Long-range interacting systems in the unconstrained ensemble

Completely open systems can exchange heat, work, and matter with the environment. While energy, volume, and number of particles fluctuate under completely open conditions, the equilibrium states of the system, if they exist, can be specified using the temperature, pressure, and chemical potential as control parameters. The unconstrained ensemble is the statistical ensemble describing completely open systems and the replica energy is the appropriate free energy for these control parameters from which the thermodynamics must be derived. It turns out that macroscopic systems with short-range interactions cannot attain equilibrium configurations in the unconstrained ensemble, since temperature, pressure, and chemical potential cannot be taken as a set of independent variables in this case. In contrast, we show that systems with long-range interactions can reach states of thermodynamic equilibrium in the unconstrained ensemble. To illustrate this fact, we consider a modification of the Thirring model and compare the unconstrained ensemble with the canonical and grand canonical ones: the more the ensemble is constrained by fixing the volume or number of particles, the larger the space of parameters defining the equilibrium configurations.


I. INTRODUCTION
Systems with long-range interactions may display a notable thermodynamic behavior that distinguish them from those where the interactions are short-ranged [1][2][3]. Long-range interactions are those for which the range is comparable with the size of the system, regardless of how large the system is, as it occurs, for instance, in selfgravitating systems [4][5][6][7][8][9][10][11], plasmas [12,13], fluid dynamics [14,15], and some spin systems [16]. Because of these interactions, the system may remain trapped in nonequilibrium quasi-stationary states [17] whose lifetimes depend on the number of particles and diverge in the limit N → ∞. For large but finite number of particles in a very long time limit, however, the system eventually evolves towards states of thermodynamic equilibrium [18] that can be described within the usual framework of ensemble theory. At equilibrium, such systems may present ensemble inequivalence [6,[20][21][22] associated to anomalies in the concavity of the thermodynamic potentials [22,23]. In particular, a negative heat capacity in the microcanonical ensemble [5,6,19] is due to an anomaly in the concavity of the entropy as a function of the energy. In addition, systems with long-range interactions, in general, do not obey the usual Gibbs-Duhem equation [24,25] (see Ref. [26] for a discussion regarding self-gravitating systems). This fact is deeply related with the nonadditive character that long-range interactions confer to these systems [25]. Actually, nonadditivity is responsible for the remarkable behavior observed in long-range interacting systems [1], as well as for the physics behind the subject we discuss in this paper: the appearance of a new equilibrium statistical ensemble in the thermodynamic limit.
In this paper we focus on the thermodynamics of longrange interacting systems under completely open conditions. A completely open system can exchange heat, work, and matter with its surroundings. That is, the energy, volume, and number of particles of the system fluctuate under completely open conditions. Thus, the control parameters that specify the thermodynamic state of the system are temperature, pressure, and chemical potential. These control parameters are properties of a suitable reservoir that weakly interacts with the system, in the sense that, by means of some mechanism, it supplies heat, work, and matter to the system, but can be assumed not to be coupled by long-range interactions. It is worth mentioning that under these conditions the system is not enclosed by rigid walls. Instead, the system could be confined by an external field exerting pressure on it; e.g., tidal forces produced by surrounding objects in self-gravitating systems or magnetic fields acting on plasmas [27]. Furthermore, the unconstrained ensemble is the statistical ensemble that describes a completely open system (in the literature, this ensemble is also termed generalized [28]). It was introduced by Guggenheim [29], but it did not receive much attention due to its lack of application in standard macroscopic systems. If the interactions in the system are short-ranged, the free energy associated to this ensemble is vanishingly small when the number of particles is large [29]. This is a consequence of the fact that temperature, pressure, and chemical potential cannot be treated as independent variables because they are related by the Gibbs-Duhem equation. Thus they are not, taken together, suitable control parameters for macroscopic systems with short-range interactions.
In case that the limit of large number of particles is not assumed, however, the situation changes, as pointed out by Hill [30]. Small systems may have an extra degree of freedom which permits that equilibrium configurations in completely open conditions can be realized [30][31][32][33]. Nevertheless, when the size of the system is increased, the usual behavior of macroscopic systems is obtained [30,34,35], as long as the interactions remain short-ranged.
