fluctuation-dissipation relations, Onsager's reciprocal relations, and Prigogine's minimum entropy production--are limited to near-equilibrium processes. Max Cal is more general. While it can readily derive these results under those limits, Max Cal is also applicable far from equilibrium. We give examples of Max Cal as a method of inference about trajectory distributions from limited data, finding reaction coordinates in bio-molecular simulations, and modeling the complex dynamics of non-thermal systems such as gene regulatory networks or the collective firing of neurons. We also survey its basis in principle and some limitations. [ABSTRACT FROM AUTHOR]

Quasi-classical trajectory (QCT) calculations are used to study state-specific ro-vibrational energy exchange and dissociation in the O2 + O system. Atom-diatom collisions with energy between 0.1 and 20 eV are calculated with a double many body expansion potential energy surface by Varandas and Pais [Mol. Phys. 65, 843 (1988)]. Inelastic collisions favor mono-quantum vibrational transitions at translational energies above 1.3 eV although multi-quantum transitions are also important. Post-collision vibrational favoring decreases first exponentially and then linearly as Δv increases. Vibrationally elastic collisions (Δv = 0) favor small ΔJ transitions while vibrationally inelastic collisions have equilibrium post-collision rotational distributions. Dissociation exhibits both vibrational and rotational favoring. New vibrational-translational (VT), vibrational-rotational-translational (VRT) energy exchange, and dissociation models are developed based on QCT observations and maximum entropy considerations. Full set of parameters for state-to-state modeling of oxygen is presented. The VT energy exchange model describes 22 000 state-to-state vibrational cross sections using 11 parameters and reproduces vibrational relaxation rates within 30% in the 2500.20 000 K temperature range. The VRT model captures 80 × 106 state-to-state ro-vibrational cross sections using 19 parameters and reproduces vibrational relaxation rates within 60% in the 5000.15 000 K temperature range. The developed dissociation model reproduces state-specific and equilibrium dissociation rates within 25% using just 48 parameters. The maximum entropy framework makes it feasible to upscale ab initio simulation to full nonequilibrium flow calculations. [ABSTRACT FROM AUTHOR]

The study of correlated mutations in alignments of homologous proteins proved to be successful not only in the prediction of their native conformation but also in the development of a two-body effective potential between pairs of amino acids. In the present work, we extend the effective potential, introducing a many-body term based on the same theoretical framework, making use of a principle of maximum entropy. The extended potential performs better than the two-body one in predicting the energetic effect of 308 mutations in 14 proteins (including membrane proteins). The average value of the parameters of the many-body term correlates with the degree of hydrophobicity of the corresponding residues, suggesting that this term partly reflects the effect of the solvent. [ABSTRACT FROM AUTHOR]

There has been interest in finding a general variational principle for non-equilibrium statistical mechanics. We give evidence that Maximum Caliber (Max Cal) is such a principle. Max Cal, a variant of maximum entropy, predicts dynamical distribution functions by maximizing a path entropy subject to dynamical constraints, such as average fluxes. We first show that Max Cal leads to standard near-equilibrium results--including the Green-Kubo relations, Onsager's reciprocal relations of coupled flows, and Prigogine's principle of minimum entropy production--in a way that is particularly simple. We develop some generalizations of the Onsager and Prigogine results that apply arbitrarily far from equilibrium. Because Max Cal does not require any notion of "local equilibrium," or any notion of entropy dissipation, or temperature, or even any restriction to material physics, it is more general than many traditional approaches. It also applicable to flows and traffic on networks, for example. [ABSTRACT FROM AUTHOR]

An issue of general interest in computer simulations is to incorporate information from experiments into a structural model. An important caveat in pursuing this goal is to avoid corrupting the resulting model with spurious and arbitrary biases. While the problem of biasing thermodynamic ensembles can be formulated rigorously using the maximum entropy method introduced by Jaynes, the approach can be cumbersome in practical applications with the need to determine multiple unknown coefficients iteratively. A popular alternative strategy to incorporate the information from experiments is to rely on restrained-ensemble molecular dynamics simulations. However, the fundamental validity of this computational strategy remains in question. Here, it is demonstrated that the statistical distribution produced by restrained-ensemble simulations is formally consistent with the maximum entropy method of Jaynes. This clarifies the underlying conditions under which restrained-ensemble simulations will yield results that are consistent with the maximum entropy method. [ABSTRACT FROM AUTHOR]

