Canonical Ensemble

The curvature dependence of the surface tension is related to the excess equimolar radius of liquid drops, i.e., the deviation of the equimolar radius from that defined with the macroscopic capillarity approximation. Based on the Tolman [J. Chem. Phys. 17, 333 (1949)] approach and its interpretation by Nijmeijer et al. [J. Chem. Phys. 96, 565 (1991)], the surface tension of spherical interfaces is analysed in terms of the pressure difference due to curvature. In the present study, the excess equimolar radius, which can be obtained directly from the density profile, is used instead of the Tolman length. Liquid drops of the truncated-shifted Lennard-Jones fluid are investigated by molecular dynamics simulation in the canonical ensemble, with equimolar radii ranging from 4 to 33 times the Lennard-Jones size parameter sigma. In these simulations, the magnitudes of the excess equimolar radius and the Tolman length are shown to be smaller than sigma/2. Other methodical approaches, from which mutually contradicting findings have been reported, are critically discussed, outlining possible sources of inaccuracy.

The cosmological many-body problem is effectively an infinite system of gravitationally interacting masses in an expanding universe. Despite the interactions' long-range nature, an analytical theory of statistical mechanics describes the spatial and velocity distribution functions which arise in the quasi-equilibrium conditions that apply to many cosmologies. Consequences of this theory agree well with the observed distribution of galaxies. Further consequences such as thermodynamics provide insights into the physical properties of this system, including its robustness to mergers, and its transition from a grand canonical ensemble to a collection of microcanonical ensembles with negative specific heat.

The relation between chaotic dynamics of nonlinear Hamiltonian systems and equilibrium statistical mechanics in its canonical ensemble formulation has been investigated for two different nonlinear Hamiltonian systems. We have compared time averages obtained by means of numerical simulations of molecular dynamics type with analytically computed ensemble averages. The numerical simulation of the dynamic counterpart of the canonical ensemble is obtained by considering the behavior of a small part of a given system, described by a microcanonical ensemble, in order to have fluctuations of the energy of the subsystem. The results for the Fermi-Pasta-Ulam model (i.e., a one-dimensional anharmonic solid) show a substantial agreement between time and ensemble averages independent of the degree of stochasticity of the dynamics. On the other hand, a very different behavior is observed for a chain of weakly coupled rotators, where linear exchange effects are absent. In the high-temperature limit (weak coupling) we have a strong disagreement between time and ensemble averages for the specific heat even if the dynamics is chaotic. This behavior is related to the presence of spatially localized chaos, which prevents the complete filling of the accessible phase space of the system. Localized chaos is detected by the distribution of all the characteristic Liapunov exponents.

In this paper, we consider the volume enclosed by the microcanonical ensemble in phase space as a statistical ensemble. This can be interpreted as an intermediate image between the microcanonical and the canonical pictures. By maintaining the ergodic hypothesis over this ensemble, that is, the equiprobability of all its accessible states, the equivalence of this ensemble in the thermodynamic limit

We study the dynamical and statistical behavior of the Hamiltonian mean field (HMF) model in order to investigate the relation between microscopic chaos and phase transitions. HMF is a simple toy model of N fully coupled rotators which shows a second-order phase transition. The solution in the canonical ensemble is briefly recalled and its predictions are tested numerically at finite N. The Vlasov stationary solution is shown to give the same consistency equation of the canonical solution and its predictions for rotator angle and momenta distribution functions agree very well with numerical simulations, A link is established between the behavior of the maximal Lyapunov exponent and that of thermodynamical fluctuations, expressed by kinetic energy fluctuations or specific heat. The extensivity of chaos in the N → ∞ limit is tested through the scaling properties of Lyapunov spectra and of the Kolmogorov-Sinai entropy. Chaotic dynamics provides the mixing property in phase space necessary for obtaining equilibration; however, the relaxation time to equilibrium grows with N, at least near the critical point. Our results constitute an interesting bridge between Hamiltonian chaos in many degrees of freedom systems and equilibrium thermodynamics.