Our aim here is to show that equilibrium configurations with a large number of particles may be realized in the unconstrained ensemble if the interactions in the system are long-ranged. Moreover, we argue that the replica energy [25] is the appropriate free energy defining such configurations in this ensemble. Thus, in Sec. II, we will first describe this ensemble and make the appropriate connection with the thermodynamics. In Sec. III we introduce a solvable model, whose thermodynamics in the unconstrained ensemble is discussed in Sec. III A. In Secs. III B and III C, this model is studied in the grand canonical and canonical ensembles, respectively. In addition, some thermodynamic relations associated with the replica energy are analyzed in Sec. IV, and our conclusions are presented in Sec. V.

II. THE UNCONSTRAINED ENSEMBLE
To make our point clear, it is worth to briefly review the thermostatistics of completely open systems by establishing its connection to the unconstrained ensemble. The description of the unconstrained ensemble presented in this section is based on Refs. [28] and [30], which the reader is referred to for an extended discussion (see also [29,32,36,37]).
The energy, volume, and number of particles of a completely open system are quantities that fluctuate due the interaction of the system with its surroundings. Let us consider the probability p i (V, N ) of the configuration of a system that is found in the state i with energy E i , and possesses N particles in a volume V . This probability is an exponential of the form [28] where α, β, and γ are parameters that will be identified below and Υ is the associated partition function given by Here discrete variables are used for simplicity. The ensemble averageĒ of the internal energy E is thus ob- while the average number of particlesN and the average volumeV  readN respectively. Now let us consider an infinitesimal change of the average internal energy in terms of changes in the probability, where the coefficients E i (V, N ) are assumed constant. Hence, using Eq. (1) to express E i (V, N ) and the condition i,V,N dp i (V, N ) = 0, Eq. (3) can be rewritten as (4) If Eq. (4) is compared with the thermodynamic equation where T is the temperature, P is the pressure, and µ is the chemical potential, one then recognizes k B T = 1/β, µ = −α/β, P = γ/β, and the entropy where k B is the Boltzmann constant. Therefore, the probability (1) leads to the correct thermostatistical description of completely open systems. Furthermore, substituting Eq. (1) in Eq. (6) with the previous identifications, one gets where we have introduced the free energy called replica energy [25] or subdivision potential [30]. By differentiation of Eq. (7) and using Eq. (5), one obtains Notice that, as a consequence of Eq. (9), one has Moreover, the unconstrained partition function can be written as where is the canonical partition function. From Eq. (13), it is straightforward to connect Υ with the partition function of another ensemble, e.g., one has where Ξ(T, V, µ) is the grand canonical partition function.
For macroscopic systems with short-range interactions in the thermodynamic limit, the internal energy and the other thermodynamic potentials are linear homogeneous functions of the extensive variables and thus, from Eq. (7), one obtains E = 0, since in this case the Gibbs free energy G =Ē − T S + PV is equal to µN . We note that in this case it does not mean that Υ = 1 but that E is negligible in the thermodynamic limit [28]. Thus, for this kind of systems, Eq. (9) reduces to the wellknown Gibbs-Duhem equation. The interesting fact is that equation (7) indicates the route to deal with the more general situation in which the internal energy is not a linear homogeneous function of S, V , and N , as it happens with systems with long-range interactions. This is the case in which here we are concerned with, for which the replica energy is, in general, different from zero.
The above considerations are directly related to the fact that for an ensemble of N replicas of the system with total energy E t = N E, entropy S t = N S, volume V t = N V , and number of particles N t = N N , E can be interpreted as the energy gained by the ensemble when the number of replicas changes holding S t , V t and N t constant. Formally [30], When the single-system energy E is a linear homogeneous function of S, V , and N (additive systems), one and therefore, Eq. (16) leads to E = 0. If E is not a linear homogeneous function of S, V , and N (nonadditive systems), Eq. (16) then requires E = 0 [25]. Finally, we want to stress that Eqs. (7) and (9)-(12) are relations at a thermodynamic level. That is, for any other ensemble, corresponding to the actual physical conditions specified by the given control parameters, an analogous set of equations can be obtained from the appropriate free energy [30]. The replica energy will emerge also in those cases, although it will not be the free energy associated to the partition function for that ensemble. However, since in any case E vanishes when the system is additive, it can always be seen as a measure of the nonadditivity of the system [25].

