Variational formula for the relaxation time in the Boltzmann equation

The relaxation time approximation (RTA) is commonly employed in nonequilibrium statistical mechanics to approximate solutions to the Boltzmann equation in terms of an exponential relaxation to equilibrium. Despite its common use, the RTA suffers from the drawback that relaxation times commonly employed are independent of initial conditions. We derive a variational principle for solutions to the Boltzmann equation, which allows us to extend the standard RTA using relaxation times that depend on the initial distribution. Tests of the approach on a calculation of the mobility for a one-dimensional (1D) tight-binding band indicate that our analysis typically provides a better approximation than the standard RTA.     

A statistical model for instable thermodynamical systems

A generic model is presented for statistical systems which display thermodynamic features in contrast to our everyday experience, such as infinite and negative heat capacities. Such system are instable in terms of classical equilibrium thermodynamics. Using our statistical model, we are able to investigate states of instable systems which are undefined in the framework of equilibrium thermodynamics. We show that a region of negative heat capacity in the adiabatic environment, leads to a first order like phase transition when the system is coupled to a heat reservoir. This phase transition takes place without a phase coexistence. Nevertheless, all intermediate states are stable due to fluctuations. When two instable system are brought in thermal contact, the temperature of the composed system is lower than the minimum temperature of the individual systems. Generally, the equilibrium states of instable system cannot be simply decomposed into equilibrium states of the individual systems. The properties of instable system depend on the environment, ensemble equivalence is broken.     

Density-dependent analysis of nonequilibrium paths improves free energy estimates II. A Feynman-Kac formalism

The nonequilibrium fluctuation theorems have paved the way for estimating equilibrium thermodynamic properties, such as free energy differences, using trajectories from driven nonequilibrium processes. While many statistical estimators may be derived from these identities, some are more efficient than others. It has recently been suggested that trajectories sampled using a particular time-dependent protocol for perturbing the Hamiltonian may be analyzed with another one. Choosing an analysis protocol based on the nonequilibrium density was empirically demonstrated to reduce the variance and bias of free energy estimates. Here, we present an alternate mathematical formalism for protocol postprocessing based on the Feynmac-Kac theorem. The estimator that results from this formalism is demonstrated on a few low-dimensional model systems. It is found to have reduced bias compared to both the standard form of Jarzynski's equality and the previous protocol postprocessing formalism.     

Measurement of nonequilibrium entropy from space-time thermodynamic integration

The entropy of a system transiently driven out of equilibrium by a time-inhomogeneous stochastic dynamics is first expressed as a transient response function generalizing the nonlinear Kawasaki-Crooks response. This function is then reformulated into three statistical averages defined over ensembles of nonequilibrium trajectories. The first average corresponds to a space-time thermodynamic perturbation relation, while the two following ones correspond to space-time thermodynamic integration relations. Provided that trajectories are initiated starting from a distribution of states that is analytically known, the ensemble averages are computationally amenable to Markov chain Monte Carlo methods. The relevance of importance sampling in path ensembles is confirmed in practice by computing the nonequilibrium entropy of a driven toy system. We finally study a situation where the dynamics produces entropy. In this case, we observe that space-time thermodynamic integration still yields converged estimates, while space-time thermodynamic perturbation turns out to converge very slowly.     

Heat capacity, enthalpy fluctuations, and configurational entropy in broken ergodic systems

A common assumption in the glass science community is that the entropy of a glass can be calculated by integration of measured heat capacity curves through the glass transition. Such integration assumes that glass is an equilibrium material and that the glass transition is a reversible process. However, as a nonequilibrium and nonergodic material, the equations from equilibrium thermodynamics are not directly applicable to the glassy state. Here we investigate the connection between heat capacity and configurational entropy in broken ergodic systems such as glass. We show that it is not possible, in general, to calculate the entropy of a glass from heat capacity curves alone, since additional information must be known related to the details of microscopic fluctuations. Our analysis demonstrates that a time-average formalism is essential to account correctly for the experimentally observed dependence of thermodynamic properties on observation time, e.g., in specific heat spectroscopy. This result serves as experimental and theoretical proof for the nonexistence of residual glass entropy at absolute zero temperature. Example measurements are shown for Corning code 7059 glass.     

