Microcanonical analysis of small systems

Abstract

The basic quantity for the description of the statistical properties of physical systems is the density of states or equivalently the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are for example related to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions already in finite systems. In this paper we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities which announce a continuous phase transition in the infinite system are characterised by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analyzing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two-and three-dimensional Ising systems.

Structural Basis of Folding Cooperativity in Model Proteins: Insights from a Microcanonical Perspective

“…The advantage is that we essentially directly analyze the probability density p(E) rather than merely looking at its lowest moments, such as the specific heat. Such a microcanonical analysis has been applied to a wide variety of problems, e.g., spin systems [7,8,9,10,11,12,13], nuclear fragmentation [14,15], colloids [16], gravitating systems [17,18], off-lattice homo-and heteropolymer models [20,19], and protein folding [21,22,23,24,25,26,27]. Two remarks are worthwhile:…”

“…This can best be observed by defining the quantity ∆S(E) = H(E) − S(E), where H(E) corresponds to the (double-)tangent to S(E) in the transition region [8,9,23,24,10]. In a finite system the existence of a barrier in ∆S(E) will imply a non-zero microcanonical latent heat ∆Q, defined by the interval over which S(E) departs from its convex hull, and in turn leads to a "backbending" effect (akin to a van-der-Waals loop) in the inverse microcanonical temperature T −1 µc (E) = ∂S/∂E (e.g., [8,9,10,23,12,13]). A non-zero ∆Q demarcates a transition region, whereas a downhill folder (continuous transition) will only exhibit a transition point, where the concavity of S(E) is minimal.…”

Ising metamagnets in thin film geometry: Equilibrium properties

“…Whereas in (a) we consider a system with periodic boundary conditions composed of N = 8 × 8 × 8 spins, with J xy = J z , in (b) and (c) we show two systems of N = 8 × 8 × 8 sites with free boundary condition in z direction, the different systems having different relative strengths of the interactions: In a microcanonical analysis one infers the physical property of a system from a direct study of the microcanonical entropy S(e, m). 36,37 Most investigations of this type focused on spin models with ferromagnetic nearest neighbor interactions, as for example the standard Ising or Potts models, [38][39][40][41][42][43][44][45][46] or on polymer models. 47,48 Due to its complicated interactions, the microcanonical entropy of the Ising metamagnet is more complex as, for example, that of the standard nearest neighbor Ising model, 37 see Fig.…”

Wang-Landau Monte Carlo formalism applied to ferroelectrics

“…when mimicking ferroelectric nanosystems 41,42 . We are therefore confident that our present work will motivate the use of the WL algorithm in ferroelectrics, both in their bulk and nanostructure forms, and can thus lead to a deeper understanding of this important class of materials as well as to the design of optimized or even novel properties.…”