Microcanonical entropy inflection points: key to systematic understanding of transitions in finite systems

Microcanonical Analysis of Semiflexible Homopolymers with Variable-Width Bending Potential

Understanding the structural dynamics of semiflexible polymers in an implicit solvent under varying conditions provides valuable insights into their behavior in diverse environments. In this work, we systematically investigate the effect of the angular width of the bending potential on structural state behavior and conformational variability using microcanonical analysis. A range of angular widths is explored, with the widest value corresponding directly to the classic semiflexible polymer model, which exhibits a diverse set of structural states, including Two-Strand, Three-Strand, Four-Strand, Ring, Random Coil, and Globule configurations. As the angular width narrows, structural variability within states decreases, overlap between structural states is reduced, and conformations become more stable, leading to an expansion of the parameter space dominated by individual structures. By examining microcanonical entropy and its derivatives, we identify transitions analogous to first-, second-, and third-order thermodynamic transitions, providing a deeper understanding of the configurational landscape of semiflexible polymers.

The Glass Transition: A Topological Perspective

Resorting to microcanonical ensemble Monte Carlo simulations, we study the geometric and topological properties of the state space of a model of a network glass-former. This model, a Lennard-Jones binary mixture, does not crystallize due to frustration. We have found two peaks in specific heat at equilibrium and at low energy, corresponding to important changes in local ordering. These singularities were accompanied by inflection points in geometrical markers of the potential energy level sets-namely, the mean curvature, the dispersion of the principal curvatures, and the variance of the scalar curvature. Pinkall's and Overholt's theorems closely relate these quantities to the topological properties of the accessible state-space manifold. Thus, our analysis provides strong indications that the glass transition is associated with major changes in the topology of the energy level sets. This important result suggests that this phase transition can be understood through the topological theory of phase transitions.

A density of states-based approach to determine temperature-dependent aggregation rates

Here, we establish an approach to determine temperature-dependent aggregation rates in terms of thermostatistical quantities, which can be obtained directly from flat-histogram and statistical temperature algorithms considering the density of states of the system. Our approach is validated through simulations of an Ising-like model with anisotropically interacting particles at temperatures close to its first-order phase transition. Quantitative comparisons between the numerically obtained forward and reverse rates to approximate analytical expressions corroborate its use as a model-independent approach.

Thermodynamic stability at phase coexistence

The main point we address in this paper is the question of thermodynamic stability for phase-separating systems, at coexistence in equilibrium. It has long been known that numerical simulations of different statistical models may yield "Van der Waals-like" isotherms in the coexistence region. Such "inverted" convexity segments of thermodynamic fields, known as unstable, are forbidden by the second law of thermodynamics on entropy, and their presence is not justified in exact results. In numerical experiments, their origin has been associated with the interface between the two coexisting phases. Nonetheless, the violation of the second law by entropy has not yet, to our knowledge, been rationalized. In this work, we introduce the thermodynamics of the interface between coexisting phases and give an alternative interpretation to the theory developed by Hill in the 1960s. Our approach points to a misinterpretation of the usual measurements of thermodynamic potentials in simulations. Correct interpretation eliminates the unstable regions of the true potentials. Our adapted theory is verified for the 2D lattice gas through carefully planned simulations. The thermodynamic description of the interface behavior inside the coexistence region restores the proper convexity of the true chemical potential isotherms. As a bonus, our interpretation allows direct calculation of surface tension in very good accordance with Onsager's analytic prediction.

Microcanonical Analysis of Helical Homopolymers: Exploring the Density of States and Structural Characteristics

This study investigates the density of states and structural characteristics of helical homopolymers. Comprising repeating identical units, the model enables the exploration of complex behaviors arising from a simple, yet generalized, set of potentials. Utilizing microcanonical analysis, transitions between helical structures are identified and categorized. Through a systematic comparison of results under varying conditions, we develop a nuanced understanding of the system's general behavior. A two-dimensional plot illustrates the relative distribution of different structural types, effectively showcasing their prevalence. The findings of this study substantially advance our understanding of the density of states and structural transformations of helical homopolymers across a range of conditions. Additionally, the prevalence plot offers valuable insights into the occurrence of suppressed intermediate states, particularly in models featuring stiff helix segments. This research significantly enhances our understanding of the complex interactions governing helix bundling phenomena within the context of helical homopolymers.