平衡分布与最概然分布

以二项分布为例,阐述了平衡分布与最概然分布间的关系。通过子数很多时,二项分布可以变为正态分布以及它的相对涨落与子数的平方根成反比的统计力学原理,阐明了热力学系统的平衡分布可用其最概然分布来代表。     

Estimating equilibrium ensemble averages using multiple time slices from driven nonequilibrium processes: theory and application to free energies, moments, and thermodynamic length in single-molecule pulling experiments

Recently discovered identities in statistical mechanics have enabled the calculation of equilibrium ensemble averages from realizations of driven nonequilibrium processes, including single-molecule pulling experiments and analogous computer simulations. Challenges in collecting large data sets motivate the pursuit of efficient statistical estimators that maximize use of available information. Along these lines, Hummer and Szabo developed an estimator that combines data from multiple time slices along a driven nonequilibrium process to compute the potential of mean force. Here, we generalize their approach, pooling information from multiple time slices to estimate arbitrary equilibrium expectations. Our expression may be combined with estimators of path-ensemble averages, including existing optimal estimators that use data collected by unidirectional and bidirectional protocols. We demonstrate the estimator by calculating free energies, moments of the polymer extension, the thermodynamic metric tensor, and the thermodynamic length in a model single-molecule pulling experiment. Compared to estimators that only use individual time slices, our multiple time-slice estimators yield substantially smoother estimates and achieve lower variance for higher-order moments.     

Heterogeneous dynamics, ageing, and rejuvenating in van der Waals liquids

It has been shown over the past ten years that the dynamics close to the glass transition is strongly heterogeneous: fast domains coexist with domains three or four decades slower, the size of these regions being about 3 nm at T(g). The authors extend here a model that has been proposed recently for the glass transition in van der Waals liquids. The authors describe in more details the mechanisms of the alpha relaxation in such liquids. It allows then to interpret physical ageing in van der Waals liquids as the evolution of the density fluctuation distribution towards the equilibrium one. The authors derive the expression of macroscopic quantities (volume, compliance, etc.). Numerical results are compared with experimental data (shape, times to reach equilibrium) for simple thermal histories (quenches, annealings). The authors explain the existence of a "Kovacs memory effect" and the temporal asymmetry between down jump and up jump temperatures experiments, even for systems for which there is no energy barriers. Their model allows also for calculating the evolution of small probe diffusion coefficients during ageing.     

Inhomogeneous multiscale dynamics in harmonic lattices

We use projection operators to address the coarse-grained multiscale problem in harmonic systems. Stochastic equations of motion for the coarse-grained variables, with an inhomogeneous level of coarse graining in both time and space, are presented. In contrast to previous approaches that typically start with thermodynamic averages, the key element of our approach is the use of a projection matrix chosen both for its physical appeal in analogy to mechanical stability theory and for its algebraic properties. We show that thermodynamic equilibrium can be recovered and obtain the fluctuation dissipation theorem a posteriori. All system-specific information can be computed from a series of feasible molecular dynamics simulations. We recover previous results in the literature and show how this approach can be used to extend the quasicontinuum approach and comment on implications for dissipative particle dynamics type of methods. Contrary to what is assumed in the latter models, the stochastic process of all coarse-grained variables is not necessarily Markovian, even though the variables are slow. Our approach is applicable to any system in which the coarse-grained regions are linear. As an example, we apply it to the dynamics of a single mesoscopic particle in the infinite one-dimensional harmonic chain.     

Relative Boltzmann entropy, evolution equations for fluctuations of thermodynamic intensive variables, and a statistical mechanical representation of the zeroth law of thermodynamics

Generalized thermodynamics or extended irreversible thermodynamics presumes the existence of thermodynamic intensive variables (e.g., temperature, pressure, chemical potentials, generalized potentials) even if the system is removed from equilibrium. It is necessary to properly understand the nature of such intensive variables and, in particular, of their fluctuations, that is, their deviations from those defined in the extended irreversible thermodynamic sense. The meaning of temperature is examined by means of a kinetic theory of macroscopic irreversible processes to assess the validity of the generalized (or extended) thermodynamic method applied to nonequilibrium phenomena. The Boltzmann equation is used for the purpose. Since the relative Boltzmann entropy has been known to be intimately related to the evolution of the aforementioned fluctuations in the intensive thermodynamic variables, we derive the evolution equations for such fluctuations of intensive variables to lay the foundation for investigating the physical implications and evolution of the relative Boltzmann entropy, so that the range of validity of the thermodynamic theory of irreversible processes can be elucidated. Within the framework of this work, we examine a special case of the evolution equations for the aforementioned fluctuations of intensive variables, which also facilitate investigation of the molecular theory meaning of the zeroth law of thermodynamics. We derive an evolution equation describing the relaxation of temperature fluctuations from its local value and present a formula for the temperature relaxation time.     

