Equilibrium Ensemble Research Articles

General theory for discrete symmetry-breaking equilibrium states in quantum systems.

Spontaneous symmetrybreaking in phase transitions occurs when the system Hamiltonian is symmetric under a certain transformation, but the equilibrium states observed in nature are not. Here we build two noncommuting quantities from the order parameter of the transition and the symmetry operator that are constants of motion when such equilibrium states exist. Then, we derive a general equilibrium ensemble for the ordered phase and show that equilibrium states consisting of superpositions of different symmetry-breaking states, like positive and negative magnetized states, may exist. We propose an experimental realization of such equilibrium states with the state-of-the-art quantum technologies, and test it by means of numerical calculations. Finally, we show that a small symmetry-breaking perturbation in the Hamiltonian stabilizes the conservation of one of the two former quantities, implying that symmetry-breaking equilibrium states become stable even in small quantum systems.

Performance at maximum figureof merit for a Brownian Carnot refrigerator.

This paper focuses on the coefficient of performance (COP) at maximum χ^{R} figureof merit for a Brownian Carnot-like refrigerator, within the context of the low-dissipation approach. Our proposal is based on the Langevin equationfor a Brownian particle bounded to a harmonic potential trap, which can perform Carnot-like cycles at finite time. The theoretical approach is related to the equilibrium ensemble average of 〈x^{2}〉_{eq} which plays the role of a statelike equation, x being the Brownian particle position. This statelike equationcomes from the macroscopic version of the corresponding Langevin equationfor a Brownian particle. We show that under quasistatic conditions the COP has the same expression as the macroscopic Carnot refrigerator, while for irreversible cycles at finite time and under symmetric dissipation the optimal COP is the counterpart of Curzon-Ahlborn efficiency as also obtained for irreversible macroscopic refrigerators.

Molecular dynamics modelling of interacting magnetic nanoparticles for investigating equilibrium and dynamic ensemble properties

Magnetic nanoparticles have the potential to be used in various biomedical applications, including magnetic drug targeting and hyperthermia for cancer treatment. In both cases, precise prediction of the local particle concentration is required to plan a successful therapy, which is a challenging task due to several complex flow phenomena. Computational macroscopic and microscopic models have been developed to support this process, but they have limitations in terms of interactions implementation, micromagnetic dynamics or computational costs. This study aims to develop a model that can represent micromagnetic dynamics and being employed to study equilibrium properties of particle ensembles in viscous media. Therefore, a kinetic Monte Carlo method in combination with Langevin equations is used. So, magnetization dynamics and mechanical motion can be simulated in combination very efficiently. The new model is validated with another model based on the stochastic Landau-Lifshitz-Gilbert equation. Various interaction potentials like Van der Waals, steric and electrostatic interactions are included to study their specific influence on structural properties. Also, methods to control the errors while integrating the equations of motion and using the Ewald method for calculating interaction quantities are implemented. As a validation we studied equilibrium and time dependent properties of non-interacting particle ensembles as well as errors made by the Ewald-method used for interaction quantities calculation. In all studies we found very good agreement of the simulation results with the theoretical predictions. The developed models can now be used for investigating equilibrium and dynamic properties of ferrofluids, or more general for biomedical research for magnetic drug targeting, hyperthermia, or imaging techniques. Furthermore, the new hybrid model can be also used for simulations in the range of milliseconds with reasonable computational effort.

AlphaFold2 Predictions of Conformational Ensembles and Atomistic Simulations of the SARS-CoV-2 Spike XBB Lineages Reveal Epistatic Couplings between Convergent Mutational Hotspots that Control ACE2 Affinity.

