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 towards 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.PACS numbers: 05.20. Gg,36.40.Ei,82.60.Nh Structure formation processes are typically accompanied by nucleation transitions, where crystalline shapes form out of a liquid or vapor phase. Thus, nucleation is governed by finite-size and surface effects. For small physical systems, it is difficult to understand thermodynamic transitions of this type, as they strongly depend on system size.Cooperativity refers to collective changes in a statistically significant fraction of the degrees of freedom in a system, which transforms the system into a new macrostate. In the thermodynamic limit of an infinitely large system, the ensemble of macrostates sharing similar thermodynamic properties would be called a "phase" and the transformation a "phase transition". The description of such a transformation in a finite system is more subtle, as it cannot be described in the traditional Ehrenfest scheme of singularities in response quantities. However, statistical physics and thus thermodynamics are also valid for systems with no thermodynamic limit. Examples include the structure formation in small atomic clusters and all biomolecules. This is particularly striking for proteins, i.e., heterogeneous linear chains of amino acids. The fact that the individual biological function is connected with the geometrical shape of the molecule makes it necessary to discriminate unfolded (non-functional) and folded (functional) states. Although these systems are finite, they undergo a structural transition by passing a single (or more) free-energy barrier(s). Since these finite-system transitions exhibit strong similarities compared to phase transitions, we extend the terminology once defined in the thermodynamic limit to all systems exhibiting cooperative behavior.In this paper, we introduce a commonly applicable and simple method for the identification and classification of cooperative behavior in systems of arbitrary size by means of microcanonical thermodynamics [1]. It also includes the precise and straightforward analysis of the finite-size effects, which are important to a general understanding of the onset of phase transitions. This is in contrast to canonical approaches, where detailed information is lost by averaging out thermal fluctuations. Re-gaining information about finite-size effects in canonical schemes, e.g., by the investigation of the distribution of Lee-Yang zeros in the complex temperature plane [2] or by inverse Laplace transform [3...

Recent tests performed on the D-Wave Two quantum annealer have revealed no clear evidence of speedup over conventional silicon-based technologies. Here, we present results from classical parallel-tempering Monte Carlo simulations combined with isoenergetic cluster moves of the archetypal benchmark problem-an Ising spin glass-on the native chip topology. Using realistic uncorrelated noise models for the D-Wave Two quantum annealer, we study the best-case resilience, i.e., the probability that the ground-state configuration is not affected by random fields and random-bond fluctuations found on the chip. We thus compute classical upper-bound success probabilities for different types of disorder used in the benchmarks and predict that an increase in the number of qubits will require either error correction schemes or a drastic reduction of the intrinsic noise found in these devices. We restrict this study to the exact ground state, however, the approach can be trivially extended to the inclusion of excited states if the success metric is relaxed. We outline strategies to develop robust, as well as hard benchmarks for quantum annealing devices, as well as any other (black box) computing paradigm affected by noise.

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.

Investigating thermodynamic properties of liquid-solid transitions of flexible homopolymers with elastic bonds by means of multicanonical Monte Carlo simulations, we find crystalline conformations that resemble groundstate structures of Lennard-Jones clusters. This allows us to set up a structural classification scheme for finite-length flexible polymers and their freezing mechanism in analogy to atomic cluster formation. Crystals of polymers with "magic length" turn out to be perfectly icosahedral.

We investigate solid-solid and solid-liquid transitions of elastic flexible off-lattice polymers with Lennard-Jones monomer-monomer interaction and anharmonic springs by means of sophisticated variants of multicanonical Monte Carlo methods. We find that the low-temperature behavior depends strongly and nonmonotonically on the system size and exhibits broad similarities to unbound atomic clusters. Particular emphasis is dedicated to the classification of icosahedral and nonicosahedral low-energy polymer morphologies.