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We revisit the expansion recently proposed by Pulvirenti and Tsagkarogiannis for a system of $N$ continuous particles in the canonical ensemble. Under the sole assumption that the particles interact via a tempered and stable pair potential and are subjected to the usual free boundary conditions, we show the analyticity of the Helmholtz free energy at low densities and, using the Penrose tree graph identity, we establish a lower bound for the convergence radius which happens to be identical to the lower bound of the convergence radius of the virial series in the grand canonical ensemble established by Lebowitz and Penrose in 1964. We also show that the (Helmholtz) free energy can be written as a series in power of the density whose $k$ order coefficient coincides, modulo terms $o(N)/N$, with the $k$-order virial coefficient divided by $k+1$, according to its expression in terms of the $m$-order (with $m\le k+1$) simply connected cluster integrals first given by Mayer in 1942. We finally give an upper bound for the $k$-order virial coefficient which slightly improves, at high temperatures, the bound obtained by Lebowitz and Penrose.

The lengthy discussion and derivations given in this paper have been corrected, and completed with additional results and greater simplicity. We show that the high temperature limits of the heterotic E8xE8 and Spin(32)/Z2 strings and their Type I A/B superstring duals are finite and convergent. The Hagedorn growth of the degeneracies in the string mass level expansion is suppressed by an exponential linear in the mass level number for both heterotic strings, and suppressed by the exponential of the negative square root of the mass level number for the Type IB superstring. However, in the massless gauge field theoretic limit of the Type IB open and closed superstring, we find clear evidence for the thermal deconfinement phase transition at the self-dual temperature by examining the annulus graph alone. Above the self-dual temperature, there is a discontinuity in the first derivative with respect to temperature of both the free energy, and the heavy quark potential, leading to a deconfined thermal gluon ensemble, with universal 1/r potential, and temperature dependent corrections, as predicted by Luscher and Weisz. A number of essential aspects of the worldsheet formalism of the heterotic strings are derived in an appendix, deducing thereby the O8-D0-D8brane Type IA duals of all of the heterotic CHL island universe moduli spaces., Comment: 52 pages

This paper reviews basic results from a theory of the classical probabilities in state-averaged complete active space self-consistent field wave functions. It is proposed that a canonical (Boltzmann) weighting scheme makes the self-consistency condition invariant with respect to configurational transformations that act locally on the support of the equilibrated SA-CASSCF ensemble. This is illustrated with a complete active space valence-bond (CASVB) analysis of low-energy electronic states in a monomethine cyanine dye., Comment: Put up to share during ACTC July 20-15 2014

It is well known that charged black $Dp$ branes of type II string theory share a universal phase structure of van der Waals$-$Maxwell liquid-gas type except $D5$ and $D6$ branes. Interestingly, the phase structure of $D5$ and $D6$ branes can be changed to the universal form with the inclusion of particular delocalized charged lower-dimensional branes. For $D5$ branes one needs to introduce delocalized $D1$ branes, and for $D6$ branes, one needs to introduce delocalized $D0$ branes to obtain the universal structure. In a previous paper [J. High Energy Phys. {\bf 04} (2013) 100] Lu with Wei study the phase structure of black $D6$ branes with the introduction of delocalized $D0$ branes in a special case when their charges are equal and the dilaton charge vanishes. In this paper, we look at the phase structure of the black $D6/D0$ system with the generic values of the parameters, which makes the analysis more involved but the structure more rich. We also provide reasons why the respective modifications of the phase structures to the universal form for the black $D5$ and $D6$ branes occur when specific delocalized lower-dimensional branes are introduced., Comment: 36 pages, 11 Figures, added discussion on the possible dual field theory, references added, minor change of the title, publised in PRD

The usual formulation of the BCS ansatz for superconductivity in the grand canonical ensemble makes the handling of the Pauli exclusion principle between paired electrons straightforward. It however tends to mask that many-body effects between Cooper pairs interacting through the reduced BCS potential are entirely controlled by this exclusion. To show it up, one has to work in the canonical ensemble. Pauli blocking between a fixed number of composite bosons is however known to be difficult to handle. To do it, we here develop a commutator formalism for Cooper pairs, along the line we used for excitons. We then rederive, within the $N$-pair subspace, a few results of BCS superconductivity commonly derived in the grand canonical ensemble, to evidence their Pauli blocking origin. We end by discussing what should be called "Cooper pair wave function".

