Microcanonical Partition Function Research Articles

Gauged permutation invariant matrix quantum mechanics: partition functions

The Hilbert spaces of matrix quantum mechanical systems with N × N matrix degrees of freedom X have been analysed recently in terms of SN symmetric group elements U acting as X → UXUT. Solvable models have been constructed uncovering partition algebras as hidden symmetries of these systems. The solvable models include an 11-dimensional space of matrix harmonic oscillators, the simplest of which is the standard matrix harmonic oscillator with U(N) symmetry. The permutation symmetry is realised as gauge symmetry in a path integral formulation in a companion paper. With the simplest matrix oscillator Hamiltonian subject to gauge permutation symmetry, we use the known result for the micro-canonical partition function to derive the canonical partition function. It is expressed as a sum over partitions of N of products of factors which depend on elementary number-theoretic properties of the partitions, notably the least common multiples and greatest common divisors of pairs of parts appearing in the partition. This formula is recovered using the Molien-Weyl formula, which we review for convenience. The Molien-Weyl formula is then used to generalise the formula for the canonical partition function to the 11-parameter permutation invariant matrix harmonic oscillator.

Stability of the microcanonical ensemble in Euclidean Quantum Gravity

This work resolves a longstanding tension between the physically-expected stability of the microcanonical ensemble for gravitating systems and the fact that the known negative mode of the asymptotically flat Schwarzschild black hole decays too rapidly at infinity to affect the ADM energy boundary term at infinity. The key to our study is that we fix an appropriate off-shell notion of energy, which we obtain by constructing the microcanonical partition function as an integral transform of the canonical partition function. After applying the rule-of-thumb for Wick rotations from our recent companion paper to deal with the conformal mode problem of Euclidean gravity, we find a positive definite action for linear perturbations about any Euclidean Schwarzchild (-AdS) black hole. Most of our work is done in a cavity with reflecting boundary conditions, but the cavity wall can be removed by taking an appropriate limit.

Euclidean de Sitter black holes and microcanonical equilibrium

Schwarzschild-de Sitter (SdS) black holes do not admit a completely smooth Euclidean continuation. We discuss some modifications of the gravitational path integral that give Euclidean SdS a semiclassical equilibrium interpretation. First we consider "gravity in a cavity," defining the canonical ensemble in a box that excises one horizon. However, this standard approach does not work for positive cosmological constant: the solution of lowest free energy has a negative heat capacity, which is inconsistent if it is to provide the leading semiclassical contribution to a canonical partition function. Instead we modify the boundary conditions in the path integral to construct the microcanonical partition function, which appears to be well-defined. We then bring two ensembles into contact and remove the boundary, producing states of a larger microcanonical ensemble that contain, for example, both a black hole and a cosmological horizon at once. These systems are closed and have no boundary, but they must possess some form of mild metric discontinuity. We discuss the case where the discontinuity is equivalent to the insertion of a thin, rigid membrane, separating two systems that can exchange energy and are at local equilibrium. Equilibrium configurations obtained in this way are found to be thermodynamically unstable if they contain a black hole.

Condensation transition and ensemble inequivalence in the discrete nonlinear Schrödinger equation.

The thermodynamics of the discrete nonlinear Schrödinger equation in the vicinity of infinite temperature is explicitly solved in the microcanonical ensemble by means of large-deviation techniques. A first-order phase transition between a thermalized phase and a condensed (localized) one occurs at the infinite-temperature line. Inequivalence between statistical ensembles characterizes the condensed phase, where the grand-canonical representation does not apply. The control over finite-size corrections of the microcanonical partition function allows to design an experimental test of delocalized negative-temperature states in lattices of cold atoms.