The analytical continuation average spectrum method (ASM) and maximum entropy (MaxEnt) method are applied to the dynamic response of a noninteracting resonant level model within the framework of the Kubo formula for electric conductivity. The frequency dependent conductivity is inferred from the imaginary time current-current correlation function for a wide range of temperatures, gate voltages, and spectral densities representing the leads, and compared with exact results. We find that the MaxEnt provides more accurate results compared to the ASM over the full spectral range. [ABSTRACT FROM AUTHOR]

Generalized hydrodynamic moment hierarchies are derived which explicitly include nonequilibrium two-particle and higher-order correlations. The approach is adapted to strongly correlated media and nonequilibrium processes on short time scales which necessitate an explicit treatment of time-evolving correlations. Closure conditions for the extended moment hierarchies are formulated by a maximum-entropy approach, generalizing related closure procedures for kinetic equations. A self-consistent set of nonperturbative dynamical equations are thus obtained for a chosen set of single-particle and two-particle (and possibly higher-order) moments. Analytical results are derived for generalized Gaussian closures including the dynamic pair distribution function and a two-particle correction to the current density. The maximum-entropy closure conditions are found to involve the Kirkwood superposition approximation. [ABSTRACT FROM AUTHOR]

We study the mechanisms of the homogeneous crystal nucleation from the supercooled liquid to the crystal phase in the Gaussian core model for colloidal suspensions with the aim to find optimal reaction coordinates. We introduce a set of novel collective variables based on the local structure of particles. By applying likelihood maximization of the committor function for the reweighted path ensemble constructed by replica exchange transition interface sampling, we select the optimal reaction coordinates from the set of collective variables. We find that the size of the cloud of prestructured particles surrounding the crystalline nucleus enhances the description of the transition. Further, we show that the rearrangement of the inner core of the nucleus according to Ostwald's step rule is a separate process, independent of the growth of the nucleus. [ABSTRACT FROM AUTHOR]

Thirty years ago, Jaynes suggested a general theoretical approach to nonequilibrium statistical mechanics, called maximum caliber (MaxCal) [Annu. Rev. Phys. Chem. 31, 579 (1980)]. MaxCal is a variational principle for dynamics in the same spirit that maximum entropy is a variational principle for equilibrium statistical mechanics. Motivated by the success of maximum entropy inference methods for equilibrium problems, in this work the MaxCal formulation is applied to the inference of nonequilibrium processes. That is, given some time-dependent observables of a dynamical process, one constructs a model that reproduces these input data and moreover, predicts the underlying dynamics of the system. For example, the observables could be some time-resolved measurements of the folding of a protein, which are described by a few-state model of the free energy landscape of the system. MaxCal then calculates the probabilities of an ensemble of trajectories such that on average the data are reproduced. From this probability distribution, any dynamical quantity of the system can be calculated, including population probabilities, fluxes, or waiting time distributions. After briefly reviewing the formalism, the practical numerical implementation of MaxCal in the case of an inference problem is discussed. Adopting various few-state models of increasing complexity, it is demonstrated that the MaxCal principle indeed works as a practical method of inference: The scheme is fairly robust and yields correct results as long as the input data are sufficient. As the method is unbiased and general, it can deal with any kind of time dependency such as oscillatory transients and multitime decays. [ABSTRACT FROM AUTHOR]

The entropy theory of glass formation, which has previously been developed to describe general classes of polymeric glass-forming liquids, is extended here to model the thermodynamic and dynamic properties of poly(α-olefins). By combining this thermodynamic theory with the Adam–Gibbs model (which relates the configurational entropy to the rate of structural relaxation), we provide systematic computations for all four characteristic temperatures (TA, Tc, Tg, T0), governing the position and breadth of the glass transition, and the fragility parameters (D,m) describing the strength of the temperature dependence of the structural relaxation time, where TA is the temperature below which the relaxation is non-Arrhenius, Tc is the crossover or empirical mode-coupling temperature, Tg is the glass transition temperature, and T0 is the temperature at which the extrapolated relaxation time diverges. These temperatures and fragility parameters are evaluated as a function of molar mass, pressure, and the length n of the α-olefin side chains. The nearest neighbor interaction energy and local chain rigidities are found to strongly influence the four characteristic temperatures and the low temperature fragility. We also observe an “internal plasticization” of the poly(α-olefins) wherein the fragility decreases as the number n of “flexible” side group units increases. Our computations provide solid support for a pressure counterpart of the Vogel–Fulcher–Tammann relation. The entropy theory of glass formation predicts systematic changes in fragility with chain stiffness, cohesive energy, polymerization index, and side chain length, and qualitative trends in these parameters are discussed. [ABSTRACT FROM AUTHOR]