When the phase structure of the black brane in grand canonical ensemble is discussed, a new bubble phase with the same boundary data is found to exist in this structure. As such, the phase transitions among bubble, black brane and "hot flat space" are possible, therefore giving a much enriched phase structure. We argue that under some conditions, either the grand canonical ensemble itself is unstable or there are some unknown new phases., Comment: v2. 12 pages. Expanded discussion, references added, typos corrected, published version

We investigate the isobar of an ideal Bose gas confined in a cubic box within the grand canonical ensemble, for a large yet finite number of particles, N. After solving the equation of the spinodal curve, we derive precise formulae for the supercooling and the superheating temperatures which reveal an N^{-1/3} or N^{-1/4} power correction to the known Bose-Einstein condensation temperature in the thermodynamic limit. Numerical computations confirm the accuracy of our analytical approximation, and further show that the isobar zigzags on the temperature-volume plane if N is greater than or equal to 14393. In particular, for the Avogadro's number of particles, the volume expands discretely about 10^5 times. Our results quantitatively agree with a previous study on the canonical ensemble within 0.1% error., Comment: 6 pages, 2 figures; Reference added. Accepted for publication in Phys. Rev. A

The Thermal Hartree-Fock-Bogoliubov approximation with reprojection to good quantum numbers is analyzed in a canonical ensemble. Simple recipes are given for the evaluation of traces, gradients and expectation values in the canonical ensemble., Comment: 20 pages

With the objective of demonstrating usefulness of thermostats in the study of dynamic critical phenomena in fluids, we present results for transport properties in a binary Lennard-Jones fluid that exhibits liquid-liquid phase transition. Results from the molecular dynamics simulations in canonical ensemble, with various thermostats, are compared with those from microcanonical ensemble. It is observed that the Nos\'{e}-Hoover and dissipative particle dynamics thermostats are useful for the calculations of mutual diffusivity and shear viscosity. The Nos\'{e}-Hoover thermostat, however, appears inadequate for the study of bulk viscosity., Comment: 5 pages, 4 figures in European Physical Journal E 2015

We present here how the gravothermal or Antonov's instability, which was originally formulated in the microcanonical ensemble, is modified in the presence of a cosmological constant and in the canonical ensemble. In contrast to the microcanonical ensemble, there is a minimum, and not maximum, radius for which metastable states exist. In addition this critical radius is decreasing, and not increasing, with increasing cosmological constant. The minimum temperature for which metastable states exist is decreasing with increasing cosmological constant, while above some positive value of the cosmological constant, there appears a second critical temperature. For lower temperatures than the second critical temperature value, metastable states reappear, indicating a typical reentrant phase transition. The two critical temperatures merge when the cosmological density equals one half the mean density of the system., Comment: Talk given at the 1st International Conference on Mathematical Modeling in Physical Sciences, IC-MSQUARE 2012, September, Budapest

This is a companion paper of our previous work [1] where we studied the thermodynamics and phase structure of asymptotically flat black $p$-branes in a cavity in arbitrary dimensions $D$ in a canonical ensemble. In this work we study the thermodynamics and phase structure of the same in a grand canonical ensemble. Since the boundary data in two cases are different (for the grand canonical ensemble boundary potential is fixed instead of the charge as in canonical ensemble) the stability analysis and the phase structure in the two cases are quite different. In particular, we find that there exists an analog of one-variable analysis as in canonical ensemble, which gives the same stability condition as the rather complicated known (but generalized from black holes to the present case) two-variable analysis. When certain condition for the fixed potential is satisfied, the phase structure of charged black $p$-branes is in some sense similar to that of the zero charge black $p$-branes in canonical ensemble up to a certain temperature. The new feature in the present case is that above this temperature, unlike the zero-charge case, the stable brane phase no longer exists and `hot flat space' is the stable phase here. In the grand canonical ensemble there is an analog of Hawking-Page transition, even for the charged black $p$-brane, as opposed to the canonical ensemble. Our study applies to non-dilatonic as well as dilatonic black $p$-branes in $D$ space-time dimensions., Comment: 32 pages, 2 figures, various points refined, discussion expanded, references updated, typos corrected, published in JHEP 1105:091,2011