In this study, we combined AlphaFold-based atomistic structural modeling, microsecond molecular simulations, mutational profiling, and network analysis to characterize binding mechanisms of the SARS-CoV-2 spike protein with the host receptor ACE2 for a series of Omicron XBB variants including XBB.1.5, XBB.1.5+L455F, XBB.1.5+F456L, and XBB.1.5+L455F+F456L. AlphaFold-based structural and dynamic modeling of SARS-CoV-2 Spike XBB lineages can accurately predict the experimental structures and characterize conformational ensembles of the spike protein complexes with the ACE2. Microsecond molecular dynamics simulations identified important differences in the conformational landscapes and equilibrium ensembles of the XBB variants, suggesting that combining AlphaFold predictions of multiple conformations with molecular dynamics simulations can provide a complementary approach for the characterization of functional protein states and binding mechanisms. Using the ensemble-based mutational profiling of protein residues and physics-based rigorous calculations of binding affinities, we identified binding energy hotspots and characterized the molecular basis underlying epistatic couplings between convergent mutational hotspots. Consistent with the experiments, the results revealed the mediating role of the Q493 hotspot in the synchronization of epistatic couplings between L455F and F456L mutations, providing a quantitative insight into the energetic determinants underlying binding differences between XBB lineages. We also proposed a network-based perturbation approach for mutational profiling of allosteric communications and uncovered the important relationships between allosteric centers mediating long-range communication and binding hotspots of epistatic couplings. The results of this study support a mechanism in which the binding mechanisms of the XBB variants may be determined by epistatic effects between convergent evolutionary hotspots that control ACE2 binding.

Generation of conformational ensembles of small molecules via surrogate model-assisted molecular dynamics

The accurate prediction of thermodynamic properties is crucial in various fields such as drug discovery and materials design. This task relies on sampling from the underlying Boltzmann distribution, which is challenging using conventional approaches such as simulations. In this work, we introduce surrogate model-assisted molecular dynamics (SMA-MD), a new procedure to sample the equilibrium ensemble of molecules. First, SMA-MD leverages deep generative models to enhance the sampling of slow degrees of freedom. Subsequently, the generated ensemble undergoes statistical reweighting, followed by short simulations. Our empirical results show that SMA-MD generates more diverse and lower energy ensembles than conventional MD simulations. Furthermore, we showcase the application of SMA-MD for the computation of thermodynamical properties by estimating implicit solvation free energies.

Footprints of loop extrusion in statistics of intra-chromosomal distances: An analytically solvable model.

Active loop extrusion-the process of formation of dynamically growing chromatin loops due to the motor activity of DNA-binding protein complexes-is a firmly established mechanism responsible for chromatin spatial organization at different stages of a cell cycle in eukaryotes and bacteria. The theoretical insight into the effect of loop extrusion on the experimentally measured statistics of chromatin conformation can be gained with an appropriately chosen polymer model. Here, we consider the simplest analytically solvable model of an interphase chromosome, which is treated as an ideal chain with disorder of sufficiently sparse random loops whose conformations are sampled from the equilibrium ensemble. This framework allows us to arrive at the closed-form analytical expression for the mean-squared distance between pairs of genomic loci, which is valid beyond the one-loop approximation in diagrammatic representation. In addition, we analyze the loop-induced deviation of chain conformations from the Gaussian statistics by calculating kurtosis of probability density of the pairwise separation vector. The presented results suggest the possible ways of estimating the characteristics of the loop extrusion process based on the experimental data on the scale-dependent statistics of intra-chromosomal pair-wise distances.

Thermodynamics and stochastic thermodynamics of strongly coupled systems.

We further develop the strong-coupling theory of thermodynamics and stochastic thermodynamics for continuous systems, constructed in the previous work [Phys. Rev. Res. 4, 013015 (2022)2643-156410.1103/PhysRevResearch.4.013015]. A small system strongly interacting with a its environment, the dynamics of the system is assumed to be much slower than that of the bath. The system Hamiltonian is defined to be the Hamiltonian of mean force, whereas the system entropy is defined as the Gibbs-Shannon entropy. Equilibrium ensemble theories and thermodynamic theories are established for the system. Variations of three types of parameters are considered: (i) the system parameter λ which couples to the system and to the interaction, (ii) the bath parameter λ^{'} which couples to the bath only, and (iii) the temperature T=1/β. The work done to the system consists of three parts, proportional to dλ,dλ^{'}, and dβ respectively. The part proportional to dβ can be understood as the work done by the bath. As long as λ^{'} and β are not fixed, the work is not the change of total energy of the joint system. The differences between our strong-coupling equilibrium thermodynamics and the classical thermodynamics are discussed. The thermodynamic theory is promoted to the nonequilibrium level. Both the first and second laws of thermodynamics, as well as fluctuation theorems, are established for nonequilibrium processes. For processes with varying temperatures, fluctuation theorems cannot be expressed in terms of integrated work alone. Regardless of various subtleties, however, the stochastic thermodynamic theory is formulated in terms of system variables only, and dS-βd[over ¯]Q is the change of total entropy. Thermodynamic quantities of the system are related to those of the joint system, and the equivalence of theories at two levels of coarse-graining is explicitly demonstrated. Finally we show that there are infinite numbers of equivalent strong-coupling theories, each determined by its definition of system Hamiltonian. Our theory is distinguished by its maximal similarity with the weak-coupling theory.

