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Phase Transitions in Small Systems: Microcanonical vs. Canonical Ensembles

We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g. temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function.


Introduction
One of the most intriguing thermodynamic properties of various macroscopic systems is their ability to undergo phase transitions (PTs) if one or more control parameters pass certain critical values [1,2]. The first systematic classification scheme for macroscopic PTs was proposed by P. and T. Ehrenfest [3] in 1912 already. After further pioneering work by Mayer et al. [4][5][6][7], Yang and Lee [8,9] elucidated the mathematical essence underlying PTs in the grandcanonical ensemble by studying the distribution of complex zeros (DOZ) of the grandcanonical partition function. Later on, Fisher [10] and Grossmann et al. [11][12][13] employed a very similar approach to analyze the temperature dependence of phase transitions in the canonical ensemble. Recently, further significant progress in the understanding of critical phenomena has been achieved by studying the connection between PTs and phase (or configuration) space topology [14][15][16][17][18][19][20].
Simultaneously, interest began to focus on the question how to identify and classify the finite-size analogues of macroscopic PTs. Import results in this regard were obtained by Wales et al. [23,24], who considered necessary and sufficient criteria for phase coexistence in finite systems. A general classification scheme for smooth canonical PTs in small systems was proposed by Borrmann et al. [34]. Pursuing an approach similar to that of Yang and Lee [8,9], Fisher [10] and Grossmann et al. [11][12][13], these authors suggest to use the DOZ in order to characterize transitions in the canonical ensemble of small systems (cf. Sec. 3.2 below). Mülken et al. [35] and Alves et al. [36] compare the DOZ classification scheme with alternative proposals made by Gross [25,37] and by Janke and Kenna [38], respectively. Such progress notwithstanding, there still exist some open questions regarding which types of non-analytic PTs can occur in small systems. For(grand-)canonical ensembles, it is well established that truly singular PTs can be observed in the thermodynamic limit only, corresponding to a system with formally infinite particle number N → ∞ [8-13, 34, 39]. In contrast to this, the microcanonical thermodynamic functions (TDFs) may exhibit non-analytic behavior even at finite N. For example, recently, Pleimling and Behringer [40] have found singularities in microcanonical quantities of finite three-dimensional spin models, which announce a continuous macroscopic PT of the infinite systems. Additionally, as we intend to demonstrate here by means of very simple examples, the microcanonical TDFs of a small system can also exhibit non-analytic microscopic PTs 1 , characterized by well-defined critical energy values and typically accompanied by strong variations of thermodynamic observables (e.g. oscillations of temperature, heat capacity and pressure). From the physical point of view, such microscopic PTs correspond to transitions between different dissociation states of the system. Hence, they are important indicators for essential structural changes in the small system under consideration (quite analogous to singular points indicating macroscopic PTs). For reasons of simplicity, the discussion in the present paper will be restricted to one-dimensional (1D) models, but the general mechanism responsible for the singular and oscillatory behavior of microcanonical thermodynamic observables works analogously in two and three spatial dimensions. Hence, it should in principle be possible to observe such microscopic oscillations in suitably designed evaporation/dissociation experiments (which must, of course, realize the physical preconditions of the microcanonical ensemble).
The paper is organized as follows: Section 2 is dedicated to microscopic PTs in the microcanonical ensemble (MCE) of small model systems. As examples, we will consider isolated 1D chains with Lennard-Jones (LJ) pair interactions and also the Takahashi gas [44]. It will be shown that these simple systems exhibit singular microscopic PTs, separating different microcanonical dissociation states. Subsequently, the Takahashi gas will be used in Sec. 3 to investigate the relation between singular microscopic PTs in the MCE and smooth PTs in the canonical ensemble (CE) as defined by the DOZ scheme [34]. The paper concludes with a summary of the main results in Sec. 4.

Microscopic phase transitions in the microcanonical ensemble
Classical microcanonical thermodynamics refers to an ensemble of thermally isolated systems, completely described by their Hamiltonian dynamics. Due 1 The appearance of such microcanonical singular points even in the 1D case is not in conflict with van Hove's theorem for the canonical ensemble [41,42], since for most small systems (as well as for many 'large' systems) the microcanonical and canonical ensembles are generally not equivalent (see, e.g., Costeniuc et al. [43]).
to the fact that the systems are decoupled from the environment, the energy E is a conserved quantity, i.e., there are no energy fluctuations in the MCE. In particular, as will be demonstrated in Secs. 2.3 and 2.4, the microcanonical TDFs may exhibit singular points even in the case of small systems.

