Quantum Thermodynamics with Missing Reference Frames: Decompositions of Free Energy Into Non-Increasing Components

If an absolute reference frame with respect to time, position, or orientation is missing one can only implement quantum operations which are covariant with respect to the corresponding unitary symmetry group G. Extending observations of Vaccaro et al., I argue that the free energy of a quantum system with G-invariant Hamiltonian then splits up into the Holevo information of the orbit of the state under the action of G and the free energy of its orbit average. These two kinds of free energy cannot be converted into each other. The first component is subadditive and the second superadditive; in the limit of infinitely many copies only the usual free energy matters.Refined splittings of free energy into more than two independent (non-increasing) terms can be defined by averaging over probability measures on G that differ from the Haar measure.Even in the presence of a reference frame, these results provide lower bounds on the amount of free energy that is lost after applying a passive covariant channel. If the channel properly decreases one of these quantities, it decreases the free energy necessarily at least by the same amount, since it is unable to convert the different forms of free energies into each other. For instance, if an electrical, optical, or acoustical signal loses some time accuracy after it has passed a passive time-invariant device, the results provide lower bounds on the free energy lost in the latter.     

Free energy computations by minimization of Kullback–Leibler divergence: An efficient adaptive biasing potential method for sparse representations

The present paper proposes an adaptive biasing potential technique for the computation of free energy landscapes. It is motivated by statistical learning arguments and unifies the tasks of biasing the molecular dynamics to escape free energy wells and estimating the free energy function, under the same objective of minimizing the Kullback–Leibler divergence between appropriately selected densities. It offers rigorous convergence diagnostics even though history dependent, non-Markovian dynamics are employed. It makes use of a greedy optimization scheme in order to obtain sparse representations of the free energy function which can be particularly useful in multidimensional cases. It employs embarrassingly parallelizable sampling schemes that are based on adaptive Sequential Monte Carlo and can be readily coupled with legacy molecular dynamics simulators. The sequential nature of the learning and sampling scheme enables the efficient calculation of free energy functions parametrized by the temperature. The characteristics and capabilities of the proposed method are demonstrated in three numerical examples.     

The role of Frank elasticity in cholesteric elastomers

The effect of Frank elasticity on deformations of cholesteric elastomers by mechanical stress applied perpendicular to the helix axis is studied by numerical minimization of the free energy. Above a critical strain, a solution with an only oscillating director is found to be stable in comparison to a distorted helix. At the critical strain, the contractions perpendicular to stress change discontinuously. The critical strain is found to increase with increasing Frank elasticity contribution to the free energy density, and to diverge when the conformation anisotropy of the polymer backbone vanishes. The results are compared with recent experiments which indicated that, in case of weak conformation anisotropies, the Frank elasticity contribution to the free energy cannot be neglected.     

The generalized van Laar approximation for free energy

The van Laar equation for the uniform substance phase is analyzed. Based on this, an expression for free energy is found. The structure of the expression allowed us to create a method for determining the free energy in an arbitrary order of the perturbation theory. A generalized equation for free energy is derived based on the using thermodynamic perturbation theory and the accelerated method of convergence of the series of the perturbation theory. This expression agrees well with the asymptotic behavior of the free energy and with the known experimental data. It can be successfully used to describe the behavior of the substance in a super-critical region, as well as to investigate the metastable phase. It is shown that the expression for free energy can be used for a broad class of substances.     

采用加权直方图分析和蒙特卡洛重采样法对自由能级的校正研究

本文采用加权直方图分析方法和蒙特卡洛重采样方法进行自由基校正研究. 生成的自由能表面几乎可以收敛到具有足够采样情况的精准表面,并且比常规的加权直方图分析校正方法能更稳定地处理采样不足的情况,为检测能级校正表面的不确定性提供指南,并且定义了明确的标准用以确定能改善其视觉效果的自由能表面平滑程度. 本文通过水中的丙氨酸二肽和KillerRed蛋白质子转移的自由能图证明该方法的优势,说明蒙特卡洛重采样法可以作为在产生自由能表面实际的系统中的实用工具.     

A mean-field free energy lattice Boltzmann model for multicomponent fluids

We propose a mean-field free energy approach to simulate multi- component fluids. The model has been validated in terms of the Laplace equation of capillarity and dispersion relation of interfacial waves. Simulations of a ternary system shows that the total free energy decreases and reaches a minimum after phase separation has occurred. Different drop shapes can be obtained by adjusting the interaction strengths between individual components. These results demonstrate that both macroscopic free energy and microscopic fluid-fluid interactions have been well described in our multicomponent model.     