AlphaFold2-Enabled Atomistic Modeling of Structure, Conformational Ensembles, and Binding Energetics of the SARS-CoV-2 Omicron BA.2.86 Spike Protein with ACE2 Host Receptor and Antibodies: Compensatory Functional Effects of Binding Hotspots in Modulating Mechanisms of Receptor Binding and Immune Escape.

The latest wave of SARS-CoV-2 Omicron variants displayed a growth advantage and increased viral fitness through convergent evolution of functional hotspots that work synchronously to balance fitness requirements for productive receptor binding and efficient immune evasion. In this study, we combined AlphaFold2-based structural modeling approaches with atomistic simulations and mutational profiling of binding energetics and stability for prediction and comprehensive analysis of the structure, dynamics, and binding of the SARS-CoV-2 Omicron BA.2.86 spike variant with ACE2 host receptor and distinct classes of antibodies. We adapted several AlphaFold2 approaches to predict both the structure and conformational ensembles of the Omicron BA.2.86 spike protein in the complex with the host receptor. The results showed that the AlphaFold2-predicted structural ensemble of the BA.2.86 spike protein complex with ACE2 can accurately capture the main conformational states of the Omicron variant. Complementary to AlphaFold2 structural predictions, microsecond molecular dynamics simulations reveal the details of the conformational landscape and produced equilibrium ensembles of the BA.2.86 structures that are used to perform mutational scanning of spike residues and characterize structural stability and binding energy hotspots. The ensemble-based mutational profiling of the receptor binding domain residues in the BA.2 and BA.2.86 spike complexes with ACE2 revealed a group of conserved hydrophobic hotspots and critical variant-specific contributions of the BA.2.86 convergent mutational hotspots R403K, F486P, and R493Q. To examine the immune evasion properties of BA.2.86 in atomistic detail, we performed structure-based mutational profiling of the spike protein binding interfaces with distinct classes of antibodies that displayed significantly reduced neutralization against the BA.2.86 variant. The results revealed the molecular basis of compensatory functional effects of the binding hotspots, showing that BA.2.86 lineage may have evolved to outcompete other Omicron subvariants by improving immune evasion while preserving binding affinity with ACE2 via through a compensatory effect of R493Q and F486P convergent mutational hotspots. This study demonstrated that an integrative approach combining AlphaFold2 predictions with complementary atomistic molecular dynamics simulations and robust ensemble-based mutational profiling of spike residues can enable accurate and comprehensive characterization of structure, dynamics, and binding mechanisms of newly emerging Omicron variants.

Free-energy measuring nanopore device.

Free energies (FEs) in molecular sciences can be used to quantify the stability of folded molecules. In this article, we introduce nanopores for measuring FEs. We pull DNA hairpin-forming molecules through a nanopore, measure work, and estimate the FE change in the slow limit, and with the Jarzynski fluctuation theorem (FT) at fast pulling times. We also pull our molecule with optical tweezers, compare it to nanopores, and explore how sampling single molecules from equilibrium or a folded ensemble affects the FE estimate via the FT. The nanopore experiment helps us address and overcome the conceptual problem of equilibrium sampling in single-molecule pulling experiments. Only when molecules are sampled from an equilibrium ensemble do nanopore and tweezer FE estimates mutually agree. We demonstrate that nanopores are very useful tools for comparing FEs of two molecules at finite times and we propose future applications.

Threading Subunits for Polymers to Predict the Equilibrium Ensemble of Solid Polymer Electrolytes.