Partial Order-Disorder Transition Driving Closure of Band Gap: Example of Thermoelectric Clathrates

In the quest for efficient thermoelectrics, semiconducting behavior is a targeted property. Yet, this is often difficult to achieve due to the complex interplay between electronic structure, temperature, and disorder. We find this to be the case for the thermoelectric clathrate Ba_{8}Al_{16}Si_{30}: Although this material exhibits a band gap in its ground state, a temperature-driven partial order-disorder transition leads to its effective closing. This finding is enabled by a novel approach to calculate the temperature-dependent effective band structure of alloys. Our method fully accounts for the effects of short-range order and can be applied to complex alloys with many atoms in the primitive cell, without relying on effective medium approximations.

Comparison of Conformational Phase Behavior for Flexible and Semiflexible Polymers

We employ the recently introduced generalized microcanonical inflection point method for the statistical analysis of phase transitions in flexible and semiflexible polymers and study the impact of the bending stiffness upon the character and order of transitions between random-coil, globules, and pseudocrystalline conformations. The high-accuracy estimates of the microcanonical entropy and its derivatives required for this study were obtained by extensive replica-exchange Monte Carlo simulations. We observe that the transition behavior into the compact phases changes qualitatively with increasing bending stiffness. Whereas the Θ collapse transition is less affected, the first-order liquid-solid transition characteristic for flexible polymers ceases to exist once bending effects dominate over attractive monomer-monomer interactions.

Exploring the landscape of model representations

The success of any physical model critically depends upon adopting an appropriate representation for the phenomenon of interest. Unfortunately, it remains generally challenging to identify the essential degrees of freedom or, equivalently, the proper order parameters for describing complex phenomena. Here we develop a statistical physics framework for exploring and quantitatively characterizing the space of order parameters for representing physical systems. Specifically, we examine the space of low-resolution representations that correspond to particle-based coarse-grained (CG) models for a simple microscopic model of protein fluctuations. We employ Monte Carlo (MC) methods to sample this space and determine the density of states for CG representations as a function of their ability to preserve the configurational information, I, and large-scale fluctuations, Q, of the microscopic model. These two metrics are uncorrelated in high-resolution representations but become anticorrelated at lower resolutions. Moreover, our MC simulations suggest an emergent length scale for coarse-graining proteins, as well as a qualitative distinction between good and bad representations of proteins. Finally, we relate our work to recent approaches for clustering graphs and detecting communities in networks.

Geometrical Aspects in the Analysis of Microcanonical Phase-Transitions

In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviours of thermodynamic observables at a phase transition point are rooted in more fundamental changes of the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of phase-transitions are shaped in the special case of ϕ 4 models with either nearest-neighbours and mean-field interactions.

Clustering of Neural Activity: A Design Principle for Population Codes

We propose that correlations among neurons are generically strong enough to organize neural activity patterns into a discrete set of clusters, which can each be viewed as a population codeword. Our reasoning starts with the analysis of retinal ganglion cell data using maximum entropy models, showing that the population is robustly in a frustrated, marginally sub-critical, or glassy, state. This leads to an argument that neural populations in many other brain areas might share this structure. Next, we use latent variable models to show that this glassy state possesses well-defined clusters of neural activity. Clusters have three appealing properties: (i) clusters exhibit error correction, i.e., they are reproducibly elicited by the same stimulus despite variability at the level of constituent neurons; (ii) clusters encode qualitatively different visual features than their constituent neurons; and (iii) clusters can be learned by downstream neural circuits in an unsupervised fashion. We hypothesize that these properties give rise to a "learnable" neural code which the cortical hierarchy uses to extract increasingly complex features without supervision or reinforcement.

