Continuous phase transitions with a convex dip in the microcanonical entropy

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.

Nuclear multifragmentation time scale and fluctuations of the largest fragment size

Distributions of the largest fragment charge, Zmax, in multifragmentation reactions around the Fermi energy can be decomposed into a sum of a Gaussian and a Gumbel distribution, whereas at much higher or lower energies one or the other distribution is asymptotically dominant. We demonstrate the same generic behavior for the largest cluster size in critical aggregation models for small systems, in or out of equilibrium, around the critical point. By analogy with the time-dependent irreversible aggregation model, we infer that Zmax distributions are characteristic of the multifragmentation time scale, which is largely determined by the onset of radial expansion in this energy range.

Calculation of microcanonical entropy differences from configurational averages

A simple expression is derived, enabling the calculation of the entropy difference between two microcanonical equilibrium states at different energies in atomistic computer simulations. This expression only requires potential energy samples from molecular dynamics or Monte Carlo simulations at the relevant energies. This presents an alternative to switching methods such as thermodynamic integration or nonequilibrium work relations, as well as flat-histogram random walks, all of which involve sampling in between the relevant states. The method is especially suited for small (nanoscopic) systems such as clusters and proteins, and is applicable to first-principles data directly.

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

We introduce a systematic classification method for the analogs of phase transitions in finite systems. This completely general analysis, which is applicable to any physical system and extends toward the thermodynamic limit, is based on the microcanonical entropy and its energetic derivative, the inverse caloric temperature. Inflection points of this quantity signal cooperative activity and thus serve as distinct indicators of transitions. We demonstrate the power of this method through application to the long-standing problem of liquid-solid transitions in elastic, flexible homopolymers.

Generalized replica exchange method

We present a powerful replica exchange method, particularly suited to first-order phase transitions associated with the backbending in the statistical temperature, by merging an optimally designed generalized ensemble sampling with replica exchanges. The key ingredients of our method are parametrized effective sampling weights, smoothly joining ordered and disordered phases with a succession of unimodal energy distributions by transforming unstable or metastable energy states of canonical ensembles into stable ones. The inverse mapping between the sampling weight and the effective temperature provides a systematic way to design the effective sampling weights and determine a dynamic range of relevant parameters. Illustrative simulations on Potts spins with varying system size and simulation conditions demonstrate a comprehensive sampling for phase-coexistent states with a dramatic acceleration of tunneling transitions. A significant improvement over the power-law slowing down of mean tunneling times with increasing system size is obtained, and the underlying mechanism for accelerated tunneling is discussed.

Microcanonical finite-size scaling in second-order phase transitions with diverging specific heat

A microcanonical finite-size ansatz in terms of quantities measurable in a finite lattice allows extending phenomenological renormalization (the so-called quotients method) to the microcanonical ensemble. The ansatz is tested numerically in two models where the canonical specific heat diverges at criticality, thus implying Fisher renormalization of the critical exponents: the three-dimensional ferromagnetic Ising model and the two-dimensional four-state Potts model (where large logarithmic corrections are known to occur in the canonical ensemble). A recently proposed microcanonical cluster method allows simulating systems as large as L=1024 (Potts) or L=128 (Ising). The quotients method provides accurate determinations of the anomalous dimension, eta, and of the (Fisher-renormalized) thermal nu exponent. While in the Ising model the numerical agreement with our theoretical expectations is very good, in the Potts case, we need to carefully incorporate logarithmic corrections to the microcanonical ansatz in order to rationalize our data.

Microcanonical analyses of homopolymer aggregation processes

Using replica-exchange multicanonical Monte Carlo simulation, the aggregates of two homopolymers were numerically investigated through the microcanonical analysis method. The microcanonical entropy showed one convex function in the transition region, leading to a negative microcanonical specific heat. The origin of temperature backbending was the rearrangement of the segments during the process of aggregation; this aggregation process proceeded via a nucleation and growth mechanism. It was observed that the segments with a sequence number from 10 to 13 in the polymer chain have leading effects on the aggregation.

Microcanonical versus canonical analysis of protein folding

The microcanonical analysis is shown to be a powerful tool to characterize the protein folding transition and to neatly distinguish between good and bad folders. An off-lattice model with parameter chosen to represent polymers of these two types is used to illustrate this approach. Both canonical and microcanonical ensembles are employed. The required calculations were performed using parallel tempering Monte Carlo simulations. The most revealing features of the folding transition are related to its first-order-like character, namely, the S-bend pattern in the caloric curve, which gives rise to negative microcanonical specific heats, and the bimodality of the energy distribution function at the transition temperatures. Models for a good folder are shown to be quite robust against perturbations in the interaction potential parameters.

Thermodynamics of peptide aggregation processes: an analysis from perspectives of three statistical ensembles

We employ a mesoscopic model for studying aggregation processes of proteinlike hydrophobic-polar heteropolymers. By means of multicanonical Monte Carlo computer simulations, we find strong indications that peptide aggregation is a phase separation process, in which the microcanonical entropy exhibits a convex intruder due to non-negligible surface effects of the small systems. We analyze thermodynamic properties of the conformational transitions accompanying the aggregation process from the multicanonical, canonical, and microcanonical perspective. It turns out that the microcanonical description is particularly advantageous as it allows for unraveling details of the phase-separation transition in the thermodynamic region, where the temperature is not a suitable external control parameter anymore.

Microcanonical analysis of association of hydrophobic segments in a heteropolymer

Using the replica-exchange multicanonical Monte Carlo simulation, the intra-association of hydrophobic segments in a heteropolymer was numerically investigated by the microcanonical analysis method. We demonstrated that the microcanonical entropy shows the features of one or multiple convexes in the association transition region depending on the number and distribution of hydrophobic segments in the chain. We found that one or multiple negative specific heats imply a first-order-like transition with the coexistence of multiple phases.

Microcanonical analyses of peptide aggregation processes

We propose the use of microcanonical analyses for numerical studies of peptide aggregation transitions. Performing multicanonical Monte Carlo simulations of a simple hydrophobic-polar continuum model for interacting heteropolymers of finite length, we find that the microcanonical entropy behaves convex in the transition region, leading to a negative microcanonical specific heat. As this effect is also seen in first-order-like transitions of other finite systems, our results provide clear evidence for recent hints that the characterization of phase separation in first-order-like transitions of finite systems profits from this microcanonical view.