Phenomenological theory of the Potts model evaporation-condensation transition
We present a phenomenological theory describing the finite-size evaporation-condensation transition of the -state Potts model in the microcanonical ensemble. Our arguments rely on the existence of an exponent , 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, , of the inverse of the system size. For the 2D Potts model we show, by means of efficient simulations up to and sites, that the exponent is compatible with , 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 , in agreement with previous theoretical guesses. The connection with the phenomenon of metastability in the canonical ensemble is also discussed.
The equilibrium and dynamics of phase coexistence in first-order phase transitions is a crucial and longstanding topic in statistical physics [1, 2, 3]. Since the eighties there has been a considerable interest in the so called micro-ensemble, in which the extensive thermodynamic variable, the magnetization in the para/ferromagnetic transition, or the particle density in the vapor/liquid case, is fixed. While systems with short-range interactions present ensemble equivalence in the thermodynamic limit, at finite sizes they exhibit a variety of equilibrium phenomenology in the micromagnetic or microcanonical ensembles, absent in the magnetic or canonical ensembles, as negative specific heat and reentrance of the entropy-temperature curve.
Of remarkable relevance is the evaporation-condensation (EC) transition [4, 5, 6, 7, 8, 9], that we now illustrate in the microcanonical ensemble. Consider a system presenting a first-order transition at fixed energy density in phase coexistence (, being the low and high energies at the transition in the thermodynamic limit), whose thermodynamic potential is the entropy density , satisfying (see fig. 1). The situation is equivalent for magnets with the free energy, the magnetization and the magnetic field , or for liquids with the free energy, the intensive volume, minus the pressure, playing the role of , and , respectively. In the thermodynamic limit the phase coexistence is given by the lever rule: for , the minority (low-) phase condensates in a connected droplet whose volume equals a fraction of the system volume. In the presence of the droplet, the (high- phase) bulk energy surrounding it is larger than the global energy , hence its formation increases the entropy. This effect, however, is attenuated for finite sizes for which the droplet/bulk interface energy is not negligible and lowers the bulk energy. Through considerations on the energetics of the droplet (originally in the system ) it results that, given a finite system size there is an energy above which there is no droplet but a supersaturated (homogeneous) phase called evaporated. Below , the condensed phase is characterized by the coexistence between the droplet and the supersaturated bulk. The EC transition is such that and converge to the phase transition values , for large sizes, recovering the Maxwell construction (see fig. 1). The convergence is characterized by exponents that we will call and :
When expressed in terms of the size-rescaled control parameter, the EC transition can be considered as a first-order phase transition under some respects, such that the discontinuity is presented by the intensive parameter . According to the Ising EC theory for the , system, it is , in dimensions. Moreover, the emerging minority phase droplet at the EC transition is known to exhibit a universal fraction of the system volume, .
After the seminal work , there has been a quite intense research activity concerned with the topic. The EC transition and its finite-size rounding have been rigorously characterized [5, 6, 8]. Numerical tests of the EC theory have been performed for different systems in two and three dimensions: the lattice gas at fixed temperature [6, 7, 10] and density [11, 12]; the Ising model in the micromagnetic ensemble [13, 14]; the Lennard-Jones gas at fixed temperature [15, 16, 17] and density . The general outcome is that the theory provides an accurate description of the EC transition and its finite-size rounding in the Ising-liquid/vapor paradigm.111Although not analyzed in therms of the Ising EC theory, the EC transition has also been observed in the crystallization of hard spheres .
There has also been a variety of studies concerning the Potts model (PM) first-order transition in the microcanonical ensemble, although the situation in this case is much less clear. Differently from the Ising case, the transition is order/disorder, temperature (not field)-driven. In the nineties, the transition temperatures and interface tension [19, 20] were analyzed in 2D with the Metropolis algorithm. Using multicanonical simulations, these quantities were also studied in 3D, up to linear sizes and for up to  and, for the first time, they estimated the “spinodal” interval , claimed to shrink as , compatible with . In 2007, an efficient cluster algorithm was developed and tested in 2D for up to  (we analyze this data in the present work). Ref.  focuses on the calculation of the interface free energy in the EC transition, and on the related exceptionally large finite-size effects. Finally, in refs. [24, 25], the Wang-Landau algorithm was applied in 2D with and , up to and respectively. By scaling of the quantity versus , they concluded the validity of the Ising EC exponents . Moreover, the fraction of system volume occupied by the droplet at the transition is claimed to be , again in agreement with the Ising theory.