The microcanonical ensemble
For the sake of simplicity we will confine ourselves to examples, where the thermodynamic state is completely characterized by two control parameters, namely, energy E and volume number V (generalizations to problems with additional macroscopic variables, e.g. the strength of external magnetic fields, are straightforward). More precisely, we will consider N identical point-like particles of mass m, moving in D equivalent spatial dimensions; i.e., the number of DOF reads d = DN and the volume is given by V = L D , where L is the length of the confining cube (volume interval). The deterministic dynamics of the system is assumed to be governed by a Hamiltonian of the standard form where q = (q 1 , . . . , q d ) and p = (p 1 , . . . , p d ) are generalized coordinates and momenta, respectively. The kinetic energy K and the potential energy U are given by where U pair (q) represents the pair interactions of the particles, and the box potential is defined by The primary thermodynamic potential of the MCE is the entropy S = S(E, V ), which is related to the microcanonical 'partition' function Z M (E, V ) by [45] with k denoting the Boltzmann constant. The equations of state (EOS) for the temperature T and pressure P are obtained by [25,[45][46][47][48][49] 1 That is, the temperature T of the MCE is a derived quantity, which is in contrast to the CE (see Sec. 3.1 below), where the temperature is one of the adjustable external control parameters.
During the past century, various different definitions for the microcanonical partition function Z M , or the entropy S, have been proposed and investigated. For classical systems as described by the Hamiltonian function (1a), the two most commonly used definitions for Z M (E, V ) read [25,[45][46][47][48][49] Z M = Ω, (4a) where the phase volume Ω is given by (h denotes the Planck constant) The Heaviside unit step function Θ(x), appearing in Eq. (5), is defined by Θ(x) = 0 for x < 0 and Θ(x) = 1 for x ≥ 0. The additional parameter ǫ 0 in Eq. (4b) is a small energy constant that quantifies the thickness of a thin energy shell around the phase space surface defined by H(q, p, V ) = E and is formally required to make Z M dimensionless.
It is well-known that the definitions (4a) and (4b) may yield (almost) identical results [39,45,47,48] in the thermodynamic limit, i.e., if N is large. However, for small systems they lead to essentially different TDFs. To briefly illustrate this, let us consider an ideal gas with N non-interacting particles, moving in the D-dimensional volume V . In this case, the phase volume is given by [45] Ω where Γ(x) is Euler's Gamma-function. Definition (4a) then yields the EOS whereas one obtains from definition (4b) For systems with d ≫ 1 the difference in the energy equations is negligible, but for small systems it becomes relevant. In particular, for d = 1 Eq. (8) yields a negative temperature at positive energy. Obviously, a similarly unreasonable result is obtained for d = 2. This indicates that the partition function (4b) is inappropriate for systems with low-dimensional phase space.
More generally speaking, only Eq. (4a) reproduces correctly the well-known laws of thermodynamics and also yields the correct equipartition theorem for an arbitrary number d of DOF, whereas the Eq. (4b) leads to inconsistencies if d is small. This important aspect was first realized by Hertz [50,51], and, later on, also emphasized by Becker [45], Berdichevsky and v. Alberti [52,53] and Adib [54]. In particular, denoting the ensemble average with respect to the microcanonical probability density function f (q, p) ∝ δ(E − H(q, p)) by · MCE , it can be shown [45,46] that the (equipartition) identity holds, only if one employs the Hertz entropy definition Due to these reasons, all subsequent considerations will be based on Eq. (10). 2 For Hamiltonians as in Eq. (1a) one can still perform the momentum integration in Eq. (5) using d-dimensional spherical coordinates, yielding where O d = 2π d/2 /Γ(d/2) denotes the surface of the d-dimensional unit sphere.

