Thermodynamic Ensemble Research Articles

Equivalence of matrix product ensembles of trajectories in open quantum systems.

The equivalence of thermodynamic ensembles is at the heart of statistical mechanics and central to our understanding of equilibrium states of matter. Recently, a formal connection has been established between the dynamics of open quantum systems and statistical mechanics in an extra dimension: an open system dynamics generates a matrix product state (MPS) encoding all possible quantum jump trajectories which allows to construct generating functions akin to partition functions. For dynamics generated by a Lindblad master equation, the corresponding MPS is a so-called continuous MPS which encodes the set of continuous measurement records terminated at some fixed total observation time. Here, we show that if one instead terminates trajectories after a fixed total number of quantum jumps, e.g., emission events into the environment, the associated MPS is discrete. The continuous and discrete MPS correspond to different ensembles of quantum trajectories, one characterized by total time, the other by total number of quantum jumps. Hence, they give rise to quantum versions of different thermodynamic ensembles, akin to "grand canonical" and "isobaric," but for trajectories. Here, we prove that these trajectory ensembles are equivalent in a suitable limit of long time or large number of jumps. This is in direct analogy to equilibrium statistical mechanics where equivalence between ensembles is only strictly established in the thermodynamic limit. An intrinsic quantum feature is that the equivalence holds only for all observables that commute with the number of quantum jumps.

Thermophoresis of an antiferromagnetic soliton

We study dynamics of an antiferromagnetic soliton under a temperature gradient. To this end, we start by phenomenologically constructing the stochastic Landau-Lifshitz-Gilbert equation for an antiferromagnet with the aid of the fluctuation-dissipation theorem. We then derive the Langevin equation for the soliton's center of mass by the collective coordinate approach. An antiferromagentic soliton behaves as a classical massive particle immersed in a viscous medium. By considering a thermodynamic ensemble of solitons, we obtain the Fokker-Planck equation, from which we extract the average drift velocity of a soliton. The diffusion coefficient is inversely proportional to a small damping constant $\alpha$, which can yield a drift velocity of tens of m/s under a temperature gradient of $1$ K/mm for a domain wall in an easy-axis antiferromagnetic wire with $\alpha \sim 10^{-4}$.

Measuring the differential stoichiometry and energetics of ligand binding to macromolecules by single-molecule force spectroscopy: an extended theory.

Many chemical techniques exist for measuring the stoichiometry of ligand binding to a macromolecule; however, these techniques are often specific to certain ligands or require the presumption of specific binding models. Here, we further develop a previously reported, general, thermodynamic method for extracting the change in number of ligands bound to a macromolecule as that macromolecule undergoes a conformational transition driven by mechanical stretching, for example, by magnetic tweezers or optical trapping. We extend the theory of this method to consider systems with many ligands, experiments conducted in different thermodynamic ensembles (e.g., constant-force, constant-extension), and experiments in which the system is not at equilibrium. Further, we show that analysis of the same single-molecule mechanical manipulation data yields a measure of the differential free energy of stabilization due to ligand binding, that is, the free energy contribution by which ligand binding favors one conformation of the macromolecule over another. We interpret an existing data set measuring ion binding to RNA and DNA in terms of this free energy.

Thermodynamic-ensemble independence of solvation free energy.

Solvation free energy is the fundamental thermodynamic quantity in solution chemistry. Recently, it has been suggested that the partial molar volume correction is necessary to convert the solvation free energy determined in different thermodynamic ensembles. Here, we demonstrate ensemble-independence of the solvation free energy on general thermodynamic grounds. Theoretical estimates of the solvation free energy based on the canonical or grand-canonical ensemble are pertinent to experiments carried out under constant pressure without any conversion.

Thermodynamic curvature and ensemble nonequivalence

In this work we consider thermodynamic geometries defined as Hessians of different potentials and derive some useful formulae that show their complementary role in the description of thermodynamic systems with two degrees of freedom that show ensemble nonequivalence. From the expressions derived for the metrics, we can obtain the curvature scalars in a very simple and compact form. We explain here the reason why each curvature scalar diverges over the line of divergence of one of the specific heats. This application is of special interest in the study of changes of stability in black holes as defined by Davies. From these results we are able to prove on a general footing a conjecture first formulated by Liu, L\"u, Luo and Shao stating that different Hessian metrics can correspond to different behaviors in the various ensembles. We study the case of two thermodynamic dimensions. Moreover, comparing our result with the more standard turning point method developed by Poincar\'e, we obtain that the divergence of the scalar curvature of the Hessian metric of one potential exactly matches the change of stability in the corresponding ensemble.

