The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble

This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C², then universal equivalence of ensembles holds with g taken from a class of quadratic functions, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization. linebreak Generalizing the paper by Ellis et al. [J. Stat. Phys. 101:999–1064 (2000)], we analyze the equivalence of the microcanonical and generalized canonical ensembles both at the level of equilibrium macrostates and at the thermodynamic level. A neat but not quite precise statement of one of our main results is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the generalized microcanonical entropy s–g is concave. This generalizes the work of Ellis et al., who basically proved that the microcanonical and canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the microcanonical entropy s is concave.     

Essay: The Holography of Gravity Encoded in a Relation Between Entropy, Horizon Area, and Action for Gravity

I provide a general proof of the conjecture that one can attribute an entropy to the area of any horizon. This is done by constructing a canonical ensemble of a subclass of spacetimes with a fixed value for the temperature T = ^–1 and evaluating the exact partition function Z(). For spherically symmetric spacetimes with a horizon at r = a, the partition function has the generic form Z exp[S – E], where S = (1/4)4 a ² and |E| = (a/2). Both S and E are determined entirely by the properties of the metric near the horizon. This analysis reproduces the conventional result for the black-hole spacetimes and provides a simple and consistent interpretation of entropy and energy for De Sitter spacetime. For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. Further, I show that the relationship between entropy and area allows one to construct the action for the gravitational field on the bulk and thus the full theory. In this sense, gravity is intrinsically holographic.     

Self-gravitating radiation in anti-de sitter space

Equilibrium configurations of self-gravitating massless thermal radiation inside spherical boxes of radiusR in asymptotically anti-de Sitter space (A = -3/b ₂) are constructed numerically for a range of central densities. For each box radius considered (R/b = 0, 1/2, 1, 2, 4, ), there is a unique configuration with maximal total mass and entropy, and another (at a lower central density) with maximum asymptotic red-shifted temperature. With the box removed toR=, the maximum total mass and entropy of self-gravitating thermal radiation areM _max 0.4598b0.7964(–A)^–1/2 andS _max1.3560a ^1/4 b ^3/2 3.0910a ^1/4(–A)^–3/4, and the maximum red-shifted temperature is     

Additivity of the entropy and definition of the temperature for quantum systems

A closed quantum systemL is considered which is described by a microcanonical ensemble.L consists of two rather weakly interacting subsystemsL ₁,L ₂. In a rigorous way, the additivity of the entropy is proved by deriving an expression for the entropy density ofL in terms of the entropy densities ofL ₁ andL ₂. Rigorous implies that the thermodynamic limit is taken. In the second part, it is shown how a microcanonical state of the composite system — provided this limit exists — gives rise to a canonical state , when restricted toL ₁, providedL ₁ is very small as compared toL ₂; is defined as a limit of Gibbs states. This yields a definition of the equilibrium temperature ^–1.On leave of absence from the Institut für Theoretische Physik, Universität Göttingen.     

Statistical mechanics and dynamics of solvable models with long-range interactions

For systems with long-range interactions, the two-body potential decays at large distances as V(r)1/r^α, with α≤d, where d is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive: the sum of the energies of macroscopic subsystems is not equal to the energy of the whole system. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties of the thermodynamics of short-range systems is at the origin of ensemble inequivalence. In turn, this inequivalence implies that specific heat can be negative in the microcanonical ensemble, and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity allows us to easily spot regions of parameter space where ergodicity may be broken. Historically, negative specific heat had been found for gravitational systems and was thought to be a specific property of a system for which the existence of standard equilibrium statistical mechanics itself was doubted. Realizing that such properties may be present for a wider class of systems has renewed the interest in long-range interactions. Here, we present a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of solvable systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Remarkably, the entropy of all these models can be obtained using the method of large deviations. Long-range interacting systems display an extremely slow relaxation towards thermodynamic equilibrium and, what is more striking, the convergence towards quasi-stationary states. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation. A statistical approach, founded on a variational principle introduced by Lynden-Bell, is shown to explain qualitatively and quantitatively some features of quasi-stationary states. Generalizations to models with both short and long-range interactions, and to models with weakly decaying interactions, show the robustness of the effects obtained for mean-field models.     