Macroscopic vs. microscopic phase transitions
Conventionally, macroscopic PTs are singularities (non-analyticities) of TDFs that arise in the thermodynamic limit, corresponding to an infinitely large system. The first systematic classification of macroscopic PTs goes back to Ehrenfest [3,55]. According to the Ehrenfest scheme, a PT is indicated by a non-analyticity of the Gibbs free enthalpy G (T, P, . . .), assumed to be a function of the temperature T , pressure P and other external control parameters. The order of the PT is determined by the lowest order at which any of the derivatives of G (T, P, . . .) becomes non-continuous. Since the Ehrenfest scheme has turned out to be too narrow in many cases, it is nowadays often preferred to merely distinguish discontinuous (first-order ) and continuous (second-order ) transitions.
Extending this concept to small systems, any non-analyticity of the thermodynamic potential as function of the external control parameters may be called a PT. However, to avoid ambiguities, we shall speak of microscopic PTs when discussing singular (non-analytic) points in the microcanonical TDFs of small systems. Furthermore, we will adopt the following terminology to classify microscopic PTs in the MCE: If the primary thermodynamic potential, the Hertz entropy S, is discontinuous, then we will call the PT discontinuous; if S is non-analytic but continuous, the microscopic PT is called continuous.
Let us next discuss how the formal (Ehrenfest-type) order of a microscopic PT depends on the number of DOF. For systems described by Hamiltonian (1a), the phase volume (11) is related to the admissible configuration space volume If, at a given critical energy E c (V ), ω(E, V ) has continuous derivatives up to order j, but a discontinuous (j + 1)st derivative, then Ω(E, V ) and hence S(E, V ) have continuous derivatives up to order (j + d/2), but a discontinuity in the (j + d/2 + 1)st derivative. Consequently, the formal order of the PT equals (j+d/2+1), i.e., the order increases with an increasing number of DOF.
For the examples discussed below, the differentiability class of the admissible configuration space volume ω(E, V ) does not change with particle number (j = 0 for Lennard-Jones chains and j = −1 for the Takahashi gas), and we indeed observe such an increasing order with increasing particle number (cf. results of Sec. 2.3.2). 4

Singular microscopic phase transitions in Lennard-Jones chains
To demonstrate the appearance of non-analytic microscopic PTs in the MCE, we consider a 1D Lennard-Jones (LJ) chain, moving freely in a 1D box volume [−L/2, L/2]. In this case, the pair potential in Hamiltonian (1a) reads: where a, σ > 0 are positive parameters. To simplify subsequent formulae, we will measure energy and length in units of the parameters a and r 0 = 2 1/6 σ, where r 0 is the position of the minimum of U LJ (r). With respect to these units the LJ potential (14) is given by For small volumes L ≤ 1 the LJ-force is always repulsive, whereas in the more interesting case of sufficiently large volumes, L > 1, the LJ-force may also become attracting.