Thermodynamic perturbation theory of simple liquids. A hierarchy of equations for expansion coefficients

The properties of an expansion of the statistical sum of a simple liquid with respect to the potential in thermodynamic perturbation theory are analyzed. The coefficients of this expansion are determined by the unperturbed potential, depend on temperature and density, and can be calculated by means of mathematical modeling. It is shown here that the derivatives of these coefficients with respect to temperature and density are expressed through the higher expansion coefficient (these relations are usually called a hierarchy of equations). These coefficients determine the expansion of the Helmholtz free energy and RDF with respect to the perturbation potential. The thermodynamic characteristics of the system (entropy, internal energy, pressure) are expressed through both the differential relations for the Helmholtz free energy and the integral expressions containing RDF. It is found that the hierarchy of equations obtained in this work makes these different methods equivalent. This is important for the application of thermodynamic perturbation theory because it becomes unnecessary to model any other equilibrium properties of the system apart from the expansion coefficients.     

Thermodynamics for single-molecule stretching experiments

We show how to construct nonequilibrium thermodynamics for systems too small to be considered thermodynamically in a traditional sense. Through the use of a nonequilibrium ensemble of many replicas of the system which can be viewed as a large thermodynamic system, we discuss the validity of nonequilibrium thermodynamics relations and analyze the nature of dissipation in small systems through the entropy production rate. We show in particular that the Gibbs equation, when formulated in terms of average values of the extensive quantities, is still valid, whereas the Gibbs-Duhem equation differs from the equation obtained for large systems due to the lack of the thermodynamic limit. Single-molecule stretching experiments are interpreted under the prism of this theory. The potentials of mean force and mean position, now introduced in these experiments in substitution of the thermodynamic potentials, correspond respectively to our Helmholtz and Gibbs energies. These results show that a thermodynamic formalism can indeed be applied at the single-molecule level.     

Application of the projection operator formalism to non-hamiltonian dynamics

Reconstruction of equations of motion from incomplete or noisy data and dimension reduction are two fundamental problems in the study of dynamical systems with many degrees of freedom. For the latter, extensive efforts have been made, but with limited success, to generalize the Zwanzig-Mori projection formalism, originally developed for hamiltonian systems close to thermodynamic equilibrium, to general non-hamiltonian systems lacking detailed balance. One difficulty introduced by such systems is the lack of an invariant measure, needed to define a statistical distribution. Based on a recent discovery that a non-hamiltonian system defined by a set of stochastic differential equations can be mapped to a hamiltonian system, we develop such general projection formalism. In the resulting generalized Langevin equations, a set of generalized fluctuation-dissipation relations connect the memory kernel and the random noise terms, analogous to hamiltonian systems obeying detailed balance. Lacking of these relations restricts previous application of the generalized Langevin formalism. Result of this work may serve as the theoretical basis for further technical developments on model reconstruction with reduced degrees of freedom. We first use an analytically solvable example to illustrate the formalism and the fluctuation-dissipation relation. Our numerical test on a chemical network with end-product inhibition further demonstrates the validity of the formalism. We suggest that the formalism can find wide applications in scientific modeling. Specifically, we discuss potential applications to biological networks. In particular, the method provides a suitable framework for gaining insights into network properties such as robustness and parameter transferability.     