Statistical mechanics of rotating systems

A microcanonical distribution function depending on the total energyE and thez-componentM of the total angular momentum of a rotating system is examined. ForM=0 the generalized microcanonical ensemble is found to give the same entropy as the usual microcanonical ensemble. The moment of inertia of a rotating gas is calculated, and the kinetic energy of rotation is given as a power series in the small parameterM ²/2I ₀E_int, whereI ₀ is the moment of inertia of the gas at rest andE _int the internal energy.     

Ensemble average of an arbitrary number of pairs of different eigenvalues using Grassmann integration

An identity satisfied by the eigenvalues of a real-symmetric matrix and an integral representation of a determinant using Grassmann variables are used to show that the ensemble average ofS different pairs of eigenvalues of a GOE is given by (–1) ^S 2^{–S –1/2}(S+1/2).     

The effect of a constant number of particles on the pair correlation function and the density profile in a one-dimensional lattice gas

A one-dimensional lattice gas (Ising model) of lengthL and with nearest-neighbor couplingJ is considered in a canonical ensemble with fixed number of particlesN=L/2. Exact expressions and asymptotic forms for largeL are derived for the density-density correlation function, using periodic boundary conditions, and for the density (magnetization) profile, using antisymmetric boundary conditions. The density-density correlation function,g, assumes for temperaturesT> T, withT = 2J(_BlnL)^–1 and forL large, the formg(x) =g ^gc(x) +BL ^–1 +a(x)L ^–1 +O(L^–2) wherex is a distance between considered lattice sites,B is known from earlier work of Lebowitz and Percus,(1b) anda(x) decays exponentially forx . For TT, the correlation function and the density profile behave differently, the latter exhibiting a step in the middle of the interface.     

Rate of abstraction of hydrogen atoms from ethane by muonium

Thermal reaction rates for the gas-phase reaction Mu+C₂H₆MuH+C₂H₅ have been measured bySR over the temperature range 510–730 K. The usual Arrhenius expression,k=Aexp(–E _a /RT), fits the data well, giving parametersA=1.0×10^–9 cm³ molecule^–1 s^–1 andE _a =15.35 kcal/mol. The activation energyE _a is 5.5 kcal/mol higher than for the H atom variant of this reaction, indicating a marked difference in reaction dynamics. Preliminary analysis indicates a still greater difference between Mu and H for the corresponding CH₄ reaction.     

Thermodynamic studies of successive phase transitions in BaZnGeO4 and a new solid phase below 186.1 K

The successive phase transitions of BaZnGeO₄ have been studied on meltsolidified samples. A new solid phase (named phase VI) has been found below 186.1 K in samples of large particle size (diameter:D0.1 mm). The higher temperature crystalline phase V can be supercooled easily down to liquid helium temperature. On heating, however, it transforms into phase VI above 95 K in a slow exothermic process. Heat capacities have been measured by adiabatic calorimetry between 14 and 300 K. The enthalpy and entropy of the V–VI phase transition are 187.1 Jmol^–1 and 0.971 J K^–1 mol^–1, respectively. The corresponding data for the IV–V phase transition at 199.8 K are 229.3 J mol^–1 and 1.168 JK^–1 mol^–1. The phase VI does not appear in samples of smaller particle size (D0.1 mm).     

Observation of the decaya 2 − (1320)→π − η′

The reaction ^– C ^– C has been studied at 36 GeV/c. A clear signal for the decaya ₂ ^– (1320) ^– · is observed in the ^– mass spectrum. The measured ratio of branchings is BR (a ₂ ^{– –} )/BR(a ₂ ^{– –} );(3.4±0.8±0.5)·10^–2.     