We present a computational method for polymer growth called "threading subunits for polymers (TSP)" that can efficiently sample solid polymer electrolyte structures with extended conformations. The TSP method involves equilibrating subunit (e.g., monomer) conformations that form favorable solvation ion shells, followed by consecutively connecting the subunits and minimizing the structures. The TSP method can sample polymers with good solvent-like conformations and from near-equilibrium structures in which ions are well-dispersed, avoiding unusual ion clustering under ambient conditions. Using the TSP method, the equilibration time can be reduced significantly by effectively sampling the polymer conformations near equilibrium. We anticipate that the TSP method can be applied to simulate various polymer electrolytes.

Canonical Density Matrices from Eigenstates of Mixed Systems.

One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (N→∞) limit of large quantum many-body systems, canonical density matrices emerge for small subsystems from almost all pure states. This notion of canonical typicality is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. For individual eigenstates, it has been shown that local observables show thermal properties provided the eigenstate thermalization hypothesis holds, which requires the system to be quantum-chaotic. In the present paper, we study the emergence of thermal states in the regime of a quantum analog of a mixed phase space. Specifically, we study the emergence of the canonical density matrix of an impurity upon reduction from isolated energy eigenstates of a large but finite quantum system the impurity is embedded in. Our system can be tuned by means of a single parameter from quantum integrability to quantum chaos and corresponds in between to a system with mixed quantum phase space. We show that the probability for finding a canonical density matrix when reducing the ensemble of energy eigenstates of the finite many-body system can be quantitatively controlled and tuned by the degree of quantum chaos present. For the transition from quantum integrability to quantum chaos, we find a continuous and universal (i.e., size-independent) relation between the fraction of canonical eigenstates and the degree of chaoticity as measured by the Brody parameter or the Shannon entropy.

A Note on the Effects of Linear Topology Preservation in Monte Carlo Simulations of Knotted Proteins.

Monte Carlo simulations are a powerful technique and are widely used in different fields. When applied to complex molecular systems with long chains, such as those in synthetic polymers and proteins, they have the advantage of providing a fast and computationally efficient way to sample equilibrium ensembles and calculate thermodynamic and structural properties under desired conditions. Conformational Monte Carlo techniques employ a move set to perform the transitions in the simulation Markov chain. While accepted conformations must preserve the sequential bonding of the protein chain model and excluded volume among its units, the moves themselves may take the chain across itself. We call this a break in linear topology preservation. In this manuscript, we show, using simple protein models, that there is no difference in equilibrium properties calculated with a move set that preserves linear topology and one that does not. However, for complex structures, such as those of deeply knotted proteins, the preservation of linear topology provides correct equilibrium results but only after long relaxation. In any case, to analyze folding pathways, knotting mechanisms and folding kinetics, the preservation of linear topology may be an unavoidable requirement.

Tools for analyzing protonation states and for tracing proton transfer pathways with examples from the Rb. sphaeroides photosynthetic reaction centers.

Protons participate in many reactions. In proteins, protons need paths to move in and out of buried active sites. The vectorial movement of protons coupled to electron transfer reactions establishes the transmembrane electrochemical gradient used for many reactions, including ATP synthesis. Protons move through hydrogen bonded chains of waters and hydroxy side chains via the Grotthuss mechanism and by proton binding and release from acidic and basic residues. MCCE analysis shows that proteins exist in a large number of protonation states. Knowledge of the equilibrium ensemble can provide a rational basis for setting protonation states in simulations that fix them, such as molecular dynamics (MD). The proton path into the QB site in the bacterial reaction centers (RCs) of Rb. sphaeroides is analyzed by MD to provide an example of the benefits of using protonation states found by the MCCE program. A tangled web of side chains and waters link the cytoplasm to QB. MCCE analysis of snapshots from multiple trajectories shows that changing the input protonation state of a residue in MD biases the trajectory shifting the proton affinity of that residue. However, the proton affinity of some residues is more sensitive to the input structure. The proton transfer networks derived from different trajectories are quite robust. There are some changes in connectivity that are largely restricted to the specific residues whose protonation state is changed. Trajectories with QB•- are compared with earlier results obtained with QB [Wei et. al Photosynthesis Researchvolume152,pages153-165 (2022)] showing only modest changes. While introducing new methods the study highlights the difficulty of establishing the connections between protein conformation.

Precise determination of pair interactions from pair statistics of many-body systems in and out of equilibrium.