Diatomic Lennard-Jones molecule
We start by discussing the simplest non-trivial example N = 2 and D = 1, where we can calculate the microcanonical TDFs exactly. 5 In this case, the energy E can take values E 0 (L) ≤ E < ∞, where the groundstate energy is given by A straightforward calculation of the phase volume, based on relative and center-of-mass coordinates, yields where the boundary values are given by using the convenient abbreviations X ≡ 1 + In the case L ≤ 1 we have r max = L for all energies E ≥ E 0 , and, hence, the phase volume Ω(E) is a smooth function for all permitted energy values. For L > 1, however, the boundary value r max changes its energy dependence at E = E c (L) in a non-analytic manner, and hence the phase volume Ω(E) is not , L ≥ 1 separates a gas-like phase (dissociated state) from the molecular phase (bound state) in the (L, E)-parameter plane. This is illustrated in Fig. 1 and will become particularly evident from the expansions presented in the next two paragraphs. It is worthwhile to stress again that the critical curve E c (L) arises naturally due to the sudden change in the energy dependence of the phase volume, occurring when the energy E passes the critical curve. The microcanonical caloric curve T (E) is continuous but not differentiable along the critical transition curve E c (L), which is located in the region of an S-bend or van der Waals-type loop, respectively. Formally, this corresponds to a fourth-order transition.
Super-critical energy values (dissociated phase). Using the result (17) with r 2 = L, corresponding to supercritical energy values E > E c (L) -or region '(1)' in Fig. 1, respectively -we can derive from Eqs. (3) the microcanonical EOS, yielding where Z ≡ 55L 2 E + 11L −4 − L −10 . Taking the high-energy limit at constant volume V = L one finds corresponding to the laws for the ideal 1D two-particle gas. Hence, the parameter region E > E c (L) can be identified as two-particle gas state or dissociated phase, respectively.
Sub-critical energy values (bound phase). Because of Eq. (16), the opposite case E < E c (L) -corresponding to region '(3)' in Fig. 1 (a) -can only be realized, if L > 1 holds; i.e., if the box volume is larger than the distance corresponding to the potential minimum r 0 . Again using Eqs. (17), but this time with r 2 = Y −1/6 , we obtain where we have made use of the abbreviations Expanding EOS (20b) near the groundstate energy E 0 = −1 yields Equation (21a) indicates that, at low energy, each momentum variable as well as an approximately harmonic excitation of the relative coordinate carries on average the energy amount kT /2. Obviously, this is in agreement with the equipartition theorem for harmonic DOF. Equation (21b) corresponds to the pressure law for an ideal one-particle gas in the reduced 1D volume V eff = L−1, reflecting the fact that, at sufficiently low energy values, the two particles form a bound molecule with distance r ≈ 1 between each other.
In Fig. 2 (b) the caloric curves and the pressure law are shown for different fixed values of L and E, respectively. When the volume is large enough, L ≥ 1, the caloric curves exhibit a characteristic convex region (S-bend) and, in particular, also a non-differentiable point (see solid and dotted curves). These kinks occur when the energy passes through the critical value E c (L).

Lennard-Jones chains with N > 2 particles
Analogous microscopic PTs do also occur in LJ molecules with larger particle numbers. For N > 2 it is very difficult or, perhaps, even impossible to express the phase volume (11) in terms of closed functions. Usually, one can perform only one or two of the N integrations analytically, and the remaining integrals have to be calculated numerically, using e.g. Monte-Carlo methods. We employed the Divonne algorithm of the Cuba library [59] to calculate the phase volume for LJ molecules with 3 and 4 particles. We used at least 1 million sample points and partially increased the sample size up to 100 million points for testing. We also cross-checked the results with other deterministic and probabilistic integration algorithms of the Cuba library and found excellent agreement. Figure 3 shows the numerically calculated microcanonical caloric curve T (E) of the three-particle LJ chain (N = 3) and its first derivative for different values of L. For L = 1.9, the caloric curve appears smooth and almost linear. For L = 6 and = 40, T (E) is still continuous, but shows two S-bend regions around energies E c1 ≈ −1 and E c2 ≈ 0. At these energies, the first derivative of T (E) exhibits a (negative) 'lambda peak', whereas the second derivative diverges.
As in the two-particle case, the phase volume Ω(E, L) can be written as an integral with an integrand which is analytic for all values of E and L, and integration boundaries which are analytic except for certain values of E and L. A detailed analysis yields the two critical curves where r c2 (L) is given by a polynomial equation of degree eighteen, with r c2 (L) ≈ −1 for L ≫ 1. The resulting phase diagram is shown in Fig. 1 (b). One readily identifies three microscopic phases separated by the critical curves E c1 (L) and E c2 (L).
At high energies E > E c1 (L), the system is in a gas-like, fully dissociated phase. The relative positions r 21 = |q 2 − q 1 | and r 32 = |q 3 − q 2 | of the particles are only restricted by the hard-core repulsive part of the interaction potential, but apart from this constraint the particles can move independently inside the remaining volume. In the high-energy limit, one finds E ≈ 3 2 kT , corresponding to a quasi-ideal 1D three-particle gas.
For L > 2 13 7 1/6 and low energies E 0 (L) < E < E c2 (L), the system is in a bound molecule phase. The relative positions r 21 and r 32 are restricted by the interaction potential to be close to the equilibrium position.
For L > 2 13 7 1/6 and intermediate energy values E c2 (L) < E < E c1 (L), the system is in a partially dissociated phase. One of the relative positions r 21 and r 32 is restricted to be close to its equilibrium value, whereas the other is only restricted by the hard-core repulsive part of the interaction potential and the box volume. Accordingly, one of the three particles may move rather independently inside the box, whereas the other two remain bound to each other.
Although more complicated in detail, the calculations for the four-particle LJ chain are in principle the same as for N = 3. We briefly list the main results: For N = 4 (and only taking nearest-neighbour interaction into account), three critical lines E c1 (L), E c2 (L) and E c3 (L) divide the (L, E)-plane into four different phases. At high energies or at low volumes L < L 0 , L 0 = 3, the system is in a gas-like phase. At low energies and large volume, the system is in a molecule phase, where the particles can only move as a whole molecule inside the volume. Between the gas phase and the molecule phase, there are now two intermediate, or partly dissociated, phases, where the LJ molecule is broken up into two or three parts consisting of one or two particles. For large volumes, the caloric curve T (E) shows three PTs with a continuous first, but discontinuous second derivative. This confirms that the formal order of the microscopic PTs increases with particle number as discussed in Sec. 2.2.