Recognition of polymer configurations by unsupervised learning

Unsupervised learning as an important branch of machine learning is commonly adopted to discover patterns, with the purpose of conducting data clustering without being labeled in advance. In this study, we elucidate the striking ability of unsupervised learning techniques in exploring the phase transitions of polymer configurations. In order to extract the low-dimensional representation of polymer configurations, principal component analysis and diffusion map are applied to distinguish the coiled state and collapsed states and further detect the delicate distinction among collapsed states, respectively. These dimensionality reduction techniques not only identify the distinct states in the feature space, but also offer significant insights to understand the relation between salient features and order parameters in physics. In addition, a hybrid neural network scheme combining the supervised learning and unsupervised learning is utilized to precisely detect the critical point of phase transition between polymer configurations. Our study demonstrates a promising strategy based on the unsupervised learning, particularly in the exploration of phase transition in polymeric systems.

Diagrams of States of Single Flexible-Semiflexible Multi-Block Copolymer Chains: A Flat-Histogram Monte Carlo Study

The combination of flexibility and semiflexibility in a single molecule is a powerful design principle both in nature and in materials science. We present results on the conformational behavior of a single multiblock-copolymer chain, consisting of equal amounts of Flexible (F) and Semiflexible (S) blocks with different affinity to an implicit solvent. We consider a manifold of macrostates defined by two terms in the total energy: intermonomer interaction energy and stiffness energy. To obtain diagrams of states (pseudo-phase diagrams), we performed flat-histogram Monte Carlo simulations using the Stochastic Approximation Monte Carlo algorithm (SAMC). We have accumulated two-Dimensional Density of States (2D DoS) functions (defined on the 2D manifold of macrostates) for a SF-multiblock-copolymer chain of length N=64 with block lengths b = 4, 8, 16, and 32 in two different selective solvents. In an analysis of the canonical ensemble, we calculated the heat capacity and determined its maxima and the most probable morphologies in different regions of the state diagrams. These are rich in various, non-trivial morphologies, which are formed without any specific interactions, and depend on the block length and the type of solvent selectivity (preferring S or F blocks, respectively). We compared the diagrams with those for the non-selective solvent and reveal essential changes in some cases. Additionally, we implemented microcanonical analysis in the “conformational” microcanonical (NVU, where U is the potential energy) and the true microcanonical (NVE, where E is the total energy) ensembles with the aim to reveal and classify pseudo-phase transitions, occurring under the change of temperature.

Influence of bonded interactions on structural phases of flexible polymers

We introduce a novel coarse-grained bead-spring model for flexible polymers to systematically examine the effects of an adjusted bonded potential on the formation and stability of structural macrostates in a thermal environment. The density of states obtained in advanced replica-exchange Monte Carlo simulations is analyzed by employing the recently developed generalized microcanonical inflection-point analysis method, which enables the identification of diverse structural phases and the construction of a suitably parameterized hyperphase diagram. It reveals that icosahedral phases dominate for polymers with asymmetric and narrow bond potentials, whereas polymers with symmetric and more elastic bonds tend to form amorphous structures with non-icosahedral cores. We also observe a hierarchy in the freezing transition behavior associated with the formation of the surface layer after nucleation.

Entropy Inflection and Invisible Low-Energy States: Defensive Alliance Example

Lower temperature leads to a higher probability of visiting low-energy states. This intuitive belief underlies most physics-inspired strategies for addressing hard optimization problems. For instance, the popular simulated annealing (SA) dynamics is expected to approach a ground state if the temperature is lowered appropriately. Here, we demonstrate that this belief is not always justified. Specifically, we employ the cavity method to analyze the minimum strong defensive alliance problem and discover a bifurcation in the solution space, induced by an inflection point in the entropy-energy profile. While easily accessible configurations are associated with the lower-free-energy branch, the low-energy configurations are associated with the higher-free-energy branch within the same temperature range. There is a discontinuous phase transition between the high-energy configurations and the ground states, which generally cannot be followed by SA. We introduce an energy-clamping strategy to obtain superior solutions by following the higher-free-energy branch, overcoming the limitations of SA.

Classification of Phase Transitions by Microcanonical Inflection-Point Analysis

By means of the principle of minimal sensitivity we generalize the microcanonical inflection-point analysis method by probing derivatives of the microcanonical entropy for signals of transitions in complex systems. A strategy of systematically identifying and locating independent and dependent phase transitions of any order is proposed. The power of the generalized method is demonstrated in applications to the ferromagnetic Ising model and a coarse-grained model for polymer adsorption onto a substrate. The results shed new light on the intrinsic phase structure of systems with cooperative behavior.

