Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1511.04609 · Europhysics Letters (EPL)

Phenomenological theory of the Potts model evaporation-condensation transition

Miguel Ibáñez-Berganza

Published Jan 1, 2016• 0 citations
arXivHTMLPDF

Abstract

We present a phenomenological theory describing the finite-size evaporation-condensation transition of the $q$-state Potts model in the microcanonical ensemble. Our arguments rely on the existence of an exponent $σ$, relating the surface and the volume of the condensed phase droplet. The evaporation-condensation transition temperature and energy converge to their infinite-size values with the same power, $a=(1-σ)/(2-σ)$, of the inverse of the system size. For the 2D Potts model we show, by means of efficient simulations up to $q=24$ and $1024^2$ sites, that the exponent $a$ is compatible with $1/4$, in disagreement with previous studies. While this value cannot be addressed by the evaporation-condensation theory developed for the Ising model, it is obtained in the present scheme if $σ=2/3$, in agreement with previous theoretical guesses. The connection with the phenomenon of metastability in the canonical ensemble is also discussed.

DOI
10.1209/0295-5075/113/26002

Related papers

Microcanonical analysis of small systems

Michel Pleimling, Behringer, Hans

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.

arXiv:cond-mat/0504018•Apr 1, 2005•1 citation

Finite-size behaviour of the microcanonical specific heat

H Behringer, M Pleimling, A Hüller

For models which exhibit a continuous phase transition in the thermodynamic limit a numerical study of small systems reveals a non-monotonic behaviour of the microcanonical specific heat as a function of the system size. This is in contrast to a treatment in the canonical ensemble where the maximum of the specific heat increases monotonically with the size of the system. A phenomenological theory is developed which permits us to describe this peculiar behaviour of the microcanonical specific heat and allows in principle the determination of microcanonical critical exponents.

arXiv:cond-mat/0411153•Journal of Physics A Mathematical and General•Jan 20, 2005•24 citations

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

L. A. Fernández, A. Gordillo-Guerrero, Víctor Martín-Mayor +1 more

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.

arXiv:0905.4029•Physical Review E•Nov 6, 2009•8 citations

Microcanonical analysis of small systems

Michel Pleimling, H. Behringer

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 related, for example, to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions in finite systems. In this article we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities, which signal a continuous phase transition in infinite systems, are characterized by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analysing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.

Phase Transitions•Sep 1, 2005•18 citations

Finite Size Scaling of Metastability in First Order Transitions

Smita Ota, S. B. Ota

Microcanonical Monte Carlo simulations have been carried out for the extended 2d XY-model and the 2d q=10 state Potts model undergoing first order transitions. It has been found that the temperature dependence of the physical quantities exhibit "S"-shaped nature at the first order transition. The excursion into the metastable branch is found to increase by reducing the system size. The metastability is found to scale with the linear dimension of the system.

International Journal of Modern Physics B•Aug 10, 1998•3 citations

Phase Transitions in Disordered Systems

A. Gordillo-Guerrero

We face the problem of phase transitions in diluted systems both from theoretical and numerical sides. We study the effects of quenched site-dilution in classical models (Heisenberg, Ising and Potts) in 2, 3, and 4 dimensions both by using the Renormalization Group and numerical simulations in the canonical and microcanonical ensembles. We propose and check a new formulation of the Finite Size Scaling ansatz (FSS) inside the microcanonical ensemble. We use microcanonical simulations to obtain a clear fist-order behavior for the diluted Potts model in 3D, estimating the tricritical dilution. We perform large-scale simulations of the 3D diluted Heisenberg model, checking its self-averaging properties. Finally we study the 4D diluted Ising model obtaining from the FSS of the specific heat a clear differentiation between the existing conflicting theories. We also compiled a large number of appendix that we expect to be used as future reference.

arXiv:1004.1579•Mar 11, 2010•0 citations
We present a phenomenological theory describing the finite-size evaporation-condensation transition of the qqq-state Potts model in the microcanonical ensemble. Our arguments rely on the existence of an exponent σσσ, relating the surface and the volume of the condensed phase droplet. The evaporation-condensation transition temperature and energy converge to their infinite-size values with the same power, a=(1−σ)/(2−σ)a=(1-σ)/(2-σ)a=(1−σ)/(2−σ), of the inverse of the system size. For the 2D Potts model we show, by means of efficient simulations up to q=24q=24q=24 and 102421024^210242 sites, that the exponent aaa is compatible with 1/41/41/4, in disagreement with previous studies. While this value cannot be addressed by the evaporation-condensation theory developed for the Ising model, it is obtained in the present scheme if σ=2/3σ=2/3σ=2/3, in agreement with previous theoretical guesses. The connection with the phenomenon of metastability in the canonical ensemble is also discussed.
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.
For models which exhibit a continuous phase transition in the thermodynamic limit a numerical study of small systems reveals a non-monotonic behaviour of the microcanonical specific heat as a function of the system size. This is in contrast to a treatment in the canonical ensemble where the maximum of the specific heat increases monotonically with the size of the system. A phenomenological theory is developed which permits us to describe this peculiar behaviour of the microcanonical specific heat and allows in principle the determination of microcanonical critical exponents.
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.
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 related, for example, to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions in finite systems. In this article we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities, which signal a continuous phase transition in infinite systems, are characterized by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analysing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.
We face the problem of phase transitions in diluted systems both from theoretical and numerical sides. We study the effects of quenched site-dilution in classical models (Heisenberg, Ising and Potts) in 2, 3, and 4 dimensions both by using the Renormalization Group and numerical simulations in the canonical and microcanonical ensembles. We propose and check a new formulation of the Finite Size Scaling ansatz (FSS) inside the microcanonical ensemble. We use microcanonical simulations to obtain a clear fist-order behavior for the diluted Potts model in 3D, estimating the tricritical dilution. We perform large-scale simulations of the 3D diluted Heisenberg model, checking its self-averaging properties. Finally we study the 4D diluted Ising model obtaining from the FSS of the specific heat a clear differentiation between the existing conflicting theories. We also compiled a large number of appendix that we expect to be used as future reference.