Equivalence of the Two Results for the Free Energy of the Chiral Potts Model

The free energy of the chiral Potts model has been obtained in two ways. The first used only the star-triangle relation, symmetries, and invariances, and led to a system of equations that implicitly define the free energy, and from which the critical behavior can be obtained The second used the functional relations derived by Bazhanov and Stroganov, solving them to obtain the free energy explicitly as a double integral. Here we obtain, for the first time, a direct verification that the two results are identical at all temperatures.     

Free Energy as a Geometric Invariant

The free energy plays a fundamental role in statistical and condensed matter physics. A related notion of free energy plays an important role in the study of hyperbolic dynamical systems. In this paper we introduce and study a natural notion of free energy for surfaces with variable negative curvature. This geometric free energy encodes a new type of marked length spectrum of closed geodesics, which lies between the well-known marked length spectrum (marked by the corresponding element of the fundamental group) and the unmarked length spectrum. We prove that the free energy parametrizes the boundary of the domain of convergence of a Poincaré series which also encodes this spectrum. We also show that this new length spectrum, or equivalently the geometric free energy, is not an isometry invariant. In the final section we use tools from multifractal analysis to effect a fine asymptotic comparison of word length and geodesic length of closed geodesics. We hope that our geometric understanding of free energy will provide new insight into this fundamental physical and dynamical quantity. The work of the second author was partially supported by a National Science Foundation grant DMS-0355180. This work was completed during a visit by the first author to Penn State as a Shapiro Fellow.     

Cyclic Heating-Annealing and Boltzmann Distribution of Free Energies in a Spin-Glass System

Ergodicity of a spin-glass is broken at low temperatures; the system is trapped in one of many ergodic configurational domains. Transitions between different ergodic domains are achievable through a heating-annealing procedure. If this experiment is repeated infinite times, all ergodic configurational domains will be visited with frequences that decreasing exponentially with their free energies. The mean free energy density of a spin-glass system on a random graph is calculated based on this free energy Boltzmann distribution in the present work, by means of the cavity approach.     

The determination of energy-level shifts which accompany chemisorption

Considerable confusion exists concerning a reasonable procedure for determining energy shifts of atomic levels on adsorption of a free gas atom to a solid surface and concerning the basic concepts involved. This paper discusses the ionization limit with respect to which energy levels of the adsorbed complex should be referenced, how this limit may be defined and what the best method is for determining its energy level. It is also demonstrated that one cannot sidestep the determination of the ionization limit by measuring in the same apparatus the kinetic energies of electrons ejected from the free atom and from the adsorbed atom.     

Cyclic Heating-Annealing and Boltzmann Distribution of Free Energies in a Spin-Glass System

Ergodicity of a spin-glass is broken at low temperatures; the system is trapped in one of many ergodic configurational domains. Transitions between different ergodic domains are achievable through a heating-annealing procedure. If this experiment is repeated infinite times, all ergodic configurational domains will be visited with frequences that decreasing exponentially with their free energies. The mean free energy density of a spin-glass system on a random graph is calculated based on this free energy Boltzmann distribution in the present work, by means of the cavity approach.     

Theoretical investigation on isomer formation probability and free energy of small C clusters

Molecular dynamics simulations and free energy calculations are employed to investigate the evolution,formation probability,detailed balance,and isomerization rate of small C cluster isomer at 2500 K.For C10,the isomer formation probability predicted by free energy is in good agreement with molecular dynamics simulation.However,for C20,C30,and C36,the formation probabilities predicted by free energy are not in agreement with molecular dynamics simulations.Although the cluster systems are in equilibrium,detailed balance is not reached.Such results may be attributed to high transformation barriers between cage,bowl,and sheet isomers.In summary,for mesoscopic nanosystems the free energy criterion,which commonly holds for macroscopic systems in dynamic equilibrium,may not provide a good prediction for isomer formation probability.New theoretical criterion should be further investigated for predicting the isomer formation probability of a mesoscopic nanosystem.     

能量公设与自由能判据的普遍化形式

从能量公设的角度论述了自由能及其判据的普遍化形式,并对其本质内涵进行了讨论．讨论认为：热力学系统中目由能的一般形式是与N种功类型相关的各种能形式之和,对孤立物体系在温度不变的情形下,不可逆过程总是导致总自由能的减少．通过实例说明了普遍化形式自由能判据的应用．     

Functional Integrals and Variational-Cumulant Expansion in sine-Gordon-Thirring Model

The free energy in 1D sine-Gordon-Thirring model with impurity coupling is studied by means of functional integrals and variational-cumulant expansion methods. Two variational parameters are introduced to evaluate free energy and statistical averages. It is shown that the non-perturbation method of functional integrals can be applied to strong-coupling range of fermion systems.     