We will show, however, that both the outcome of novel simulations for a larger value of , and the data of ref. , seem to be incompatible with the exponent . Motivated by such a controversy, we discuss a possible alternative to the Ising EC exponents for the 2D PM. We first expose simple arguments to describe the EC phenomenology in terms of Potts quantities and of an exponent relating the droplet volume and its interface area. We claim how, in 2D, may not assume its geometrical value, , but rather . Finally we analyze the numerical data and show how, up to the simulated sizes, it is compatible with and incompatible with and hence with the Ising EC scaling. The relationship of the EC phenomena with metastability in the canonical ensemble will be finally discussed.
Model and notations
We consider the -color Potts model in a -dimensional lattice with sites. It is defined by the Hamiltonian , where the sum is over the bonds of the lattice, and the degree of freedom takes one out of values. The intensive entropy , inverse temperature, , and the (disordered) finite-size energy probability density (EPD) at the transition, , as functions of , are related in the following way: , . For a sufficiently high value of , the transition is first-order ; the quantities , , behave in the coexistence interval qualitatively as sketched in fig. 1. We will also considered the order and disorder entropies at the transition, . The quantities , , , , , are , , and lattice-dependent.
Evaporation-condensation in the Potts Model
Our arguments are very similar to that describing the EC-transition in the Ising/Lattice gas model, in the spirit of that of ref. . We will focus on the disordered phase, i.e., with near the disordered energy .
Our first working hypothesis is that the energy difference in the coexistence region can be split into two main contributions: one coming from a single, connected droplet of the ordered phase with energy and extensive mass , ; the second one comes from a disordered, supersaturated fraction of the system, the bulk, which surrounds the droplet. The equivalent assumption in the Ising case has been rigorously proven , and for the Potts case it appears to be quite reasonable, at least in two dimensions, in light of the snapshots of the condensed phase configurations shown in the refs. [22, 24]. The energy difference in the bulk is obtained by an increasing of the correlation length with respect to that at . Our second crucial hypothesis is that the average interface energy between the droplet and the disordered bulk, proportional to the droplet interface perimeter, is in its turn proportional to a power of the droplet volume: , being the proportionality constant, and being a -dependent exponent, of possible non-geometrical nature [1, 27]:222 The ambiguity in the droplet definition  reflects in its interface perimeter and area. In the case of geometrical droplets, the interface energy is precisely equal to the interface perimeter of the droplet. We will assume that for different cluster definitions (as the clusters in the Fortuin-Kasteleyn representation) the energy contribution is proportional to the droplet interface, the proportionality constant being and (slowly) -dependent, such that it can be absorbed into , not affecting the present discussion. In any case, we expect the droplet definition being less and less influent on the interface energy, the larger the value of .
In this circumstance, the conservation of energy is expressed as
where is the energy of the disordered bulk, depending on the droplet volume fraction , with . Due to the presence of the droplet, the bulk energy may be larger than the global energy , such an energy excess decreases with decreasing size, when the relative contribution of the interface perimeter increases.
The total entropy presents contributions from the droplet, bulk and droplet surface, . The droplet entropy is simply . The surface entropy is proportional to the droplet perimeter (the contour entropy being the proportionality constant), with a logarithmic correction that we neglect333The number of self-avoiding polygons with perimeter is expected to behave as . In any case, neglecting the surface entropy do not alter our conclusions, as we will see.[28, 29, 30], hence . The bulk entropy takes the form , with . It is obtained by expanding up to second order in around the disordered state at , where is the specific heat at the transition . Finally, using , where , and eq. (Evaporation-condensation in the Potts Model), the total -dependent intensive entropy reads:
We will call the value maximising , such that and become the finite-size thermodynamic functions, and . It is expected to be a solution of in the condensed phase for below a threshold , and in the evaporated phase for .
Thermodynamic limit. Maximizing in eq. ( ‣ Evaporation-condensation in the Potts Model) with , one gets , i.e., the Maxwell construction, required by van Hove’s theorem. The maximizing corresponds to the lever rule expected in the thermodynamic limit: .