After deriving the isothermal Hellmann-Feynman theorem (IHFT) that is suitable for mixed states in a canonical ensemble, we use this theorem to obtain the isothermal magnetic-field sweep theorems for the free, average and trapping energies, and for the entropy, specific heat, pressure and atomic compressibility of strongly-correlated ultra-cold quantum gases. In particular, we apply the sweep theorems to two-component Fermi gases in the weakly-interacting BCS and BEC limits, showing that the temperature dependence of the contact parameter can be determined by the variation of either the entropy or specific heat with respect to the scattering length. We also use the IHFT to obtain the Virial theorem in a canonical ensemble, and discuss its implications for quantum gases., Comment: 6 pages (added more discussion)

We have studied a bulk electrolyte, and polyelectrolyte solutions with surfactants or multivalent salt with the explicit presence of counterions and solvent molecules by means of the mesoscopic dissipative particle dynamics (DPD) method in the Grand Canonical ensemble. The electrostatic interactions are calculated using the Ewald sum method and the structure of the fluid is analyzed through the radial distribution function between charged particles. The results are in very good agreement with those reported in the literature using a different method for the calculation of the electrostatic forces, and with those obtained using DPD in the canonical ensemble. We also studied the salt dependent conformation of polyelectrolyte solutions as a function of the solvent quality, and analyzed the electrostatic interaction strength dependence of dilute flexible polyelectrolytes in solution. For the complex systems mentioned above, the electrostatic interactions and the solvent quality play a key role in understanding phenomena that do not occur in noncharged systems., Comment: Sended to the Journal of Chemnical Physics

The existing thermodynamics of the cosmological horizon in de-Sitter spacetime is established in the micro-canonical ensemble, while thermodynamics of black hole horizons are established in the canonical ensemble. Generally in the ordinary thermodynamics and statistical mechanics, both of the micro-canonical and canonical ensembles yield the same equation of state for any thermodynamic system. This implies the existence of a formulation of de-Sitter thermodynamics based on the canonical ensemble. This paper reproduces the de-Sitter thermodynamics in the canonical ensemble. The procedure is as follows: We put a spherical wall at the center of de-Sitter spacetime, whose mass is negligible and perfectly reflects the Hawking radiation coming from the cosmological horizon. Then the region enclosed by the wall and horizon settles down to a thermal equilibrium state, for which the Euclidean action is evaluated and the partition function is obtained. The integration constant (subtraction term) of Euclidean action is determined to reproduce the equation of state (e.g. entropy-area law) verified already in the micro-canonical ensemble. Our de-Sitter canonical ensemble is well-defined to preserve the "thermodynamic consistency", which means that the state variables satisfy not only the four laws of thermodynamics but also the appropriate differential relations with thermodynamic functions; e.g. partial derivatives of the free energy give the entropy, pressure, and so on. The special role of cosmological constant in de-Sitter thermodynamics is also revealed., Comment: 28 pages, 2 figures, Accepted for publication in Prog.Theor.Phys, Typos are corrected

We solve the time-dependent Schrodinger equation for the combination of a spin system interacting with a spin bath environment. In particular, we focus on the time development of the reduced density matrix of the spin system. Under normal circumstances we show that the environment drives the reduced density matrix to a fully decoherent state, and furthermore the diagonal elements of the reduced density matrix approach those expected for the system in the canonical ensemble. We show one exception to the normal case is if the spin system cannot exchange energy with the spin bath. Our demonstration does not rely on time-averaging of observables nor does it assume that the coupling between system and bath is weak. Our findings show that the canonical ensemble is a state that may result from pure quantum dynamics, suggesting that quantum mechanics may be regarded as the foundation of quantum statistical mechanics., Comment: 12 pages, 4 figures, accepted for publication by J. Phys. Soc. Jpn

