Erratum: Different couplings of the chemical potential with an identical partition function in QCD on a lattice [Phys. Rev. D69, 074508 (2004)]

2006 ◽  
Vol 74 (7) ◽  
Author(s):  
Fabrizio Palumbo
Keyword(s):  
Partition Function ◽  
Chemical Potential

scholarly journals Modular invariance in finite temperature Casimir effect

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich
Keyword(s):  
Partition Function ◽  
Chemical Potential ◽  
Casimir Effect ◽  
Temperature Inversion ◽  
Massless Scalar ◽  
Partition Functions ◽  
Parallel Plates ◽  
Modular Transformations ◽  
Real Analytic ◽  
Set Up

Abstract The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell’s theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.

scholarly journals Notes on massless scalar field partition functions, modular invariance and Eisenstein series

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich ◽  
Martin Bonte
Keyword(s):  
Scalar Field ◽  
Partition Function ◽  
Low Temperature ◽  
Eisenstein Series ◽  
Chemical Potential ◽  
Proper Time ◽  
Series Representation ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Spacetime Manifold

Abstract The partition function of a massless scalar field on a Euclidean spacetime manifold ℝd−1 × 𝕋2 and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL(2, ℤ) Eisenstein series with s = (d + 1)/2. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series representation, the operator approach yields its Fourier series, while the proper time/heat kernel/world-line approach shows that it is the Mellin transform of a Riemann theta function. High/low temperature duality is generalized to the case of a non-vanishing chemical potential. By clarifying the dependence of the partition function on the geometry of the torus, we discuss how modular covariance is a consequence of full SL(2, ℤ) invariance. When the spacetime manifold is ℝp × 𝕋q+1, the partition function is given in terms of a SL(q + 1, ℤ) Eisenstein series again with s = (d + 1)/2. In this case, we obtain the high/low temperature duality through a suitably adapted dual parametrization of the lattice defining the torus. On 𝕋d+1, the computation is more subtle. An additional divergence leads to an harmonic anomaly.

scholarly journals Algorithmic thermodynamics

2012 ◽  
Vol 22 (5) ◽  
pp. 771-787 ◽  
Author(s):  
JOHN BAEZ ◽  
MIKE STAY
Keyword(s):  
Partition Function ◽  
Chemical Potential ◽  
Algorithmic Information Theory ◽  
Thermodynamic Relation ◽  
Thermodynamic Cycles ◽  
Heat Engines ◽  
Algorithmic Information ◽  
The Subject ◽  
Expected Values ◽  
Special Case

Algorithmic entropy can be viewed as a special case of the entropy studied in statistical mechanics. This viewpoint allows us to apply many techniques developed for use in thermodynamics to the subject of algorithmic information theory. In particular, suppose we fix a universal prefix-free Turing machine and let X be the set of programs that halt for this machine. Then we can regard X as a set of ‘microstates’, and treat any function on X as an ‘observable’. For any collection of observables, we can study the Gibbs ensemble that maximises entropy subject to constraints on the expected values of these observables. We illustrate this by taking the log runtime, length and output of a program as observables analogous to the energy E, volume V and number of molecules N in a container of gas. The conjugate variables of these observables allow us to define quantities we call the ‘algorithmic temperature’ T, ‘algorithmic pressure’ P and ‘algorithmic potential’ μ, since they are analogous to the temperature, pressure and chemical potential. We derive an analogue of the fundamental thermodynamic relation dE = TdS − PdV + μdN, and use it to study thermodynamic cycles analogous to those for heat engines. We also investigate the values of T, P and μ for which the partition function converges. At some points on the boundary of this domain of convergence, the partition function becomes uncomputable – indeed, at these points the partition function itself has non-trivial algorithmic entropy.

scholarly journals Generalised partition functions: inferences on phase space distributions

2016 ◽  
Vol 34 (6) ◽  
pp. 557-564 ◽  
Author(s):  
Rudolf A. Treumann ◽  
Wolfgang Baumjohann
Keyword(s):  
Phase Space ◽  
Partition Function ◽  
Bessel Functions ◽  
Chemical Potential ◽  
Temperature Limit ◽  
Large Temperature ◽  
Partition Functions ◽  
Functional Dependencies ◽  
Boltzmann Factor ◽  
Nonextensive Statistical Mechanics

Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs–Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1∕|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ ≠ ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel–Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs–Boltzmann partition function is fundamental not only to Gibbs–Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.

scholarly journals CHIRAL SYMMETRY BREAKING AT NONZERO CHEMICAL POTENTIAL

2006 ◽  
Vol 21 (04) ◽  
pp. 859-864 ◽  
Author(s):  
J. C. Osborn ◽  
K. Splittorff ◽  
J. J. M. Verbaarschot
Keyword(s):  
Partition Function ◽  
Symmetry Breaking ◽  
Dirac Operator ◽  
Chiral Symmetry ◽  
Chemical Potential ◽  
Chiral Symmetry Breaking ◽  
Chiral Condensate ◽  
Zero Temperature ◽  
Dirac Spectrum

We consider chiral symmetry breaking at nonzero chemical potential and discuss the relation with the spectrum of the Dirac operator. We solve the so called Silver Blaze Problem that the chiral condensate at zero temperature does not depend on the chemical potential while this is not the case for the Dirac spectrum and the weight of the partition function.

A GENERAL METHOD FOR CALCULATING PARTITION FUNCTION OF QCD AT FINITE CHEMICAL POTENTIAL

2007 ◽  
Vol 22 (19) ◽  
pp. 3201-3209 ◽  
Author(s):  
WEI-MIN SUN ◽  
HONG-SHI ZONG
Keyword(s):  
Partition Function ◽  
Chemical Potential ◽  
Baryon Number ◽  
Quark Propagator ◽  
General Situation ◽  
Multiplicative Constant ◽  
First Case ◽  
Free Quark ◽  
Quark Theory ◽  
General Method

In this paper we propose a general method for calculating the partition function of QCD at finite chemical potential. It is found that the partition function is totally determined by the dressed quark propagator at finite chemical potential up to a multiplicative constant. From this a criterion for the phase transition between the Nambu and the Wigner phase is obtained. This general method are applied to two specific cases: the free quark theory and QCD with a model dressed quark propagator proposed in H . Pagels and S. Stokar, Phys. Rev. D20, 2947 (1979). In the first case, the standard Fermi distribution at T = 0 are reproduced. In the second case, a particular form of baryon number distribution is obtained. It is found that when μ is below a critical value, the baryon number density is identically zero, which agrees with the general conclusion in M. A. Halasz et al., Phys. Rev. D58, 096007 (1998). All the results in the present paper are obtained under the condition T = 0 and μ ≠ 0. However, they can be generalized to the the general situation T ≠ 0 and μ ≠ 0 without fundamental difficulty.

On one-dimensional regular assemblies

1948 ◽  
Vol 44 (2) ◽  
pp. 263-271 ◽  
Author(s):  
G. S. Rushbrooke ◽  
H. D. Ursell
Keyword(s):  
Partition Function ◽  
Simple Formula ◽  
Chemical Potential ◽  
Absolute Temperature ◽  
Grand Partition Function ◽  
One Dimensional ◽  
Image Position

The grand partition function of any statistical assembly may be defined by the equationwhere E denotes any value of the energy of the assembly, k is Boltzmann's constant, T the thermodynamic absolute temperature and λi a parameter which is later to be connected with the chemical potential, μi, of the ith species in the assembly by the simple formula

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