Possible explanations for fine-tuning of the universe

The Froggatt–Nielsen mechanism and the multilocal field theory are interesting and promising candidates for solving the naturalness problem in the universe. These theories are based on the different physical principles: The former assumes the microcanonical partition function [Formula: see text], and the latter assumes the partition function [Formula: see text] where [Formula: see text] is the multilocal action [Formula: see text] Our main purpose is to show that they are equivalent in the sense that they predict the same fine-tuning mechanism. In order to clarify our argument, we first study (review) the similarity between the Froggatt–Nielsen mechanism and statistical mechanics in detail, and show that the dynamical fine-tuning in the former picture can be understood completely in the same way as the determination of the temperature in the latter picture. Afterward, we discuss the multilocal theory and the equivalence between it and the Froggatt–Nielsen mechanism. Because the multilocal field theory can be obtained from physics at the Planck/String scale, this equivalence indicates that the microcanonical picture can also originate in such physics. As a concrete example, we also review the IIB matrix model as an origin of the multilocal theory.

Microcanonical finite-size scaling of an ideal Bose gas in a box

We derive an exact recursive scheme to determine exactly the microcanonical partition function of a finite Bose system. Such a recursive approach is identical to that previously obtained within the context of counting statistics. Within the exact microcanonical ensemble, we study microcanonical finite-size scaling behaviors of condensate fraction and specific heat around the critical energy ϵ c for the finite ideal Bose system. We show that the microcanonical scaling functions governing the various critical behaviors are universal in the ideal Bose-Einstein condensates.

Finite size effect on classical ideal gas revisited

Finite size effects on classical ideal gas are revisited. The micro-canonical partition function for a collection of ideal particles confined in a box is evaluated using Euler–Maclaurin’s as well as Poisson's summation formula. In Poisson's summation formula there are some exponential terms which are absent in Euler–Maclaurin’s formula. In the thermodynamic limit the exponential correction is negligibly small but in the macro/nano dimensions and at low temperatures they may have a great significance. The consequences of finite size effects have been illustrated by redoing the calculations in one and three dimensions keeping the exponential corrections. Global and local thermodynamic properties, diffusion driven by the finite size effect, and effect on speed of sound have been discussed. Thermo-size effects, similar to thermoelectric effects, have been described in detail and may be a theoretical basis with which to design nano-scaled devices. This paper can also be very helpful for undergraduate and graduate students in physics and chemistry as an instructive exercise for a good course in statistical mechanics.

On the relativistic micro-canonical ensemble and relativistic kinetic theory for N relativistic particles in inertial and non-inertial rest frames

A new formulation of relativistic classical mechanics allows a reconsideration of old unsolved problems in relativistic kinetic theory and in relativistic statistical mechanics. In particular a definition of the relativistic micro-canonical partition function is given strictly in terms of the Poincaré generators of an interacting N-particle system both in the inertial and non-inertial rest frames. The non-relativistic limit allows a definition of both the inertial and non-inertial micro-canonical ensemble in terms of the Galilei generators.

Condensate fluctuations of interacting Bose gases within a microcanonical ensemble

Based on counting statistics and Bogoliubov theory, we present a recurrence relation for the microcanonical partition function for a weakly interacting Bose gas with a finite number of particles in a cubic box. According to this microcanonical partition function, we calculate numerically the distribution function, condensate fraction, and condensate fluctuations for a finite and isolated Bose-Einstein condensate. For ideal and weakly interacting Bose gases, we compare the condensate fluctuations with those in the canonical ensemble. The present approach yields an accurate account of the condensate fluctuations for temperatures close to the critical region. We emphasize that the interactions between excited atoms turn out to be important for moderate temperatures.

Dynamics of a single polymer chain: Ergodicity and conformation of a rotating chain

The dynamics of an isolated polymer chain has irreversible aspects that lead to the decorrelation of configurational properties over time. For simple mechanical models with time reversible equations of motion, the irreversibility is a consequence of the chaotic nature of the dynamics which, for a many body system, is expected to result in ergodic mixing. Here we study a fixed bond length N-mer interaction-site chain with fixed total energy and angular momentum. For N>or=4 the equations of motion for such a chain are nonintegrable and chaotic dynamics is expected. We directly assess the ergodicity of short repulsive Lennard-Jones chains by comparing phase space and time averages for structural and energetic properties. The phase space averages are determined from the exact microcanonical partition function while the time averages are obtained from molecular dynamics (MD) simulations. For N=4 and 5 we find that our exact phase space averages agree with the MD time averages, as expected for an ergodic system. The N=3 system is integrable and thus displays regular dynamics for which time averages are found to depend on initial conditions. In all cases, the total angular momentum is found to have a large effect on both the average chain conformation and the partitioning of the total energy between potential, vibrational, and rotational contributions. Compared to a nonrotating chain, a small to moderate angular momentum slightly speeds up the internal chain dynamics, while a large angular momentum dramatically slows the internal dynamics.