In this paper, a canonical ensemble model for the black hole quantum tunneling radiation is introduced. With this model the probability distribution function corresponding to the emission shell is calculated. Comparing with this function, the statistical significance of the quantum tunneling radiation spectrum of black holes is investigated. Moreover, by calculating the entropy of the emission shell, a discussion about the mechanism of information flowing out from the black hole is given too., Comment: 6 pages

We study the bond percolation problem under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We show via an analytical approach that at criticality, the constraint can induce new finite-size corrections with exponent y_{can}=2y_t-d both in energy-like and magnetic quantities, where y_t=1/{\nu} is the thermal renormalization exponent and d is the spatial dimension. Furthermore, we find that while most of universal parameters remain unchanged, some universal amplitudes, like the excess cluster number, can be modified and become non-universal. We confirm these predictions by extensive Monte Carlo simulations of the two-dimensional percolation problem which has y_{can}=-1/2., Comment: 19 pages, 4 figures; v2 includes small edits

It was shown that in the canonical ensemble the simple exactly soluble statistical model of nuclei decay into nucleons, which is a limiting case of the statistical multifragmentation model, predicts the nuclear first order phase transition associated with the Helmholtz free energy different from the first order phase transition of the liquid-gas type associated with the Gibbs free energy. The main thermodynamic properties of this phase transition were explored on the basis of the method of the thermodynamic potential and its first and second derivatives in the thermodynamic limit. It was established that the thermodynamic potential $F$ is a piecewise smooth function and its first order partial derivatives with respect to variables of state are piecewise continuous functions. At the points of phase transition, the energy in the caloric curve is discontinuous at the constant temperature and fixed values of the specific volume, while the pressure and the chemical potential in the equations of state are discontinuous at the constant specific volume and fixed values of the temperature., Comment: 10 pages, 5 figures

In this paper we generalize a methodology [T. E. Ouldridge, A. A. Louis, and J. P. K. Doye, J. Phys.: Condens. Matter {\bf 22}, 104102 (2010)] for dealing with the inference of bulk properties from small simulations of self-assembling systems of characteristic finite size. In particular, schemes for extrapolating the results of simulations of a single self-assembling object to the bulk limit are established in three cases: for assembly involving multiple particle species, for systems with one species localized in space and for simulations in the grand canonical ensemble. Furthermore, methodologies are introduced for evaluating the accuracy of these extrapolations. Example systems demonstrate that differences in cluster concentrations between simulations of a single self-assembling structure and bulk studies of the same model under identical conditions can be large, and that convergence on bulk results as system size is increased can be slow and non-trivial., Comment: Accepted by J. Chem. Phys

A new formulation of relativistic classical mechanics allows a revisiting 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 Poincar\'e 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 strictly in terms of the Galilei generators. Also the one-particle relativistic distribution function is defined and a new approach to the relativistic Boltzmann equation is delineated. Finally there are some comments on relativistic dissipative fluids., Comment: 33 pages

We use the maximum information principle to include particle-interaction into the grand-canonical theory of BECs. The inclusion of the particle-interaction elucidates why thermodynamic calculations for BECs by the grand-canonical ensemble for the non-interacting case are in coincidence with up to date experimental results. However, in our generalized theory we can show that a BEC experiences a real and abrupt phase-transition in contrast to the smooth phase-transition predicted by the non-interacting grand-canonical ensemble. In addition, we discuss possible effects due to a thermal environment and the experimental probing from a general point of view., Comment: 6 pages, two figures