Antenna entropy in plant photosystems does not reduce the free energy for primary charge separation

We have investigated the concept of the so-called “antenna entropy” of higher plant photosystems. Several interesting points emerge:1.In the case of a photosystem which harbours an excited state, the “antenna entropy” is equivalent to the configurational (mixing) entropy of a thermodynamic canonical ensemble. The energy associated with this parameter has been calculated for a hypothetical isoenergetic photosystem, photosystem I and photosystem II, and comes out in the range of 3.5 - 8% of the photon energy considering 680nm.2.The “antenna entropy” seems to be a rather unique thermodynamic phenomenon, in as much as it does not modify the free energy available for primary photochemistry, as has been previously suggested.3.It is underlined that this configurational (mixing) entropy, unlike heat dispersal in a thermal system, does not involve energy dilution. This points out an important difference between thermal and electronic energy dispersal.

Elucidating the effects of adsorbent flexibility on fluid adsorption using simple models and flat-histogram sampling methods.

Using flat-histogram Monte Carlo methods, we investigate the adsorptive behavior of the square-well fluid in two simple slit-pore-like models intended to capture fundamental characteristics of flexible adsorbent materials. Both models require as input thermodynamic information about the flexible adsorbent material itself. An important component of this work involves formulating the flexible pore models in the appropriate thermodynamic (statistical mechanical) ensembles, namely, the osmotic ensemble and a variant of the grand-canonical ensemble. Two-dimensional probability distributions, which are calculated using flat-histogram methods, provide the information necessary to determine adsorption thermodynamics. For example, we are able to determine precisely adsorption isotherms, (equilibrium) phase transition conditions, limits of stability, and free energies for a number of different flexible adsorbent materials, distinguishable as different inputs into the models. While the models used in this work are relatively simple from a geometric perspective, they yield non-trivial adsorptive behavior, including adsorption-desorption hysteresis solely due to material flexibility and so-called "breathing" of the adsorbent. The observed effects can in turn be tied to the inherent properties of the bare adsorbent. Some of the effects are expected on physical grounds while others arise from a subtle balance of thermodynamic and mechanical driving forces. In addition, the computational strategy presented here can be easily applied to more complex models for flexible adsorbents.

On the thermodynamic stability of rotating black holes in higher dimensions—a comparision of thermodynamic ensembles

Thermodynamic potentials relevant to the microcanonical, the canonical and the grand canonical ensembles, associated with rotating black holes in D-dimensions, are analysed and compared. Such black holes are known to be thermodynamically unstable, but the instability is a consequence of a subtle interplay between specific heats and the moments of inertia and it manifests itself differently in the different ensembles. A simple relation between the product of the specific heat and the determinant of the moment of inertia in both the canonical and the grand canonical ensembles is derived. Myers–Perry black holes in arbitrary dimension are studied in detail. All temperature extrema in the microcanonical ensemble are determined and classified. The specific heat and the moment of inertia tensor are evaluated in both the canonical and the grand canonical ensembles in any dimension. All zeros and poles of the specific heats, as a function of the angular momenta, are determined and the eigenvalues of the isentropic moment of inertia tensor are studied and classified. It is further shown that many of the thermodynamic properties of a Myers–Perry black hole in dimensions can be obtained from those of a black hole in D dimensions by sending one of the angular momenta to infinity.

Designing and analyzing the structure of Tat-BoNT/A(1-448) fusion protein: An in silico approach.