MICROCANONICAL MODEL FOR A GAS OF EVAPORATING BLACK HOLES AND STRINGS, SCATTERING AMPLITUDES AND MASS SPECTRUM

We study the system formed by a gas of black holes and strings within a microcanonical formulation. The density of mass levels grows asymptotically as [Formula: see text], (i = 1,…,N). We derive the microcanonical content of the system: entropy, equation of state, number of components N, temperature T and specific heat. The pressure and the specific heat are negative reflecting the gravitational unstability and a nonhomogeneous configuration. The asymptotic behavior of the temperature for large masses emerges as the Hawking temperature of the system (classical or semiclassical phase) in which the classical black hole behavior dominates, while for small masses (quantum black hole or string behavior) the temperature becomes the string temperature which emerges as the critical temperature of the system. At low masses, a phase transition takes place showing the passage from the classical (black hole) to quantum (string) behavior. Within a microcanonical field theory formulation, the propagator describing the string–particle–black hole system is derived and from it the interacting four-point scattering amplitude of the system is obtained. For high masses it behaves asymptotically as the degeneracy of states ρ(m) of the system (i.e. duality or crossing symmetry). The microcanonical propagator and partition function are derived from a (Nambu–Goto) formulation of the N-extended objects and the mass spectrum of the black hole–string system is obtained: for small masses (quantum behavior) these yield the usual pure string scattering amplitude and string–particle spectrum [Formula: see text]; for growing mass the spectrum describes all the intermediate states up to the pure black hole behavior. The different black hole behaviors according to the different mass ranges: classical, semiclassical and quantum or string behaviors are present in the model.

The microcanonical ensemble of the ideal relativistic quantum gas with angular momentum conservation

We derive the microcanonical partition function of the ideal relativistic quantum gas with fixed intrinsic angular momentum as an expansion over fixed multiplicities. We developed a group theoretical approach by generalizing known projection techniques to the Poincaré group. Our calculation is carried out in a quantum field framework and applies to particles with any spin. It extends known results in the literature in that it does not introduce any large volume approximation, and it takes particle spin fully into account. We provide expressions of the microcanonical partition function at fixed multiplicities in the limiting classical case of large volumes and large angular momenta and in the grand-canonical ensemble. We also derive the microcanonical partition function of the ideal relativistic quantum gas with fixed parity.

The microcanonical ensemble of the ideal relativistic quantum gas

We derive the microcanonical partition function of the ideal relativistic quantum gas of spinless bosons in a quantum field framework as an expansion over fixed multiplicities. Our calculation generalizes well known expressions in the literature in that it does not introduce any large-volume approximation and it is valid at any volume. We discuss the issues concerned with the definition of the microcanonical ensemble for a free quantum field at volumes comparable with the Compton wavelength and provide a consistent prescription for calculating the microcanonical partition function that is finite at finite volume and yielding the correct thermodynamic limit. Besides an immaterial overall factor, the expression obtained turns out to be the same as in the non-relativistic multi-particle approach. This work is an introduction to the derivation of the most general expression of the microcanonical partition function fixing the maximal set of observables of the Poincare group.

Statistical Mechanics of Many Interacting Point Masses in Spherical Geometry

The long terms dynamics of point masses in a trapping potential interacting gravitationally cannot be derived from the usual ergodic assumption. The microcanonical partition function diverges for more than two particles because of the large volume allowed in momentum space when the potential energy of a close pair becomes very large negative. In the long term the trajectories explore by diffusion a phase space of infinite volume. This does not say what happens for a large number of self gravitating masses, like the globular clusters or even the galaxies perhaps. I reexamine below this question and try to point out the various difficulties faced by a rational approach. There is a small parameter, the inverse number of masses. At the lowest order in this parameter, the dynamics is given by solutions of the mean field Newton–Vlasov (N–V) kinetic equation. This yields well defined predictions for the steady state under given initial conditions, at least without initial angular momentum, because, besides the usual mechanical invariants, the N–V equation has as an invariant function, the histogram of the probability distribution in phase space. The next step is far more difficult: it requires to take into account the irreversible effects due to ‘collisions’. Actually, these irreversible effects are mostly coming from the resonant interactions between orbits of commensurate or close to commensurate period, not from close pair encounters as usually assumed. The evolution there is the one of slow changes of the steady states of N–V. I discuss briefly the possible solutions on the long time scale linked to the ‘collisions’.