Finite sizes. When the system size is finite, the entropy gets maximized creating a droplet of size smaller than due to the term in the square bracket in ( ‣ Evaporation-condensation in the Potts Model), and to a further penalization for large ’s in the term (since , as we will see). Looking for the solution of one obtains, to second order in , the equation:
where we have defined shortcuts for our variables, , and:
At fixed , there is a value of (a sufficiently small value of ) above which there is no real solution; in other words, for , (the evaporated phase). Conversely, for (the condensed phase), is a solution of ( ‣ Evaporation-condensation in the Potts Model). From now on, we will refer to and to and as functions of the reduced energy and size variables, .
In the evaporated phase with , the -dependent EPD is entirely composed by Gaussian thermal fluctuations . In the condensed phase for one has , being a function of approaching one for large or low . The EPD presents in this regime an stretched exponential extra term which dominates for large :
An estimation of the -dependence of is obtained equating in both regimes and, neglecting terms of , one gets (the proportionality constant includes , so that it must be for a solution to exist), which gives the exponent in eq. (Introduction). is obtained inserting in the expression of (derivating with respect to ). In substance,
or for and for , which is our main prediction. Taking the geometrical value of the sigma exponent, , one recovers the standard Ising EC exponents . There is, indeed, essentially the same nucleation-like competition in both cases, between (higher) bulk fluctuations, and (lower) bulk fluctuations plus droplet fluctuations (see for example eq. (7) of )444The difference is that, in the Ising case, the surface and bulk free energy terms are independent, while in the Potts case the surface enters as an entropic term by its own, and indirectly in the -dependence of the bulk entropy through eq. (Evaporation-condensation in the Potts Model)..
Analysis of the solution in two dimensions
The value of the exponent
In the rest of the article we will focus in the 2D case. Ref.  presents an effective droplet (Fisher) series for the 2D PM disordered phase. The results of this work suggest that the exponent could be different from its geometrical value, at least in two dimensions. In the droplet series, the free energy of clusters of area is assumed to be proportional to , with , and being an effective surface tension. The value of is obtained requiring the matching with exact results on the free energy cumulants for , or fitting it from the numerical free energy cumulants for hight . The resulting Fisher series accurately describes the energy histograms at (see [31, 32, 33]). The result is compatible with setting both the energy and the entropy of droplets proportional to the 2/3-th power of their area at fixed energy, as we do. The analogy between the analysis in  and the EC case is, however, not straightforward since the droplet theory is concerned with droplets of the metastable state, whose area is of order , while the ones involved in the EC transition are expected to be of order .
Droplets in the Ising/Lattice gas model at low temperatures are compact since the free energy is minimized by minimizing the perimeter of the droplet/bulk interface. In the Potts model, condensed droplets of the ordered phase at fixed energy, i.e., at fixed perimeter interface, are those maximizing the entropy. It is possible, although we are not aware of such a result, that the most probable two-dimensional lattice polygon with fixed perimeter exhibits an area proportional to . It is known that the average area of self-avoiding lattice polygons and loops with perimeter is [34, 35].555We note, however, that the average perimeter of self-avoiding polygons with area is proportional to .
The EC transition for and
For , eq. ( ‣ Evaporation-condensation in the Potts Model) becomes a third-order equation in terms of , which can be solved analytically (see the details in ). There is no real solution for , and , hence , as we anticipated. In the whole evaporated region for small enough , the scaling holds, in terms of the scaling variable . Moreover, the condensed phase fraction converges as to .
In the case, the solution is qualitatively identical to that of the case except by the values , as anticipated, and by . Eq. ( ‣ Evaporation-condensation in the Potts Model) takes the form of a fourth order equation in . Numerically finding the solution of , one finds , with . The discontinuity of the droplet volume fraction turns out to be (see the scaling function in fig. 2).
We note that both and are different from the value of the equivalent quantity in the Ising case, .