III. THE MODIFIED THIRRING MODEL
We introduce here a model that, as discussed in the following, can attain equilibrium configurations under completely open conditions. Consider a system of N particles of mass m enclosed in a volume V with a Hamiltonian of the form where p i ∈ R 3 and q i ∈ R 3 are the momentum and position of particle i, respectively, φ(q i , q j ) is the interaction potential, p = (p 1 , . . . , p N ), and q = (q 1 , . . . , q N ). The interactions in this model are defined by where ν > 0 and b are constants, and we have introduced the functions Here V 0 is the volume of an internal region of the system such that V 0 < V , and V 1 is the volume of the region outside V 0 so that V 1 = V −V 0 . In this way, V 0 is a parameter of the interaction potential that does not depend on the state of the system. Hence, the total potential energy in the large N limit is given bŷ where N 0 is the number of particles in V 0 and N 1 = N − N 0 is the number of particles in V 1 for a given configuration. Notice that N 0 = N 0 (q) and N 1 = N 1 (q), so thatŴ depends implicitly on the position of all particles. This model is a modification of the Thirring model [6], which is obtained in the particular case b = 0. Furthermore, the model is solvable in the different statistical ensembles, and below we focus on the unconstrained ensemble and compare it with the canonical and grand canonical cases. In the following it will become clear why we consider this modification of the model.

A. The unconstrained ensemble
Here we study the modified Thirring model in the unconstrained ensemble. Thus, we will assume that the system is in contact with a reservoir characterized by fixed temperature T , pressure P , and chemical potential µ. Consider the canonical partition function for this model, which is given by where λ T = h/ √ 2πmT is the thermal wave length, h being a constant, and hereafter we take units where k B = 1. Thus, using Eq. (13), the unconstrained partition function becomes (we take V as a continuous variable) Following Thirring's method [6], the partition function can be computed by replacing the integral over coordinates with a sum over the occupation numbers in each region of the system, where the sum runs over all possible values of N 0 and N 1 and the Kronecker δ enforces the condition N = N 0 +N 1 . This leads to where we have introduced (from now on we omit writing down explicitly the dependence on the control parameters)Ê with k = 0, 1, and we have used Stirling's approximation.
Since the replica energy is given by E = −T ln Υ, using the saddle-point approximation one obtains Thus, minimization with respect to V , N 0 , and N 1 requires that whereV ,N 0 , andN 1 are the values of the volume and the number of particles in each region that minimizes the replica energy. It is clear thatV ,N 0 , andN 1 are functions of T , P , and µ, whose dependence is implicit through Eqs. (27)- (29). The mean value of the total number of particles is then given byN = kN k . Moreover, the total potential energy for equilibrium configurations in the unconstrained ensemble becomes Hence, notice that from Eqs. (28) and (29) one obtains with V 1 =V − V 0 . Therefore, using Eqs. (25) and (26), the replica energy takes the form as given in [25], where P (e) = P −N T /V is the excess pressure.
We introduce the dimensionless variables which will be denoted asv,x 0 , andx 1 when evaluated at V ,N 0 , andN 1 , respectively. In addition, we define the reduced pressure p and the relative fugacity z by where Since controlling T , P , and µ is equivalent to controlling T , p, and z, the latter set of variables can be taken as the set of control parameters in this ensemble. Using these variables, from Eqs. (27)- (29) one obtains where Eq. (36) defines implicitlyx 0 =x 0 (z). Furthermore, let us consider the reduced replica energy ϕ U = νE /T 2 and the functionφ U = νÊ /T 2 , where, using Eqs. (33) and (34), the latter can be written aŝ We note that with the dimensionless quantities, i.e., in the reduced replica energy and in Eqs. (36)- (38), the temperature does not appear; therefore T acts as a simple scaling factor. The condition (26) becomes Since the Hessian matrix H U associated toφ U at the stationary point (v,x 0 ,x 1 ) takes the form one infers thatv,x 0 , andx 1 lead to a minimum of replica energy whenx The last two inequalities guarantee, from Eqs. (37) and (38), thatx 1 > 0 andv > 0; the latter inequality assures that the average volume actually contains V 0 in any possible equilibrium configuration. Moreover, Eq. (36) has two positive solutions if 0 < z < z 0 with z 0 = 1/(2e) ≈ 0.1839, while no real solution exists for z > z 0 . Notice that the smallest of the roots of Eq. (36) is that corresponding to 0 <x 0 < 1/2. This implies that the fugacity cannot be arbitrarily large and that Therefore, we conclude that equilibrium configurations in the unconstrained ensemble can indeed be realized.
