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 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 Magnetic field corrections to the repulsive Casimir effect at finite temperature

2016 ◽  
Vol 31 (07) ◽  
pp. 1650018 ◽  
Author(s):  
Andrea Erdas
Keyword(s):  
Magnetic Field ◽  
Free Energy ◽  
High Temperature ◽  
Finite Temperature ◽  
Casimir Effect ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Parallel Plates ◽  
The Magnetic Field ◽  
Plate Distance

I investigate the finite temperature Casimir effect for a charged and massless scalar field satisfying mixed (Dirichlet–Neumann) boundary conditions on a pair of plane parallel plates of infinite size. The effect of a uniform magnetic field, perpendicular to the plates, on the Helmholtz free energy and Casimir pressure is studied. The [Formula: see text]-function regularization technique is used to obtain finite results. Simple analytic expressions are obtained for the zeta function and the free energy, in the limits of small plate distance, high temperature and strong magnetic field. The Casimir pressure is obtained in each of the three limits and the situation of a magnetic field present between and outside the plates, as well as that of a magnetic field present only between the plates is examined. It is discovered that, in the small plate distance and high temperature limits, the repulsive pressure is less when the magnetic field is present between the plates but not outside, than it is when the magnetic field is present between and outside the plates.

scholarly journals Casimir Effect in Domain Wall Formation

2003 ◽  
Vol 18 (23) ◽  
pp. 4285-4293 ◽  
Author(s):  
M. R. Setare
Keyword(s):  
Domain Wall ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Casimir Effect ◽  
Mixed Boundary Conditions ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Parallel Plates ◽  
De Sitter ◽  
Casimir Forces

The Casimir forces on two parallel plates in conformally flat de Sitter background due to conformally coupled massless scalar field satisfying mixed boundary conditions on the plates is investigated. In the general case of mixed boundary conditions formulae are derived for the vacuum expectation values of the energy–momentum tensor and vacuum forces acting on boundaries. Different cosmological constants are assumed for the space between and outside of the plates to have general results applicable to the case of domain wall formations in the early universe.

scholarly journals SCALAR CASIMIR EFFECT BETWEEN TWO CONCENTRIC SPHERES

2012 ◽  
Vol 27 (16) ◽  
pp. 1250082 ◽  
Author(s):  
MUSTAFA ÖZCAN
Keyword(s):  
Scalar Field ◽  
Casimir Effect ◽  
Casimir Energy ◽  
Outer Radius ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Parallel Plates ◽  
New Approach ◽  
Concentric Spheres ◽  
Spherical Boundary

The Casimir effect giving rise to an attractive force between the closely spaced two concentric spheres that confine the massless scalar field is calculated by using a direct mode summation with contour integration in the complex plane of eigenfrequencies. We developed a new approach appropriate for the calculation of the Casimir energy for spherical boundary conditions. The Casimir energy for a massless scalar field between the closely spaced two concentric spheres coincides with the Casimir energy of the parallel plates for a massless scalar field in the limit when the dimensionless parameter η, ([Formula: see text] where a(b) is inner (outer) radius of sphere), goes to zero. The efficiency of new approach is demonstrated by calculation of the Casimir energy for a massless scalar field between the closely spaced two concentric half spheres.

scholarly journals Casimir effect in an axially symmetric spacetime with unparticles

2019 ◽  
Vol 79 (10) ◽  
Author(s):  
V. B. Bezerra ◽  
C. R. Muniz ◽  
H. S. Vieira
Keyword(s):  
Energy Density ◽  
Characteristic Length ◽  
Casimir Effect ◽  
Casimir Energy ◽  
Einstein Equations ◽  
Kerr Solution ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Parallel Plates ◽  
Axially Symmetric

Abstract We investigate the Casimir effect of the massless scalar field in a cavity formed by ideal parallel plates in the spacetime generated by a rotating axially symmetric distribution of vector or scalar (tensor) unparticles, around which the plates orbit. The presence of the unparticles is incorporated to the background by means of a correction to the Kerr solution of the Einstein equations, in which the characteristic length and the scale dimension associated to the unparticle theory are taken into account. We show that the Casimir energy density depends also on these parameters. The analysis of the “ungravity” limit for the Casimir energy density, in which the characteristic length is very large in comparison to the horizon radius, is made, too. At zero temperature, we show that such a limit implies the instability of the system, since the Casimir energy density becomes an imaginary quantity. The general result is compared to the current terrestrial experiments of the Casimir effect. Thermal corrections also are investigated and the ungravity limit again examined, with the aforementioned instability disappearing at high temperatures.

scholarly journals Casimir effect in the Kerr spacetime with quintessence

2016 ◽  
Vol 32 (01) ◽  
pp. 1750005 ◽  
Author(s):  
V. B. Bezerra ◽  
M. S. Cunha ◽  
L. F. F. Freitas ◽  
C. R. Muniz ◽  
M. O. Tahim
Keyword(s):  
Casimir Effect ◽  
Polar Angle ◽  
State Parameter ◽  
Spherical Body ◽  
Casimir Energy ◽  
Radial Coordinate ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Parallel Plates ◽  
Kerr Spacetime