Rapid sampling of all‐atom peptides using a library‐based polymer‐growth approach

We adapted existing polymer growth strategies for equilibrium sampling of peptides described by modern atomistic forcefields with a simple uniform dielectric solvent. The main novel feature of our approach is the use of precalculated statistical libraries of molecular fragments. A molecule is sampled by combining fragment configurations—of single residues in this study—which are stored in the libraries. Ensembles generated from the independent libraries are reweighted to conform with the Boltzmann‐factor distribution of the forcefield describing the full molecule. In this way, high‐quality equilibrium sampling of small peptides (4–8 residues) typically requires less than one hour of single‐processor wallclock time and can be significantly faster than Langevin simulations. Furthermore, approximate, clash‐free ensembles can be generated for larger peptides (up to 32 residues in this study) in less than a minute of single‐processor computing. We discuss possible applications of our growth procedure to free energy calculation, fragment assembly protein‐structure prediction protocols, and to “multi‐resolution” sampling. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Kinetic Control over Pathway Complexity in Supramolecular Polymerization through Modulating the Energy Landscape by Rational Molecular Design

Far‐from‐equilibrium thermodynamic systems that are established as a consequence of coupled equilibria are the origin of the complex behavior of biological systems. Therefore, research in supramolecular chemistry has recently been shifting emphasis from a thermodynamic standpoint to a kinetic one; however, control over the complex kinetic processes is still in its infancy. Herein, we report our attempt to control the time evolution of supramolecular assembly in a process in which the supramolecular assembly transforms from a J‐aggregate to an H‐aggregate over time. The transformation proceeds through a delicate interplay of these two aggregation pathways. We have succeeded in modulating the energy landscape of the respective aggregates by a rational molecular design. On the basis of this understanding of the energy landscape, programming of the time evolution was achieved through adjusting the balance between the coupled equilibria.     

Equilibrium free energy estimates based on nonequilibrium work relations and extended dynamics

Jarzynski's relation and the fluctuation theorem have established important connections between nonequilibrium statistical mechanics and equilibrium thermodynamics. In particular, an exact relationship between the equilibrium free energy and the nonequilibrium work is useful for computer simulations. In this paper, we exploit the fact that the free energy is a state function, independent of the pathway taken to change the equilibrium ensemble. We show that a generalized expression is advantageous for computer simulations of free energy differences. Several methods based on this idea are proposed. The accuracy and efficiency of the proposed methods are evaluated with a model problem.     

Small Systems and Limitations on the Use of Chemical Thermodynamics

Limitations on using chemical thermodynamics to describe small systems are formulated. These limitations follow from statistical mechanics for equilibrium and nonequilibrium processes and reflect (1) differences between characteristic relaxation times in momentum, energy, and mass transfer in different aggregate states of investigated systems; (2) achievements of statistical mechanics that allow us to determine criteria for the size of smallest region in which thermodynamics can be applied and the scale of the emergence of a new phase, along with criteria for the conditions of violating a local equilibrium. Based on this analysis, the main thermodynamic results are clarified: the phase rule for distorted interfaces, the sense and area of applicability of Gibbs’s concept of passive forces, and the artificiality of Kelvin’s equation as a result of limitations on the thermodynamic approach to considering small bodies. The wrongness of introducing molecular parameters into thermodynamic derivations, and the activity coefficient for an activated complex into the expression for a reaction rate constant, is demonstrated.     

Maximum entropy formulation of the Kirkwood superposition approximation

Using a variational formulation, we derive the Kirkwood superposition approximation for systems at equilibrium in the thermodynamic limit. We define the entropy of the triplet correlation function and show that the Kirkwood closure brings the entropy to its maximal value. This approach leads to a different interpretation for the Kirkwood closure relation, usually explained by probabilistic considerations of dependence and independence of particles. The Kirkwood closure is generalized to finite volume systems at equilibrium by computing the pair correlation function in finite domains. Closure relations for high order correlation functions are also found using a variational approach. In particular, maximizing the entropy of quadruplets leads to the high order closure g(1234)=g(123)g(124)g(134)g(234)/[g(12)g(13)g(14)g(23)g(24)g(34)] used in the Born-Green-Yvon 2 equations which are a pair of integral equations for the triplet and pair correlation functions.     

Thermodynamics and structure of the ordered amorphous phase in polymers

Experimental data are analysed to show that the activation enthalpy and the structural entropy appear to follow empirical relations for structural relaxation of polymeric systems. The relations indicate that melting temperature, glass transition temperature, relaxation time, coefficient of thermal expansion of free volume are interrelated in polymers. The parameters of structural relaxations, measured by mechanical and dielectric spectroscopies, are reviewed for polyethylene, poly(4-methyl-1-pentene) and a liquid-crystalline polynorbornene derivative. The thermodynamic parameters obtained from calorimetric measurements, are reported for zero heating rate extrapolation and they are used in the empirical relation, which combines the Arrhenius and the Vogel-Fulcher formulae.