In addition, we stress that there are no different states minimizing the replica energy for the same control parameters in the equilibrium region, hence the model has no phase transitions in the unconstrained ensemble. Furthermore, it is clear that equations (37) and (38) are not well defined when b = 0. As a consequence, the Thirring model (b = 0) does not attain equilibrium states in this ensemble; in this case, T , P , and µ cannot be taken as independent control parameters, just as it happens for macroscopic systems with short-range interactions. We will come back to this point in Sec. IV. Interestingly, we note that the condition b < 0 means that the interactions within the outer parts of the system must by repulsive to guarantee equilibrium configurations. This prevents the system from collapsing under completely open conditions in the appropriate range of control parameters. In order to get further insight in the nature of the system, let us introduce the reduced number of particles x and reduced densityȳ defined as The behavior of these thermodynamic functions in the unconstrained ensemble can be quantitatively described for the model we are studying. Thus, the reduced number of particlesx(p, z) is shown in Fig. 1(a) as a function of the reduced pressure p with constant fugacity z, while in Fig. 1(b) is plotted as a function of the fugacity with constant reduced pressure. In Fig. 2(a), the reduced densitȳ y(p, z) is represented as a function of the reduced pres-  sure p with constant fugacity z, and, in Fig. 2(b), as a function of the fugacity with constant reduced pressure. Furthermore, in Fig. 3(a), we plot the reduced volumē v(p, z), as given by Eq. (38), as a function of the reduced pressure p for fixed values of the fugacity z. In Fig. 3(b), we showv as a function of z holding the pressure constant. In all these plots we have set the parameter b of the model to b = −1.
Let us also consider the dimensionless response functions q V and q N defined according to These response functions will be useful to compare the unconstrained ensemble with the canonical ensemble (see below), and we note that q V is related to the isothermal compressibility κ T , according to In fact, here we are only interested in the signs of q V and q N . Since V and P , and N and µ are conjugate variables, respectively, and both the number of particles and the volume fluctuate here, the response functions q V and q N must be positive in this ensemble. Moreover, the signs of q V and q N can be seen by plotting the curves v vs. p with constantx, andȳ vs. z with constantv, respectively. Sincex is not a control parameter, the curvē v vs. p with constantx represents the evolution ofv as a function of p through a series of equilibrium states where the actual control parameters, p and z in this case, are chosen in such a way thatx takes the same value in all these states. This can be achieved if the reduced pressure is parametrized as for the givenx, where we have used Eqs. (37) and (44). Moreover, since the reduced pressure is always lower than z in equilibrium configurations, Eq. (49) must be restricted only to values of z satisfying the condition b[x −x 0 (z)] < 0. Hence, the curvev vs. p with constantx is obtained by parametrically plotting px(z) and v(px(z), z) using z as a parameter. The curveȳ vs. z with constantv can be obtained in analogous manner by choosing p and z in such a way thatv takes always the same value. In this case the reduced pressure is given by with fixed values of the reduced volumev. As the reduced densityȳ is given by Eq. (45) as a function of p and z, the curveȳ vs. z with constantv is obtained from Eq. (45) using p = pv(z).
According to the previous discussion, in Fig. 4(a)-(b) we observe thatv is a decreasing function of p whenx is fixed. This means that q V is positive for these configurations, as expected. Furthermore, as can be seen in Fig. 5(a)-(b),ȳ increases for increasing z with fixed v, which indicates that the response function q N is also positive. Below we will show that the response functions in this model can be negative in the canonical ensemble, where V and N are fixed control parameters.