Fluctuations of the instantaneous local Lagrangian strain $\epsilon_{ij}(\bf{r},t)$, measured with respect to a static ``reference'' lattice, are used to obtain accurate estimates of the elastic constants of model solids from atomistic computer simulations. The measured strains are systematically coarse- grained by averaging them within subsystems (of size $L_b$) of a system (of total size $L$) in the canonical ensemble. Using a simple finite size scaling theory we predict the behaviour of the fluctuations $<\epsilon_{ij}\epsilon_{kl}>$ as a function of $L_b/L$ and extract elastic constants of the system {\em in the thermodynamic limit} at nonzero temperature. Our method is simple to implement, efficient and general enough to be able to handle a wide class of model systems including those with singular potentials without any essential modification. We illustrate the technique by computing isothermal elastic constants of the ``soft'' and the hard disk triangular solids in two dimensions from molecular dynamics and Monte Carlo simulations. We compare our results with those from earlier simulations and density functional theory.

We summarize previous works on the exact ground state and the elementary excitations of the exactly solvable BCS model in the canonical ensemble. The BCS model is solved by Richardson equations, and, in the large coupling limit, by Gaudin equations. The relationship between this two kinds of solutions are used to classiffy the excitations.

QCD at non-zero baryon density is expected to have a critical point where the zero-density cross-over turns into a first order phase transition. To identify this point we scan the density-temperature space using a canonical ensemble method. For a given temperature, we plot the chemical potential as a function of density looking for an &quot;S-shape&quot; as a signal for a first order transition. We carried out simulations using Wilson fermions with $m_\pi \approx 1{GeV}$ on $6^3\times 4$ lattices. As a benchmark, we ran four flavors simulations where we observe a clear signal. In the two flavors case we do not see any signal for temperatures as low as $0.83 T_c$. Preliminary results for the three flavor case are also presented.

We study the three-dimensional Ising model at the critical point in the fixed-magnetization ensemble, by means of the recently developed geometric cluster Monte Carlo algorithm. We define a magnetic-field-like quantity in terms of microscopic spin-up and spin-down probabilities in a given configuration of neighbors. In the thermodynamic limit, the relation between this field and the magnetization reduces to the canonical relation M(h). However, for finite systems, the relation is different. We establish a close connection between this relation and the probability distribution of the magnetization of a finite-size system in the canonical ensemble.

The dynamics and thermodynamics of particles/spins interacting via long-range forces display several unusual features compared with systems with short-range interactions. The Hamiltonian mean field (HMF) model, a Hamiltonian system of N classical inertial spins with infinite-range interactions represents a paradigmatic example of this class of systems. The equilibrium properties of the model can be derived analytically in the canonical ensemble: in particular, the model shows a second-order phase transition from a ferromagnetic to a paramagnetic phase. Strong anomalies are observed in the process of relaxation towards equilibrium for a particular class of out-of-equilibrium initial conditions. In fact, the numerical simulations show the presence of quasi-stationary states (QSS’s), i.e. metastable states that become stable if the thermodynamic limit is taken before the infinite time limit. The QSS’s differ strongly from Boltzmann-Gibbs equilibrium states: they exhibit negative specific heats, vanishing Lyapunov exponents and weak mixing, non-Gaussian velocity distributions and anomalous diffusion, slowly decaying correlations, and aging. Such a scenario provides strong hints for the possible application of Tsallis generalized thermostatistics. The QSS’s have recently been interpreted as a spin-glass phase of the model. This link indicates another promising line of research, which does not preclude to the previous one.

By means of the Lanczos method we analyze superconducting correlations in ultrasmall grains at fixed particle number. We compute the ground state properties and the excitation gap of the pairing Hamiltonian as a function of the level spacing $\delta$. Both quantities turn out to be parity dependent and universal functions of the ratio $\delta/\Delta$ ($\Delta$ is the BCS gap). We then characterize superconductivity in the canonical ensemble from the scaling behavior of correlation functions in energy space.

We calculate the scale of the critical fluctuations in the quantum parametric oscillator at threshold. This al- lows us to asymptotically calculate the size of both the depletion of the pump mode and the amount of squeezing produced in the signal mode for large system size. In this driven, open quantum system, there is no known solu- tion to the equilibrium density matrix. For this reason these results cannot be obtained using standard canonical ensemble methods. The calculations are compared to time dependent quantum stochastic simulations of the fully nonlinear coupled mode equations. We find that the critical squeezing variance in the output spectrum will scale as N 1 2, giving the novel feature that these fluctuations are reduced rather than enhanced near the critical point.

We derive Green-Kubo relations for the viscosities of a nematic liquid crystal. The derivation is based on the application of a Gaussian constraint algorithm that makes the director angular velocity of a liquid crystal a constant of motion. Setting this velocity equal to zero means that a director-based coordinate system becomes an inertial frame and that the constraint torques do