Ultrasonic determination of electron-phonon scattering rates in copper

Electron-phonon scattering rates in ultrapure single crystals of copper have been determined from the temperature dependence of the magnetoacoustic oscillation amplitudes for various orbits on the Fermi surface using both longitudinal and transverse waves. The central belly orbit seattering rate is found to be (6.0±0.3)×10⁶ T ³ sec^–1. Additionally, a rate of (2.9±0.2)×10⁶ T ³ sec^–1 is found which is attributed to belly orbits displaced from the zone center by about 1.25/a ₀, wherea ₀ is the lattice constant. Geometric oscillations associated with the [111]—directed open orbit are observed at low fields forq [113] and the rate for this orbit is found to be (4.8±0.3)×10⁶ T ³ sec^–1. Geometric oscillations for the dog's bone and neck orbits are observable but rates for these orbits are believed to be unreliable. Our measured rates are compared with those of other workers.     

Possible generalization of Boltzmann-Gibbs statistics

With the use of a quantity normally scaled in multifractals, a generalized form is postulated for entropy, namelyS _q k [1 – _i=1 ^W p _i ^q ]/(q-1), whereq characterizes the generalization andp _i are the probabilities associated withW (microscopic) configurations (W). The main properties associated with this entropy are established, particularly those corresponding to the microcanonical and canonical ensembles. The Boltzmann-Gibbs statistics is recovered as theq1 limit.     

Resonant shielding of an inhomogeneous gyrotropic plasma slab

Conclusions The problem of the incidence of a plane TM electromagnetic wave on an isotropic, symmetric, resonance plasma slab (n=2, 4, 6, ...), discussed in [1], was solved by an iteration method in a recent paper [6]. The physical results found there are the same as those of [1]. Zhivulin and Makarov [6] then applied the iteration method to the analogous problem of a gyrotropic resonance plasma slab [7]. The analysis in these papers furnishes a clearer mathematical justification of the results of [1] and the present paper and thus of the method used there. The present method, which satisfies only a physical condition of rigor, is preferable to the mathematically more rigorous methods (in particular, the iteration method) because of its simplicity, its graphic nature, and its clear physical meaning. It also answers many questions which cannot be answered in the more rigorous approach because of the serious difficulties which arise (and which have not yet been overcome).N. G. Denisov has called our attention to the fact that complete shielding was actually found previously by Rytov and Yudkevich [8], who treated the problem of the incidence of a plane TE electromagnetic wave on a slab with a dielectric constant (x)=₁ – A₁/(a – x)² for x<0. (x)=₂ – A₂/(b + x)² for x>0, and ₁ – A₁/(a² = ₂ – A₂/b² in the plane x=0. In the limita0, b0, they found results corresponding to a slab with a dielectric constant having a first-order pole; it is in this case that complete shielding is achieved. This method for obtaining the corresponding results is analogous to the method used in [1] and the present paper. We also note that the distribution of the effective dielectric constant (6) in the immediate vicinity of the pole—where the contribution of the last term. (1–u)x², can be neglected—is the same as the distribution adopted in [8] if we seta-b-0, ₁ = ₂, A₁=A₂=u.Scientific-Research Institute of Radiophysics. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 19, No. 8, pp. 1130–1141, August, 1976.     

Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems

Classical spin systems with nonadditive long-range interactions are studied in the microcanonical ensemble. It is expected that the entropy of such a system is identical to that of the corresponding mean-field model, which is called “exactness of the mean-field theory”. It is found out that this expectation is not necessarily true if the microcanonical ensemble is not equivalent to the canonical ensemble in the mean-field model. Moreover, necessary and sufficient conditions for exactness of the mean-field theory are obtained. These conditions are investigated for two concrete models, the α-Potts model with annealed vacancies and the α-Potts model with invisible states.     

