Extended canonical Monte Carlo methods: Improving accuracy of microcanonical calculations using a reweighting technique

Velazquez and Curilef [J. Stat. Mech. (2010) P02002; J. Stat. Mech. (2010) P04026] have proposed a methodology to extend Monte Carlo algorithms that are based on canonical ensemble. According to our previous study, their proposal allows us to overcome slow sampling problems in systems that undergo any type of temperature-driven phase transition. After a comprehensive review about ideas and connections of this framework, we discuss the application of a reweighting technique to improve the accuracy of microcanonical calculations, specifically, the well-known multihistograms method of Ferrenberg and Swendsen [Phys. Rev. Lett. 63, 1195 (1989)]. As an example of application, we reconsider the study of the four-state Potts model on the square lattice $L\times L$ with periodic boundary conditions. This analysis allows us to detect the existence of a very small latent heat per site $q_L$ during the occurrence of temperature-driven phase transition of this model, whose size dependence seems to follow a power law $q_L\left(L\right)\propto {(1/L)}^z$ with exponent $z\simeq 0$.$26\pm 0$.02. Discussed is the compatibility of these results with the continuous character of temperature-driven phase transition when $L\to +\infty$.

Figure 1

Behavior of energy distributions within canonical ensemble (2) along the occurrence of phase coexistence phenomenon of the 10-state Potts model on the square lattice $25\times 25$ with periodic boundary conditions ($u=U/N$ is the energy per site) (after [23]). This study clearly illustrates the bimodal character of energy distribution functions when the inverse temperature parameter $\beta$ of the canonical ensemble takes values around the critical value ${\beta }_c\simeq 1$.421 of temperature-driven discontinuous PT. Notice that the branch of microcanonical caloric curve $\beta \left(u\right)=\partial s\left(u\right)/\partial u$ (open squares) with states with negative heat capacities $C<0$ is poorly populated by using a bath with constant temperature since these states are canonically unstable. The values of bath inverse temperature $\beta$ are represented here by horizontal lines. Intersection points of these horizontal lines with microcanonical inverse temperature correspond to the energies where the energy distribution function exhibits its local maxima and minima.

Figure 2

The use of a bath with a finite heat capacity $C_{\omega }$ enables a direct study of the branch of microcanonical caloric curve (open squares) with negative heat capacities, which is shown for the same model system of Fig. 1 (after [23]). Here, energy distribution exhibits a single Gaussian peak that is located inside the region where microcanonical caloric curve exhibits negative heat capacities $C<0$. Both the bath inverse temperature ${\beta }_{\omega }$ (thick red line) and the system energy $U$ exhibit thermal fluctuations around their equilibrium values (coordinates of the red circle that mark the interception point between microcanonical caloric curve and the bath inverse temperature curve). In the past [31], Gerling and Hüller proposed this type of arguments to obtain microcanonical caloric curve considering the expectation values $〈{\beta }_{\omega }〉$ and $〈U〉$. The analysis of their thermal fluctuations $\langle \delta U^2\rangle$ and $〈\delta {\beta }_{\omega }\delta U〉$ enables a direct derivation of a negative value of microcanonical heat capacity $C$ at the equilibrium energy considering fluctuation relation (1).

Figure 3

Correlated thermal fluctuations analogous to the one described by fluctuation relation (1) can also be observed among other pairs of conjugated thermodynamical variables. Left: Thermal fluctuations of the total magnetization $M$ of a magnetic sample (red rectangle) induce thermal fluctuations in its associated total magnetic flux (red dash lines) through the Helmholtz coils (whose cross sections are represented here by four gray circles). As consequence of electromagnetic induction, the total magnetic field $H_{\omega }$ of these Helmholtz coils (blue lines) experiences correlated thermal fluctuations $〈\delta H_{\omega }\delta M〉$ with the total magnetization $M$ of the sample. Right: Schematic representation of two finite fluid systems A and B that are separated by a moving wall or piston (red rectangle). The total volume $V$ of fluid system A experiences correlated thermal fluctuations $〈\delta p_{\omega }\delta V〉$ with the external pressure $p_{\omega }$ of fluid system B.