B. The grand canonical ensemble
We now constrain the system by fixing the volume, so that the control parameters in this case are T , V , and µ, which corresponds to the grand canonical ensemble. Using Eq. (21), the grand canonical partition function Ξ = N e βµN Z can be written as where, using Striling's approximation in the large N limit,Ω Using the saddle-point approximation, the grand potential is given by Ω = inf {N0,N1}Ω (N 0 , N 1 ). The minimization with respect to N 0 and N 1 leads to where nowN 0 andN 1 , being functions of T , V , and µ, are the number of particles in each region that minimize the grand potential. The total mean number of particles is then given byN =N 0 +N 1 . In addition, in the grand canonical ensemble, the pressure is given by where the expression containingΩ must be evaluated at N 0 =N 0 and N 1 =N 1 . Thus, for the modified Thirring model, one obtains In addition, the replica energy in the grand canonical ensemble is given by [30] This means that E = 0 if Ω(T, V, µ) is not a linear homogeneous function of V , and, therefore E vanishes only when the system is additive. From Eq. (58), using Eq. (53) evaluated at N 0 =N 0 and N 1 =N 1 , one obtains with the excess pressure P (e) = P −N T /V . We stress that, in this case, the replica energy is a function of T , V , and µ.
In order to study the equilibrium states of the system, we introduce the reduced grand potential ϕ G = νΩ/T 2 and the associated functionφ G = νΩ/T 2 , which are related through Here x 0 = νN 0 /T and x 1 = νN 1 /T , so thatx 0 = νN 0 /T andx 1 = νN 1 /T are the corresponding quantities that minimizeφ G , defining thus the equilibrium configurations in the grand canonical ensemble. From Eq. (53), we obtainφ with the relative fugacity z = e (µ−µ0)/T and the reduced volume v = (V − V 0 )/V 0 . Here the variables z and v can be taken as control parameters together with T . We note that, analogously to what occurred for the reduced replica energy in Sec. III A, the temperature T does not appear explicitly in the reduced grand potential. In dimensionless variables, Eqs. (54) and (55) can be rewritten asx Furthermore, the Hessian matrix H G associated toφ G at the stationary point (x 0 ,x 1 ) takes the form and, therefore, one finds thatφ G can be minimized if We thus observe that, as before, Eq. (62) has two solutions if 0 < z < z 0 with z 0 = 1/(2e), and that the smallest of the roots of Eq. (62) is that corresponding to 0 <x 0 < 1/2. Besides that, on the one hand, Eq. (63) has always one solution if b ≤ 0. On the other hand, when b > 0, Eq. (63) has two solutions if 0 < z < z 1 (v), where in such a way that the smallest of these solutions satisfies the condition (65). At z = z 1 (v), the only solution is given byx 1 = 1/(2b) and hence, it corresponds to an unstable state. Therefore, Eqs. (62) and (63) can be solved simultaneously to give equilibrium configurations if where We note that for given values of the control parameters, the saddle-point equations define only one state, and, hence, there are no phase transitions in the grand canonical ensemble. We also remark that the Thirring model (b = 0) attains equilibrium states in the grand canonical ensemble if 0 < z < z 0 .
In the grand canonical ensemble, we now introduce the reduced number of particles, density, and pressure given byx respectively. In addition, the response functions are given by which for constant z defines the same relation as in the unconstrained ensemble. Moreover, the equilibrium conditions are the same in the two ensembles for b < 0. Therefore, in the modified Thirring model, the grand canonical and unconstrained ensembles are equivalent when b < 0.