Comparison with numerical data in the microcanonical ensemble
We have simulated the 2D PM in the microcanonical ensemble with the algorithm presented in ref.  (a Monte Carlo cluster algorithm based on the Fortuin-Kasteleyn representation, with Metropolis acceptance probability), for systems with , and up to . The details of our numerical methods will be presented in . We also analyze the data from ref.  for up to . For we have detected the position of the EC transition point as the maximum of a c-spline fitting the data. For , for which the EC transition becomes sharp, with the prominence of strong metastable phenomena, we have estimated the lower (upper) bound for as the highest (lowest) at which one observes the condensed (evaporated) phase in a simulated annealing decreasing (increasing) , and as the value in the evaporated phase. Plotting the quantity versus , we observe (see fig. 3) that, up to the simulated sizes, the data are, in principle, compatible with the exponent (the linear fit predicts a vanishing , as required by the theory), while they exclude the value . Moreover, only for the quantity stays constant for the three largest sizes. For , up to the situation is qualitatively similar. However, longer simulations are needed for a conclusive confirmation .
The quantity reveals more ambiguous results, yet a slight preference for is observed . We conclude that the data exhibit a scaling not compatible with the exponent , although more precise measurements are needed to safely determine the exponents , .
We have also analyzed the statistics of Fortuin-Kasteleyn cluster areas and perimeters in the condensed phase, where the last quantity is defined as the Potts interface energy of each Wolff cluster  (both quantities are naturally accessed in the cluster algorithm). The area-perimeter dependence (shown in fig. 4 for , ) provides a more direct indication that the value is more suited for an effective description.
Evaporated versus metastable
The equilibrium curve in the evaporated phase (illustrated in fig. 1) resembles the metastable continuation in the canonical ensemble. Nevertheless, they are, in general, different. The EC transition converges to the thermodynamical transition in the large-system size limit, while the spinodal point, or the metastable limit, does not in general. In fact, in the Ising model -driven transition, the metastable interval and the lifetime of the metastable phase become independent on size for large sizes [38, 39].
An opposite example is the 2D Potts model for which, more trivially, metastable states at can be obtained from a reweighting of the disordered state at . The thermodynamics of the metastable state (for ) is, hence, determined by the finite-size potential, , which is the one fully characterizing also the evaporated phase. We consequently argue that, in this case, the thermodynamic quantities in the evaporated (microcanonical) phase and in the metastable (canonical) state coincide. Based on such an evaporated/metastable connection, one can estimate the value of and related quantities in the metastable state, if one has a method to systematically exclude out-of equilibrium phenomena (the dynamics of the nucleating droplets), as in ref. . As a confirmation of this fact, we anticipate that the data for computed with the present method coincides with that of ref.  for . Increasing one can in principle approach as much as one likes the point and sample more easily the inflexion point of the EPD. The efficiency of such a method will be analyzed in a forthcoming communication.
Conclusions and Perspectives
We have presented a phenomenological theory describing the EC transition of the 2D PM. In the presence of minority phase droplets with fractal interface, the EC phenomena result different from that of the known Ising paradigm. This hypothesis is supported by the data for the 2D PM up to , , that we have estimated with the help of a cluster algorithm, although the numerical evidences are still not conclusive (a more accurate estimation of the quantity is needed).
As further perspectives, we propose a quantitative comparison between data and eq. ( ‣ Evaporation-condensation in the Potts Model): in 2D the relevant quantities , , are known , and , could be estimated from geometrical measures of self-avoiding random walks, from measures of the cluster size distribution, or related with the known interface tension of the 2D PM. A more direct test would be a numerical estimation of , to discriminate between the values , , and (see ). A different challenge for future research could be that of clarifying the nature of the 3D system, for which the conventional EC exponents were proposed . A final controversy regarding metastability: we suggest that the thermodynamic metastable spinodal  can be identified with the EC transition; this would contradict the results based on the pseudo-critical divergence [42, 43, 44], which indicate that there is a “thermodynamic” endpoint of the metastable phase, not vanishing for large system sizes.
A fundamental question, already put forward in ref. , is under what general conditions the metastable behavior of a system and its evaporated phase in the micro-ensemble coincide. In this work we have proposed (as suggested in ) that both coincide in the case of the Potts model, differently with respect to the field-driven Ising transition.
I acknowledge Fabrizio Antenucci, Kurt Binder and Alberto Petri for their comments on the manuscript and bibliographic advices, and Tomoaki Nogawa for sharing with me the scalings of his data. Particular thanks to Víctor Martín-Mayor for useful conversations on the very genesis of the article and for the raw data of reference .
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