Plaquette expansion proof and interpretation

The plaquette expansion, a general non-perturbative method for calculating the properties of lattice Hamiltonian systems, is established up to the first two orders for an arbitrary system. This method employs an expansion of the Lanczos coefficients, the tridiagonal Hamiltonian matrix elements or equivalently the continued fraction coefficients of the resolvent, in a descending series in the size of the system. The coefficients of this series are formed from the low order cumulants or connected Hamiltonian moments. The lowest order approximation in the plaquette expansion corresponds to a gaussian model which is a consequence of the central limit theorem. The first nontrivial order yields a model with a spectrum on a bounded energy interval, becoming asymptotically uniform in the thermodynamic limit.     

Size-dependent thermodynamic criterion for the thermal stability of binary immiscible metallic multilayers

With the aim of understanding the thermal stability of binary immiscible metallic multilayers, we propose a generally size-dependent thermodynamic criterion for determining the interface alloying in multilayers, with respect to the size-dependent interface energy of binary metal systems. Taking the copper/tungsten bilayer as an example, we obtain the interfacial alloying phase diagram based on the proposed thermodynamic model. Our theoretical predictions are consistent with experiments, implying that the size-dependent thermodynamic criterion of the thermal stability could be expected to be applicable to many multilayers.     

The statistical mechanics of turbo codes

The “turbo codes”, recently proposed by Berrou et al. [1] are written as a disordered spin Hamiltonian. It is shown that there exists a threshold such that for signal to noise ratios the error probability per bit vanishes in the thermodynamic limit, i.e. the limit of infinitely long sequences. The value of the threshold has been computed for two particular turbo codes. It is found that it depends on the code. These results are compared with numerical simulations. Received 14 March 2000 and Received in final form 17 July 2000     

Asymptotics of the partition function of a general Markov random field on an infinite rectangular lattice

We investigate theoretically and numerically the asymptotics of the partition function of a general Markov random field (MRF) on an infinite rectangular lattice. We first propose the general local energy function (LEF)-parameterized MRF. Then we prove that the thermodynamic limit of the free energy of the MRF can be exactly characterized by the Perron root of the fundamental transfer matrix of a particular Markov additive process (MAP). This matrix possesses a special structure and many interesting properties that enable parallel computation of the Perron root and may be beneficial for deriving an analytical form of the free energy. We also develop another transfer matrix for numerical computation of the desired Perron root. Specifically, the former is a site-to-site transfer matrix on a twisted cylindrical lattice, while the latter is the one associated with a row-to-row transition on a vertical strip. Numerical results show that our methods exhibit consistent finite-size scaling behavior even for small values of the lattice width. This study reveals that the fundamental transfer matrix is an alternative direction of research on the analysis of the partition function of general MRFs within the scope of matrix algebra.     

Phase transitions in “small” systems

Traditionally, phase transitions are defined in the thermodynamic limit only. We discuss how phase transitions of first order (with phase separation and surface tension), continuous transitions and (multi)-critical points can be seen and classified for small systems. “Small” systems are systems where the linear dimension is of the characteristic range of the interaction between the particles; i.e. also astrophysical systems are “small” in this sense. Boltzmann defines the entropy as the logarithm of the area of the surface in the mechanical N-body phase space at total energy E. The topology of S(E,N) or more precisely, of the curvature determinant allows the classification of phase transitions without taking the thermodynamic limit. Micro-canonical thermo-statistics and phase transitions will be discussed here for a system coupled by short range forces in another situation where entropy is not extensive. The first calculation of the entire entropy surface S(E,N) for the diluted Potts model (ordinary (q=3)-Potts model plus vacancies) on a square lattice is shown. The regions in {E,N} where D>0 correspond to pure phases, ordered resp. disordered, and D<0 represent transitions of first order with phase separation and “surface tension”. These regions are bordered by a line with D=0. A line of continuous transitions starts at the critical point of the ordinary (q=3)-Potts model and runs down to a branching point P_m. Along this line vanishes in the direction of the eigenvector of D with the largest eigen-value . It characterizes a maximum of the largest eigenvalue . This corresponds to a critical line where the transition is continuous and the surface tension disappears. Here the neighboring phases are indistinguishable. The region where two or more lines with D=0 cross is the region of the (multi)-critical point. The micro-canonical ensemble allows to put these phenomena entirely on the level of mechanics. Received 18 October 1999 and received in final form 17 November 1999     