Clostridium botulinum type A (BoNT/A) produces a neurotoxin recently found to be useful as an injectable drug for the treatment of abnormal muscle contractions. The catalytic domain of this toxin which is responsible for the main toxin activity is a zinc metalloprotease that inhibits the release of neurotransmitter mediators in neuromuscular junctions. A cell penetrating cationic peptide, Tat, which is a truncated N-terminal part of the Tat protein from human immunodeficiency virus, can help the toxin penetrate the skin uninvasively. This study aimed at an in silico analyses of the Tat-BoNT/A(1-448) fusion protein structure. A genomic construct was designed and optimized based on E. coli codon usage. The structure of mRNA as well as the properties of hypothetical chimeric protein was then analyzed by bioinformatic tools. Afterwards, the secondary and tertiary structures of the fusion protein were predicted by GOR4 and I-TASSER online web servers. The interaction with synaptosomal associated protein 25kDa (SNAP-25) was also analyzed as a natural substrate for the toxin. Based on the studied secondary and tertiary structures of the protein, the selected order of fusion proteins provides the natural activity of each peptide. Energy calculating data show that the acquired thermodynamic ensemble related to the mRNA structure was-1473.2 kJ/mol (-352.10 kcal/mol) and both total protein energy (Etotal) and shape related energy(Eshape) were calculated as -2294.2kJ/mol (-548.32 kcal/mol). The stability index of TAT-BoNT/A was computed to be 27.22 which has an acceptable stability as compared to that of native BoNT/A (22.39).

Holographic Renormalization in Dense Medium

The holographic renormalization of a charged black brane with or without a dilaton field, whose dual field theory describes a dense medium at finite temperature, is investigated in this paper. In a dense medium, two different thermodynamic descriptions are possible due to an additional conserved charge. These two different thermodynamic ensembles are classified by the asymptotic boundary condition of the bulk gauge field. It is also shown that in the holographic renormalization regularity of all bulk fields can reproduce consistent thermodynamic quantities and that the Bekenstein-Hawking entropy is nothing but the renormalized thermal entropy of the dual field theory. Furthermore, we find that the Reissner-Nordström AdS black brane is dual to a theory with conformal matter as expected, whereas a charged black brane with a nontrivial dilaton profile is mapped to a theory with nonconformal matter although its leading asymptotic geometry still remains as AdS space.

Accelerated Monte-Carlo Simulations for All-Atom Protein Folding

Recent advances in the computational description of proteins facilitate the atomic-level characterization of fast folding proteins using molecular-dynamics simulations[1]. However, due to the femtosecond timestep of an MD simulation, 10 ˆ 10 computationally expensive energy evaluations are required to reach timescales of the order of single folding events even for ultrafast-folding proteins. Stochastic simulation techniques can overcome this limiting factor by generating thermodynamic conformational ensembles without incorporating atomic vibrations, but most of these methods are inherently sequential. Here we investigate a parallel extension to the conventional Metropolis-Hastings algorithm, the Multiple-Try-Metropolis, enabling parallel simulation of reversible Markov-chains[2,3]. We successfully characterize the thermodynamic landscape of the Villin-headpiece-subdomain using a generalized MTM approach adapted to protein systems. We parameterized the polypeptide using the AMBER99SB-STAR-ILDN-forcefield with an implicit solvent model. In our simulations we could observe approximately two folding transitions per day on standard hardware, with a total of 109 transition events during 3∗10 ˆ 9 energy evaluations. The MTM approach thus facilitates the efficient thermodynamic characterization of proteins on common computer architectures. The comparison with experimental resolved folding times of 8µs translates to a time equivalent of 16ps per MTM step.[1] Shaw, Science-2010.[2] Liu, JASA-2000.[3] Pandolfi, JMLR-2010.View Large Image | View Hi-Res Image | Download PowerPoint Slide

The Effective Temperature of Mutations: A General Mechanism for the Congruent Evolution of Robustness

Genetic robustness is critical to the understanding of evolution, as phenotypically expressed variation is the fuel of natural selection. The origin of robustness, whether it evolves directly by natural selection or it is a correlated byproduct of other phenotypic traits, is unresolved. Examining microRNA (miRNA) genes of several eukaryotic species, Borenstein and Ruppin [1] showed that the structure of miRNA precursor stem-loops exhibits significantly increased mutational robustness in comparison with random RNA sequences with the same stem-loop structure. Introducing a novel measure of robustness based on the equilibrium thermodynamic ensemble of secondary structures of miRNA precursor sequences, we demonstrated [2] that, (i) miRNA are significantly more tolerant with respect to thermal and mutational perturbations (exhibit enhanced thermodynamic and mutational robustness) than samples of random RNA sequences with the same structure, (ii) the biophysics of RNA folding induces a high level of correlation between the responses to mutational and thermodynamic perturbations. Motivated by the striking similarity between the effects of mutational and thermodynamic perturbations that the observed high level of correlation implies, here we attempt to quantify this similarity. Extending our study to lattice proteins we demonstrate that the effects of mutations can very generally and with unanticipated precision be described as an effective temperature increase of the thermodynamic ensemble of structures [3]. This result presents a general mechanism for the congruent evolution of thermodynamic and mutational robustness in the context of molecular phenotypes, and provides an explanation of how increased stability can facilitate opportunities for molecular innovation by facilitating increased neutral variation. [1] Borenstein & Ruppin (2006) Direct evolution of genetic robustness in microRNA. PNAS. [2] Szollosi & Derenyi (2009) Congruent evolution of genetic and environmental robustness in microRNA. Mol. Biol. Evol. [3] Szollosi & Derenyi (in preperation) The effective temperature of mutations and the congruent evolution of robustness.