Subphase transitions in first-order aggregation processes

In this paper, we investigate the properties of aggregation transitions in the context of generic coarse-grained homopolymer systems. By means of parallel replica-exchange Monte Carlo methods, we perform extensive simulations of systems consisting of up to 20 individual oligomer chains with five monomers each. Using the tools of the versatile microcanonical inflection-point analysis, we show that the aggregation transition is a first-order process consisting of a sequence of subtransitions between intermediate structural phases. We unravel the properties of these intermediate phases by collecting and analyzing their individual contributions towards the density of states of the system. The central theme of this systematic study revolves around translational entropy and its role in the striking phenomena of missing intermediate phases. We conclude with a brief discussion of the scaling properties of the transition temperature and the latent heat.

Identifying polymer states by machine learning

The ability of a feed-forward neural network to learn and classify different states of polymer configurations is systematically explored. Performing numerical experiments, we find that a simple network model can, after adequate training, recognize multiple structures, including gaslike coil, liquidlike globular, and crystalline anti-Mackay and Mackay structures. The network can be trained to identify the transition points between various states, which compare well with those identified by independent specific-heat calculations. Our study demonstrates that neural networks provide an unconventional tool to study the phase transitions in polymeric systems.

Canonical free-energy barrier of particle and polymer cluster formation

A common approach to study nucleation rates is the estimation of free-energy barriers. This usually requires knowledge about the shape of the forming droplet, a task that becomes notoriously difficult in macromolecular setups starting with a proper definition of the cluster boundary. Here we demonstrate a shape-free determination of the free energy for temperature-driven cluster formation in particle as well as polymer systems. Combined with rigorous results on equilibrium droplet formation, this allows for a well-defined finite-size scaling analysis of the effective interfacial free energy at a fixed density. We first verify the theoretical predictions for the formation of a liquid droplet in a supersaturated particle gas by generalized-ensemble Monte Carlo simulations of a Lennard-Jones system. Going one step further, we then generalize this approach to cluster formation in a dilute polymer solution. Our results suggest an analogy with particle condensation, when the macromolecules are interpreted as extended particles.

On the Pseudo Phase Diagram of Single Semi-Flexible Polymer Chains: A Flat-Histogram Monte Carlo Study

Local stiffness of polymer chains is instrumental in all structure formation processes of polymers, from crystallization of synthetic polymers to protein folding and DNA compactification. We present Stochastic Approximation Monte Carlo simulations-a type of flat-histogram Monte Carlo method-determining the density of states of a model class of single semi-flexible polymer chains, and, from this, their complete thermodynamic behavior. The chains possess a rich pseudo phase diagram as a function of stiffness and temperature, displaying non-trivial ground-state morphologies. This pseudo phase diagram also depends on chain length. Differences to existing pseudo phase diagrams of semi-flexible chains in the literature emphasize the fact that the mechanism of stiffness creation matters.

Dilute Semiflexible Polymers with Attraction: Collapse, Folding and Aggregation

We review the current state on the thermodynamic behavior and structural phases of self- and mutually-attractive dilute semiflexible polymers that undergo temperature-driven transitions. In extreme dilution, polymers may be considered isolated, and this single polymer undergoes a collapse or folding transition depending on the internal structure. This may go as far as to stable knot phases. Adding polymers results in aggregation, where structural motifs again depend on the internal structure. We discuss in detail the effect of semiflexibility on the collapse and aggregation transition and provide perspectives for interesting future investigations.

First-order phase transitions in the real microcanonical ensemble

We present a simulation and data analysis technique to investigate first-order phase transitions and the associated transition barriers. The simulation technique is based on the real microcanonical ensemble where the sum of kinetic and potential energy is kept constant. The method is tested for the droplet condensation-evaporation transition in a Lennard-Jones system with up to 2048 particles at fixed density, using simple Metropolis-like sampling combined with a replica-exchange scheme. Our investigation of the microcanonical ensemble properties reveals that the associated transition barrier is significantly lower than in the canonical counterpart. Along the line of investigating the microcanonical ensemble behavior, we develop a framework for general ensemble evaluations. This framework is based on a clear separation between system-related and ensemble-related properties, which can be exploited to specifically tailor artificial ensembles suitable for first-order phase transitions.