The maximum entropy analytic continuation (MEAC) method is used to extend the range of accuracy of the linearized semiclassical initial value representation (LSC-IVR)/classical Wigner approximation for real time correlation functions. LSC-IVR provides a very effective “prior” for the MEAC procedure since it is very good for short times, exact for all time and temperature for harmonic potentials (even for correlation functions of nonlinear operators), and becomes exact in the classical high temperature limit. This combined MEAC+LSC/IVR approach is applied here to two highly nonlinear dynamical systems, a pure quartic potential in one dimensional and liquid para-hydrogen at two thermal state points (25 and 14 K under nearly zero external pressure). The former example shows the MEAC procedure to be a very significant enhancement of the LSC-IVR for correlation functions of both linear and nonlinear operators, and especially at low temperature where semiclassical approximations are least accurate. For liquid para-hydrogen, the LSC-IVR is seen already to be excellent at T=25 K, but the MEAC procedure produces a significant correction at the lower temperature (T=14 K). Comparisons are also made as to how the MEAC procedure is able to provide corrections for other trajectory-based dynamical approximations when used as priors. [ABSTRACT FROM AUTHOR]

The maximum entropy analytic continuation (MEAC) and ring polymer molecular dynamics (RPMD) methods provide complementary approaches to the calculation of real time quantum correlation functions. RPMD becomes exact in the high temperature limit, where the thermal time βh tends to zero and the ring polymer collapses to a single classical bead. MEAC becomes most reliable at low temperatures, where βh exceeds the correlation time of interest and the numerical imaginary time correlation function contains essentially all of the information that is needed to recover the real time dynamics. We show here that this situation can be exploited by combining the two methods to give an improved approximation that is better than either of its parts. In particular, the MEAC method provides an ideal way to impose exact moment (or sum rule) constraints on a prior RPMD spectrum. The resulting scheme is shown to provide a practical solution to the “nonlinear operator problem” of RPMD, and to give good agreement with recent exact results for the short-time velocity autocorrelation function of liquid parahydrogen. Moreover these improvements are obtained with little extra effort, because the imaginary time correlation function that is used in the MEAC procedure can be computed at the same time as the RPMD approximation to the real time correlation function. However, there are still some problems involving long-time dynamics for which the RPMD+MEAC combination is inadequate, as we illustrate with an example application to the collective density fluctuations in liquid orthodeuterium. [ABSTRACT FROM AUTHOR]

Synchrotron x-ray diffraction measurements were performed on liquid SnI4 up to a scattering vector of 25 A-1, utilizing a horizontal two-axis diffractometer installed at the SPring-8 bending magnet beam line BL04B2 in Japan. An effective method based on the maximum entropy method was devised to transform the measured total structure factor to the reduced radial distribution function. The reliability of the density estimation is discussed. [ABSTRACT FROM AUTHOR]

Synchrotron x-ray diffraction measurements were performed on liquid SnI4 up to a scattering vector of 25 A-1, utilizing a horizontal two-axis diffractometer installed at the SPring-8 bending magnet beam line BL04B2 in Japan. An effective method based on the maximum entropy method was devised to transform the measured total structure factor to the reduced radial distribution function. The reliability of the density estimation is discussed. [ABSTRACT FROM AUTHOR]

