Microcanonical entropy for small magnetizations

Abstract

Physical quantities obtained from the microcanonical entropy surfaces of classical spin systems show typical features of phase transitions already in finite systems. It is demonstrated that the singular behaviour of the microcanonically defined order parameter and susceptibility can be understood from a Taylor expansion of the entropy surface. The general form of the expansion is determined from the symmetry properties of the microcanonical entropy function with respect to the order parameter. The general findings are investigated for the four-state vector Potts model as an example of a classical spin system.

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.…”

“…These features are accompanied by singular physical quantities [18,19,20,21,22,23,24,25,26,27]. The associated singularities, however, are not the consequence of a non-analytic microcanonical entropy and can therefore be characterized by classical critical exponents [19,22]. Similarly, intriguing features are also revealed in the microcanonical entropy of small systems with a discontinuous transition in the infinite volume limit [5,28,29,30,31,32,33].…”

Microcanonical scaling in small systems

“…The entropy surface exhibits a well-defined transition point at an energy E c and magnetization M c although in finite systems S N (E, M) is everywhere perfectly analytic. The microcanonical analysis also shows that typical features of symmetry breaking, as for example the abrupt onset of several order parameter branches when the transition point is crossed from above, are already encountered in small systems [5,6,7]. With regard to these intriguing effects, it is tempting to ask whether a direct analysis of the microcanonical entropy also allows the determination of critical quantities from finite-size data.…”

“…Here β, α, and ν are the canonical critical exponents. As the microcanonical order parameter varies like a square root in the vicinity of ε c,N for all finite system sizes N [5,7] the scaling function W is asymptotically given as a square root W (x) ∼ √ x for small scaling variables x. One remarkable feature of equation (5) is the absence of any non-universal quantity related to the infinite system.…”

“…The expected universality of the scaling function for a given universality class has been demonstrated. The microcanonical finite-size scaling theory, presented for the order parameter in the present work, can also be formulated more generally in terms of scaling relations of the entropy of finite systems considered as a function of the energy, the magnetization and the inverse system size [17].…”