System-Size Dependence of Helix-Bundle Formation for Generic Semiflexible Polymers

Helical polymer bundles are an important fixture in biomolecular systems. The particular structural geometry of helix bundles is dependent on many factors including the length of the polymer chain. In this study, we performed Monte Carlo simulations of a coarse-grained model for helical polymers to determine the influence of polymer length on tertiary structure formation. Helical structures of semiflexible polymers are analyzed for several chain lengths under thermal conditions. Structural hyperphase diagrams, parametrized by torsion strength and temperature, are constructed and compared.

Microcanonical analysis of a finite-size nonequilibrium system

Microcanonical analysis is a powerful method that can be used to generalize the concept of phase transitions to finite-size systems. However, microcanonical analysis has only been applied to equilibrium systems. I show that it is possible to conduct the microcanonical analysis of a finite-size nonequilibrium system by generalizing the concept of microcanonical entropy. A one-dimensional asymmetric diffusion process is studied as an example for which such a generalized entropy can be explicitly found, and the microcanonical method is used to define a generalized phase transition for the finite-size nonequilibrium system.

Structural phases of adsorption for flexible polymers on nanocylinder surfaces

By means of generalized-ensemble Monte Carlo simulations, we investigate the thermodynamic behavior of a flexible, elastic polymer model in the presence of an attractive nanocylinder. We systematically identify the structural phases that are formed by competing monomer-monomer and monomer-substrate interactions. The influence of the relative surface attraction strength on the structural phases in the hyperphase diagram, parameterized by cylinder radius and temperature, is discussed as well. In the limiting case of the infinitely large cylinder radius, our results coincide with previous outcomes of studies of polymer adsorption on planar substrates.

Towards an Optimal Flow: Density-of-States-Informed Replica-Exchange Simulations

Replica exchange (RE) is one of the most popular enhanced-sampling simulations technique in use today. Despite widespread successes, RE simulations can sometimes fail to converge in practical amounts of time, e.g., when sampling around phase transitions, or when a few hard-to-find configurations dominate the statistical averages. We introduce a generalized RE scheme, density-of-states-informed RE, that addresses some of these challenges. The key feature of our approach is to inform the simulation with readily available, but commonly unused, information on the density of states of the system as the RE simulation proceeds. This enables two improvements, namely, the introduction of resampling moves that actively move the system towards equilibrium and the continual adaptation of the optimal temperature set. As a consequence of these two innovations, we show that the configuration flow in temperature space is optimized and that the overall convergence of RE simulations can be dramatically accelerated.

Confinement effects upon the separation of structural transitions in linear systems with restricted bond fluctuation ranges

By means of advanced parallel replica-exchange Monte Carlo methods we examine the influence of elasticity and confinement on the structural transitions of linear systems with restricted bonded interaction. For this purpose, we adopt a model for coarse-grained flexible polymers of finite length in the dilute regime. Hyperphase diagrams are constructed using energy-dependent canonical quantities to demonstrate the effects of the changes in the range of the confined interaction on the liquid and solid structural phases. With increasing bonded interaction range we observe the disappearance of the liquid phase and the fusion of the gas-liquid (or Θ) and the liquid-solid transitions. One of the most remarkable features, the liquid-gas transition, changes from second to first order if the confined interaction range exceeds a threshold that separates polymeric from nonpolymeric systems. The notoriously difficult sampling of the entropically suppressed conformations in the region of very strong first-order transitions is improved by using multiple Gaussian modified ensembles.

