Phase Transitions In Small Systems Research Articles

Principles of the equilibrium theory of small multicomponent systems in three aggregate states

Principles of the molecular statistical theory of small multicomponent drops/microcrystals in a three-dimensional bulk and in two-dimensional adsorption systems are developed. Equations of the theory are derived using the cluster approach. The theory describes discrete distributions of molecules in space (on a size scale comparable to the molecular size) and continuous molecular distributions (at short distances inside cells) upon their translational and vibrational motions. The theory provides a unified description of the equilibrium molecular distributions in three aggregate states and at their interfaces. Pair intermolecular interaction potentials (such as the Mie potential) in several coordination spheres that determine lattice structure compressibility are taken into account. For simplicity, it is considered that the sizes of mixture components are virtually the same. Structural cell distribution functions for the transition region of curved interfaces are derived. Expressions for the pressure tensor components inside small bodies are obtained, allowing us to calculate the thermodynamic characteristics of a vapor–liquid interface, including surface tension. Questions regarding the consistency between the theory of phase transitions in small systems and the traditional theory of associate (cluster) formation and the transition to systems limited in the total volume value are discussed.

Density of Yang-Lee zeros for the Ising ferromagnet

The densities of Yang-Lee zeros for the Ising ferromagnet on the L x L square lattice are evaluated from the exact grand partition functions (L=3 approximately 16). The properties of the density of Yang-Lee zeros are discussed as a function of temperature T and system size L. The three different classes of phase transitions for the Ising ferromagnet--first-order phase transition, second-order phase transition, and Yang-Lee edge singularity--are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite-size systems. The divergence of the density of zeros at Yang-Lee edge in high temperatures (Yang-Lee edge singularity), which has been detected only by the series expansion until now for the square-lattice Ising ferromagnet, is obtained from the finite-size data. The identification of the orders of phase transitions in small systems is also discussed using the density of Yang-Lee zeros.

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.

Phase transitions in systems small enough to be clusters

We analyze peculiarities of phase transitions in small systems. We focus on the formation of a tiny liquid drop in a small N, V, T system, and develop an “extended modified liquid drop” model, which incorporates the effect of the fluctuations relevant for small systems. We compare the predictions for the p−v isotherms and the density profiles with the results of Monte-Carlo simulations of a Lennard-Jones system, and Density Functional Theory.

Nuclear Fragmentation: a Paradigm for the Study of Phase Transitions in Small Systems

Nuclei are finite representatives of infinite nuclear matter which is expected to undergo a phase transition similar to the macroscopic liquid-gas transition. We rely on a simple classical spin model in order to study the characterization of phase transitions whose existence can be observed in small systems. It is shown that finite systems may undergo transitions whose order is not, in specific cases, the effective order which is observed in the corresponding infinite (large) system. We show that scaling properties of order parameter fluctuation distributions related to the fragment content of the system may allow to signal the existence of a thermodynamic transition.

Numerical comparison of two approaches for the study of phase transitions in small systems.

We compare two recently proposed methods for the characterization of phase transitions in small systems. The validity and usefulness of these approaches are studied for the cases of the q=4 and q=5 Potts model, i.e., systems where a thermodynamic limit and exact results exist. Guided by this analysis we then discuss the helix-coil transition in polyalanine, an example of structural transitions in biological molecules.

Structural transitions in biomolecules - a numerical comparison of two approaches for the study of phase transitions in small systems

We compare two recently proposed methods for the characterization of phase transitions in small systems. The usefulness of these techniques is evaluated for the case of structural transition in alanine-based peptides.

Origins of phase transitions in small systems.

The identification and classification of phases in small systems, e.g., nuclei, social and financial networks, clusters, and biological systems, where the traditional definitions of phase transitions are not applicable, is important to obtain a deeper understanding of the phenomena observed in such systems. Within a simple statistical model, we investigate the validity and applicability of different classification schemes for phase transtions in small systems. We show that the whole complex temperature plane contains necessary information in order to give a distinct classification.

Classification of phase transitions in small systems

We present a classification scheme for phase transitions in finite systems like atomic and molecular clusters based on the Lee-Yang zeros in the complex temperature plane. In the limit of infinite particle numbers the scheme reduces to the Ehrenfest definition of phase transitions and gives the right critical indices. We apply this classification scheme to Bose-Einstein condensates in a harmonic trap as an example of a higher order phase transition in a finite system and to small Ar clusters.

Fragmentation phase transition in atomic clusters IV

Within the micro-canonical ensemble it is well possible to identify phase-transitions in small systems. The consequences for the understanding of phase transitions in general are discussed by studying three realistic examples. We present micro-canonical calculations of the fragmentation phase transition in Na-, K-, and Fe- clusters of N = 200 to 3000 atoms at a constant pressure of 1 atm. The transition is clearly of first order with a backbending micro-canonical caloric curve T P (E, V (E,P)) = {∂S(E, V (E,P))/∂E P}−1. From the Maxwell construction of βP (E/N,P) = 1/T P one can simultaneously determine the transition temperature T tr, the specific latent heat q lat, and the specific entropy-loss Δs surf linked to the creation of intra-phase surfaces. T trΔs surf*N/(4πr ws 2 N eff 2/3 ) = γ gives the surface tension γ. Here 4πr ws 2 N eff 2/3 = ΣN i*4πr ws 2 m i 2/3 is the combined surface area of all fragments with a mass m i ≥ 2 and multiplicity N i. All these characteristic parameters are for ∼1000 atoms similar to their experimentally known bulk values. This finding shows clearly that within micro-canonical thermodynamics phase transitions can unambiguously be determined without invoking the thermodynamic limit. However, one has carefully to distinguish observables which are defined for each phase-space point, like the values of the conserved quantities, from thermodynamic quantities like temperature, pressure, chemical potential, and also the concept of pure phases, which refer to the volume of the energy shell of the N-body phase-space and thus do not refer to a single phase-space point. At the same time we present here the first successful microscopic calculation of the surface tension in liquid sodium, potassium, and iron at a constant pressure of 1 atm.