Figure 4

Behavior of distributions of magnetization per site $m=M/N$ of the Ising model on the square lattice with periodic boundary conditions for different constant values of the external magnetic field $H$ at constant temperature $T<T_c$, where $T_c$ is the temperature critical value of ferromagnetic-paramagnetic continuous PT of this model (after [26]). Open circles are a MC estimation of microcanonical dependence of magnetization vs the external magnetic field (analogous to microcanonical caloric curve). Dotted line represents dependence of average magnetization $〈m〉$ when the intensity of the external magnetic field $H$ is smoothly varying from negative to positive values at constant temperature. Clearly, this canonical dependence (constant values of control parameters $T$ and $H$) fails to describe the S bend of its microcanonical counterpart. Moreover, the region with negative values of isothermal magnetic susceptibility ${\chi }_T<0$ is poorly populated by magnetization distributions for constant values of the external magnetic field $H$ and temperature $T$. In full analogy as the energy-temperature fluctuation relation (1) enables the study of systems with negative heat capacities $C<0$, fluctuation relation (9) was employed in this MC study to obtain anomalous values of isothermal magnetic susceptibility ${\chi }_T$.

Figure 5

Entropy per site $s\left(u\right)$ and microcanonical inverse temperature $\beta \left(u\right)$ of 2D four-state Potts model estimated from the Wang-Landau multicanonical algorithm. Here, the variable $u$ denotes the energy per site $u=U/N$, with $N=L^2$.

Figure 6

Microcanonical caloric curve of the four-state Potts model system on the square lattice $32\times 32$ with periodic boundary conditions, which was estimated from two different realizations of the Wang-Landau method and extended versions of canonical MC algorithms of the Metropolis importance sample and Wang-Landau and Wolff cluster algorithms (after [9]). The agreement among all these MC methods is very good, although some small discrepancies are clearly evidenced in the inset panel, where these same dependencies were represented with lower energy and inverse temperature scales in order to appreciate better the mathematical behavior of these curves near PT.

Figure 7

Comparison among results obtained from MC simulations using extended Wolff and usual Swendsen-Wang cluster algorithms. Results from the Wang-Landau method are employed here as reference. (a), (b) Microcanonical dependencies $\beta \left(u\right)=\partial s\left(u\right)/\partial u$ and $\kappa \left(u\right)=-{\partial }^{2}s\left(u\right)/\partial u^2$ estimated using point statistical estimation (22), where $u=U/N$ and $s=S/N$ are energy and entropy per site, respectively. Squares and circles are the punctual values of these dependencies using a single energy histogram, while solid and dashed-dotted lines are smooth estimations using recalculation procedure once obtained microcanonical entropy per site $\widehat{s}\left(u\right)$ derived from the multihistograms method. To check prediction of the multihistograms method for states with minimum curvature, the blue star point corresponds to a point statistical estimation using a single histogram obtained from a very long simulation run with $M=8.1\times {10}^7$ MC steps using extended Wolff algorithm. Additionally, we have included an estimation of microcanonical curvature by direct numerical differentiation of microcanonical caloric curve estimated from the Wang-Landau method in (a). (c) Energy histograms near critical point for each MC method. (d) Decorrelation time $\tau$ (open squares and circles) and efficiency factor $\eta =\tau {\Delta }_T^2$ (solid squares and circles) vs the most likely value of energy per particle $u_e$ for each simulation run.

Figure 8

Some microcanonical dependencies of the four-state Potts model on the square lattice $L\times L$ with different $L$, which were estimated considering the multihistograms method combined with extended Wolff algorithm. (a) Microcanonical inverse temperature $\beta \left(u\right)=\partial s\left(u\right)/\partial u$. (b) Dependency $s^{*}\left(u\right)=s\left(u\right)-{\beta }_{c}u-s_c$ obtained from entropy per site $s\left(u\right)$ and inverse temperature ${\beta }_c$ corresponding to the temperature-driven PT. This auxiliary function reveals better the existence of a convex intruder for entropy per site $s\left(u\right)$, which is directly related to the existence of a branch with negative curvature $\kappa \left(u\right)=-{\partial }^{2}s\left(u\right)/\partial u^2$ or, equivalently, states with negative heat capacities $C<0$. Typical energies $(u_1,u_2,u_3)$ and entropy defect $\Delta s=s^{*}\left(u_1\right)-s^{*}\left(u_2\right)$ are employed to characterize this convex intruder region, in particular, to estimate the latent heat per site $q_L=u_3-u_1$.

Figure 9

Dependence of inverse temperature ${\beta }_c$ (black circles) corresponding to temperature-driven PT and latent heat per site $q_L$ (red open circles) on the inverse lattice size $1/L$. Additionally, we have shown the extrapolation of these data when $1/L\to 0$ using polynomial fits (black line) for the inverse temperature ${\beta }_c\left(L\right)$ and a power-law fit for latent heat per site as $q_L\left(L\right)\propto {\left(1/L\right)}^z$ with $z=0.26\pm 0.02$, which is also shown in the inset panel using the $log-log$ scale.