Thermodynamics and signatures of criticality in a network of neurons

The activity of a neural network is defined by patterns of spiking and silence from the individual neurons. Because spikes are (relatively) sparse, patterns of activity with increasing numbers of spikes are less probable, but, with more spikes, the number of possible patterns increases. This tradeoff between probability and numerosity is mathematically equivalent to the relationship between entropy and energy in statistical physics. We construct this relationship for populations of up to N = 160 neurons in a small patch of the vertebrate retina, using a combination of direct and model-based analyses of experiments on the response of this network to naturalistic movies. We see signs of a thermodynamic limit, where the entropy per neuron approaches a smooth function of the energy per neuron as N increases. The form of this function corresponds to the distribution of activity being poised near an unusual kind of critical point. We suggest further tests of criticality, and give a brief discussion of its functional significance.

Entropic Description of Gas Hydrate Ice-Liquid Equilibrium via Enhanced Sampling of Coexisting Phases

Metastable β ice holds small guest molecules in stable gas hydrates, so its solid-liquid equilibrium is of interest. However, aqueous crystal-liquid transitions are very difficult to simulate. A new molecular dynamics algorithm generates trajectories in a generalized NPT ensemble and equilibrates states of coexisting phases with a selectable enthalpy. With replicas spanning the range between β ice and liquid water, we find the statistical temperature from the enthalpy histograms and characterize the transition by the entropy, introducing a general computational procedure for first-order transitions.

Identifying transitions in finite systems by means of partition function zeros and microcanonical inflection-point analysis: a comparison for elastic flexible polymers

For the estimation of transition points of finite elastic, flexible polymers with chain lengths from 13 to 309 monomers, we compare systematically transition temperatures obtained by the Fisher partition function zeros approach with recent results from microcanonical inflection-point analysis. These methods rely on accurate numerical estimates of the density of states, which have been obtained by advanced multicanonical Monte Carlo sampling techniques. Both the Fisher zeros method and microcanonical inflection-point analysis yield very similar results and enable the unique identification of transition points in finite systems, which is typically impossible in the conventional canonical analysis of thermodynamic quantities.

Effects of the interaction range on structural phases of flexible polymers

We systematically investigate how the range of interaction between non-bonded monomers influences the formation of structural phases of elastic, flexible polymers. Massively parallel replica-exchange simulations of a generic, coarse-grained model, performed partly on graphics processing units and in multiple-gaussian modified ensembles, pave the way for the construction of the structural phase diagram, parametrized by interaction range and temperature. Conformational transitions between gas-like, liquid, and diverse solid (pseudo) phases are identified by microcanonical statistical inflection-point analysis. We find evidence for finite-size effects that cause the crossover of "collapse" and "freezing" transitions for very short interaction ranges.

Examining the phase transition behavior of amphiphilic lipids in solution using statistical temperature molecular dynamics and replica-exchange Wang-Landau methods

Two different techniques - replica-exchange Wang-Landau (REWL) and statistical temperature molecular dynamics (STMD) - were applied to systematically study the phase transition behavior of self-assembling lipids as a function of temperature using an off-lattice lipid model. Both methods allow the direct calculation of the density of states with improved efficiency compared to the original Wang-Landau method. A 3-segment model of amphiphilic lipids solvated in water has been studied with varied particle interaction energies (ε) and lipid concentrations. The phase behavior of the lipid molecules with respect to bilayer formation has been characterized through the calculation of the heat capacity as a function of temperature, in addition to various order parameters and general visual inspection. The simulations conducted by both methods can go to very low temperatures with the whole system exhibiting well-ordered structures. With optimized parameters, several bilayer phases are observed within the temperature range studied, including gel phase bilayers with frozen water, mixed water (i.e., frozen and liquid water), and liquid water, and a more fluid bilayer with liquid water. The results obtained from both methods, STMD and REWL, are consistently in excellent agreement with each other, thereby validating both the methods and the results.

From flexible to stiff: systematic analysis of structural phases for single semiflexible polymers

Inspired by recent studies revealing unexpected pliability of semiflexible biomolecules like RNA and DNA, we systematically investigate the range of structural phases by means of a simple generic polymer model. Using a two-dimensional variant of Wang-Landau sampling to explore the conformational space in energy and stiffness within a single simulation, we identify the entire diversity of structures existing from the well-studied limit of flexible polymers to that of wormlike chains. We also discuss, in detail, the influence of finite-size effects in the formation of crystalline structures that are virtually inaccessible via conventional computational approaches.