We calculate the Casimir energy of a massless scalar field in a cavity formed by nearby parallel plates orbiting a rotating spherical body surrounded by quintessence, investigating the influence of the gravitational field on that energy, at zero temperature. This influence includes the effects due to the spacetime dragging caused by the source rotation as well as those ones due to the quintessence. We show that the energy depends on all the involved parameters, as source mass, angular momentum and quintessence state parameter, for any radial coordinate and polar angle. We show that at the north pole the Casimir energy is not influenced by the quintessential matter. At the equatorial plane, when the quintessence is canceled, the result obtained in the literature is recovered. Finally, constraints in the quintessence parameters are obtained from the uncertainty in the current measurements of Casimir effect.

scholarly journals Gravitons in a Casimir box

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich ◽  
Martin Bronte
Keyword(s):  
Partition Function ◽  
Boundary Conditions ◽  
Chemical Potential ◽  
Finite Size ◽  
Neumann Boundary ◽  
Scalar Fields ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Analytical Control ◽  
Slab Geometry

Abstract The partition function of gravitons with Casimir-type boundary conditions is worked out. The simplest box that allows one to achieve full analytical control consists of a slab geometry with two infinite parallel planes separated by a distance d. In this setting, linearized gravity, like electromagnetism, is equivalent to two free massless scalar fields, one with Dirichlet and one with Neumann boundary conditions, which in turn may be combined into a single massless scalar with periodic boundary conditions on an interval of length 2d. When turning on a chemical potential for suitably adapted spin angular momentum, the partition function is modular covariant and expressed in terms of an Eisenstein series. It coincides with that for photons. At high temperature, the result provides in closed form all sub-leading finite-size corrections to the standard (gravitational) black body result. More interesting is the low-temperature/small distance expansion where the leading contribution to the partition function is linear in inverse temperature and given in terms of the Casimir energy of the system, whereas the leading contribution to the entropy is proportional to the area and originates from gravitons propagating parallel to the plates.

scholarly journals Casimir effect in space-times of rotating wormholes

2021 ◽  
Vol 81 (3) ◽  
Author(s):  
C. R Muniz ◽  
V. B. Bezerra ◽  
J. M. Toledo
Keyword(s):  
Black Hole ◽  
Energy Density ◽  
Casimir Effect ◽  
Kerr Black Hole ◽  
Casimir Energy ◽  
Vacuum Energy Density ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Thermodynamic Quantities ◽  
Parallel Plates

AbstractWe investigate the Casimir effect between parallel plates placed along a circular trajectory around the rotating Damour–Solodkhin (D–S) and Teo wormholes. This is made through the calculation of the renormalized quantum vacuum energy density of a massless scalar field obeying the Dirichlet boundary conditions, initially at zero temperature. We use the zero tidal approximation inside the cavity. Then, we compare our results with those ones previously obtained in the literature with respect to the Kerr black hole. We also compare the computed Casimir energy density in a static D–S wormhole spacetime with that one recently found for a static Ellis wormhole. In what follows, we investigate the effect around the rotating Teo wormhole by calculating the Casimir energy density between the plates, and compare it with the same quantities obtained previously. Finally, we investigate the phenomenon at finite temperature, obtaining some Casimir thermodynamic quantities in the rotating D–S wormhole spacetime, comparing them with the ones valid in the Kerr black hole spacetime. With this, the ways as gravito-inertial and frame dragging effects influence the vacuum quantum fluctuations inside the Casimir apparatus allows to distinct among the different types of rotating wormholes and black holes.

Development of a Capacitor Probe to Detect Subsurface Deterioration in Concrete

1997 ◽  
Vol 503 ◽  
Author(s):  
B. K. Diefenderfer ◽  
I. L. Al-Qadi ◽  
J. J. Yoho ◽  
S. M. Riad ◽  
A. Loulizi
Keyword(s):  
Dielectric Constant ◽  
Portland Cement ◽  
Electromagnetic Waves ◽  
Low Cost ◽  
Complex Dielectric Constant ◽  
Life Cycle Costs ◽  
Parallel Plates ◽  
Portland Cement Concrete ◽  
Cement Concrete ◽  
Set Up

ABSTRACTPortland cement concrete (PCC) structures deteriorate with age and need to be maintained or replaced. Early detection of deterioration in PCC (e.g., alkali-silica reaction, freeze/thaw damage, or chloride presence) can lead to significant reductions in maintenance costs. However, it is often too late to perform low-cost preventative maintenance by the time deterioration becomes evident. By developing techniques that would enable civil engineers to evaluate PCC structures and detect deterioration at early stages (without causing further damage), optimization of life-cycle costs of the constructed facility and minimization of disturbance to the facility users can be achieved.Nondestructive evaluation (NDE) methods are potentially one of the most useful techniques ever developed for assessing constructed facilities. They are noninvasive and can be performed rapidly. Portland cement concrete can be nondestructively evaluated by electrically characterizing its complex dielectric constant. The real part of the dielectric constant depicts the velocity of electromagnetic waves in PCC. The imaginary part, termed the “loss factor,” describes the conductivity of PCC and the attenuation of electromagnetic waves.Dielectric properties of PCC have been investigated in a laboratory setting using a parallel plate capacitor operating in the frequency range of 0.1 to 40.1MIHz. This capacitor set-up consists of two horizontal-parallel plates with an adjustable separation for insertion of a dielectric specimen (PCC). While useful in research, this approach is not practical for field implementation. A new capacitor probe has been developed which consists of two plates, located within the same horizontal plane, for placement upon the specimen to be tested. Preliminary results show that this technique is feasible and results are promising; further testing and evaluation is currently underway.

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