Tricriticality and the Blume-Capel model: A Monte Carlo study within the microcanonical ensemble

The microcanonical partition function ${\ensuremath{\Omega}}_{\ensuremath{\alpha},N}(E)$ of a three-dimensional ferromagnetic simple cubic Blume-Capel model is calculated for several coupling ratios $\ensuremath{\alpha}$ close to and on the first-order side of the tricritical point. To this end a single Monte Carlo simulation of a suitably extended partition function for systems of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ spins with $L\ensuremath{\in}{8,10,12,14,18}$ was performed. A finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well. An empirical scaling of its coordinates to the thermodynamic limit yields ${\ensuremath{\alpha}}^{(t)}=2.84479\ifmmode\pm\else\textpm\fi{}0.00030$ for the tricritical coupling ratio, and ${k}_{B}{T}^{(t)}/J=1.4182\ifmmode\pm\else\textpm\fi{}0.0055$ for the tricritical transition temperature. The results are compared with values obtained on fcc lattices.

The evaluation of NE(R) within a variably defined reaction coordinate framework

A new algorithm is presented for evaluating the microcanonical partition function for transition state dividing surfaces defined in terms of a fixed value for an arbitrarily defined reaction coordinate. The basis of this algorithm involves the formal integration of the momenta for arbitrary quadratic kinetic energy expressions. Specific application is made to a recently derived variable reaction coordinate formalism of current utility in the study of barrierless reactions. The algorithm is found to provide a more efficient means for evaluating the number of states function at fixed energy and also provides new physical insight regarding this variational formalism

Statistics of two-dimensional point vortices and high-energy vortex states

Results from the theory ofU-statistics are used to characterize the microcanonical partition function of theN-vortex system in a rectangular region for largeN, under various boundary conditions, and for neutral, asymptotically neutral, and nonneutral systems. Numerical estimates show that the limiting distribution is well matched in the region of major probability forN larger than 20. Implications for the thermodynamic limit are discussed. Vortex clustering is quantitatively studied via the average interaction energy between negative and positive vortices. Vortex states for which clustering is generic (in a statistical sense) are shown to result from two modeling processes: the approximation of a continuous inviscid fluid by point vortex configurations; and the modeling of the evolution of a continuous fluid at high Reynolds number by point vortex configurations, with the viscosity represented by the annihilation of close positive-negative vortex pairs. This last process, with the vortex dynamics replaced by a random walk, reproduces quite well the late-time features seen in spectral integration of the 2d Navier-Stokes equation.

The problem of equivalence in statistical mechanics of equilibrium

The physical problem of the equivalence of certain partition function in statistical mechanics leads to the following mathematical problem: Under what conditions is the logarithm of a Laplace–Stieltjes transform of the so-called microcanonical partition function equal to a Legendre transform of the logarithm of the latter? The conditions we shall formulate turn out to be physically reasonable.

Statistical mechanics of a neurobiological dynamical system: The spectrum of local entropies (S(α)) applied to cocaine-perturbed behavior

The singularity spectrum of local entropies ( S(α)) computed from the microcanonical dynamical partition function is used to describe the sequential dynamical structure of rat locomotor paths under the influence of cocaine. The approach used was analogous to the microcanonical ensemble of statistical mechanics and permitted the sensitive detection of “phase-transition”-like behavior. The results suggest that the behavioral sequences exhibited by the rats are determined by two processes, each with its own entropy (α i, α h). The processes are changed simultaneously by cocaine, suggesting that they coexist as local repellers with overlapping insets and undergo a change to first order phase transition.