Magnetocaloric effect: Overcoming the magnetic limit

We have studied anomalous peaks observed in magnetocaloric −ΔS(T) curves for systems that undergo first-order magnetostructural transitions. The origin of those peaks, which can exceed the conventional magnetic limit, R ln(2J+1), is discussed on thermodynamic bases by introducing an additional-exchange contribution (due to exchange constant variation arising from magnetostructural transition). We also applied a semiphenomenological model to include this additional-exchange contribution in Gd₅Si₂Ge₂- and MnAs-based systems, obtaining excellent results for the observed magnetocaloric effect.     

One-dimensional statistical mechanics models with application to peapods

Quasi one-dimensional systems of molecules of C₆₀ encapsulated in (10/10) nanotubes were studied by both lattice-gas and Takashi–Gursey configurational integral methods of statistical mechanics for both open and capped finite nanotubes as well as infinite nanotubes. From well-established potentials, the energy, heat capacity compressibility, equation of state and absorption isotherms were computed as a function of temperature and molecular density. The existing theories were extended to include the calculation of clustering, and the number of clusters as a function of size was computed for a variety of temperatures and densities. For both models, all molecules are frozen into a single cluster, and increasing the temperature results in a break-up into smaller clusters. The corresponding heat capacity has a broad maximum, which is lower for the T–G model than for the lattice-gas model. The equations of state have a similar form in both models and are identical at low temperatures. The absorption isotherms show that filling of the tubes can take place at all temperatures of practical interest. Peapods are nearly ideal realizations of one-dimensional systems whose thermodynamic and structural properties can be accurately obtained by statistical mechanics. Received: 15 November 2001 / Accepted: 25 October 2002 / Published online: 10 March 2003 RID="*" ID="*"Corresponding author. Fax: +1-215/573-2128, E-mail: lag@sol1.lrsm.upenn.edu RID="**" ID="**"Present address: Dept. of Physics, North Carolina State University, Raleigh, NC 27695, USA     

Possible canonical distributions for finite systems with nonadditive energy

It is shown that a small system in thermodynamic equilibrium with a finite thermostat can have a q-exponential probability distribution which closely depends on the energy nonextensivity and the particle number of the thermostat. The distribution function will reduce to the exponential one at the thermodynamic limit. However, the nonextensivity of the system should not be neglected.     

An evolution theory in finite size systems

A new model of evolution is presented for finite size systems. Conditions under which a minority species can emerge, spread and stabilize to a macroscopic size are studied. It is found that space organization is instrumental in addition to a qualitative advantage. Some peculiar topologies ensure the overcome of the initial majority species. However the probability of such local clusters is very small and depend strongly on the system size. A probabilistic phase diagram is obtained for small sizes. It reduces to a trivial situation in the thermodynamic limit, thus indicating the importance of dealing with finite systems in evolution problems. Results are discussed with respect to both Darwin and punctuated equilibria theories. Received 25 June 2000     

Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators

We analyze a minimal model of a population of identical oscillators with a nonlinear coupling—a generalization of the popular Kuramoto model. In addition to well-known for the Kuramoto model regimes of full synchrony, full asynchrony, and integrable neutral quasiperiodic states, ensembles of nonlinearly coupled oscillators demonstrate two novel nontrivial types of partially synchronized dynamics: self-organized bunch states and self-organized quasiperiodic dynamics. The analysis based on the Watanabe-Strogatz ansatz allows us to describe the self-organized bunch states in any finite ensemble as a set of equilibria, and the self-organized quasiperiodicity as a two-frequency quasiperiodic regime. An analytic solution in the thermodynamic limit of infinitely many oscillators is also discussed.     