Takahashi gas
As another example, let us consider the piecewise linear pair potential where a and r 0 are positive parameters. In the literature this model is known as the Takahashi gas [44]. The potential u from Eq. (22) is qualitatively similar to the LJ potential, but, unlike U LJ , it allows to calculate the exact TDFs for both MCE and CE in the case N = 2. The Takahashi gas is, therefore, particularly well suited for studying the differences between the two ensembles.
To keep subsequent calculations as simple as possible, we will measure energy and length in units of a and r 0 from now on (r 0 now defines the range of the potential). With respect to these units, the potential in Eq. (22) simplifies to and, in contrast to Eq. (16), the groundstate energy is now given by E 0 (L) = 0. A straightforward calculation of the phase volume yields: where the critical energy curve is given by E c (L) ≡ u(L). Note that, compared with LJ potential from Sec. 2.3.1, we must now additionally distinguish the cases L ≤ 1 and L > 1, but this is only because the piecewise linear potential is not differentiable at r = 1. However, analogous to the case of the LJ potential, the critical curve E c (L) = u(L) separates a gas-like phase (dissociated state) from the molecule phase (bound state) in the (L, E)-parameter plane. This becomes evident from the microcanonical EOS: Taking the high-energy limits of Eqs. (25) at constant volume V = L one finds corresponding to the laws for the ideal 1D two-particle gas. Hence, the parameter region E > E c (L) can be referred to as gas-like or dissociated phase, respectively.
In the opposite case, E ≤ E c (L), a low-energy expansion of the EOS near the groundstate energy E 0 = −1 yields Neglecting terms of higher order, Eq. (27) corresponds to the pressure law for an ideal one-particle gas. This reflects that at sufficiently low energy the two particles form a bound LJ-type molecule. Compared with the results of the preceding section, a slight difference is given by the fact that in the LJ case the particles have a non-vanishing distance in the groundstate, see Eq. (21b).
In Fig. 4 (a) and (b) the caloric curves and the pressure law are shown for different fixed values of L and E, respectively. In particular, in Fig. 4 (a) one The local minimum is observable only if E/a 1. In the limit L → ∞ the particles behave as an ideal two-particle (one-particle) gas, if they are in the dissociated phase E > a (bound phase E < a).
can see that the caloric curves exhibit a singularity, when the energy passes through the critical value E c (L). More exactly, for L ≤ 1 the caloric curves are continuous but not differentiable at E = E c (L), corresponding to a thirdorder PT. By contrast, in the complementary case L > 1, the caloric curves become discontinuous at E c (L), corresponding to a second-order PT (which is a consequence of the additional singularity at r = 1 and the vanishing gradient of the piecewise linear potential at r > 1).
In the next section, we are going to study the relationship between this singular microcanonical PT and the critical behavior of the corresponding systems in the CE.

Smooth phase transitions in the canonical ensemble
Although referring to different physical conditions, MCE and CE may yield (almost) identical TDFs for well-behaved systems with a large number of DOF d → ∞. However, this 'equivalence' between the different statistical ensembles does usually not hold for systems with a small number of DOF [25,43,45,60,61]. Hence, one has to describe the thermodynamics of small systems by the statistical ensemble which actually corresponds to the given physical conditions.