Since also the class of models with b ≥ 0 attains equilibria in the grand canonical ensemble, the phenomenology in this case is richer than in the unconstrained case. For instance, equilibrium configurations with p ≥ z or with negative isothermal compressibility cannot be observed under completely open conditions, while these configurations can be realized with fixed volume. In Fig 6(a), we show p as a function of z with fixed v for different values of the parameter b. It can be seen that p > z when b > 0, while p < z when b < 0 as it happens in the unconstrained ensemble. In addition, in this plot we observe that p = z for b = 0, a general feature of the Thirring model. All the curves in Fig 6(a) finish at z = z G (v), since beyond this critical fugacity the stability is lost in the grand canonical ensemble. In Fig 6(b), p is plotted as a function of v by holdingx constant, where also different values of b are chosen. Since holdinḡ x constant determines the value of the fugacity when v is varied, the curves start at the minimum value of v for which the condition 0 < z < z G (v) is satisfied, ensuring thus the equilibrium of the configurations. As a remarkable fact, a region where q V < 0 is observed for b = 0.5, which corresponds to the points of the curve with positive slope. Configurations in such a region have negative isothermal compressibility.

C. The canonical ensemble
In order to understand better the behavior of the system in the unconstrained ensemble, it is instructive to compare its equilibrium states with the corresponding ones in the canonical ensemble. Hence, we consider now that the control parameters are T , V , and N . Using (23) and the integral representation with Re[ζ] = α = −µ/T , the canonical partition function (21) can be written as where, for large N , The canonical Helmholtz free energy is thus given by F = inf {α,N0,N1}F (α, N 0 , N 1 ), where ζ is evaluated at its real part α. Moreover, minimization with respect to α enforces thatN 1 = N −N 0 , while minimization with respect to N 0 and N 1 leads to where we have used that α = −µ/T , and where the bars denote that the corresponding quantity minimizes the canonical free energy. In view of with the expression containingF evaluated at N 0 =N 0 and N 1 =N 1 , we note that µ is indeed the chemical potential of the system, that in the canonical ensemble is no more a control parameter. Equating the right hand sides of Eqs. (78) and (79) we get an equation forN 0 as a function of T , V and N . Furthermore, the pressure is given by so that We now turn our attention to the replica energy in the canonical ensemble. It is given by [30] so that now E = 0 if F (T, V, N ) is not a linear homogeneous function of V and N , and it vanishes only when the system is additive. From Eqs. (83) and (77), we again obtain where P (e) = P − N T /V . Furthermore, taking x = νN/T and v = (V − V 0 )/V 0 as control parameters, and, as before, introducingx 0 = νN 0 /T in the canonical ensemble, equations (78) and (79) can be combined into Notice thatx 0 /x =N 0 /N represents the fraction of particles inside the volume V 0 , so that 0 ≤x 0 /x ≤ 1. Equation (85) definesx 0 =x 0 (x, v) in this ensemble, and, depending on the parameters, it can have two solutions. Again, in reduced variables the temperature does not appear. The solution determining the equilibrium states corresponds to x 0 =x 0 that minimizes the canonical free energy, or, equivalently, the reduced free energy ϕ C = νF/T 2 . From Eq. (77) and according to the variational problem for this case, one obtains withφ where in Eq. (87) we have used thatN 1 = N −N 0 , and omitted terms that do not depend on x 0 . It is interesting to observe that Eq.(85) can have more than one solution, so that the model may exhibit phase transitions. In the Appendix we discuss how to obtain the critical point for the modified Thirring model in the canonical ensemble. Once (85) is solved satisfying (86), the relative fugacity z = e (µ−µ0)/T can be computed from Eq. (78), since this equation can be rewritten as Furthermore, the reduced pressure p = νV 0 P/T 2 takes the form The response functions, as defined in Sec. III A, in the canonical ensemble are given by where we have introduced the reduced density As discussed in Sec. III A, the sign of the response functions q V and q N can be be inferred from the slopes of the curves p(v) at constant x and z(y) at constant v, respectively. Here we want to compare these curves with the corresponding ones in the unconstrained ensemble. We do this for negative values of b, for which the unconstrained and the grand canonical ensembles are equivalent. Then, the curves represent also the comparison between the canonical and grand canonical ensembles when b < 0. Thus, in Fig. 7(a) we show p as a function ofv in the unconstrained ensemble with b = −1, wherex is held constant. The corresponding curves in the canonical ensemble are also shown in this plot, where v and x are fixed to the same values ofv andx, respectively. In Fig. 7(b), these curves are represented for b = −0.1 and different values ofx, where it can be appreciated that the model presents first-order phase transitions in the canonical ensemble, as indicated by the jumps in the pressure. In addition, in Fig. 8(a) we represent z as a function ofȳ in the unconstrained ensemble, wherev is held constant and we set b = −1. We also show the analogous curves in the canonical ensemble, where y and v are fixed to the same values ofȳ andv, respectively. In Fig. 8(b), these curves are shown for b = −0.1. The jumps in the fugacity correspond to first-order phase transitions in the canonical ensemble. Thus, the canonical and unconstrained ensembles are nonequivalent at the macrostate level [38,39], since the equilibirum states in the two ensembles are not in one-to-one correspondence. In particular, we highlight that the response functions can be negative in the canonical ensemble. Finally, we briefly note that a nonequivalence between the microcanonical and canonical ensembles is expected for the modified Thirring model, since this happens for the particular case b = 0 [40].