Irreversibility, entropy and incomplete information

The open system has been proved to be a system with perfect accessibility represented as a probability space in which is defined a PA-measure. But, the PA-measure is not yet known; consequently, it is difficult to develop the statistical thermodynamics for an irreversible system. Here its integral expression is obtained in order to its use in the statistical thermodynamic analysis of the complex and irreversible systems.     

Stability and instability in a class of car following model on a closed loop

A velocity-matching car following model is modified to represent the motion of n vehicles travelling on a closed loop. Each vehicle is given a preferred velocity profile, which it attempts to achieve while also attempting to maintain a zero relative velocity between itself and the vehicle in front. The crucial distinctive of the looped model, as opposed to ‘non-looped’ models, is that the last vehicle in the stream is itself being followed by the lead (first) vehicle. The model gives rise to a system of n coupled time delay differential equations which are solved approximately (using a Taylor series expansion in time delay) and numerically using a fourth-order Runge-Kutta routine.The stability of the model is considered and an analytic form of the stable region in parameter space is found in the limit as n approaches infinity.     

Transition to superfluidity in strongly coupled fermion systems: low density limit

The rôle of interacting bound pairs in strongly coupled fermion systems is considered in connection with the transition to superfluidity. Model calculations are performed for finite-range separable interaction potentials. The results of a cluster-Hartree-Fock approximation are compared with recent approaches improving the Nozières and Schmitt-Rink theory. In the low-density strong coupling limit, a first order transition to the superfluid phase is obtained.     

Finite temperature corrections to the Casimir effect in rectangular cavities with perfectly conducting walls

The thermodynamical properties of a quantized electromagnetic field inside a box with perfectly conducting walls are studied using a regularization scheme that permits to obtain finite expressions for the thermodynamic potentials. The source of ultraviolet divergences is directly isolated in the expression for the density of modes, and the logarithmic infrared divergences are regularized imposing the uniqueness of vacuum and, consequently, the vanishing of the entropy in the limit of zero temperature. We thus obtain corrections to the Casimir energy and pressures, and to the specific heat; these results suggest effects that could be tested experimentally.     

Phase transitions from saddles of the potential energy landscape

The relation between saddle points of the potential of a classical many-particle system and the analyticity properties of its thermodynamic functions is studied. For finite systems, each saddle point is found to cause a nonanalyticity in the Boltzmann entropy, and the functional form of this nonanalytic term is derived. For large systems, the order of the nonanalytic term increases unboundedly, leading to an increasing differentiability of the entropy. Analyzing the contribution of the saddle points to the density of states in the thermodynamic limit, our results provide an explanation of how, and under which circumstances, saddle points of the potential energy landscape may (or may not) be at the origin of a phase transition in the thermodynamic limit. As an application, the puzzling observations by Risau-Gusman et al. [Phys. Rev. Lett. 95, 145702 (2005)] on topological signatures of the spherical model are elucidated.     

Quantum boundary layer: a non-uniform density distribution of an ideal gas in thermodynamic equilibrium

Density distribution of an ideal Maxwellian gas confined in a finite domain is not uniform even in thermodynamic equilibrium. Near to the boundaries, there is a layer in which the density goes to zero. Existence of this boundary layer explains the shape and size dependence of the thermodynamic quantities in nano scale.     

On the thermodynamic stability of odd-frequency superconductors

The thermodynamic stability of odd-frequency pairing states is investigated within an Eliashberg-type framework. We find the rigorous result that in the weak coupling limit a continuous transition from the normal state to a spatially homogeneous odd-in-ω superconducting state is forbidden, irrespective of details of the pairing interaction and of the spin symmetry of the gap function. For isotropic systems, it is shown that the inclusion of strong coupling corrections does not invalidate this result. We discuss a few scenarios that might escape these thermodynamic constraints and permit stable odd-frequency pairing states.