Explicit Gibbs free energy equation of state for solids

Semi-empirical equations of state (EOS) are used for interpolation and extrapolation of experimental data and/or electronic structure calculations. For calculation of phase equilibria, it is preferable to use an explicit Gibbs free energy EOS, that is, to express the Gibbs free energy directly as a function of the pressure and temperature. Existing explicit Gibbs free energy EOS formulations often give unphysical predictions at high pressures. The origins of these problems are internal inconsistencies and uncontrolled extrapolations. A set of conditions is put forward, that should be fulfilled by semi-empirical EOS formulations in order to constrain them to known physical behaviour, e.g., to the Thomas-Fermi and quasi-harmonic models at high pressures. A new alternative integration path is devised that eliminates the need for the problematic extrapolation of the heat capacity to high temperatures at low pressures. Based on these developments, a new explicit Gibbs free energy EOS is formulated which is suitable for computational applications. The new EOS may be fitted to represent the thermophysical properties of solids with a reasonably small number of adjustable parameters. A sample application for MgO is presented.     

Energy Fluctuations Shape Free Energy of Nonspecific Biomolecular Interactions

Understanding design principles of biomolecular recognition is a key question of molecular biology. Yet the enormous complexity and diversity of biological molecules hamper the efforts to gain a predictive ability for the free energy of protein-protein, protein-DNA, and protein-RNA binding. Here, using a variant of the Derrida model, we predict that for a large class of biomolecular interactions, it is possible to accurately estimate the relative free energy of binding based on the fluctuation properties of their energy spectra, even if a finite number of the energy levels is known. We show that the free energy of the system possessing a wider binding energy spectrum is almost surely lower compared with the system possessing a narrower energy spectrum. Our predictions imply that low-affinity binding scores, usually wasted in protein-protein and protein-DNA docking algorithms, can be efficiently utilized to compute the free energy. Using the results of Rosetta docking simulations of protein-protein interactions from Andre et al. (Proc. Natl. Acad. Sci. USA 105:16148, 2008), we demonstrate the power of our predictions.     

Free energy change of a dislocation due to a Cottrell atmosphere

The free energy reduction of a dislocation due to a Cottrell atmosphere of solutes is computed using a continuum model. We show that the free energy change is composed of near-core and far-field components. The far-field component can be computed analytically using the linearized theory of solid solutions. Near the core the linearized theory is inaccurate, and the near-core component must be computed numerically. The influence of interactions between solutes in neighbouring lattice sites is also examined using the continuum model. We show that this model is able to reproduce atomistic calculations of the nickel–hydrogen system, predicting hydride formation on dislocations. The formation of these hydrides leads to dramatic reductions in the free energy. Finally, the influence of the free energy change on a dislocation’s line tension is examined by computing the equilibrium shape of a dislocation shear loop and the activation stress for a Frank–Read source using discrete dislocation dynamics.     

Experimental free energy surface reconstruction from single-molecule force spectroscopy using Jarzynski's equality

We used the atomic force microscope to manipulate and unfold individual molecules of the titin I27 domain and reconstructed its free energy surface using Jarzynski's equality. The free energy surface for both stretching and unfolding was reconstructed using an exact formula that relates the nonequilibrium work fluctuations to the molecular free energy. In addition, the unfolding free energy barrier, i.e., the activation energy, was directly obtained from experimental data for the first time. This Letter demonstrates that Jarzynski's equality can be used to analyze nonequilibrium single-molecule experiments, and to obtain the free energy surfaces for molecular systems, including interactions for which only nonequilibrium work can be measured.     

Lifshitz tricritical point and its relation to the FFLO superconducting state

We study the phase diagram of spatially inhomogeneous Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superconducting state using the Ginzburg–Landau (GL) free energy, derived from the microscopic Hamiltonian of the system, and notice that it has a very clear Lifshitz tricritical point. We find the specific heat jumps abruptly near the first-order line in the emergent phase diagram which is very similar to the recent experimental observation in layered organic superconductor. Comparison with experimental data allows us to obtain quantitative relations between the parameters of phenomenological free energy. The region of the phase diagram where the specific heat jumps can be probed by doing a dynamical analysis of the free energy.     

A stable static universe?

Starting from the assumption that general relativity might be an emergent phenomenon showing up at low energies from an underlying microscopic structure, we reanalyze the stability of a static closed universe filled with radiation. In this scenario, it is sensible to consider the effective general-relativistic configuration as in a thermal contact with an “environment” (the role of the environment can be played, for example, by a higher-dimensional bulk or by the trans-Planckian degrees of freedom). We calculate the free energy at a fixed temperature of this radiation-filled static configuration. Then, by looking at the free energy, we show that the static Einstein configuration is stable under the stated condition.