The determination of the pair potential v(r) that accurately yields an equilibrium state at positive temperature T with a prescribed pair correlation function g_{2}(r) or corresponding structure factor S(k) in d-dimensional Euclidean space R^{d} is an outstanding inverse statistical mechanics problem with far-reaching implications. Recently, Zhang and Torquato [Phys. Rev. E 101, 032124 (2020)2470-004510.1103/PhysRevE.101.032124] conjectured that any realizable g_{2}(r) or S(k) corresponding to a translationally invariant nonequilibrium system can be attained by a classical equilibrium ensemble involving only (up to) effective pair interactions. Testing this conjecture for nonequilibrium systems as well as for nontrivial equilibrium states requires improved inverse methodologies. We have devised an optimization algorithm to precisely determine effective pair potentials that correspond to pair statistics of general translationally invariant disordered many-body equilibrium or nonequilibrium systems at positive temperatures. This methodology utilizes a parameterized family of pointwise basis functions for the potential function whose initial form is informed by small-, intermediate- and large-distance behaviors dictated by statistical-mechanical theory. Subsequently, a nonlinear optimization technique is utilized to minimize an objective function that incorporates both the target pair correlation function g_{2}(r) and structure factor S(k) so that the small intermediate- and large-distance correlations are very accurately captured. To illustrate the versatility and power of our methodology, we accurately determine the effective pair interactions of the following four diverse target systems: (1) Lennard-Jones system in the vicinity of its critical point, (2) liquid under the Dzugutov potential, (3) nonequilibrium random sequential addition packing, and (4) a nonequilibrium hyperuniform "cloaked" uniformly randomized lattice. We found that the optimized pair potentials generate corresponding pair statistics that accurately match their corresponding targets with total L_{2}-norm errors that are an order of magnitude smaller than that of previous methods. The results of our investigation lend further support to the Zhang-Torquato conjecture. Furthermore, our algorithm will enable one to probe systems with identical pair statistics but different higher-body statistics, which will shed light on the well-known degeneracy problem of statistical mechanics.

Information-Guided Dynamic Nuclear Polarization

A spin ensemble in thermal equilibrium continuously undergoes random fluctuations analogous to those observed in Brownian motion, a process known as spin noise. Here, we investigate the dynamics of a system comprising a spin-1 paramagnetic center and a hyperfine-coupled spin-1/2 nucleus in the vicinity of a level crossing. We theoretically show that nuclear spins polarize efficiently under the combined action of thermal fluctuations and a closed-loop feedback protocol. The latter articulates periodic observations of the electronic magnetization and radio-frequency pulses connecting hybrid states with opposite nuclear spin alignment. Since nuclear polarization emerges from electronic spin fluctuations, not spin order, this microwave-free technique generically benefits from warmer, not colder, operation temperatures. Furthermore, because the spin dynamics at play near a level crossing is rather insensitive to the absolute value of the magnetic field, our work promises opportunities for high-field dynamic nuclear polarization, which is difficult to attain through present methods.

Linear stochastic thermodynamics

We study the thermodynamics of open systems weakly driven out-of-equilibrium by nonconservative and time-dependent forces using the linear regime of stochastic thermodynamics. We make use of conservation laws to identify the potential and nonconservative components of the forces. This allows us to formulate a unified near-equilibrium thermodynamics. For nonequilibrium steady states, we obtain an Onsager theory ensuring nonsingular response matrices that is consistent with phenomenological linear irreversible thermodynamics. For time-dependent driving protocols that do not produce nonconservative forces, we identify the equilibrium ensemble from which Green–Kubo relations are recovered. For arbitrary periodic drivings, the averaged entropy production (EP) is expressed as an independent sum over each driving frequency of non-negative contributions. These contributions are bilinear in the nonconservative and conservative forces and involve a novel generalized Onsager matrix that is symmetric. In the most general case of arbitrary time-dependent drivings, we advance a novel decomposition of the EP rate into two non-negative contributions—one solely due to nonconservative forces and the other solely due to deviation from the instantaneous steady-state—directly implying a minimum EP principle close to equilibrium. This setting reveals the geometric structure of near-equilibrium thermodynamics and generalizes previous approaches to cases with nonconservative forces.

Annealing glasses by cyclic shear deformation.