The crystal structure of deuterated methane hydrate (structure I, space group: Pm3n) was investigated by neutron powder diffraction at temperatures of 7.7–185 K. The scattering amplitude density distribution was examined by a combination of Rietveld method and maximum entropy method (MEM). The distribution of the D atoms in both D2O and CD4 molecules was clarified from three-dimensional graphic images of the scattering amplitude density. The MEM results showed that there were low-density sites for the D atom of D2O in a particular location within the D2O cage at low temperatures. The MEM provided more reasonable results because of the decrease in the R factor that is attainable by this method. Accordingly, the low-density sites for the D atom of D2O probably exist within the D2O cage. This suggests that a spatial disorder of the D atom of D2O occurs at these sites and that hydrogen bonds between D2O molecules become partially weakened. With regard to the CD4 molecules, there were high-density sites for the D atom of CD4, and the density distribution of the C and D atoms was observed separately in the scattering amplitude density image. Consequently, the C-D bonds of CD4 were not observed clearly because the CD4 molecules had an orientational disorder. The D atoms of CD4 were displaced from the line between the C and O atoms, and were located near the face center of the polygon in the cage. Accordingly, the D atoms of CD4 were not bonded to specific O atoms. This result is consistent with the hydrophobicity of the CD4 molecule. We also report the difference between the small and the large cages in the density distribution map and the temperature dependence of the scattering amplitude density. [ABSTRACT FROM AUTHOR]

Starting with the heat-capacity data of Breaux et al., [J. Am. Chem. Soc. 126, 8629 (2004)] we use the maximum-entropy method to calculate energy distribution functions for gallium-ion nanoclusters over a wide temperature range (100–1050 K). Specifically, we calculate energy distributions for clusters containing n=39 and n=45 gallium atoms. For the case of n=39 clusters the energy distribution gets systematically broader as a function of temperature with no indication of any marked structural change in the cluster. On the other hand, the energy distribution for the n=45 cluster first gets broader as a function of temperature but then gets narrower again as the temperature is further increased, indicating that there is some kind of structural transition taking place in this cluster species. [ABSTRACT FROM AUTHOR]

The translational kinetic energy release distribution (KERD) for the halogen loss reaction of the bromobenzene and iodobenzene cations has been reinvestigated on the microsecond time scale. Two necessary conditions of validity of the orbiting transition state theory (OTST) for the calculation of kinetic energy release distributions (KERDs) have been formulated. One of them examines the central ion-induced dipole potential approximation. As a second criterion, an adiabatic parameter is derived. The lower the released translational energy and the total angular momentum, the larger the reduced mass, the rotational constant of the molecular fragment, and the polarizability of the released atom, the more valid is the OTST. Only the low-energy dissociation of the iodobenzene ion (E≈0.45 eV, where E is the internal energy above the reaction threshold) is found to fulfill the criteria of validity of the OTST. The constraints that act on the dissociation dynamics have been studied by the maximum entropy method. Calculations of entropy deficiencies (which measure the deviation from a microcanonical distribution) show that the pair of fragments does not sample the whole of the phase space that is compatible with the mere specification of the internal energy. The major constraint that results from conservation of angular momentum is related to a reduction of the dimensionality of the dynamics of the translational motion to a two-dimensional space. A second and minor constraint that affects the KERD leads to a suppression of small translational releases, i.e., accounts for threshold behavior. At high internal energies, the effects of curvature of the reaction path and of angular momentum conservation are intricately intermeddled and it is not possible to specify the share of each effect. [ABSTRACT FROM AUTHOR]

The problem of calculating collective density fluctuations in quantum liquids is revisited. A fully quantum mechanical self-consistent treatment based on a quantum mode-coupling theory [E. Rabani and D.R. Reichman, J. Chem. Phys. 116, 6271 (2002)] is presented. The theory is compared with the maximum entropy analytic continuation approach and with available experimental results. The quantum mode-coupling theory provides semiquantitative results for both short and long time dynamics. The proper description of long time phenomena is important in future study of problems related to the physics of glassy quantum systems, and to the study of collective fluctuations in Bose fluids. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Using the maximum-entropy method, we calculate the end-to-end distance distribution of the force-stretched chain from the moments of the distribution, which can be obtained from the extension-force curves measured in single-molecule experiments. If one knows the force expansion of the extension through the (n-1)th power of a force, that is enough to calculate the n moments of the distribution. We examine the method with three force-stretched chain models: the Gaussian chain, free-joined chain, and self-avoiding chain on a two-dimensional lattice. The method reconstructs all distributions precisely. We also apply the method to force-stretched complex chain molecules: the hairpin and secondary structure conformations. We find that the distributions of homogeneous chains of the two conformational classes are quite different: there are two independent peaks in the hairpin distribution, while only one peak is observed in the distribution of secondary structure conformations. Our discussion shows that the end-to-end distance distribution, instead of extension-force curves, can reveal more important physical information. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A modified version of the maximum entropy principle, called “constrained maximum entropy” method (MEC), is revisited to combine the information obtained in computer simulations of polymers with external information in the form of configurational averages. A random-temperature molecular dynamics trajectory is being proposed as a biased random walk in configurational space to be reweighted by using the given average information. This random walk, generating a “meta” configurational probability, has been found to contain relevant information on the system. The method is compared with other computational techniques, like the generalized-ensemble and configurational-biased Monte Carlo, for simple models in the field of polymers and biopolymers. The main features of polymer configurational distribution functions of interest in polymer physics are consistent among the different methods in a wide range of temperatures and especially at room conditions. The advantage of the MEC approach is in taking into account all the degrees of freedom in the model, thus allowing applications in complicated biopolymers in the explicit solvent. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