IV. THERMODYNAMIC RELATIONS AND REPLICA ENERGY
In Sec. II, from an equation for the differential variations of the replica energy in the unconstrained ensemble, we have obtained a set of relations in terms of partial derivatives that allows one to obtain, e.g., the entropy of the system. Let us recall these relations since they will be the central issue of the following discussion: and These equations, however, are thermodynamic relations valid in any ensemble; with this in mind, we do not use here a bar over the variables to indicate whether a certain quantity fluctuates or not. The present discussion is to emphasize that a situation may exist in which the replica energy is different from zero, but T , P , and µ cannot be taken as a set of independent variables. This can happen in an ensemble different from the unconstrained one. When T , P , and µ cannot be taken as independent variables in the unconstrained ensemble, the mean-field equations will not lead to a minimum of replica energy, and, therefore, Eqs. (93)-(96) are meaningless.
Let us assume that the system is itself constrained such that, for instance, µ = µ(T, P ). In this case, Eq. (93) becomes which establishes a functional relation E = E (T, P ). Thus, one actually has which are the relations satisfied in this case instead of Eqs. (94) and (95). This is equivalent to directly consider a constraint E = E (T, P ) in Eq. (93), which yields (100) The above expression shows that the usual Gibbs-Duhem equation is not valid when the replica energy is different from zero even if T , P , and µ are not a set of independent variables. Moreover, analogous equations can be obtained if one considers any other functional relation constraining T , P , and µ.
To go further, consider an arbitrary d-dimensional system with long-range interactions whose number density at a point x ∈ R d is given by where Φ(x) is the mean-field potential. The chemical potential and the entropy of the system satisfy [25] where W = 1 2 n(x)Φ(x)d d x is the potential energy. Moreover, the local pressure is given by p(x) = n(x)T when short-range interactions are completely ignored, and the pressure P is p(x) evaluated at the boundary of the system [24]. Consider also that the mean-field potential is not completely arbitrary but it always vanishes at the boundary of the system, regardless of the thermodynamic state of the system. This will enforce the condition where P (e) = P − N T /V . Hence, with the particular constraint (104), Eqs. (105) and (106) hold in place of Eqs. (94) and (95). The interesting fact here is that the modified Thirring model can be used to test these general considerations. In Sec. III A we found that in the case b = 0 the model never attains equilibrium configurations in the unconstrained ensemble. As we shall see below, this is due to the fact that, for the case b = 0, the chemical potential satisfies Eq. (104) (with d = 3), and thus T , P , and µ cannot be taken as independent variables. But first we will check that for b = 0 the entropy can be computed from Eq. (94), showing that T , P , and µ can actually be taken as independent variables in this case. This, of course, is in agreement with the statistical mechanics description of the system obtained in Sec. III A.
