Microcanonical scaling in small systems

Abstract

A microcanonical finite-size scaling ansatz is discussed. It exploits the existence of a well-defined transition point for systems of finite size in the microcanonical ensemble. The best data collapse obtained for small systems yields values for the critical exponents in good agreement with other approaches. The exact location of the infinite system critical point is not needed when extracting critical exponents from the microcanonical finite-size scaling theory.

Finite-size behaviour of the microcanonical specific heat

“…Naturally, the back-bending of the coefficient A 2 (l) directly affects the behaviour of the microcanonical specific heat of small systems as can be seen from equations (15) and (12). Indeed, the maximum of the specific heat first decreases with growing system size before increasing again, thus yielding a divergence in the thermodynamic limit.…”

“…This theory, which is based on the analyticity of the microcanonical entropy surface, uses as a variable the distance to the pseudo-critical point e pc of a given finite system. This unusual ansatz has allowed us recently to extract the order parameter critical exponent directly from the density of states of small systems [12]. The phenomenological finite-size scaling theory should therefore be viewed in the broader context of deriving a finite-size scaling theory in the microcanonical ensemble.…”

“…Clearly, this inequivalence in finite systems makes the microcanonical analysis of possible signatures of phase transitions an important issue [2,9,10,5,11,12]. For discontinuous phase transitions, the microcanonical analysis reveals typical small system signatures, as e. g. a back-bending of the caloric curve or the appearance of a negative specific heat.…”

Continuous phase transitions with a convex dip in the microcanonical entropy

“…For more details see Section III and in particular Reference [26]. Note also that other microcanonical quantities like the spontaneous magnetization and the susceptibility also exhibit typical features of continuous phase transitions in finite systems [19,20,21,22,25,27]. These aspects are however not studied in the present work.…”

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.…”