A major challenge in simulating glassy systems is the ability to generate configurations that may be found in equilibrium at sufficiently low temperatures, in order to probe static and dynamic behavior close to the glass transition. A variety of approaches have recently explored ways of surmounting this obstacle. Here, we explore the possibility of employing mechanical agitation, in the form of cyclic shear deformation, to generate low energy configurations in a model glass former. We perform shear deformation simulations over a range of temperatures, shear rates, and strain amplitudes. We find that shear deformation induces faster relaxation toward low energy configurations, or overaging, in simulations at sufficiently low temperatures, consistently with previous results for athermal shear. However, for temperatures at which simulations can be run until a steady state is reached with or without shear deformation, we find that the inclusion of shear deformation does not result in any speed up of the relaxation toward low energy configurations. Although we find the configurations from shear simulations to have properties indistinguishable from an equilibrium ensemble, the cyclic shear procedure does not guarantee that we generate an equilibrium ensemble at a desired temperature. In order to ensure equilibrium sampling, we develop a hybrid Monte Carlo algorithm that employs cyclic shear as a trial generation step and has acceptance probabilities that depend not only on the change in internal energy but also on the heat dissipated (equivalently, work done). We show that such an algorithm, indeed, generates an equilibrium ensemble.

Free energy calculation of crystalline solids using normalizing flows

Taking advantage of the advances in generative deep learning, particularly normalizing flow, a framework, called Boltzmann generator, has recently been proposed to generate equilibrium atomic configurations from the canonical ensemble and determining the associated free energy. In this work, we revisit Boltzmann generator to motivate the construction of the loss function from the statistical mechanical point of view and to cast the training of the normalizing flow model in a purely unsupervised manner that does not require any sample of the atomic configurations from the equilibrium ensemble. We further show that the normalizing flow framework furnishes a reference thermodynamic system, very close to the real thermodynamic system under consideration, that is suitable for the well-established free energy perturbation methods to determine accurate free energy of solids. We then apply the normalizing flow to two problems: temperature-dependent Gibbs free energy of perfect crystal and formation free energy of monovacancy defect in a model system of diamond cubic Si. The results obtained from the normalizing flow are shown to be in good agreement with that obtained from independent well-established free energy methods.

A novel Random Forest integrated model for imbalanced data classification problem

In recent years, most researchers focused on the classification problems of imbalanced data sets, and these problems are widely distributed in industrial production and medical research fields. For these highly imbalanced data sets, the ensemble method based on over-sampling is one of the most competitive techniques in the present research. However, the incorrect sampling strategy easily affected the model performance, which increased the training complexity and caused an over-fitting problem. This article proposed an equilibrium ensemble method (DCI-ISSA) with two novel techniques to conquer these shortcomings. Firstly, this paper raised an over-sampling approach (Data Center Interpolation DCI) to offer a counterbalanced data set for the single learner, which can prevent the base learners from the impact of class imbalance. Additionally, we provided a parameter optimization method for Random Forest (RF), which used the Improved Sparrow Search Algorithm (ISSA) to find the optimal parameters for different imbalanced data sets dynamically. These parameters can improve the classification performance of base classifiers and adjust to all kinds of lopsided data sets with distinct sizes. Experimental results showed that the DCI-ISSA-RF model outperforms other famous approaches for the imbalanced data sets with various dimensions.

Reaction-path statistical mechanics of enzymatic kinetics.

We introduce a reaction-path statistical mechanics formalism based on the principle of large deviations to quantify the kinetics of single-molecule enzymatic reaction processes under the Michaelis-Menten mechanism, which exemplifies an out-of-equilibrium process in the living system. Our theoretical approach begins with the principle of equal a priori probabilities and defines the reaction path entropy to construct a new nonequilibrium ensemble as a collection of possible chemical reaction paths. As a result, we evaluate a variety of path-based partition functions and free energies by using the formalism of statistical mechanics. They allow us to calculate the timescales of a given enzymatic reaction, even in the absence of an explicit boundary condition that is necessary for the equilibrium ensemble. We also consider the large deviation theory under a closed-boundary condition of the fixed observation time to quantify the enzyme-substrate unbinding rates. The result demonstrates the presence of a phase-separation-like, bimodal behavior in unbinding events at a finite timescale, and the behavior vanishes as its rate function converges to a single phase in the long-time limit.