The canonical ensemble
Considering the CE is appropriate if the system under investigation is coupled to an infinite heat bath [45,[62][63][64]. The surrounding heat bath (thermostat) keeps the temperature of the particles constant, but causes energy fluctuations δE > 0 around the energy mean valueĒ. Consequently, singularities in the thermodynamic functions may only exist in the thermodynamic limit with N, E → ∞, such that e = E/N and n = N/V remain constant and δe → 0; i.e., for a finite system with N < ∞ the heat bath smoothens the singularities due to non-vanishing fluctuations δe. Nevertheless, it is possible to define and to classify 'smooth' PTs by virtue of the DOZ scheme [34], discussed below.
Given a Hamiltonian of the form (1a), the canonical partition function is defined by where β ≡ (kT ) −1 . The external control variables are now (T, V ) or (β, V ), respectively. When Z C (β, V ) is known, (mean) energy and pressure of the CE are obtained by differentiation where F ≡ −kT ln Z C is the free energy. For convenience, we are going to drop the over-bars and simply write E and P in the next section (over-bars will be reinstated occasionally, e.g., when comparing microcanonical and canonical quantities).

Classification of canonical phase transitions in the DOZ scheme
According to Yang and Lee [8,9], the distribution of zeros (DOZ) of the grandcanonical partition function determines the PTs in the grandcanonical ensemble. Later on, a similar approach has been employed by Fisher [10] and Grossmann and Rosenbauer [11] to identify and classify PTs in the canonical ensemble. They considered the distribution of complex zerosβ k of the canonical partition function Z C (β), taken as complex function of the complex inverse temperatureβ = β + iτ (β > 0) . For finite systems, one can show that there are no zeros on the positive real axis, i.e., ℑ(β k ) = 0 ∀k. In the thermodynamic limit, however, certain points β c on the real β-axis may become limiting values of the DOZ. By studying how the zeros condense near the β-axis, one can characterize the PT.
Although for small systems there are, strictly speaking, no non-analytic PTs in the CE, it may nevertheless be helpful to distinguish different thermodynamic phases [31,32]. The DOZ classification scheme of Bohrmann et al. [34] is based on the idea that the complex zerosβ k closest to the real β-axis can be employed to estimate the DOZ behavior in the thermodynamic limit. The extrapolated limiting values β c are used to define the 'smooth' canonical PT of the finite system. To be more specific, one first numbers the complex zerosβ k , k = 1, 2, . . . of the canonical partition function Z C (β) according to their distance to the real β-axis, and then calculates the quantities These quantities characterize the DOZ near the real β-axis. According to the scheme of Borrmann et al. [34], a first-order transition at temperature T c = (kβ c ) −1 appears if α = 0 and γ = 0, whereas values 1 > α > 0 correspond to second-order transitions, and α > 1 to even higher-order transitions in the original Ehrenfest classification. In order to actually observe a singular PT in the Ehrenfest sense, it is required that τ 1 → 0 in the thermodynamic limit.
A similar classification scheme for smooth canonical PTs in small systems, based on the average cumulative density of zeros of the canonical partition function, has been proposed by Janke, Kenna et al. [38,65]. Alves et al. [36] compared both approaches for a variety of model systems. In the following, however, we will refer to the DOZ-scheme by Borrmann et al. [34] in order to classify PTs in the CE of a small system. In particular, we aim to compare the DOZ-classification with the MCE results obtained for the Takahashi gas.