A counterion-only double-layer theory is developed taking a constrained entropy maximization approach based on an isothermal thermodynamic system. The double-layer interaction free energy per unit area between two charged planar surfaces is obtained in a consistent manner for the boundary conditions of constant surface charge density and charge regulation. Once applied to the nonpolar medium, it reveals that the electrical double-layer repulsion is weak and long-ranged. Asymptotically, the interaction decays as a power law, in contrast with that in the presence of an ion reservoir which decays exponentially at large surface separation. The applicability of the counterion-only model for describing colloidal stability in nonpolar media is discussed. [ABSTRACT FROM AUTHOR]

Discusses the use of the maximum-entropy method to calculate the chain-length distribution as function of time for cooperative polymerization models involving nucleation and growth. Review of the special case of reversible aggregation; Rate equations for the distribution function moments; Case of reversible growth from a fixed concentration of nuclei.

Examines spin-spin time correlation in the spin-boson model through maximum entropy moment imaging procedure. Description of time correlation functions; Graphical determination of time derivatives; Description of maximum entropy imaging; Analysis of imaging results.

Examines the performance of maximum entropy (ME) and singular value decomposition (SVD) analytic continuation methods to the problem of quantum mechanical vibrational relaxation processes. Application of ME and SVD to imaginary-time correlation functions; Details on the prototype model for studying vibrational relaxation in condensed phases.

In this article we investigate the applicability of the statistical Gamma state as following from the quasi-Gaussian entropy theory, where all thermodynamic properties at every temperature are obtained from the knowledge of the potential energy distribution at one temperature. We compared for a typically polar system (water) and an apolar one (methane) the experimental heat capacity and entropy data with the predictions of the theory at various densities, ranging from the almost ideal gas to typical liquids. Interestingly, the behavior of water and methane is quite similar. Low-density gases and fluid-liquid systems can be described as weakly perturbed Gamma states. For intermediate densities a more complex statistical state arises. In order to describe more accurately the fluid-liquid regime, we propose in this paper a confined Gamma state, based on the division of phase-space into two different regions: one of them described by an exact Gamma state and another very unstable one. We conclude that typical fluid-liquids can be described very well by this new Gamma state approximation. We also try to give a physical interpretation of the two parts of phase space that arise from the model. The high accuracy of the theory over a large temperature range makes the approach very suitable for the prediction of thermodynamical properties at, for example, supercritical conditions. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The maximum-entropy method is used to construct the end-to-end distribution function for lattice polymers when a large number of moments are known exactly. We use two-dimensional lattice polymers with a finite range of intrachain interaction as examples since, for these systems, the end-to-end distribution function and any number of moments can be calculated exactly using Toeplitz matrices. For chains with strong intrachain interactions the distributions are very non-Gaussian requiring up to six moments to reproduce the main features of the functions. © 1995 American Institute of Physics. [ABSTRACT FROM AUTHOR]