To check the validity of Eq. (94), we need to write the replica energy as a function of T , P , and µ. We note that Φ(x) = −2ν [N 0 θ V0 (x) + bN 1 θ V1 (x)] and N 1 /V 1 = P/T for the modified Thirring model. Thus, evaluating the number density (101) at any point x in V 1 and rearranging terms gives us which defines N 1 = N 1 (T, P, µ) if b = 0. Analogously, evaluating (101) at any point x in V 0 yields which defines implicitly N 0 = N 0 (T, µ). Therefore, when b = 0, we have Since the replica energy is implicitly given as a function of T , P , and µ by means of Eqs. (107) and (108). Hence, where we have rearranged terms using Eqs. (107) and (108). By comparing with Eq. (103), we thus see that the rhs of Eq. (112) is indeed −S. This confirms that when b = 0, the system has enough thermodynamic degrees of freedom to take T , P , and µ as a set of independent variables. To see whether the configurations are stable or not, however, one must perform an analysis of the second order variations of the appropriate free energy, as we have done in Sec. III A, for instance, depending on the actual physical conditions imposed by the corresponding control parameters. In view of Eq. (107), it is now obvious that if b = 0, the chemical potential is given by Eq. (104) and that now defines implicitly N 0 = N 0 (T, P ). Hence, Using Eq.(114) and since in this case E = νN 2 0 + P V 0 − N 0 T , it is easy to see that in agreement with Eq. (105). These arguments show that, although the replica energy is different from zero, in the Thirring model (b = 0) one cannot take T , P , and µ as a set of independent variables.

V. CONCLUSIONS
We have shown that systems with long-range interactions can attain configurations of thermodynamic equilibrium in the unconstrained ensemble. In this ensemble, the control parameters are temperature, pressure, and chemical potential and the appropriate free energy is the replica energy. We have presented a solvable model that is stable in this ensemble, and we have compared its equilibrium states with those of the grand canonical and canonical ensembles. From this comparison we observe that the space of parameters defining the possible stable configurations is enlarged when the system is constrained by fixing the volume and the number of particles. These quantities fluctuate in the unconstrained ensemble, as well as the energy of the system. Moreover, the model we have introduced exhibits first-order phase transitions in the canonical ensemble at which the system undergoes pressure and fugacity jumps.
On the one hand, macroscopic systems with shortrange interactions cannot attain equilibrium states if the control parameters are the temperature, pressure, and chemical potential. According to the usual Gibbs-Duhem equation, these variables cannot be taken as independent. A physical reason behind this property is that in this case temperature, pressure, and chemical potential are truly intensive properties and therefore, they cannot define the size of the system corresponding to an equilibrium state. On the other hand, however, in systems with long-range interactions temperature, pressure, and chemical potential are not intensive properties, so that controlling, e.g., the temperature T ∼ N can actually define the size of the system in an equilibrium configuration. Thus, the typical scaling of the thermodynamic variables in long-range interacting systems makes it possible to have equilibrium configurations in completely open conditions. Phase transitions in the modified Thirring model in the canonical ensemble can be studied by extending the analysis done in Ref. [40] using Landau theory of phase transitions. In [40], the expansion parameter specifying the transition is taken as the deviation of the fraction 1 − N 0 /N with respect to the value of this fraction at the transition line. Here, equivalently, we consider the fraction of particles in V 0 , given by x 0 /x, and take m = (x 0 −x 0 )/x as the expansion parameter. Accordingly, to obtain the critical point we expand the free energy (87) asφ C = ϕ 0 + ϕ 1 m + ϕ 2 m 2 + O(m 3 ), and look for a solution of the system of equations ϕ 1 (x 0 , x, v) = 0, ϕ 2 (x 0 , x, v) = 0, (A.1) with the constraint 0 ≤x 0 /x ≤ 1. The first of the equations (A.1) is exactly (85), while the second one can be written as Thus, for b > −1, Eq. (A.2) has two real solutions, , when x > 2/(1 + b) ≡ x c . The critical point is defined by the condition [40] x + (x c ) = x − (x c ), in such a way that at the critical point one has x = x c andx 0 = x ± (x c ) = x c /2. In terms of the thermodynamic variables, this means that the critical temperature is given by T c = νN (1+b)/2 and that the fractionN 0 /N at the critical point is 1/2. Moreover, we note that when b = −1, the l.h.s. of Eq. (A.2) does not vanish, and, therefore, there are no phase transitions in this case. For b < −1, the fractions x ± /x do not lie in the interval [0, 1], so that also in this case the model will not exhibit phase transitions. We therefore observe that the model may present phase transitions in the canonical ensemble, depending of the value of v, if x is larger than x c and if b > −1.
The critical value of the reduced volume v c is obtained by replacingx 0 = x ± (x c ) in Eq.