Takahashi gas
We first calculate the canonical TDFs for the Takahashi model from Sec. 2.4, corresponding to two point-like particles, N = 2, confined in a 1D volume V = L and interacting via the (rescaled) piecewise linear pair potential from Eq. (23). For L ≤ 1, we find explicitly and in the opposite case, that is for L > 1, Accordingly, one obtains the following canonical EOS In the high-temperature limit, corresponding to β → 0 at constant volume V = L, one finds i.e., the system behaves like an ideal 1D two-particle gas in this limit.  Figure 5 shows several curves, corresponding to the thermodynamic laws (32). One may notice a strong local increase in the solid caloric curve of Fig. 5 (a). This behavior is associated with a smooth canonical PT in the DOZ scheme of Borrmann et al. [34]. To see this, we next determine the complex zeros of the complex partition function Z C (β + iτ, L) for L > 1 from Eq. (31). Figure  6 (a) shows the corresponding numerical results found with Mathematica [66] for L = 20, by using a contour plot of the function |Z C (β + iτ, L)|. Since the zeros of the partition function are complex conjugate [34], only the zeros in the upper complex half-plane are shown.
To also obtain an analytic estimate for the DOZ in the more interesting case L ≫ 1, we expand the partition function (31b) near L → ∞, and find Thus, neglecting terms of O(L −2 ), the zeros of Z C (β, L) are given bỹ  Using Eq. (35) we can calculate the characteristic quantities α, β c , γ from Eqs. (30), required for the DOZ classification. Table 1 shows a summary of the results for different values of the volume parameter L. The critical temperature is given by kT c = 1/β c and E c is obtained by inserting β c into the energy equation (32a). According to the DOZ scheme, for 1 ≪ L < ∞ we find 0 < α < 1, corresponding to a canonical second-order PT. However, for L → ∞ we observe that α → 0, γ → 0 indicating that in this limit the transition converges to first order. Moreover, as evident from the last two columns in Tab. 1, for increasing volume L the critical temperature vanishes, T c → 0, while the corresponding energy values E(T c ) approach the value 0.5. However, by first taking the limit L → ∞ in Eq. (32a), we find that the corresponding asymptotic caloric curve is given by Inserting T c into the rhs. of this equation and letting T c → 0, we obtain E c = 1, which is in agreement with microcanonical result E c (L = ∞) = 1, and corresponds to the dissociation energy. In particular, the latter result means that, in the case of a very large volume, very small energy fluctuations (requiring T > 0) suffice to permanently break up the molecule. On the other hand, the system becomes deterministic at T = 0, and, correspondingly, the mean energy of the CE is then always given by the groundstate value E 0 = 0, representing the bound state [formally this corresponds first taking the limit T → 0 in Eq. (32a)]. The apparent convergence to 0.5 in the last column of Table 1 just reflects the fact that for Eq. (32a) the two limits T → 0 and L → ∞ do not commute.  Table 1 Canonical PT parameters according to the DOZ scheme [34] for the 1D diatomic molecule with piecewise linear interaction potential from Eq. (22). For L → ∞ the numerical results suggest that α → 0, γ → 0 and kT c /a → 0.

Summary and discussion
In this paper we have studied phase transition-like phenomena in small systems, characterized by a finite number of degrees of freedom (DOF). The main objective was to clarify similarities and differences that arise when considering either the microcanonical ensemble (MCE), corresponding to a thermally isolated system, or the canonical ensemble (CE), corresponding to a system in contact with an infinite heat bath.