The Pioneer Anomaly as Acceleration of the Clocks

This work proposes an explanation of the Pioneer anomaly, the unmodelled and as yet unexplained blueshift detected in the microwave signal of the Pioneer 10 and other spaceships by Anderson et al. in 1998. What they observed is similar to the effect that would have either (i) an anomalous acceleration a_P the ship towards the Sun, or (ii) an acceleration of the clocks a_t=a_P/c. The second alternative is investigated here, with a phenomenological model in which the anomaly is an effect of the background gravitational potential (t) that pervades all the universe and is increasing because of the expansion. It is shown that 2a_t=d /dt=d²_clocks/dt², evaluated at present time t₀, where t and _clocks are the coordinate time and the time measured by the atomic clocks, respectively. The result of a simple estimate gives the value a_t 1.8× 10^–18 s^–1, while Anderson et al. suggested a_t=(2.9±0.4) × 10^–18 s^–1 on the basis of their observations. The calculation are performed near the Newtonian limit but in the frame of general relativity.     

Superconductivity and processing of single Bi−O layered cuprates in the (Bi,Pb)−Sr−(Ca,Y)−Cu−O system

The effects of improved materials processing on single Bi–O layered cuprates in the (Bi, Pb)–Sr–(Ca, Y)–Cu–O system have been investigatged. For Bi-1212 we have improvedT _c to 102 K. The bulk nature of superconductivity is confirmed by the presence of superconducting volume fractions ( ZFC) around 30–40%. The critical current density is 2×10⁶ Acm^–2 at 5 K and 0 T. Moreover, indications for the presence of a second phase probably Bi-1223 with a transition to superconductivity in the range of 115–150 K have been found.     

The broken supersymmetry phase of a self-avoiding random walk

We consider a weakly self-avoiding random walk on a hierarchical lattice ind=4 dimensions. We show that for choices of the killing ratea less than the critical valuea _cthe dominant walks fill space, which corresponds to a spontaneously broken supersymmetry phase. We identify the asymptotic density to which walks fill space, (a), to be a supersymmetric order parameter for this transition. We prove that (a)(a _c–a) (–log(a _c–a))^1/2 asaa _c, which is mean-field behavior with logarithmic corrections, as expected for a system in its upper critical dimension.Research partially supported by NSF Grants DMS 91-2096 and DMS 91-96161.     

Energieverlust und Lichtausbeute bei axialem Channeling von α-Teilchen in Anthrazen

In an atomic beam experiment we measured the hyperfine structure (hfs) of the metastable states 3d ² 4s ⁴ F _3/2,9/2 of Scandium, which were populated by electron impact. A single mode cw dye laser was used to create a population difference between neighbouring hfs states by optical pumping. Rf transitions between these states, equalizing the population numbers, could then be detected by repeating the optical pumping experiment and looking for the laser induced fluorescent light. The following constants of the magnetic dipole and electric quadrupole interactions have been obtained by this method:A _3/2=–158.513(3)MHzB _3/2=–5.19(2)MHzA _9/2=285.967(9)MHzB _9/2=–15.46(70)MHzFrom these constants and the corresponding values forJ=5/2, 7/2, as measured by [1], the following parameters can be calculated: r ^–3 _d ⁰¹ =0.911(4)a ₀ ^–3 r ^–3 _d ¹² =2.395 (62)a ₀ ^–3 and:=r ^–3 _d ⁰¹ /r ^–3 _d ¹² =0.38(1).     

Quantum Spectrum of Some Anharmonic Central Potentials: Wave Functions Ansatz

Based on the ansatz to the wave functions, the quasi-exact solutions of the 2D Schrödinger equation with some anharmonic potentials are reviewed and analyzed if admitting restrictions on the parameters of the potential and the angular momentum m. These potentials are taken as the screened Coulomb potential V(r)=a/r+b/(r+), the singular one-fraction power one V(r)=ar ^–1/2+br ^–3/2 and the singular two-fraction one V(r)=ar ^2/3+br ^–2/3+cr ^–4/3. The latter one is found that the hidden symmetry exists if substituting r–ir. It will reverse the signs of E and c of quantum system, leaving the remaining parameters invariant.