It is found that a necessary completion of phase structure of D-dimensional charged black p-brane ($p > 0$) in a cavity requires two additional thermodynamical phases, the so-called "bubble of nothing" and/or the extremal brane, in canonical ensemble. This finding resolves the puzzle about the missing phases which are needed for the underlying phase diagram when $\tilde d = D - p - 3 \leq 2$ and gives a new (bubble) phase which can become globally stable when $\tilde d > 2$. An analog of Hawking-Page transition is also found among other new phase transitions, giving a complete phase structure in this setup., Comment: 18 pages, 4 figures, 1 table, the expanded version with new title and abstract, to appear in Nucl. Phys. B

We study entropy corrections due to thermal fluctuations for asymptotically AdS black holes in the grand canonical ensemble. To leading order, these can be expressed in terms of the black hole response coefficients via fluctuation moments. We also analyze entropy corrections due to mass and charge fluctuations of R-charged black holes, and our results indicate an universality in the logarithmic corrections to charged AdS black hole entropy in various dimensions., Comment: 1 + 14 Pages, one eps figure

A thermodynamic expression for the analog of the canonical ensemble for nonequilibrium systems is described based on a purely information theoretical interpretation of entropy. As an application, it is shown that this nonequilibrium canonical distribution implies some important results from nonequilibrium thermodynamics, specifically, the fluctuation theorem and the Jarzynski-equality., Comment: 4 pages, 0 figures

A canonical ensemble algorithm is employed to study the phase diagram of $N_f = 3$ QCD using lattice simulations. We lock in the desired quark number sector using an exact Fourier transform of the fermion determinant. We scan the phase space below $T_c$ and look for an S-shape structure in the chemical potential, which signals the coexistence phase of a first order phase transition in finite volume. Applying Maxwell construction, we determine the boundaries of the coexistence phase at three temperatures and extrapolate them to locate the critical point. Using an improved gauge action and improved Wilson fermions on lattices with a spatial extent of $1.8 \fm$ and quark masses close to that of the strange, we find the critical point at $T_E = 0.925(5) T_c$ and baryon chemical potential $\mu_B^E = 2.60(8) T_c$., Comment: 5 pages, 7 figures, references added, published version

Grand canonical and canonical ensembles become equivalent in the thermodynamic limit, but when the system size is finite the results obtained in the two ensembles deviate from each other. In many important cases, the canonical ensemble provides an appropriate physical description but it is often much easier to perform the calculations in the corresponding grand canonical ensemble. We present a method to compute averages in canonical ensemble based on calculations of the expectation values in grand canonical ensemble. The number of particles, which is fixed in the canonical ensemble, is not necessarily the same as the average number of particles in the grand canonical ensemble.

Some of the more powerful results of mathematical statistics are becoming of increasing importance in statistical mechanics. Here the use of the central limit theorem in conjunction with the canonical ensemble is shown to lead to an interesting and important new insight into results associated with the canonical ensemble. This theoretical work is illustrated numerically and it is shown how this numerical work can form the basis of an undergraduate laboratory experiment which should help to implant ideas of statistical mechanics in students' minds., Comment: 15 pages, 7 figures

There exists a dramatic difference in phase structure between charged black Dp-branes with $p < 5$ and charged black D5 or D6 branes. In [1], we found that the phase structure of charged black D5 system can be changed qualitatively to that of charged black Dp with $p < 5$ by adding the delocalized D1 branes but the similar change does not happen for the charged black D6 system through adding the delocalized D2 branes (Note that both D1/D5 and D2/D6 belong to the same type of D(p - 4)/Dp system). Adding the delocalized D4 branes to the black D6 branes doesn't work, either. In this paper, we consider adding the delocalized D0 branes, the only remaining lower dimensional branes, to the black D6 system for this purpose. We find that the delocalized charged black D0-branes alone share the same phase structure as the charged black D6 branes, having no van der Waals-Maxwell liquid-gas type. However, when the two are combined to form D0/D6, the resulting phase diagram finally gets changed dramatically to the wanted one, containing now the above liquid-gas type. This change arises from the interaction between D6 and the delocalized D0., Comment: 27 pages, 3 figures, references added, typos corrected, expanded version (more discussion added and results unchanged) to appear in JHEP. arXiv admin note: text overlap with arXiv:1210.0708