Microscopic phase transitions in the microcanonical ensemble
In Sec. 2 it was shown that the microcanonical thermodynamic functions (TDFs) of a small system can exhibit complex oscillatory behavior and singularities (non-analyticities). Analogous to macroscopic phase transitions (PTs), such non-analytic points can be interpreted as microscopic PTs in the MCE. 6 To illustrate the physical meaning of such microscopic PTs, we calculated the microcanonical TDFs for two slightly different 1D toy models (Lennard-Jones and Takahashi gas). For both systems one can identify critical energy curves E c (L) along which the primary microcanonical thermodynamic potentialthe Hertz entropy -is non-analytic, reflected by kinks or discontinuities in the TDFs. In these models the microscopic PTs separate energetically different dissociation states. Their number and formal order increases with increasing particle number N [57]. In general, our results indicate that, typically, microscopic PTs in the MCE are accompanied by strong qualitative changes of the thermodynamic observables, as e.g. rapid drop-offs or oscillations of temperature and heat capacities, see Fig. 3 (a). These effects should be observable in suitably designed evaporation experiments, realizing the conditions of the MCE (similar to those of Schmidt et al. [30], but without heat bath).
For the model systems considered in this paper, a non-analyticity in the TDFs is characterized by a critical energy curve E c (L). Mathematically, such critical curves E c (L) arise due to the integration over the Θ-function in the definition of the microcanonical phase volume Ω. The Θ-function in Eq. (11) effectively constrains the range of the integration to the subset in the configuration space; i.e., A is the energetically permitted configuration space region. Hence, a singularity in the microcanonical TDFs may arise whenever A changes its geometry or 'shape' in an irregular manner during a small variation of the control parameters E and V . One possible origin for this may be a change in the topology of A (as discussed by Pettini et al. [18][19][20]). For the models considered here, however, the topology of A remains unaffected and another general mechanism is at work. To illustrate this in more detail, we recall the example from Sec. 2.3.1, corresponding to the two-particle Lennard-Jones gas. In this case, the set A can be expressed as where U LJ is given by Eq. (14). The first constraint for (q 1 , q 2 ) reflects the box potential, whereas the second constraint arises from the interaction potential. For E < E c (L), A consists of the two diagonal 'strips' where r − (E) > 0 and r + (E) > 0 denote the classical turning points of the LJ potential. The 'strips' A ± are bounded by the box potential on two sides, whereas the other two boundaries are determined by the LJ interaction potential. For E > E c (L), however, the regions A ± become triangles, bounded by the box potential on two sides and by the interaction potential on the remaining side: A ± = (q 1 , q 2 ) ∈ [−L/2, L/2] 2 ∧ r − (E) ≤ ± (q 1 − q 2 ) .
Evidently, A changes its geometry dramatically, when the energy passes through the critical energy E c (L), thereby giving rise to the microscopic PT. For a sufficiently large volume L ≫ r 0 , where r 0 is the range of the pair interaction, this non-analytic transformation of A at the dissociation energy is accompanied by a change of A's effective dimensionality. Here, we mean by 'effective dimensionality' the number of orthogonal configuration space directions in which the set A extends comparably to the system size L. For energy values E ≪ E c (L), the 'strips' A ± defined by Eq. (38) are very narrow compared to the system size L, and, hence, A = A + ∪ A − is effectively one-dimensional. When the energy approaches E c (L) from below, the (average) width of the two subsets A ± grow very rapidly, and for E = E c (L), they become triangles whose size is of the order of L 2 ; i.e., the set A is effectively two-dimensional for E ≥ E c (L). In particular, it is this very rapid growth of A -or Ω, respectively -which leads to a negative slope of the caloric curve T (E, V ) in the vicinity of the dissociation energy E c (L). 7

Comparison with the canonical ensemble
In contrast to the microcanonical TDFs, singular PTs cannot occur in the CE of a finite system [8,9,[11][12][13]41,42]. This is a consequence of the two completely different physical conditions, underlying MCE and CE, respectively [25,45]. In spite of lacking sharp transitions, it is useful to distinguish different thermodynamic 'phases', when considering CEs of finite systems [31,32]. Following the proposal of Borrmann et al. [34], we determined the distribution of complex zeros (DOZ) for the canonical partition function of the two-particle Takahashi gas (Sec. 3) and found a 'smooth' canonical second-order PT (according to the DOZ classification scheme), provided the volume L is large compared to the range of the potential, but still finite; for L → ∞ the canonical PT changes to first order in the DOZ scheme.
In principle, however, there exist important differences between the smooth PTs in the CE and the singular microscopic PTs in the MCE: Smooth canonical PTs can be viewed as direct finite size counterparts of macroscopic PTs. In particular, for N particle systems as discussed in this paper the DOZ-scheme usually yields only one transition point. Furthermore, the DOZ-order of the transition point is typically close to one or two. By contrast, when considering the corresponding isolated system (i.e. the MCE), both the number and the formal order of the singular microscopic PTs increase approximately proportional to the particle number (since microscopic PTs signal each dissociation step separately). Such differences notwithstanding, canonical and microcanonical partition functions are linked by a Laplace transformation, which suggests that (not only for small systems) there might exist a direct connection between the canonical DOZ and the appearance of microscopic non-analyticities in microcanonical entropy (see also [67] for a discussion of this hypothesis).
To conclude with, if one wishes to describe small systems by means of a thermodynamic approach, then the non-equivalence of the different statistical ensembles [43] renders necessary to specify in advance, whether or not a heat bath (thermostat) is coupled to the small system under consideration. The discussion in the present paper has focussed on two extreme limit cases, corresponding to a vanishing heat bath (MCE) and an infinite heat bath (CE).
In the future, it would also be interesting to study phase transition-like phenomena in small systems coupled to a finite heat bath [68][69][70], e.g., on the basis of Tsallis' generalized statistics [71][72][73].