Estimation of Interaction Potentials through the Configurational Temperature Formalism.

Molecular interaction potentials are difficult to measure experimentally and hard to compute from first principles, especially for large systems such as proteins. It is therefore desirable to estimate the potential energy that underlies a thermodynamic ensemble from simulated or experimentally determined configurations. This inverse problem of statistical mechanics is challenging because the various potential energy terms can exhibit subtle indirect and correlated effects on the resulting ensemble. A direct approach would try to adapt the force field parameters such that the given configurations are highly probable in the resulting ensemble. But this would require a full simulation of the system whenever a parameter changes. We introduce an extension of the configurational temperature formalism that allows us to circumvent these difficulties and efficiently estimate interaction potentials from molecular configurations. We illustrate the approach for various systems including fluids and a coarse-grained protein model.

Dependence of the Population on the Temperature in the Boltzmann Distribution: A Simple Relation Involving the Average Energy

The dependence on the temperature of the population of the ith state, Pi, in the Boltzmann distribution is analyzed by studying its derivative with respect to the temperature, T. A simple expression is found, involving Pi, the energy of the state, Ei, and the average energy, ⟨E⟩. This relation is completely general (it has the same form in all the thermodynamic ensembles), and it has a relevant didactic content, given that it predicts the qualitative variation of Pi with T even in complex systems. The derivation of this relation, the discussion of its properties, and its application to simple problems is appropriate for a statistical thermodynamics course in the chemistry curriculum.

Probing protein ensemble rigidity and hydrogen–deuterium exchange

Protein rigidity and flexibility can be analyzed accurately and efficiently using the program floppy inclusion and rigid substructure topography (FIRST). Previous studies using FIRST were designed to analyze the rigidity and flexibility of proteins using a single static (snapshot) structure. It is however well known that proteins can undergo spontaneous sub-molecular unfolding and refolding, or conformational dynamics, even under conditions that strongly favor a well-defined native structure. These (local) unfolding events result in a large number of conformers that differ from each other very slightly. In this context, proteins are better represented as a thermodynamic ensemble of ‘native-like’ structures, and not just as a single static low-energy structure. Working with this notion, we introduce a novel FIRST-based approach for predicting rigidity/flexibility of the protein ensemble by (i) averaging the hydrogen bonding strengths from the entire ensemble and (ii) by refining the mathematical model of hydrogen bonds. Furthermore, we combine our FIRST-ensemble rigidity predictions with the ensemble solvent accessibility data of the backbone amides and propose a novel computational method which uses both rigidity and solvent accessibility for predicting hydrogen–deuterium exchange (HDX). To validate our predictions, we report a novel site specific HDX experiment which characterizes the native structural ensemble of Acylphosphatase from hyperthermophile Sulfolobus solfataricus (Sso AcP). The sub-structural conformational dynamics that is observed by HDX data, is closely matched with the FIRST-ensemble rigidity predictions, which could not be attained using the traditional single ‘snapshot’ rigidity analysis. Moreover, the computational predictions of regions that are protected from HDX and those that undergo exchange are in very good agreement with the experimental HDX profile of Sso AcP.

Adsorption Deformation and Structural Transitions in Metal–Organic Frameworks: From the Unit Cell to the Crystal

Much attention has recently been focused on soft porous crystals, a fascinating subclass of metal–organic frameworks that behave in a remarkable stimuli-responsive fashion, presenting structural changes of large amplitude in response to guest adsorption, mechanical pressure, or variations in temperature. In this Perspective, we summarize the recently developed thermodynamic and mechanical theoretical models for the understanding of these materials, based on the concepts of adsorption stress and osmotic thermodynamic ensemble. We show how these models provide a coherent picture of adsorption-induced deformation and structural transitions in flexible metal–organic frameworks, all the way from the length scale of the unit cell to that of the full crystal. In particular, we highlight the new perspectives opened by these models, as well as some of the important open questions in the field.