An analytical functional form for the distribution of intensities in an absorption spectrum is derived. Deviations from the purely statistical ‘‘Porter–Thomas’’ distribution are shown to be directly related to finite time information on the dynamics in phase space. The predicted distribution is wider than the purely statistical one with a higher proportion of very low intensity transitions. The derivation is based on a maximum entropy form of the spectrum. The constraints used are the values of the survival amplitude at finite number of times. The amplitude is obtainable as the Fourier transform of an observed spectrum or as the result of a dynamical computation. The optimal choice of the time points which characterize the spectrum, is discussed and a numerical algorithm is provided. Extensive spectral fluctuations occur when more than one time scale is needed to characterize the dynamics. This separation of time scales is also manifested as a clump structure in the spectrum of maximal entropy. The formalism also provides the distribution of line spacings and the ‘‘correlation hole’’ in the time autocorrelation function is discussed as an illustration. [ABSTRACT FROM AUTHOR]

Imaginary-time correlation functions calculated by quantum Monte Carlo (QMC) are analyzed using the maximum entropy method (MaxEnt) to determine the ground-state energy and spectral overlap function. In contrast to earlier applications of MaxEnt, the data is obtained from importanced-sampled zero-temperature quantum Monte Carlo simulations. The analysis includes two steps. First, that spectral overlap function and ground state energy which maximizes the entropy and agrees with the QMC correlation functions is obtained. Then the errors in the energy are evaluated by averaging over all the possible images (average MaxEnt method), the multidimensional integrals being computed using the Metropolis algorithm. The central feature of this approach is that all the information present in the correlation functions is used in the only way consistent with fundamental probabilistic hypotheses. This allows us to fully exploit the information contained in the correlation functions at small imaginary times, thus avoiding large statistical fluctuations associated with large imaginary times. In addition, the computed errors include both the statistical errors and systematic extrapolation errors. The method is illustrated with a harmonic oscillator and the four-electron LiH molecule. [ABSTRACT FROM AUTHOR]

Investigates the extension of thermodynamics in the information entropy maximization method. Derivation of evolution equations from the Liouville equation; Provision of a basis for extended thermodynamics through the method; Relationship between nonequilibrium statistical mechanics and irreversible thermodynamics.

The principle of minimum cross entropy, instead of the principle of maximum entropy, of information theory is employed to estimate a molecular radial momentum density I(p) under the constraint of a given moment . An application to the ground-state H2 system shows that the minimum cross-entropy inference is superior to the maximum entropy inference. [ABSTRACT FROM AUTHOR]

I employ the van der Waals theory of Baus and co-workers to analyze the fast, adiabatic decay of a supercooled liquid in a closed vessel with which the solidification process usually starts. By imposing a further constraint on either the system volume or pressure, I use the maximum-entropy method to quantify the fraction of liquid that is transformed into solid as a function of undercooling and of the amount of a foreign gas that could possibly be also present in the test tube. Upon looking at the implications of thermal and mechanical insulation for the energy cost of forming a solid droplet within the liquid, I identify one situation where the onset of solidification inevitably occurs near the wall in contact with the bath. [ABSTRACT FROM AUTHOR]

In order to characterise the dynamics of proteins, a well-established method is to incorporate experimental parameters as replica-averaged structural restraints into molecular dynamics simulations. Here, we justify this approach in the case of interproton distance information provided by nuclear Overhauser effects by showing that it generates ensembles of conformations according to the maximum entropy principle. These results indicate that the use of replica-averaged structural restraints in molecular dynamics simulations, given a force field and a set of experimental data, can provide an accurate approximation of the unknown Boltzmann distribution of a system. [ABSTRACT FROM AUTHOR]

The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive nth-order Markov processes and the master equation as unique solutions to an inverse problem. We find that when constraints are not enough to uniquely determine the stochastic model, an nth-order Markov process emerges as the unique maximum entropy solution to this otherwise underdetermined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from the first principles. [ABSTRACT FROM AUTHOR]

We use the maximum-entropy method to calculate molecular energy distributions from the moments of the distribution which in turn can be obtained from the temperature dependence of the heat capacity. If one knows the temperature expansion of the heat capacity through the nth power of the temperature, this then gives the exact first (n+2) energy moments. We illustrate the method for the ideal gas (the Maxwell-Boltzmann distribution of kinetic energy) and then use a model function to show that if one knows four or more moments of the energy distribution this allows one to resolve two or more distinct peaks in this function. We examine argon above the critical pressure, a one-dimensional model, and the protein barnase, all of which exhibit bimodal energy distributions. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]