Crystallites of magnetic charges in artificial spin ice

Artificial spin ice is a class of lithographically created arrays of interacting ferromagnetic nanometre-scale islands. It was introduced to investigate many-body phenomena related to frustration and disorder in a material that could be tailored to precise specifications and imaged directly. Because of the large magnetic energy scales of these nanoscale islands, it has so far been impossible to thermally anneal artificial spin ice into desired thermodynamic ensembles; nearly all studies of artificial spin ice have either treated it as a granular material activated by alternating fields or focused on the as-grown state of the arrays. This limitation has prevented experimental investigation of novel phases that can emerge from the nominal ground states of frustrated lattices. For example, artificial kagome spin ice, in which the islands are arranged on the edges of a hexagonal net, is predicted to support states with monopolar charge order at entropies below that of the previously observed pseudo-ice manifold. Here we demonstrate a method for thermalizing artificial spin ices with square and kagome lattices by heating above the Curie temperature of the constituent material. In this manner, artificial square spin ice achieves unprecedented thermal ordering of the moments. In artificial kagome spin ice, we observe incipient crystallization of the magnetic charges embedded in pseudo-ice, with crystallites of magnetic charges whose size can be controlled by tuning the lattice constant. We find excellent agreement between experimental data and Monte Carlo simulations of emergent charge-charge interactions.

Generalized Deam–Edwards approach to the statistical mechanics of randomly crosslinked systems

We address the statistical mechanics of randomly and permanently crosslinked networks. We develop a theoretical framework (vulcanization theory) which can be used to systematically analyze the correlation between the statistical properties of random networks and their histories of formation. Generalizing the original idea of Deam and Edwards, we consider an instantaneous crosslinking process, where all crosslinkers (modeled as Gaussian springs) are introduced randomly at once in an equilibrium liquid state, referred to as the preparation state. The probability that two functional sites are crosslinked by a spring exponentially decreases with their distance squared. After formally averaging over network connectivity, we obtained an effective theory with all degrees of freedom replicated 1 + n times. Two thermodynamic ensembles, the preparation ensemble and the measurement ensemble, naturally appear in this theory. The former describes the thermodynamic fluctuations in the state of preparation, while the latter describes the thermodynamic fluctuations in the state of measurement. We classify various correlation functions and discuss their physical significances. In particular, the memory correlation functions characterize how the properties of networks depend on their method of preparation, and are the hallmark properties of all randomly crosslinked materials. We clarify the essential difference between our approach and that of Deam–Edwards, and discuss the saddle-point order parameters and its physical significance. Finally we also discuss the connection between saddle-point approximation of vulcanization theory, and the classical theory of rubber elasticity as well as the neo-classical theory of nematic elastomers.

Massively parallel Monte Carlo for many-particle simulations on GPUs

Current trends in parallel processors call for the design of efficient massively parallel algorithms for scientific computing. Parallel algorithms for Monte Carlo simulations of thermodynamic ensembles of particles have received little attention because of the inherent serial nature of the statistical sampling. In this paper, we present a massively parallel method that obeys detailed balance and implement it for a system of hard disks on the GPU. We reproduce results of serial high-precision Monte Carlo runs to verify the method. This is a good test case because the hard disk equation of state over the range where the liquid transforms into the solid is particularly sensitive to small deviations away from the balance conditions. On a Tesla K20, our GPU implementation executes over one billion trial moves per second, which is 148 times faster than on a single Intel Xeon E5540 CPU core, enables 27 times better performance per dollar, and cuts energy usage by a factor of 13. With this improved performance we are able to calculate the equation of state for systems of up to one million hard disks. These large system sizes are required in order to probe the nature of the melting transition, which has been debated for the last forty years. In this paper we present the details of our computational method, and discuss the thermodynamics of hard disks separately in a companion paper.

Rigid DNA Beams for High‐Resolution Single‐Molecule Mechanics

Rigid DNA Beams for High‐Resolution Single‐Molecule Mechanics